· 2015-09-01 · ykm . % rov &@ 2/oz. @j,/ %)&. % % y#+ , ... zc c/d 9/d 0/ 54 & . 2 d dwd8d dwd dwd dwdwd d8d dwd dwdwd dwd d8dwd dwd dwd dwd8d 4 zc c/d 9/d 9 e ( . @ % dwd8d dwd

!#"$%& ' (# ) ( *+,

-.0/213547698:/<;+=>6?=A@B#=C6?4EDGFH/<F;JIK#L5MNPO8,D5QCRS0RTQCRG698URJVWFX3YFZ/[=Y\(8Z]^V_R9R@+/a`bRJV_RJ=G8cX3T6?=G8:351 d(RJ1;5V,/eRHF

f ;g=C698h6?=jib;g]^Vi&V6kF9l.m3nQCRJV!o?/<Fh;^[email protected]+=J@utCRTV!\.gs5VRT=GFvRJ=

wyx#zgk|h~|k|Uk^hk?vzg9?9T gn9~k^k!z^+|h~k|22bH A¡£¢_¤A¥¦¥§Y¥,¨¤Y2©Hq¡,ª2¥_«Z¬ubH®C¤A J¯:°H¤±&q¤A²¯:¡,³´vµ ¥, ·¶k®Y¥,²¸C¡ µZ¹H¹Zº

"!$#%&'( #")*#+ (,-.,/.0123& 4,654&2%&!873&:9<;<;<=%?><.@AB9<;<;<C/DE:.0F G!1<HI2%+J()&K&LM@LM(,N#-@ POQ%!RS#+@' ,*#/D 10<T#+@!1L<#+ UOVW1X #8%YZLM(0Z#+%[\O'1]Z.T#+^.!_!1&!1A' )`KT#,G7a(&G%()(&Z@AG,&LM&@ 23(,1#+,b%(%%(,GGdc/23&!1&ZeG']Z.T#+^.!f2 )(@S#+73`#+%A8#+gh &@ ji lkm.%<H+n.@ ,b2a&oOV!p&@@^G#"Erqjs'tTuvmA^23w%&23*#+%&2%%4)(@yxz%(K1Z%4,/.)& |dD W2%7@ &! #W0<`#,/.T#+%&@AF0M&&`#+@ %(,%,&LM&@ 2!1&^b+O #~7%M#,&8!1^,&@e' )`#+@ %&)(!K2T#%W&b]Z.T#+^.!!1&!1 (#+,$%)&@.,"!14,/\T)*#+ +O3"2%%^)(@vDwy@AK@S#+%1"K2%^)((%n,/ ,K2%+J()&y#+223*#+'YrOV!2%(%&Z%(,6&%<D G(% rZ%&,(,UOz8#+#+.,/ &)(1+O &W!$#%&W.,&^%8+O2ZA/ )(W%*#,&%+On(]^.L<#+@ &Z7T#)%40<%.,?F%)& &)(<DyX^%^,/.)& F%~!1<r23()&S#+@ %(,)()(&2% r2%L^ ,(,UFG7a&0<0Y+OG( *H7.8#[0M4^,?.,&%`#+,/0-+O]^.T#+Z.!!1()`T#+ )(G G%(]Z.%(,OVG#-%& .G%*#,/0Hw#+,%!1dc/2a& &)(~']^.T#+Z.!T#+!1 )y<%)&@@S#+%%H<OVwdc#+!K2@ "OQ%!]Z.T#+^.!2 )(*HZ I2%7T#+7@AK&@2/uOz.@vD 40WK 2%+J()&eT#7a(&KLM&Ab%&#+%,/0P#[email protected]<$!1du)(.!K0b#+,,&!$#+,/0$#+ !1(*DeE:.0K!8A[n~keT#,:0M^4,[223.A~%K#+%&,Y#8Oz&42"#+,A4!K23<S#W'%)& &^ % yT&@ ,H#+,[ T#73(&~^%&%(0K%0M&#K)&@ <`rL^ &#+")(^%&!K2a`#+A%(`*#+%)%6Y]^.T#+Z.!2ZA/ )(*D?#+ZA23(2@ nT#(LMn7a(&P&@2/OQ.@/#+,b&)(.`#+0<0r%!1n,/.0:nb#+,8'0Dl%+O|#+@@4kn.@ ,P@Mw%yT#+8!bA8.23&L^ `ly,&%54,/73&0854%&%&POzl&LM&0$+O!8A]^.( *H23&,/01!$#+Â~.%y6,/ %)&.%% #+,Ydc/2@S#+T#+ y#+,OV8&)(.`#+0<0X!1K%2T#+ )&2T#+%[]Z.T#+^.!_/OV!$#+ &LM&^%8#+,#+%.,hy &@ "i km.%<Dk3#+@ %' #+Zj%'T#+8sn w-.%)%r+OI?#cûe@S#+)`Zukm.gOQly.T#+^%&/u23H^r&!$#+ÂGOze4,/@AP%T#+0W&InK$+Oa"#+%!K )y]Z.T#+^.!f!1&!1ou (")(% ,&%(,Y( D T#+/Y -,/.F%e.0M&l@(aHW>^#)(75^&%%#+,p#+@@8F23(2@ +OG]^.T#+Z.!2 )(@S#+73`#+%A6#+"bh &@ :i ykm.%rOV&@2/Oz.@j,/ %)&.%% y#+,OV"T#+01&dc/23&'^'@ (,/0M8Ydc/2a&!1&^`#+@j]^.T#+Z.!2 )(DTY5^ A42*#+@S#+7a.Y+OG2%4+OQ%*#,/0H:OVY)& )*#+@8)(!K!1&Z%X#+,!8.)`2!K !!8A[7a&T#+@\OkI*r!$#+ZA[T#+4 %Kc#+T#P- T#/H#+,~T#+$AM.Y#+@ %%KX%&6:&@!1%(,/'OV")(%()&0$!8Ae0<@ %D

ii

!"#$%'&)(*+$-, .9/D0/ p7y+OIy.T#+^.!X()%T#+ )( DWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D =9/D0/<D0/ Er&A6y23&`#+%DWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D =9/D0/<D 9 6\c(,Y#+,e.%G5^`#+%(bDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D C9/D0/<D21 l ,&@A DD8DDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 39/D0/<D 4 eZ`#+0<@ &!1&^FDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 59/D0/<D = :#+!1 )P:%)&@@S#+%:5Â/%&!1 DWDWDDWDDWD8DDWDDWDDWDWDD8D /*;9/D 9 i'# )("+Owy.T#+Z.!km/OV!$#+ DD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D /649/D 9/D0/ y.7%#+,YX&!1 ( DWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D /649/D 9/D 9 s!K2.`#+ X#+,-s!K!b. )*#+ DWDDWD8DDWDDWDDWDWDD8D /*=9/D 9/D21 y.T#+Z.!g&@ &23`#+ DD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D /37 89%: ;=<>*+?@*+A*+B AC1/D0/ Wh: %e%7@ &! DDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D /D1/D 9 y.T#+Z.!FE&23*#+%&% DWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 9<9

1/D 9/D0/ i'# )85^.)&.%8+OGE&23*#+%&YDWDDWDDWD8DDWDDWDDWDWDD8D 9<91/D 9/D 9 E&23*#+%&''6e.\T)*#+ D8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 911/D 9/D21 Erqjs'tTuvA42a)E&23*#+%&~DDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 941/D 9/D 4 -4,&@@0D8DDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 9H3

I JK*H(BMLN$>OP:QR,SBB 7S4D0/ br&&`#+0~.)& DWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 1A/4D 9 Ty%0Kbr&&`#+0[.)& DD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 1<=

. OVUW*XYARZ*[OP??-!( 7\=/D0/ 5^`#+0[lZOz'Wl&.7T#+ DDWDWDDWDDWD8DDWDDWDDWDWDD8D 1H3=/D 9 -4,GE(,/.)& D8DDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 15=/D21 j&.7T#+ DWDD8DDWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 1D=/D 4 s,/ T#+@jl ,&@A DWDDWDWDDWDD8DDWDWDDWDDWD8DDWDDWDDWDWDD8D 4Z9

iv CONTENTS

& (%*+* <P*X? *+A*X ,SB* ; B I .C/D0/ s!K!1YOV*#+.%(-D8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 4Z=C/D 9 WErqjs'tTuv%&23*#+%& DWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 4ZCC/D 9/D0/ 54&.2 DDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 4ZCC/D 9/D 9 E(.@% DWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 4+DC/D21 E&23*#+%&"'Y+uv2T#%:qg#+!1ye%()((`0$X&!1A DDWDDWDDWD8D =4C/D21/D0/ 54&.2 DDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D =4C/D21/D 9 E(.@% DWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D C<;C/D 4 E&23*#+%&"'-yduv2T#%y-&!1A D8DDWDDWDWDDWDD8DWDDWDDWDDWD8D C4C/D 4D0/ 54&.2 DDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D C4C/D 4D 9 E(.@% DWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D C5\ ) %+Q B :% %Q$" \!734D0/ s)&@. YDDWDDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 3 134D 9 y.@ ^ DWDDWDDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 3=

U !-:Q Q*XQ (:*XQ J %# \!C5/D0/ km,*# DWDDWDDWDDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 3 D5/D 9 @0M! DDWDDWD8DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 5<95/D21 e%&@!KT#+A E(%.@%'DDWDDWDDWDWDD8DDWDDWDWDDWDD8DWDDWDDWDDWD8D 51

O 89%: ;=& *XQ*+?@$-AR \ JK: , ; * *X*+A#S C :%# Q*X; * )*+ :%)*+AMR( *+:; ? Q *X-B C7 Q %R %B L $PUW*X-YA C . O R:Q #-*XB CC RB9$L U)-#$:; B S

F

UK@S#8%($,()*#,(*H#& T&@ ,+O %)& &)([UK73%,&@(873&(&2Â4 )(HT!$#+&!$#+ )(W#+,X)(!K2.%&%)& &)(8T#:2.0~.2D 8T&@ ,-+O ! #"$%&'( ! $*)+,$%.-U #8OV.,(,H'&b #b%*#+@ `(,T#+P/OV!$#+ &)(4,(,F2Â4 )*#+@I%A4%&!1+O ]Z.T#+^.!!1()%T#+ )*#+@nT#+.%KT#r2%23& (W' )%#+%POQ./u,#+!1&^`#+@@A,/0/&%&ZPOQ%! <%~+O"/Oz!$#+ B&)(4,(,A/%&!1P,(%)&7a(,7ZA)&@S#%% )*#+@2ZA/ )(*D)(!K2.%&:7T#%(,Y6]^.T#+Z.!!1()%T#+ )(yu#12345768$95:+2%Kuw #G23%&/uS#+@@AKLM&A123*&oOQ.@3%4@vHZ73()*#+.%y@S#*" +Og]Z.T#+^.! !1()`T#+ )(')*#+$73ydc/2@ %(,%[%@LMb`!18)(!K2.`#+ T#+@e2%7@ &!1:' )%U#+%b2%(%&Z@A6)(% ,&%(,6^`#)&`#+7@ p#+Â )&@S#` )*#+@W)(!K2.%&*D km!K2a`#+^6dc#+!K2@ (Y)&@.,.;$)& &Z-!8.@S#+ +O]^.T#+Z.! !1()`T#+ )*#+@nA/%&!1*H#+,?2!1POv#)&%0Y+On@S#+0MPZ%&0M&%*DGy % )*#+@@AMHE )%T#+%,&A4!$#+ #1+O:~%%K2a(2@ %F.0<0M(%$#?)(!K2.%&17T#%(,B]^.T#+Z.!!1()%T#+ )(*H#+,K w ,*##w%W.%"eOzn!8.@S#+ +O]Z.T#+^.!A/%&!1< /1>=D GT&@ ,+O']^.T#+Z.! /OV!$#+ (AU@AF%4?0<^b /DD<CY*&LM&*H'&U5^y%*#+@ `(,6T#+:2!1WOv#)&%0+OI@S#+0M8^%&0M&%)*#+6738%@LM(,X.;$)& &Z@A7ÂX#[]Z.T#+^.!)(!K2.%& < 9D@=D ,/%&!8.)%?Oz.,/0%[8T&@ ,?73()*#+.%b!1<!14,&b)&A42%<A/%&!1g.`(,8<D 0D7Â7T#+4H2.7@ ) #+. (e#+,8bIZ%&&BA!1<'2%!K&^@A' ,(2%*#, Er54#+@0M!CA #+%r7T#%(,6W#%.!K2 T#+"2!1OV#)&%01 )(!K2.`#+ T#+@@AZ`#)&`#+7@ @D+DX% %()(&Z@AMH^!8.)`[./eT#I73(&K2.eZ%8'.,/AP+O]^.T#+Z.!f/OV!$#+ #+,-!8.)`FT#y73(&X.,&%%^4,--%&!1+OI/OV!$#+ F(A-#+,X)(!K2.`#+ D:*&LM&$2%0<%(%KT#17a(&B@S#+0M&@A(%& )*#+@vDN 2Â4 )*#+@!K2@ &!1&Z`#+ +O#1OQ.@@e%)*#+@ G]^.T#+Z.!)(!K2.%& :LM&A-,/0;$)&.@*H,/.G%[PT#+%?%(]Z.%&!1&^%% %@S#+ +O~]Z.T#+^.! A/%&!1*Hl@ * %[@ &LM&@ K#+,.;$)& &^$#+,)(^%@@S#+7@ ^%&`#)& Dwc/2a&!1&^% T#(LM:73(&~23&oOV!1(,[' )`%(n7T#% ):23&`#+ ((,(,G')(!K2.%&w.)`$#]Z.T#+^.!f%&@ &2a`#+ HZ7.I@A8LM&AG!$#+@@]Z.T#+^.! )(!Gu2.%&% T#73(&~7.@ %G,#+%Kxz:%,&'+O /*;P]^.7% dH#+,@A[#8Oz& ]Z.T#+^.!#+@0M!1wT#(LM'73(&K!K2@ &!1&Z%(,Dn6.)`1&%dOz%@@%&!$#+e%73', 1 T&@ ,H/#+,1ynK[ w( n.@ ,[7a'L4 &n(,#+0Z#+"7%M#,&7T#)%40<%.,

E:FHG>IKJLG@M&[email protected](OQOQNU>RWVXP2OQNZY@T>[>\]N>VÔS_a`QIbOQN\B`QcKM&JdJdIb`,M8c3M&cbe&V&RWIf\WG@ghGM8JLT(OQOQNijV.[>Nkl\WV^k@M8\WO&m/

2 1. INTRODUCTION

+O ]Z.T#+^.!/OV!$#+ U(AU#+,FG@ 0uv%&!0MM#+@w+O!K2@ &!1&Z`#+ B+OOQ.@@%)*#+@ ]^.T#+Z.!)(!K2.`#+ DkmF b( n1Oz4)&.82ZA/ )*#+@w!K2@ &!1&^`#+ +O @ 0u,/ `#+)(~]Z.T#+^.!)(!K!b. )*#+ DBy.T#+^.! )(!K!8. )*#+ #+@%*#,/AT#K)(!K!1&%)&S#+@:#+22@ )*#+ *H ]Z.T#+^.! )&A42%0<`#+2 )XT#+%,/#+%MHy#+,nX)*#+#+@ %&^L^ #+22@ )*#+ ~dc/2a&!1&^%w1]^.T#+Z.!f!1()`T#+ )(neK'OQ..% /OV!$#+ $dc)%T#+0M"73&(&]^.T#+Z.! )(!K2.%&%Dw 7T# )%.T#+ YnW)(% ,&" "'60DS/<D0/

A B

!"#$%#&'()'*,+-!#&.%/021'*$354

0<T#+@&)(4,(,Z%K]^.T#+Z.! 7%8x,76 ! 58 'dc)%T#+0M(,73&(&YK2T#+ ( #+,$irDM #%*#+@ )y)%T#+&@vH^&%"'@@|7a"`!1" %",/.0b`#+!K % H/#+,n%()(&LM(,K`#+%'@@/l73n ,&Z )*#+@%: n' )`K#e%&^*DjkOT0 ,%P)(%()&' %<H4r]Z.T#+@mA[+Og:%()(&LM(,0<T#+@3'@@,()&%*#%rdc423&ZS#+@@A~'$@ &0<+O"$)`T#+&@vHI#+,U[`#+%[#+b' )%/Oz!$#+ )*#+731`#+oOV&%(,Oz%!_%&,&b%Y%()(&LM&P'@@#+@ %-,/%2dc/2a&^S#+@@AMD6Pdc#+!K2@ ~]^.7%8)*#+7a&)(4,(,GK 2a@S#+ #+ $+Oa2%I' )%[#+%"`&Ze%.0<[#+12 )*#+@7%<DErdu23@S#+ `#+ &@ *#,n%K#b,&0<`#,#+ Y+Ogr0<T#+@3]Z.T#+@mA~#+,#+7%2 @ *#,%~#1,()&%*#%(,-`#+%<D:r)(!K!8. )*#+ F`#+0M(:8!$#*AA%[)(!K23&`#+% 87ZA?!K2%*L4061]^.T#+@AU+O7%<Hl7.8.)`#+#+22%M#)` Wb%)*#+@S#+7@ <D[:0<T#+@]Z.T#+@mA,/%2Gdc423&ZS#+@@A'~)(!K!8. )*#+ B@ &0<H~,&!$#+,%Gr2%23& (+O:7%W]^. )%4@A[73()(!1!K23<%7@ r%K`#+ oOzA# @ &0<Y )&%*#%(,DG Kdc/2a&^S#+@w,()&%*#%GU0<T#+@I]^.T#+@AF#+,F)(!K!8. )*#+ `#+%G !$#+K2%7@ &!f' )`1!8.%e7a *LM&%)(!1"Oze]^.T#+Z.! )(!K!b. )*#+ [%r73()(!1L4S#+7@ <DkOe#+,-iT#+%8#12T#++OI&Z`#+0<@ (,?]Z.7%H&H|#ydc42@S#+(,-Y8)`T#+2%&7T#)%40<%.,(AMH&A)*#+dc)%T#+0M~/OV!$#+ '.P@ <`87ÂU!1*#+G+O]^.T#+Z.! %&@ &23`#+ D &%dOV%F?0M&&`#+@2%7@ &! +OK)(!K!8. )*#+ LM& A1)`T#+&@ ')*#+[73"%(,/.)((,~%by%2a()&\T)y2%7@ &!+Og(`#+7@ 01#+[&^`#+0<@ (,2T#+73&(&GY&,W+O"#)%T#+&@vD:9^345 %X+:93+,(:+2%$8#+%G2%%^)(@ ' )`T#*LMb73(&?,(0<(,X%[%@LMb :2%7@ &!YDk- :( :nK#+%b)()(&(,?'F#2T#+ )&.@S#+rA423b+On]^.T#+Z.! %&23*#+%&%*H30<T#+@@A62%23<%(,-7ZAXEy.T#+H3q.4Hgsn`#)#+,$ta@@ &n < 5@=|#+,GOVe e%*#%1%&!1(,1Erqjs'tTuvmA^23"%&23*#+%&%*D "Erqjs'tTuvmA^23%&23*#+%&1 $#X2%7T#+7@ )62%%^)(@yB' )% ]Z.T#+^.! !1&!1 (-xQvD <DN!1&!1 (OV:]^.7% "2@S#(A?#+X^%&0<`#+@e%@ <DWkm-G0<T#+@2%2a< #+@l7Â-Erqjs'tG]Z.T#+^.!!1&!1 ( &%r#+%!K ):!1&!1 ( ' )`#+@@ n(,Y%%`#+0My+Oj]^.T#+Z.!`#+%( +Og@0<^*H#+,B,((,N K 1.Oz[#+@@YA/%&!1KT#+16T#+@@"@ 4B#+*D :*&LM&$;=<#OQO O.m e>mG\d\WU?> @A@=BCBCB m g M&e.IED2\WOQ`:G m `QV&g"@#m

3

!$#+&!$#+ )*#+@e,(%)&2 ?+Oe82%%4)(@g y7%M#,&r#+,6n8A%$&!K2T# %b %.0<.y( *Dkm1'(% w%(y,/0/3&%&^n]^.T#+Z.! !1&!1 (Iul#+@@|7T#%(,1#+%!K )%`#+0My+O@0<^"`#+%( uw#+%r%&%(,Z%1`#+!1rE:qls'tTuvA42a2%%4)(@vDI :%'!1&!1A~ "0<T#+@T2%2a<`#+@T7ZAKErqjs'tlHM'%()(,$!1&!1AG#I2%23<%(,K7ÂP-.%)`1+ < 9A/2=#+,8%,P!1&!1A8T#l73(&P,&!1`#+%(,GGdc/2a&!1&ê#+]Z.T#+^.!2 )(b@S#+7a`#+%A+O ~hy &@ 8i Pkm.%<Hl'%(%.@%8%&23%(, < /5@=D #+!f+Oa nGT#7a(&K%r^LM(0Z#+%'n&@@'Erqjs'tTuvmA^23'2%%4)(@2a&oOV!1OV:*#)`F+Ow(%b!1&!1 (*DWW#+@@l%(P%A4%&!1HTP!K2a`#+^r2T#+`#+!1&%&%OVW%&23*#+%&P23&`#+ #+%~#,,/%(%%(,P~]Z.T#+@mAU+O"10M&&`#+%(,&Z`#+0<@ &!1&Z#+,G'#)% &L<#+7@ ")(!K!b. )*#+ $`#+%(*D 'dc#+!K *B(%']^.T#+Z (I,&23&,r2T#+`#+!1&%&%y+Oj!1&!1 ('T#+:#+%r.%(,H#+,LM&\OQA~T#+".7/udc/2a&^S#+@%)*#+@0$'Y)%T#+&@j@ &0<6)*#+Y73W#)% &LM(,OV'736+Ol&!YDj73n#+7@ w%:#+T#+@A/%ww%(w2%%^)(@ *HM#"!$#+&!$#+ )*#+@4OQ`#+!1&WOVjErqjs'tTumA^23 %&2a*#+%&%I I,&LM&@ 2a(,D ')(%.)&n#:!1^,&@$%&!1I+OaT#+!1 )<%)&@@S#+%%#+,U,&LM&@ 2!$#+&!$#+ )*#+@n!1&4,rOV7a#+T#+@A4 )*#+@n#+,F^.!1& )*#+@n%*#+!1&^+Ogy%&23*#+%& %A4%&!1Dlkm~2T#+ )&.@S#+: ,*#b+Ol#80M&&`#+0GOQ.)& ~OVi 0M@\u.73*L`#+oOV!$#+ *Ha,/.W%1PD 5aD|54%&%&H| "!K2@ &!1&^%(,?#+,6dc4%&,(,H3#+,-#23&.7T#+LM !1^,&@/OVeErqjs'tTuvA42a"%&2a*#+%&%e.%0#:0M&&`#+@2T#+`#+!1& `(,1]Z.T#+^.!!1&!1AU b2%(%&^%(,D- $!$#+&!$#+ )*#+@'%^@ G#+%$!K2@ &!1&^%(, *)+.'( ! 6X2%0<`#+!1*DiA!1*#++Oe2%0<`#+!1*H|Z%&%(0[2%2a& (:+Oe%(W%&23*#+%&A/%&!1P#+%[#+22%M#)%(,@S#+0M&@AU7ÂFZ.!1& )*#+@'!b.@S#+ *H7.P#+@ %67ZA#+T#+@A4 )*#+@)*#+@ )&.@S#+ *De "%^@ n,&LM&@ 2a(,1Ozw#+T#+@A/ IT#*LMy#:7%M#,& #+22@ )*#+ [T#+Jo.%(bA/%&!1'n8dc#+!K<Ha#+,Y&A)(.%b#1.7`#+ZS#+@j2T#++Oen73&, '( Dkmb#,,/ P%#+T#+@A/ g+OTErqjs'tTuvmA^23I%&23*#+%&%g.%0I%(n23()&\T)I]Z.T#+^.!!1&!1 (BA#+,b@0<Z@AP#+2T#+Oz%!pn!$#+WOV^)&.j+OI( LA #+@ %y)( ,&#W!1^,/\T)*#+ +O"%&23*#+%&n2%%4)(@a%W)&@.,y!K\c40G+O2T#+`#+@@ &@a)`T#+&@ *DI ,*#b# %P23((,$.2r)(!K!8. )*#+ 6`#+%<DnsnT#+&@3!K\c40G dc42@S#+(,Y#+&,Y+Ol( *H#+,6%!12%&@!KT#+A%(.@%y#+%W#+@ `K2%(%&Z%(,D5^.!K!K0[.2H/W(%%&^S#+@n2%(%&^%(,Y ( ")( %+O

• !1^,&@yOzErqjs'tTuvA423-%&23*#+%&%#+, ]Z.T#+^.! !1&!1 (7T#`(, T#+ou!1 )-<%)&@@S#+%%H %0M&&[' !$#+&!$#+ )*#+@:%4@ KOV1!$#+40^.!1& )*#+@#+,6#+T#+@A^ )W2%(,/ )& *D• 23&.7T#+LMn#+22%M#)%b%'Erqjs'tTuvA42aw%&2a*#+%&%g.0# 0M&&`#+@M2T#+`#+!1& `(,]Z.T#+^.! !1&!1AMD• T*#)%6+Ol%(W23()&\T)W#+%!K )W]^.T#+Z.! !1&!1 (HT#+6#+T#+@A4% "+Oj23&ouOz!$#+)(W+OW0<T#+@gErqjs'tF2%%^)(@j&!K2@ *A401 "!1&!1AMD• 2%&@!KT#+A[@ 4[#+ r./3()&'+Oj)`T#+&@!K\c40POVErqjs'tTuvmA^23r%&23*#+%&%*D

4 1. INTRODUCTION

& (%*XBB %Q * r( ',/L^ ,(,YZ%$&0<^)%T#+2%&%H'YrOV@@ *'0[.)&.% (!?%*X xQ )%T#+2%& lkmZ%4,/.)((j!$#+K7/J()&LM(l+O|wP#+,G.@(( *D (!?%*XW I%*L4 ,(g%*#,&j'W()((%`#+A:7T#)%40<%.,W4*'@ (,/0MIOQ%!]^.T#+Z.!!1()%T#+ )(r#+,Y]Z.T#+^.! /OV!$#+ 6(AMD (!?%*X7 c/2@S#+P~)()(&2K+O]^.T#+Z.!R%&23*#+%&%b,&`#+@vHe2T#+ )&.@S#+WErqjs'tTuvA42aW%&2a*#+%&*D (!?%*XPI c/2@S#+"0M&&`#+0[Oz.)& Y!1&^,HT' )%- '7T#)%73#+@@.y#+T#+@A^ )*#+@#+,Z.!1& )*#+@)*#+@ )&.@S#+ *D (!?%*X . Er(%)&73(G#623&.7T#+LM#+22%M#)`B%-Erqjs'tTuvmA^23~%&2a*#+%&%P.0F#0M&&`#+@g]Z.T#+^.! !1&!1A (!?%*X Er(%)&7a(G*#)`+O"[%([]^.T#+Z.!R!1&!1 (P' )%#+%[.%(,Oz%&2a*#+%&%:#+,Y2%(%&^%8%(.@%'Oz%!#+T#+@A4 )*#+@l#+,Y^.!1& )*#+@#+T#+@A/ y+O*#)%6+O&!YD (!?%*X \ s)&@.,(r-8%(.@%"OQ%!)%T#+2%&W=$#+,-)%T#+2%&WC1#+,X.@(OQ..%aD (!?%*X Er(%)&73(~X ,*#+OP)%T#+&@!K\c/0#+, %&23% 2%&@!KT#+A%(.@%*D

y2.2a<`+O )%T#+2%& n%82%L^ ,y%*#,& '~"()((% #+A1(%& )*#+@7T#)`^0<%.,B%X.,&%`#+,* ]^.T#+Z.! %&23*#+%&%Kn#+,%?.,&%`#+,!14,&@ w#+%0M0P%8.`'OVw&"#+T#+@A4% *Dk["%%()& +Oy)`T#+2%& n,(%)&73yL#+ .&@ &!1&Z%'+Oj]^.T#+Z.! !1()`T#+ )(*DIkm$:%()(,[`()& ~0<LMr#+^%^,/.)& -%1]Z.T#+^.! )(!K2.0~#+,6]Z.T#+^.!)(!K!8. )*#+ D "! #%$'&)(*&,+ '%*#,&w w#%.!1(,K%W7aOv#+!K@S#+I'$]^.T#+Z.!!1()%T#+ )(n1"E:`#)'7`#uvM&OV!8.@S#+ N#+,B.@ ,B#+@ %XT#*LM!1&P'!1<K+O:~)(^%&Z%[+O K%()& 73dOz%<H/7.#"(%W#+%W)()(&2%'' )`6'@@73r.%(,dc4%&%LM&@A[!$#+734,/A+OT (% j !$#+M"#:%&!K,&e&%<D ,:2%(%&Ze2%4+OVe+Oa#+ZAG+OT)&@S#+!1n$ n%()& Dw5^.)%~2%^+OVI!$#*AK7a"Oz.,$1"@%&`#+.%<H/<D 0D < /*9H^9<9HZ9H3>=D-/.102.30 465879:7<;8=?>A@CBEDGFH7<IKJL@?MNIkm-b..T#+@lOV!8.@S#+ FP`#+%(r+O#$]^.T#+Z.!%A4%&! #+%bLM()&%%r?#~y@7a&2T#)([,(%)&706[A/%&! |ψ〉 ∈ H Dy7%&L<#+7@ (G#+%[:&!K%*#+23&`#+%%b 2T#)( #+,Pn`#+%(&LM@LMP!1n7ÂW.`#+A8`#+Oz!$#+ *D Ey()*#+@@4T#+l]^.T#+Z.! `#+%"'$%`#+%"LM()&% |ψ〉 !$#(A1(]^.T#+@@AGn&@@T73",(%)&73(,17Â1#+[23&`#+%*H)*#+@@ (,6OZ+.78 ! - $&9 +.%Q W$%dH|,d(,-#

ρ ≡ |ψ〉〈ψ| xV9/D0/<D0/y@@g/OV!$#+ ?#+73.:W2ZA/ )*#+@A/%&!)(^`#+(,- |ψ〉 r#+@ %[)(Z`#+(,X

ρ#+,Y&LM&A401' )`6!$#(A~73W)*#+@ )&.@S#+%(,Oz%! |ψ〉 )*#+-#+@ %G73)*#+@ )&.@S#+%(,Oz%!ρ

Dkm6#P0<LM&Y7T# |ei〉Oz H n!$#*A'%

ρ =∑

i,j

|ei〉〈ei| ρ |ej〉〈ej | =∑

i,j

|ei〉〈ei|ψ〉〈ψ|ej〉〈ej |xV9/D0/<D 9M

=

6 2. BACKGROUND THEORY

&)(<H4 7T# r,&A~2a&`#+%' %&2%(%&^%(,7ZA$ OZ+278 ! - % ! '&@ &!1&^% 〈ei| ρ |ej〉D b%T#+@@gy,/ %0<. -73&(&?823&`#+%r#+,6%!$#+\c%&2%(%&^`#+ #+,F'@@.%F?%&!1-,&A !$#+\cp#+,,&mA 23&`#+%^%&%)%T#+0M*#+7@AMD b#)& 6+Oe#+Y23&`#+%

A H 6,&A!$#+\c '0<LM&Y7Â

AρA† xV9/D0/<D21MkOgr2a&oOV! #P!1*#.%&!1&Z+Olr7%&L#+7@

O'Y&0M&^L#+@.(

oi&2%7T#+7@mAW+O/7`#+0*#)`823<%7@ e.%)(!1w#+,Ww#%%4)&S#+%(,r2a<%ouv!1*#.%&!1&Z,&mA[!$#+\c6#+%0<LM&67ZAD

P (oi) =

(Piρ)#+, PiρP

†i

(Piρ)

xV9/D0/<D 4^'&%

Pi $62%+J()& 2a&`#+%Z%6&0M&2T#)(?#%%4)&S#+%(, '

oiDpkOV@@ *"'T#+Wdc423()&`#+ YL<#[email protected]+O

O "!K2@A

(Oρ)D &G A4%%&! )(!K23<%<HM)( 0<D 0D+O|.7%A4%&!1

A#+,

BH&G,&mA~23&`#+%% OV'.7A/%&!1')*#+Y73r,&LM(,OQ%!T#+y+OjJZA/%&!7Â[`#+4012T#+S#+@`#)(

ρA =

B(ρ) ρB =g

A(ρ)xV9/D0/<D =M

G23&`#+%%ρA, ρB

#+%G#+@ `~)*#+@@ (, %X+ O368+ O OZ+278 ! -1$&93+.%,(W$%,8MD8e%(,/ )& 8#+73.!1*#.%&!1&^%K[.7%A4%&!1P)*#+7a[!$#,[OQ%!ρA, ρB

dc#)&@AU~`#+!1 #(AY#",(%)&73(,OVρ

#+7aLM<Dgn~2%23& (:)`T#+`#)&%& %Pb`&:+Ow,&mA623&`#+%% '?23&`#+%ρ

H '#b,&A~2a&`#+%#%`^)&S#+%(,['%!1r`#+%Kxzn!K\c/.%<H%(yydc/'%()& |\Oe#+,Y@A\Ol' '23<LMW#+,T#"."`#)(〈ψ|ρ|ψ〉 ≥ 0 ∀ |ψ〉 ∈ H g

(ρ) = 1xV9/D0/<D CM

km Oz@@ " Oz%! r2a<%L^mA~T#+:#+ZA,&A23&`#+%" #+@ %1:&!K%*#+D-/.102. - >'7J2; IK7 ,@ JL@ 7<=yK%*#ÙT#+Wn$.%G1,&mAF2a&`#+%8Oz!$#+@ ! T#+88#+22@ (G#+@ %%)(&%&^Y!K\c4.%(+OG`#+%(Y#+, &)(? !1%?0M&&`#+@T#+?`#+%?LM()&%OV!$#+@ !YD5^.223<%bT#+.:4*'@ (,/0MG+OwG]^.T#+Z.!A/%&! y@!K%(,?%[T#+rnK#+%.T#+7@ b%#%)&7ab%[r#[0<@ P`#+%<Ha7.y`#+&*#)%F`#+%8Oz%!P%& |ψi〉

4*'%X^)()&.G'2%7T#+7@Api

He'&% ∑i pi = 1

D &!1%~T#+B~+OE <\dRWIK` \Wc JdU(O,M#IKN>e>YH\WG>OQJdO i0V8RWg [>c M8O M8RWO P&M&cbIKk V&N@c'i0V8R U RWV*OQ`S\WIbP2O gaO,M8Jd[>RWOQgaOQN\,m OQN@O R:M&cbIbJdO,kgaO,M&Jd[ RWOQgaOQN\WJ7BLIf\WGlgaOQM&Jd[>RWOSgaOQN\ V&U(OSR:M8\WV8RWJ BLG@Ib`:GMXRWO N>V&\HN>OQ`QOQJdJWMXRWIbc V8Rd\WG@V&e.V.N@M&c(U RWV*OQ`S\WIbV.N>JLM&NkV.[ \W`QV.gaOQJ BLG@Ib`:G MXRWO N>V&\ OQIbe.OQN#P&M&cb[@OQJ V&i5M&N V.T>JdOSRWP.M8T@cbOBg M T(O k ON>O,kZY2T@[>\JdIbN@`QOBHO k>O,M8cV.N>c BLIf\WG\WG@O U RWV*OQ`S\WIbP2O `,M8JdO IKNl\WG@IbJ]\WG@OQJdIbJQYBHO \:M.O#m.m ! BM8JBV&[>R k>O"N@If\WIbV.N$# %& m

2.1. A BIT OF QUANTUM MECHANICS 7

pi

#+%/uv*&%HTWA/%&! ,(`)&7a(,7ÂY#G`#+ )*#+@g")(&%&^y!K\c/.%+O"`#+%(*D ~`#*AXT#+P$%A4%&!_ W# ! + O 8.W(:+(D &#+@@w7.P[+O'pi

L#+ %FKA4%%&! r# 9 %X+ 8.W(:+(D K,&AF23&`#+% W,d(,?OV:!K\c(,`#+%("rOz@@ '0$ #(A ρ ≡

∑

i

piρi =∑

i

pi |ψi〉〈ψi|xV9/D0/<D 3<

y@@2%23& ('@ %(,6`()<D9/D0/<D0/@ ,6#+@ %POV' ",d D5^`#+ % )*#+@|#(LM&`#+0M(n*LM&n#:!K\c/.%#+% LM&AG*#@AP)*#+@ )&.@S#+%(,KOQ%!ρ

D &&LM&n#+22@AW#'@*#+j!$#+28%8xV9/D0/<D 3<nw0M&g!K2@Ae&0<Z%(,G.!p+O j!$#+2P#+22@ (,%[*#)%?+O82.%8`#+%b,&A6!$#+ )((ρi

H|vD <D W!1*#+D'rdc#+!K2@ <HOV"dc/2a()&`#+ 6L#+@.

(Oρ) =∑

i

pi

(Oρi) =∑

i

pi 〈O〉i = 〈O〉 xV9/D0/<D25Mkm" :#+@ %1*#A%1%(%"'&&b`#+%W '2.%b!K\c(,6)(Ozy#+ÂY,&A!$#+\cH

ρ '2.%W\Oe#+,@A\Oe

(ρ2) = 1D K`(bT#+KT#(LM1#~ )(GT#+,/@ $U,0Y)(!K2.`#+ OVWA4%%&!1y' )`73&T#(LM82%7T#+7@ )*#+@@AMH#+,Y '@@j)(!1W-T#+,/A'&6873&0<-.:.,/A+O!K2@ &!1&^`#+ :+Or%&23*#+%&*D

-/.102. > 73>A@CB !1*#%.%P+Onb)&@ <%&(`:+O]Z.T#+^.!`#+%(: :`!1&0~GT#+@@((,Xdc/%&ouLM&@AMDw5^.)`6#W!1*#.%: 2%*L4 ,(,7ZA$ OZ+ ! -<D yT,&@A+Ol#+#+7`#+A~`#+%ρ

'6%(2a()&'%[#P2.%W`#+% |ψ〉 HTnW,d%17aF (ρ, ψ) ≡

(ρ|ψ〉〈ψ|) = 〈ψ|ρ|ψ〉 xV9/D0/<D2DMl ,&@A )*#+ #+@ %7a-,d(,N 6)*#%6'&%673 %`#+%(~#+%-!K\c/(,H 7.~ ,d -'@@.;$)(rOV"."2.2a<`( Dkm' ''T#+%,%12%*LMT#+

0 ≤ F (ρ, ψ) ≤ 1xV9/D0/<D0/*;M

'F (ρ, ψ) = 0

\Og#+,$@AK\O.223w+Oρ

w~#8.72T#)(y0MT#+@|% |ψ〉 H#+,F (ρ, ψ) = 1

\OI#+,6@A\Oρ = |ψ〉〈ψ| D'.&!1%<Ha'OV@@ *"'Oz%! 8)&A4)&@ )2%23&A$+Oy`#)(T#+n"T,&@mA1 I2%(`&LM(,$.,&.`#+A1`#+oOV!$#+

F (UρU †, U |ψ〉) = F (ρ, |ψ〉) xV9/D0/<D0//; V&\WO \WG@V&[@e&G YH\WG@M8\ #m&m % k VOQJ N@V&\ e&OQN@O R:M&cbIbJdO k IbRWOS`S\Wc JdIbN@`SO F R

(ρρ) < 1IfiρIbJ N@V8\U>[>RWO.mF5V OQN>Jd[>RWO

F (ρ, ρ) = 1i0V8R gaI O,k J\:M8\WOQJQY4MlRWVV&\ IKJ IbN@`Qcb[@k>O,k IKN\WG@O k>O"N@If\WIbV.NZm *N# % &

F (ρ, σ) ≡F R pρ1/2σρ1/2

m O JdGM8cKcB]RWIf\WO

F (ρ,ψ)V8RF (ρ, |ψ〉)

M8cf\WOSRWNMX\WOQc M&JLIbJLgaV&RWO `SV.N#P2OQN>IKOSN#\,m


kmYW)*#%8'&%W736`#+%(r#+%!K\c(,H7.y)*#+-7a8!8.@`#+(.@A-,/S#+0MT#+@ %(,HF (ρ, σ)

)*#+B73~0<LM&N#X0M(!1& )Z%&2%&`#+ D 0ρ =

∑

i λi|ei〉〈ei|#+,

σ =∑

i κi|ei〉〈ei|nT#*LM

F (ρ, σ) =∑

i

λiκixV9/D0/<D0/*9M

' )` 8%!K2@AX$&82%4,/.)&b+O'1LM()&%%P)( 0X+O'1&0M&ZL<#+@.(G+O~`#+%(*DXk[0M&&`#+@ )*#`$?.)`^%&2%&`#+ T#P7a(&Oz.,MH7.P!K2@ 8`#)(8,d BxV9/D0/<D2DM !1*#+yT#+WT,&@A *#%A~%$n'?#+,Y%&L<#[email protected]#+%<HT#+,Y%G'%&!$#+:#PL#[email protected]#+7@ W%^@ < 9<9>=DkmW G#+22T#+%&ZGT#+81T,&@AU bP#Y!1& )<He)([#Y!1& )$ 8!K!$#+@Oz ,&^ )*#+@j`#+%(*D &%b#+%8&y,/ `#+)(8!1*#.%("' )`X,KA4 &@ ,!1& )(:Y%&:+On,&mA62a&`#+%%#+,XT#(LMG!K2@ b0M(!1& )PZ%&2%&`#+ *H7.yb2%dOV&:T,&@mA7a()*#+.%+O%"!K2@ )&mA6#+,7a()*#+.`' 'W)(!K!1-!1*#.%@%&`#+.%W-]Z.T#+^.! %&23*#+%&% < 1HC/H:5H9A/.=D-/.102. ";,@ J2; 37 7<;,@ @ w%()& e#+73*LM ,*#+@'b!$#+&!$#+ )*#+@/OV!$#+@ !+Oa]Z.T#+^.!!1()`/u#+ )(*H/ y%()& #+73.y#P0M&Z.82ZA/ )*#+@2&!1&D &&LM&y#K]^.T#+Z.!A/%&! )( %"+Oe%&LM&`#+@2T#+%&% #G23<%7@mA~Oz<%12T#+%8%73$)(%&@S#+%(,#Y0<@AX/u)&@S#` )*#+@n!$#+&*D 12&!1& K)*#+@@ (, +24W +. +.4VDkmB]Z.T#+^.! )(!K!8. )*#+ &Z`#+0<@ &!1&^[ K)(!K2@ &%&@A(%`&ZS#+@OV'/Oz!$#+ 6`#+oOz&#+,YW&Z%2.23<%+O%&23*#+%&" '%G0M&/u&`#+%1#+,?,/ 7.%1&^`#+0<@ &!1&ZG#+!106,/0/3&%&^@ ^)*#+ Db:&)(G )()(&2 ")(&^`#+@OV"rnY '(% *DT!$#+@@A&^`#+0<@ &!1&Zr ",d(,6#'Oz@@ "

q& |ψAB〉,&%rnJo^"`#+%r+OlA/%&!1

A#+,

BDe &H/'&&LM& `#+%K)*#+W73b'%&#W#~2%^,/.)&W+O `#+%(W+OnK%.7A4%%&!1

|ψAB〉 = |ψA〉 ⊗ |ψB〉H/" '`#+ ,Y%G7a +.W + OY $ 8+:9(%Q6 +&D

`#+%b' )%F y&Z`#+0<@ (,? :)*#+@@ (, 8+:9(%Q6 +ZDWe^`#+0<@ (,F`#+%(:7a&mn(&#+ZA^.!873&y+O2T#+ )&2T#+Z%y '2a<`7@ <H/7."&%'@@g)( ,&@A~72T#+%8&^`#+/u0<@ &!1&^"uIT#+" "&^`#+0<@ &!1&Z:7a&mn(&$J .'mn[A4%%&!1*D 8*# ('#*A%K.,&%`#+,Y&^`#+0<@ &!1&Zr '7Âdc#+!K2@ <D5/#*AT#+"nT#*LMmn- ,&Z )*#+@ ]Z.T#+^.! A/%&!1b,(%)&7a(,7ZAFmn+u,/!1& T#+@"y@7a&b2T#)(( HA#+, HBD?w#)`+O(%[T#(LMY#67T# K#+,HeM(&20#YOz%(0<^%(,&AM~~%()du #+7a.~]Z.7%[7a&@ WH nX,&%Y-&@ &!1&Z% |0〉A , |1〉A

#+, |0〉B , |1〉B%(23()&LM&@AMD -)*#+ & ,d6OV.[!K23`#+Zdc#+!K2@ (~+O8&^`#+0<@ (, `#+%(*D <#V.gaO MX\d\WOQgaU>\WJHG@M,P2O T(OQOQN g M.k>O IbN \WG@O `QM&JdO V&i D[>T@If\LJ\:M8\WOQJQY>O.m e>m # %& m


W`+u)*#+@@ (,+ C82W(:+,8 ∣

∣Φ+⟩

=|00〉AB + |11〉AB√

2

∣

∣Φ−⟩ =|00〉AB − |11〉AB√

2

xV9/D0/<D0/1M∣

∣Ψ+⟩

=|01〉AB + |10〉AB√

2

∣

∣Ψ−⟩ =|01〉AB − |10〉AB√

2 w)*#+W.%II+O/wi &@@Z%`#+%(%'@@.`#+%wI)(%&@S#+ g73&(&P&^`#+0<@ (,A/%&!1*De5^.223<%"y`#+%+O)(!K23<%:A/%&!f |ψAB〉 = |Ψ−〉 #+,$ny!$#+M#P2%+J()&LM!1*#.%&!1&^:A

Z%KW7T# *H/vD <De!1*#.%7%&L<#+7@ MA = λ0 |0〉AA 〈0| + λ1 |1〉AA 〈1| λ0, λ1 ∈ R

xV9/D0/<D0/64^ 23<%7@ .%)(!1(y+Ol "!1*#.%&!1&Z#+%&0M&ZL<#+@.(

λ0, λ1DI &A*#)%4)()&.''Y2%7T#+7@A 1

2

#+,W)(%(23,/0$23<'!1*#.%&!1&Z:`#+%(#+%

λ0 :P0|Ψ−〉√

1/2= |01 〉AB λ1 :

P1|Ψ−〉√

1/2= − |10 〉AB

xV9/D0/<D0/*=M'&%

P0 = |0〉AA 〈0| #+,P1 = |1〉AA 〈1| #+%2%+J()& N23&`#+%%1Z%U&0M&2T#)((l)(%(23,/0y%λ0, λ1

Dgh:%IT#+g&%w 23&oOz()&l)(%&@S#+ b73&(&P!1*#.%&!1&^8.%)(!1(W#+,XP%`#+%P+O A/%&!B

#OQ%&rP!1*#.%&!1&Z*DPkOλ0

#:7`#+(,H|&6W23<"!1*#%.%&!1&Z`#+%8+OB

|1〉 HT\Oλ1

#:7`#+(,G`#+%P |0〉 Dyr%4?#rP!1*#.%&!1&^WA

T#:7a(&?23&oOz!1(,XG^'U)(&`#+ÂF'T#+W.@ ,F7aGK.%)(!1[+O #!K@S#+b.7%(]^.&ZW!1*#.%&!1&^B

Dkm1%%&@\O&%y I0P/u)&@S#% )*#+@3#+7a.n w4,1+O)(%&@S#+ D ".@ ,7`#+ 6`#+!16%(.@$\OW6&%6%2 )% #+[`#+,! Oz%! #U@S#+0M-&`&!87@ X+O2T#+23&b@2HjT#+@\O+O'' )`&%$@S#+73&@@ (,'/*;6#+,U~&PT#+@\Oy;A/<DYyn&LM&K!Gu@S#+K)(%&@S#+ 1#+%[Oz.,B#+@ %-'& |Ψ−〉 #+%[!1*#.%(,&G7T#%(*Hw#+,7ZA)(!K2T#+0Gdc423()&`#+ $L<#+@.(wOQ%! ,/0/&%&Z!1*#.%&!1&^%*H4w I23<%7@ "%87`#+`#+ % )(K' )% n.@ ,N7a!K23<%7@ OV[#F)&@S#` )*#+@A/%&!YDX%2%()& %&@A%+u)*#+@@ (,?sr54(]^.T#+@A-)*#+673W7%M&D"km y.% ,WP%)(23W+OI (% "%0M8Z%8,&`#+@ HZ7.(%%&ZS#+@@A$:s54 (]^.T#+@A$ IL<#+@ ,KOVIL#+S#+7@ (n' )%#+%)&@S#%% )*#+@gXb%&`WT#+:&AYT#(LMbL#+@.(:' )`?#+%8c/(,-2 :%$!1*#.%&!1&^xQ8#`.!K2 6+Oe%*#+@ ![dH|#+,Y&)(7ZA7%*#+40$"8%*""T#+ )*#+73$)*#%$OVG]^.T#+Z.!L<#+S#+7@ (*DF:&%~n#+%~%.)%0F.23[OV.,#+ +Oe]Z.T#+^.! !1()`T#+ )(*HTW(A+Oe ,,&YL<#+S#+7@ (r#+,rOv#+!1.I E2T#+`#u,*c3D Ey*#,&%'$n.@ ,-@MW4*!1%b#+73. r#+%W%dOV&%(,6%KG#+ )&@ (:+O>M-i&@@ < /2=Dn54(rOVdc#+!K2@ mi &@!$#+ y%4)%/#+,WT#+.%8+O%*#+@A:OVy#,/ %)&.` Y+O&^`#+0<@ &!1&Z*D F5V V.T \:M&IbN'\WG@O U RWV.T@M&T@IbcbIf\WIbOQJaV.N>O `,M8N N@k \WG@OlRWO,k>[>`QO,k'k>OQN>JdIf\ 'g M8\dRWI ijV&RaJ J\WOQg

A>ρA =F R

B(˛

˛Ψ−¸ ˙

Ψ−˛

˛)m4FHG>OQN

P (λ0) =F R

A(|0〉AA 〈0| ρA)M&N@kJdIbgaIbcKM8RWcijV&R

λ1

Y>`.m i*m4JdOQ`.m #m.m.m O"N@OQk \WG>IKJ BHMYZ\WG@O U>RWV OQ`S\WIbV&N V.U(OSR:MX\WV&RWJ M&`S\ V&N@c V.N J J\WOQg

A \WG>O^V.U(OSR:MX\WV&RWJ IbN#m&m % M8RWO RWO,M8cbc

P0A ⊗ B , P1A ⊗ B

m V=BHOQP2O RLIb\BIbJL`QV.N#P2OSN#\WIbV&N\WV2OSOQU\WG>O IKk>OQN\WIf\ lV.U(O R:M8\WV&RBIbgaU@cbIb`QIf\,m


5^)('&^`#+0<@ &!1&Z e+O3%.)%K!K23`#+)(<HMen.@ ,17a .%dOz.@/%rT#(LM %%w# #(A+O %(0OV8&Z`#+0<@ &!1&^G#+,%()(,#!1*#%.%1+O$,&0<%($+O'&Z`#+0<@ &!1&Z23<%%(%`(,[7Â#K0<LM&6`#+%<D 6,d #+73*LM6,4(K[@ &,B%%&@\OLM&A&@@%F%(%0H%)(2%*L40/udc/ %&)(+O~`#+%( |ψA〉 , |ψB〉 b0M&&`#+@"#6,/0;$)&.@G`#3D?.T#+%&@A&%1#+%K&W%(%%r' )%#+%G!b.)%%!K2@ &*DP`#+%G W%&2T#+`#+7@ G\O"#+,F@AX\Or%(,/.)((,-,&A!$#+ )((+O%".7%A4%&!1"#+%W2.%

|ψ〉AB

%&2T#+`#+7@ ⇔ ρA =

B(|ψ〉AB 〈ψ|) 2.% ⇔ (ρ2

A) = 1xV9/D0/<D0/*CM

'&%A,B

!$#(AK(]^.T#+@@AP&@@|73 Z%&%)`T#+0M(,D rLM&[#r!$#+\c1I n`#+0<ZoOV#+%,%K)*#+@ )&.@S#+%8r`#)(<H|#+,%G #P0M^4,2`#)& )*#+@g%(% Oz"&^`#+0<@ &!1&Z*DErd08#0M^4,P!1*#.% +Oa&^`#+0<@ &!1&ZeOV!K\c/(,G`#+%( l!8.)`KT#+%,&<HM#+,2%(%&^@AF&%$ 8-#+0<%(&!1&^G#+ZAU0<@ $!1*#.%[1@%&`#+.%<D Ey#+&#$L#+ &mA6+On,/0/&%&Zr!1*#.%(rdc/ *DrX*LM&L4 &f*LM&%!1P+Ow(%<Ha'-%dOV&ou&)((*H4!$#*A$73"OV.,~ & +2 < 1<;>=D #+%:0M0G%P((,#+&Z`#+0<@ &!1&Z!1*#.%y)('[ w( wn'@@T73#+!K0bJ .#+w0M&&`#+0G&Z`#+0<@ &!1&Z*H7.r#+0M&&`#+0#K2T#+ )&.@S#+&Z`#+0<@ (,?`#+%<D T#+8((,6&- :#K ?+O)&@ <%&(`b+O.b0M&&`#+%(,%`#+%(8%- 8 ,*#+@ `#+%<He#+, 8 b2%*L4 ,(,7ZAFT,&@mAMD?5^.;$)(~P%6`#*AUT#+*H'T#+%&LM&G!1*#.%$ G#,2%(,He~i &@@`#+%(G#+%' ! ' - +.4W + O#+,1OzImn+u]^.7'A4%%&!1n#+@@T!$#c4!$#+@@A[&Z`#+0<@ (,`#+%(#+%%&@S#+%(,K%r'i &@@`#+%(e7Âb0<@ du]^.7I.`#+AP`#+oOz!$#+ < =H1<;@=Dj:&)('&Y0M&&`#+0~&Z`#+0<@ (,-`#+%(*H/73('n8)*#+Y2a:OV' #Gi &@@g`#+%<D

IZ`#+0<@ &!1&Zy)(!1( #+73.w~!$#+ZA$#(A/ $T#+.%<Dwdc#+!K2@ r#+%!1n' )`^%&`#)&'X@0<^$xz<D 0Dr%.0<#+7%2 #+,?23Z`#+(.W&!K %% |"0M&&`#+@@A73()(!1&Z`#+0<@ (,6'Y@0<Z"T&@ ,Dns@@ "73&(&X#+%!1")*#+X#+@ %G0<LMr %%-&Z`#+0<@ (,`#+%(*DYkm]^.T#+Z.!R)(!K2.0U#+,)(!K!b. )*#+ %)%&!1(G~+OK0<%*#+b)%T#+@@ &0M(b %M(&2U1%A4%&!Oz%(1+O .,(`#+7@ $&^`#+0<@ &!1&ZG'~&ZL4%!1&Z*Hw'@ $2%(%&L40?)(Z%@@ (,B&Z`#+0<@ (,`#+%(P'DXg?)&%*#+%&^`#+0<@ &!1&Z'w n2%dOV&`#+7@ %8.%]^.T#+Z.!A/%&!1I' )`Z%&`#)& %0<@A1'*#)`&*Hg7.G#+W1`#+!11!1KnK((,KZ%&`#)& 'U1%.%.,/0M%73G!K!$#+@vHg&)(1#+ZAF,(0<U&%1!8.W731#~`#,du@/B7a&mn(&(%K./()&%*DIr&&`#+0G&Z`#+0<@ &!1&Z #)(Z%@@ (,~!$#+&w wn*#AMH/#+,1T#+.`#+@@Ar!1&4,.%(,~!8.",&23&,Yr2ZA/ )*#+@T#+.%W+Ol%&.2Z%G' )`Y' ^%&0<`#+%(,D-/.102. J)I MN; > D = > JL@ M I ,B = @ 7 =j#+T#+@A/%bP2a&oOV!$#+)(K+OnErqjs'tTuvA423b%&23*#+%&%rnP'@@!14,&@l&!7Â6T#+ou!1 )[<%)&@@S#+%%*DYkmU P%()& 1^%^,/.)(~)()(&2%bOQ%!_1(AU+O"T#+ou!1 )Y<`)&@@S#+%%$T#+16T#+@@",B((,BOV*DN ,*#$7a&@ #+%.`(,BN.

V8RBU@[>RWO J\:M8\WOQJLV.N>O g M [@JdO \WG@O P2V.N OQ[>g M&N@NlOQN\dRWV.U m


!14,&@n+O"~Erqjs'tTuvA42a[%&23*#+%&*HI#+,#+%~#+22@ (,[)`T#+2D%4?[0M&&ou#+0~Oz.)& #y&@@w#rF.7%(]^.&Z%()& r' )%U,*#+@e'F23()&\T)P2ZA/ )*#+@A/%&!1*DEy()*#+@@eT#+K`#+%K2T#)(K+O K]Z.T#+^.!du,/!1& T#+@!K2@ GT#+!1 )<%)&@@S#+% "2T#+(,7Â~ 4 6X+.% 8.W(:+A8 |n〉 H

n = 0, 1, 2, ... De 8#+@S#+ #+,Y)&%*#+ -2a&`#+%%a, a†

T#(LMWrOz@@ '0$2%23& ( [a, a†] = 1

xV9/D0/<D0/3<a|0〉 = 0, a|n〉 =

√n|n− 1〉, n > 0 a†|n〉 =

√n+ 1|n+ 1〉

^.!873&`#+%(y)*#+7adc/2%(%%(,%&!1y+Oa†

# |n〉 =

1√n!

(a†)n|0〉 xV9/D0/<D0/5M `#+% |0〉 %!1&!1(e)*#+@@ (,K#L#)&..! `#+%<DlkP ]Z.T#+^.! ,(%)&2 H<@0<Zxzw#+ÂP&e73<% )T&@ ,Tl %&2%(%&^%(,K7ZAG'e!1%'T#+!1 )"<`)&@@S#+%%*HZ#+,:L<#)&..!&Y)(%(2a, %Gr`#+%:'&%r&%W#+%rP2%':T&@ ,D5^)(XX7T# `#+%()*#+ 7aXdc/2%(%%(, &Z%&@A %&!1+O

a†#+, |0〉 H:#+@@&"`#+%(+OlrT#+!1 )W<`)&@@S#+%"!$#(A#+@ %G73dc42%(%`(,[(`r%&!1*DI!K23`#+Z:)&@S#%%('+Oe`#+%(y#+%'6X$)+.%X+.4 8.W :+,8b#+, 822 +X+>+ O 82W(:+,8&D P)(&%&Z`#+%(r#+%b80<^r&0M&%`#+%(:+OIG#+@S#+ F2a&`#+% 'Oz#+Â8)(!K2@ dc8Z.!b7a&

α&% dc/ %l#`#+% |α〉 .)%PT#+

a|α〉 = α|α〉 De )(&%&Z`#+%(:#+%0MT#+@l7.,db#+X*LM&%)(!K2@ &%b7T# 'OV"W%`#+%W2T#)(<Dk%&!1"+OW)&%*#+ X2a&`#+%")(&%&^:%`#+%(#+%

|α〉 = e−|α|2

2

∞∑

n=0

αn

√n!

(a†)n|0〉 = D(α)|0〉 xV9/D0/<D0/DM'&%6O ! 8:9 68+. +.4^$&93+2%Q(W$(%

D ",d(,7Â

D(α) ≡ eαa†−α∗a xV9/D0/<D 9<;My@@n2%23& (1+O)(&%&^$`#+%(G!$#(A7a~,&LM(,Oz%! ,/ 2@S#)(&!1&^$2/u&`#+%*Hj'<%K2%23& (W!$#*AX.73K,&LM(,FOz%!(]Z.T#+ ~xV9/D0/<D0/3<dD[kmF0M&&`#+0OQ.)& !1&^, +OW%()<D 4D0/<H 6!$#+M-dc4%&LM6.%Y+O8,/ 2@S#)(&!1&^23&`#+%%*DIZA/ )*#+@@AMH)(&%&ZG`#+%(b#+%K !1<b)&@S#%% )*#+@ `#+%(8+O'KT#+!1 )~<ou)&@@S#+%*DK &AX%&2%(%&ZbK22a<%%G#(LMduv@M[)*#%G%K^.!873&8`#+%(*H' )`#+%$1!1<W2T#+ )&@ duv@M`#+%(D~D 0D$)*#`1+O'@0<Z*H1^.!873&P%`#+% |1〉%&2%(%&^%~#0<@ Y2% '@ Y-)(&%&Z`#+%($0<LM-#F0M44,,(%)&2 +O.2.GOz%! #X@S#%&K' )%2%NZ.!873&1 KGc(,7.GOz@@ "[#23 %%S#+Y,/ 7. < /*9/HT9<;>=D,FHG>O N [>g T(OSR J\:M8\WOQJ M8RWO M8cbJdVa`,M&cbcbO,k V` J\:M8\WOQJQm


8%]Z.(&*(,-`#+%(y#+%8/u)&@S#% )*#+@vD &AY#+%8`#+%("Oz' )%XW]Z.T#+^.!.)(&`#+Â1+Oa`!1'7%&L#+7@ I7a&@ B'`#+,#+%,K@!Ke0<LM&17Âb':& %&Z73&0.)(&`#+Â%&@S#+ D:&%?nU)( ,&-]Z.T#,/`#+.%U`]Z.(&(0D 23KO%,(5%,+$&9 +.%Q W$%$8G#+%,d(,Y7ZA

X =1√2(a† + a) P =

i√2(a† − a)

xV9/D0/<D 9A/kmBY)*#%Y+Or@0<Z*H &AB)(%(23,B%F66#+,)(<-)(!K2a&^%[+Or&@ ()& )T&@ ,D U# !$#%LM2T#+ )&@ f#T#+!1 )B<`)&@@S#+%F2a%&^S#+@vHb&A)(%(23,%N2a< #+,!1!1&Z.!YD &A73&A F.)(&`#+ZmAp%&@S#+ ⟨

(∆X)2⟩ ⟨

(∆P )2⟩

≥ 14

H#+,6#K%]^.(&*(,`#+% y#G`#+%:Oz' )% ⟨

(∆X)2⟩

<1

2

⟨

(∆P )2⟩

<1

2

xV9/D0/<D 9<9Mkm0D9/D0/n g)(%(%2a,l%y#,dOz!$#+ G+On,/ %:%&2%(`&Z0rI.)(&`#+Â+O`#+%<D

& 1'.%* .?' *3&! *$*$#& #& 1*X? +2+%*4! ."*,*#,#?4

$ %*$*$#& #& %1*P+%+*'4& 1* %1+*$*%#=+*,. #&3&3. %+2* %1* '=#(%1*".'*$*$#&

'() #&."3'$*$!*$"*?*,+%'+2.,4!& 1* $+%.%. #&( #$%*,. %1*0*#+?*A=%#&' )3&*A. X,P ,.- 1#21

+1* 1*$+*$ . %* |α〉 +* √2 / *

(α),√

2 0 (α)+%*A. *, %#&)'*,3/41& 1* 3+%*2 +*, . 1 - .

1* *+ *,=* ."+%*A43 )+#&' *65 +'( %1* *A 487'+ .'*$*$#& , %1* )+#$* #. %1*.!* # '331'.%* .?' *9 #&+%*A=#.$4;::1*, 1*$+* #.."*,*#,# , .%'!*< #+*, %#&'. 1? )'* +%*2 $*)+#$* 1* *+?*,.%*"#&$+%*A'.%*()+#&' *% + 1* $'=3&*$!*$2+/>3 # 1* *,+ 2#& /+*$3#5?#+*, %#&'.,4

?#+&!$#+ )*#+@@AMH/%]^.(&*(,[`#+%(n)*#+$73"0M&&`#+%(,[7ZA$#)&0P'[0822 +X+ ! $&9 +.%Q W$%

S(ξ) ≡ e−12(ξa†2−ξ∗a2) xV9/D0/<D 91M

:&%ξ = reiθ

b0M&&`#+@'#-)(!K2@ dc2T#+`#+!1&%&*D [!$#+0<.,r

,&%&!K(,&0<%(W+Oe%]^.(&(0[#+,2T#%θ

,&%&!K(yW#c4 "2T#%du2T#)(' )` b%]Z.(&*(,D54]^.(&(0?#+@ 0X,P

#+%[7`#+(,FOVθ = 0, π

%(23()&LM&@AMDkO%]^.(&(08 n'#+@ 0P:+O(%<H^y&@@2%y10D/9/D0/y'@@|73"%`#+%(,#( #(A1Oz%!W#c(*D@ RWV&g\WG@O RWOSc MX\WIbV.N

˙

(∆A)2¸ ˙

(∆B)2¸

≥ 14|〈[A,B]〉|2

Y>`.m i*m # 2A& U mCB #m


y!K23`#+ZK)*#%$ 8~%]^.(&*(,UL<#)&..! %`#+%<Hj0<LM&7ZAS(ξ)|0〉 D-Tb `#+%<H/'&

θ = 0nT#*LM⟨

(∆X)2⟩

=1

2e−2r

⟨

(∆P )2⟩

=1

2e+2r xV9/D0/<D 94^

y%&@S#+ ?#+@ `1@ ,Ozy#+Â6,/ 2@S#)(&!1&^r+Oeb`#+%<Ha#:'6680<.%<D G%]^.(&*(,6L<#)&..! `#+%P'@@j2@S#*AX#[%@ 8OzP+OwG]Z.T#+^.! !1&!1 (rn@ 4#+'-)%T#+2DTC1#+,8,&0<%(W+O`]Z.(&(01'@@g7aW#+!K23`#+Zy2T#+`#+!1&%&*D54POv#+'nWT#(LMW)( ,&%(,6@A#G0<@ rT#+!1 )b<%)&@@S#+%*H7.''T#+"T#+@@((,$Oz.n%&23*#+%&!1^,&@ #W)(@@ ()& +Oj<%)&@@S#+%%xz (]Z.L<#+@ &Z@A[#80<&ou,/!1& T#+@T#+!1 )<%)&@@S#+% dDeE:0/3&%&Î<%)&@@S#+%%e'@@%&2%(%&^w,/0/3&%&ê2T#+%+OG2ZA/ )*#+@IA/%&!YH<D 0DbK,(`)&7a(W#$T&@ ,F+On@0<^W'@ 1#+&8,(%)&73(2`#+%+Oe#+%!1*D r,dc[)(@@ ()& -+Ol23&`#+%%'7ZAiDew#)%-<%)&@@S#+% "#+@ %K)*#+@@ (,-# '$ O +MHT#+,W#+@S#+ ?#+,)&%*#+ 623&`#+%%ai, a

†i

#+%)*#+@@ (,'$ OZ+ $.93+.%,(W$%,8<D 6`#+%-2T#)(X+Or1Jo^~%A4%&! [2T#+(, 7ZAB6%&%2%4,/.)&%"+OZ.!b7a&:`#+%( |n〉i

Oz"*#)`Y!14, D DIN

!1^,(''%|n1...nN 〉 =

N⊗

i=1

|ni〉i =1√

n1!...nN !(a†1)

n1 ...(a†N )nN |vac〉 xV9/D0/<D 9<=M'&% |vac〉 ,&%(11Jo^1L<#)&..! `#+%<H'#+,

a†i#)&%[@AN!1^,

iD )(!K!b.`#+ -%&@S#+ YxV9/D0/<D0/3< '!14,/\T(,%K73

[ai, a†j ] = δij

xV9/D0/<D 9<CM:POV[#?0<@ <%)&@@S#+%*H'#+@@`#+%(1!$#(A7a,(`)&7a(,BB%&!1[+O:!14,23&`#+%%y#+,6L#)&..!YDkmY2T#+ )&.@S#+*HTWn+uv!14,P%]Z.(&*(,YL#)&..!`#+% 0<LM&Y7Â

Sij(ξ)|vac〉H'&%

Sij(ξ) = eξ∗aiaj−ξa†

i a†j

xV9/D0/<D 9H3<' )`Y!K2@ (

Sij(ξ)|vac〉 =%()`

(r)∞∑

n=0

[

−eiθ tanh r]n

|nn〉ijxV9/D0/<D 95M

km W*#AX%%(GT#+W W%`#+%K W&Z`#+0<@ (,H)(1#+@@n)(!K2a&^%WT#(LMni = nj

Dgn+uv!14,%]Z.(&(0G !K23`#+Z OV&^`#+0<@ &!1&Z0M&&`#+ H#+,~y'@@%(r#+dc#+!K2@ Y%()<DTC/D 9/D:"8@S#%2 )b6 :%()& H|nP,d $ $ ! 76X$ %,78*"2$(%&'( ! $78MD (%#+%K`#+Oz!$#+ P1`&8+O'!1^,123&`#+%%*Hj' )%B#+%K@*#+P#+,2%(%&LM)(!K!8.`#+ %&@S#+ *DEr&0F`#+oOV!1(,B23&`#+%%G'B2!1(*Hn#0M&&`#+@gi [email protected]*L`#+oOV!$#+ 6 '0<LM&67ZA a′j =

∑

i

bjiai + cjia†i

xV9/D0/<D 9DME <#IKN>`QO If\BIbJL\WG>O \WOQN@JdV8R k>IfRWOQ`S\LU RWV#k>[@` \LV&i3\WG@O IbcbT(OSRd\BJdUM&`SOQJ]i0V&R O,M8` G gaV#k O.m


bji, cji

#+%W)(!K2@ dcHT#+,W%(]Z.%&!1&^[a′j , a

†′k ] = δjk

73()(!1(2D,D bb

† − cc† =

xV9/D0/<D21<;M'&%r!$#+ )((

b, cT#(LMW&@ &!1&^%

bji, cjiDkm1 1`!1&!1([*# &[%Udc/2%(%[#+N23&`#+%$ %&!1$+O-#)& !14, 2a&`#+%%T#+KG%&!1I+OT ^.!873&I`#+%(*DlX#+Âb.`dOQ.@23&`#+ e)*#+K7adc/2%(%%(,6#[email protected]*L6`#+oOz!$#+ H)&@.,/0du#+,Ymn+uv!14,b%]^.(&(0#",d(,-#+7aLM<D

+<*& + : ! )! L*km G%()& n[Z%4,/.)(7T# )~)()(&2%K+O]Z.T#+^.! )(!K2.0U#+,)(!K!b./u )*#+ ?u 2345 ! "2$%.'( ! $?Oz*D P,~,&@LMK,(&2@A6^%]Z.T#+^.!/OV!$#+ [(AMH47. ,/ %)&.%n@AKOQ.,#+!1&^`#+@3 ,*#wT#+'@@|73"((,(,$OVw%&23*#+%& D <De y`()& ~7.@ ,2!$#+@A~ < 9<9@=#+,~)(.%%!$#+%&S#+@ Oz%! )(.%%(#+"E":X'jH3s #+!87 ,/0M<D-/. -/.30 > @ = J); 7 M I >37<= P7T# )G&ZmAF+O]^.T#+Z.! /OV!$#+ F r1276 ! VDb K]Z.7r rG]Z.T#+^.!#+T#+@ 0<.+Oy)&@S#%% )*#+@n7*Hw#+,G G,d(,B!K2@A#K#+ZA+uv@ &LM&@]Z.T#+^.!A/%&!YDe T#+" "#+ÂA4%%&!'<%W`#+%2T#)(W '+u,/!1& T#+@vD &%#+%y!$#+ZA~dc#+!K2@ ( +Og2Â4 )*#+@3A/%&!1' )%!$#(A[%&LM#n]^.7%*Dw5^2`#+%(b+O"2/u 1

2

2T#+ )&@ (*Hl23@S#+ `#+ %`#+%(b+O'2%P86Â^23&o$&&0<A@ &LM&@ e+O#+K#+%! #+% 7.w#OV&WDjkm e'+O|0<%*#+I%&0<I+Oa!$#+&!$#+ )*#+@OV!8.@S#+ G+OT]Z.T#+^.!/Oz!$#+ P(AT#+jj g,&23&,&Ze+O'T#+lA4%%&!p .%(,Oz!K2@ &!1&^`#+ DI Z.l#+7`#)&@A8# ]Z.7j ,(%)&73(,7ZAW# mn+u,/!1& T#+@y@7a&b2T#)( H D>T:.T#+@@A?&%1'@@I731%!1$7T# rOV H )(()&%(,#T#+.`#+@ #(A'Y82ZA/ )*#+@jA/%&!xQvD <D'#G7T# Y' )%X!1*#.%&!1&^%:#+%b)*#+ (,6.T#+.`#+@@AdDI 7T# )*#+@@ *)+ 68$95W ! $ 6,8 ! 88#+,~n,&%r&@ &!1&^%|0 〉 , |1 〉 D h: )(-X.%X+O8`#+@ )(%,/ 0<. ]^.7~%`#+%(~OQ%! T#+!1 )<%)&@@S#+%8Z.!873&b%`#+%(*D1 1)(@@ ()&LM[`#+%1+O !1%1]Z.7%W W,(%)&73(,F7ZA?%&%I,/%()&w2%^,/.)&w+O3&wy@7a&I%2T#)((*H^#+,G'7T#% I&@ &!1&Z%n+Oa'2%4,/.)&2T#)(-#+%Y%&%[2%^,/.)&%~+O60<T#+@7T#% $&@ &!1&Z%D 6#,#+2#?`#+ !K@S#+%xV9/D0/<D 9<=MdH#+, OV<D 0D %(F]Z.7%F)(!K2.`#+ T#+@87T# `#+%? |001 〉 = |0 〉1 ⊗ |0 〉2 ⊗ |1 〉3

D &]^.T#+Z.!_/Oz!$#+ `#+!K%(,FOz%!K@ 4)*#+ U%2T#)(K%6#+/u&*Ha:()((%`#+@AYT#:%~7ab&)(^,(,X^%~]^.7%:' )%F)*#+?!1*LMxz<D 0D:2%7.y#+@ `K&"2T#+ )&@ ( dD'yYW&"T#+,H'&-2a&oOV!K0$)(!K2.`#+ y" +Oz%&-P)*#%bT#+rr#+@@e]^.7%:#+%G23&`#+%(,X?!b.@`#+(.@AMHg#+,-P]^.7%' )`N#+%K#)&LM&BT#*LM%X7a%%(,B%!12T#+S#+@@Ac/(,A/%&!YDkEdE RWV&g #m.m ! %>

[a′j , a†′k ] =

P

i,i(bjib∗kiδii − cjic

∗kiδii) =

P

i bjib∗ki −

P

i cjic∗kiE ; V8R M \WG@V8RWV.[@e&G IbN\dRWV#k>[>`S\WIbV.N \WV D#[@M&N\W[@g IKN i0V8RWg M8\WIbV.N \WG>OQV&RY JdOQO# & m

2.2. BASICS OF QUANTUM INFORMATION 15

r%&!K@ 0<AY+Oe]Z.T#+^.! /OV!$#+ 6(A~nW23*#+[+O - ! #+, 8.W ! $ %&-]^.7%*D$yU ,*#+@ 2(4 +.'$%&-X b#Y]Z.T#+^.!_A/%&!^%Y' )%$`#+%$+O?A40-]^.7%b)*#+731'%&B#+,@S#+%&b%& &LM(,.)`T#+0M(, D ZDYhr#+.`#+@@A?OVP#+ZA%*#+@ )!1&!1AMH<%& &LM(,K`#+%'@@/7a23&oOz()&@A8 ,&Z )*#+@%: n' )` #%%(,H37.rF#+Â-)*#%bnK%LMb%~M(&2?G.2.8#r)&@ <%P%$G2.W#23<%7@ rxQvD <DlwT,&@mAb+Oaw%& &LM(,K`#+%WD *D *Dl2.e`#+% .@ ,P7a)&@ <%%9/w#+,K"23<%7@ %`#+0M"!1(#I@ 0P#w2a<`7@ <DkKy)(Z%dc/n+O]Z.T#+^.!%&23*#+%&%*H+!1&!1 (I#+%w.`dOQ.@^7a()*#+.%ww%&23*#+%&l2%%^)(@ lZLM@LM!$#+ÂW2T#+`#+@@ &@)(!K2.`#+ *Dw54!1y+O(%''@@T73 Ov#%&IT#+$&% #+,Ky.2.n]^.7%I&((,6%173W%%(,Y'@ bW%&!$#+0)(!K2.`#+ y D'y@ %$#K@ :+OI)&@S#% )*#+@)(!K!b. )*#+ `#+M(r2@S#)(K,/.0G2%%4)(@ W#+,F]Z.7%r((,?%~#+WOzr dc)%T#+0M%17a,<D-/. -/. - 6M F @?JL@K> M,; J2;6M 8; > JL@K> MN;><.%w@M"y]Z.7w%(%&!87@ ()&@S#` )*#+@T7n7ZAGT#(L40bmnb@ &LM&@ *H^".)&.%:+OW]^.T#+Z.!)(!K2.%&:#+@ %G%(%&!b7@ (8.)&.%W+Oe%,/T#+A)(!K2.%&%Ds!Gu2.`#+ 1./3()&%(,N7ZA#+22@A40#F%(]^.&)(Y+O0Z#+%(1%#U)(@@ ()& +Or7%[#+,%*#,/0.&r`#+%<D':*n&LM&*H38T#+.%b+OI80Z#+%(:#+,68%*#,.: yLM&A,/0/&%&Z $yG)*#%(*H^7a()*#+.%:]^.7%#+@@ OVn%.2a&23< +O@ &LM&@ '#+,&`#+%''@@T[0M&&`#+@T73,/ .73(,K7ÂK!1*#.%&!1&Z%*DwX#+&!$#+ )*#+@@A$)(!K!b./u )*#+ 1)*#+K73wL4 &n(,1#I#:23()&S#+@)*#% +Oa)(!K2.`#+ De )(!K!8. )*#+ [)`T#+&@LM&:' )Ù]Z.T#+^.! /Oz!$#+ FXbOz! +On]^.7%: :`#+!K%(,? r!K2@AX#)(!K2.`#+ -'&%2.y#+,Y.2.:#+%8#+@@ *(,Y%K73%&2T#+`#+%(,Y2T#)(<DypU)&@S#% )*#+@8)(!K2.%&*HW0Z#+%(-#+%F@ 0< )U23&`#+%%6%.)%#X E6:hyEGDy.T#+^.! 0Z#+%(w#+%',d(,$%r7a @*#+*HZ.`#+Ab`#+oOV!$#+ $#r%&I+O3]^.7%

U : H⊗n → H⊗n xV9/D 9/D0/ %&+Og]Z.T#+^.!0Z#+%( n`#+ ,1%b7a ! @+.%,823Oz]Z.T#+^.! )(!K2.`#+ \Og#+ZA`#+oOV!$#+ +O"OV!xV9/D 9/D0/w)*#+~7a:#+22%(c/!$#+%(,#+7`#+@A[n&@@a.0G@A0Z#+%(nOz%!r%&*D:IOVnr)&@S#%% )*#+@a)(!K2.%&"#b!$#+@@3Z.!873&'+Oj,/0/&%&Z 0Z#+%( "%.;$)& &Z'OVy#P.LM&%`#+@g`& D Dkm 0M&&`#+@vHF,/ 0<. 73&(&p0<@ du]^.7Y2a&`#+ *Hr#+@ %)*#+@@ (, $#6X 23&`#+ *HM#+, $ 0$#68 T23&`#+ *H' )%1#)&b!1%nT#+Kn]^.7#+,P)*#+73:OV#)&%%(,YZ%K@ 4)*#+@g23&`#+ *D %('%0<@ ]^.7I23&`#+ *H<' )%K'T#+@@/((,G7a&@ WHZ#+% 7 ?2H<2T#%

?2#+,$)(!87(,~7 #+,12T#% ?2$0Z#+%(*D ,&%y&!X,Z, Y

%(23()&LM&@A D E O]`,M&NaM8cbJdV \WG@IbN M&T(V&[>\4gaOQgaV&RWIbOQJ3V&[>\WJdIKk>OBM D[>T@If\4k>OQJd` RWIKU \WIbV.N m D[M&N\W[>g gaOQgaV&R IbJ4JdIbgaU@cM J J\WOQg BLG@Ib`:G M&cbcbV=BLJ J\WV8R:M&e.OlM8Nk RWO \dRWIKOSP.M8c V&i \WG@O D[M8N#\W[>g J\:MX\WOaV&iLJdV.gaOaV8\WG@OSR,Y5[@Jd[@M&cbc gaV8RWOP2V.cKMX\WIKcbO.Y@U@G JdIb`,M&cZJ J\WOQg mE \ `,M8N T(O U RWVXP2O,k \WGMX\ JdIbN>e.cbO D[@T>If\ V&U(OSR:M8\WIbV&N@J M8Nk \WG@O `QV&N\dRWV.cbcbO,k Fe2MX\WO \WV&e.OS\WG>OSR ijV&RWg M[@N>IKP.OSRWJWM&cZJdOS\ # %& mE N\WG@O `QV.gaU>[>\:M8\WIbV&NM&cZTM8JdIbJQY@\WG@O

X,Y, Z e2MX\WOQJ MXRWO RWOQU>RWOSJdOQN\WO,kT \WG@O [>Jd[M8c3M8[@cbI g MX\dRWIb`QOQJQm


#+,&AY#)&"-#+-#+7`#+A0<@ ]^.7`#+% |ψ〉 = a|0 〉 + b|1 〉 #'Oz@@ "*DX|ψ〉 = a|1 〉 + b|0 〉 Z|ψ〉 = a|0 〉 − b|1 〉 xV9/D 9/D 9M

Y |ψ〉 = −ia|1 〉 + ib|0 〉 P%PT#+X

Z%&%)`T#+0M(W0

uy#+,1

u)(!K2a&^%8+O |ψ〉 '@ Z

\OQ%:%&@S#+LM2T#`7ZAπ

D:'#+dc#+!K2@ r+Ol#b/uv@ 4)*#+@2a&`#+ nr@ 4[#+"#8n+u]^.723&`#+ DI )(^%@@ (,4uhr: 0Z#+%?xz)(hr: P#Y7oud?2[]^.7b)(,/ (,[`#+%+Ow#+&*D &6b`#+%8+Oe8)(^%@e]Z.7y |1 〉 H X :#+22@ (,-%Kb`#+0M&]^.7*DP5^)(K)(h: W#[]^.T#+Z.! 0Z#+%1#)&0YF]^.7%r :@*#+8#+,?2%(%&LM(.23&23< *Dedc#+!K2@ )(hr: [ |0 〉 + |1 〉√

2⊗ |0 〉

]

=|00 〉 + |11 〉√

2

xV9/D 9/D21Mh:%b e(]^.T#+ KT#+'@ S#+@/%`#+%n #e%&2T#+`#+7@ <H<T#+@`#+% #Pi &@@`#+%<H/vD <DI#b!$#c4!$#+@@A&Z`#+0<@ (,Y%`#+%<DI "#+7@A~%G)&%*#+%&Z`#+0<@ &!1&Z!$#+M(")(h:#PLM&A2an&oOz.@0Z#+%<HT#+,!$#+Â~2%23<%(,!K2@ &!1&^`#+ y+O]^.T#+Z.! )(!K2.%&*Hl[+O'K!$#+)`T#+@@ &0M(b W%Y!K2@ &!1&ZP W0Z#+%[8' )` G(]^.L#+@ &^b%-b.,&P@ 4)*#+@23&`#+ < /DHj9<9>=DX 8 K#6,/0;$)&.@P`#%)(W/uv@ ^)*#+@jT#+.%b+OW0Z#+%W!1*#+T#+y[]^.7%"!b."Z%&`#)&:#G)(^%@@ (,Y!$#+&*DyY!K23`#+Zr,/0/3&%&)(b73&n(&F]^.T#+Z.! #+,6)&@S#%% )*#+@g/OV!$#+ - T#+ !K23<%7@ r%K7`#+Y)(!K2@ &%4*'@ (,/0MP#+7a.'%`#+%+Ol]^.7%'7ZA~!1*#.%du!1&^*HT#+,6#+%&!K2%y%K,1%G'@@g,(%AW`#+%<DwkOgOV"dc#+!K2@ WnW!1*#.%8#+7%&L<#+7@ W'X&0M&ZL<#+@.( x #+,X#1)(!K2@ &%P%&+On#%%4)&S#+%(,Y&0M&M&% |x〉 H&FK.%)(!1K'@@e73KK+On

x W#+,FK`#+%K+OnK%A4%&!'@@e73G2%+uJ()&%(,6%~P)(%(23,/0Y&0M&M&*Dy&)(P\OwP`#+%G+OwPA4%%&!73dOz%b!1*#.%&!1&^ #y#G.23&2a<%

|ψ〉 =∑

x

cx |x〉xV9/D 9/D 4^

w,'j@ *#+P#+ZA40#+7a.jw)(4.;$)& &Z%cx

Djk80M&&`#+@vH%y,/ %)(*LM&lIL<#+@.(+OT(`)(4.;$)& &Z%n#)()&.`#+%&@AMH<n .@ ,G((,b%:!1*#.% GLM&A~xQ/%&@Aj!$#+ZA)(2 (K+O:`#+%FxV9/D 9/D 4^dDi.K]^.T#+Z.! !1()`T#+ )($,4(G$#+@@ *.G%?!$#+M)(2 (*De '$ S6 0$ ! *)+X$%,+2`#+%("T#+ k !K2a<%%7@ I%")(.)&#',&L4 )(<H' )`80<LM&G#+W.^'P]^.T#+Z.!`#+%r'@@g.2."mn1 ,&^ )*#+@vH23&oOz()&y)(2 (+OjT#+y`#+% < 9<9>=D

I+u)&@ 0(%&! g#!$#Jg7`#)&@ I'&8,(0<0y]^.T#+Z.!#+@0M!1#+,?)(!K!8. )*#+ 2%%^)(@ *Dbdc#+!K2@ GG)*#+M(&2X7T#)`^.28+On]Z.T#+^.!/OV!$#+ U' )% W%&^8*LM&b#~@ <ÀX)`T#+&@vHl#+,U'&UnK%*#,.K%(.@+Oy#-)(!K2.`#+ HI~,-G7`#+$Oz.@@`#+%~+O~]Z.7%D-:*&LM&*Hw&LM&

2.2. BASICS OF QUANTUM INFORMATION 17

.0<)(!K2@ &%~^'@ (,/0M+O"1T#+@ `#+%[!$#(AU731T#)()((%%7@ [%-.*Hln~)*#+!$#+M.%<H7Â,(0<0?#+@0M!11)*#+%dOz.@@AT#+1#6%0<@ [!1*#%.%&!1&Z6xz#FOz&%&`#+ 1,4(10<LM-.1/OV!$#+ 2a()&\T)6%6`#X#+@0M! 23&oOz!K0DkmW.@ ,F731#+22T#+%&ZPT#+W1,/ `)&2@$+O',(0<06]^.T#+Z.! #+@0M!1b %!1&'T#+!1% .7@ T#+G%)&@S#% )*#+@)(.Z%&2T#+*D n'@@/e,/ %)&.%e#+@0M!,(0<[!8.)`[1 n( *D OV4)&.w[]^.T#+Z.!%&2a*#+%&% #+,1,*#+@|'$&!2T#+@A%.0<F#G2T#+ )&.@S#+r!1^,&@jY%&!1:+OIT#+!1 )b<%)&@@S#+%%y`#+&:T#+6]^.7WOV!8.@S#+ D $'@@w((,1LM&AF!K23`#+ZG)&%)&.b.0<HlT#+!1&@AF]^.T#+Z.! %&@ &2a`#+ Y#+@0M!YDIkmw #8%&!$#+#+7@ OV#)&nT#+]^.T#+Z.! !1()`T#+ )(#+@@ "gOVgw`#+oOV&j+OT]^.T#+Z.!/Oz!$#+ G'.e#+ZA8,/%()&]Z.T#+^.!)`T#+&@73&(&1`&,&P#+,U1%()(&LM&*DinAh2345 :+ +:93$%.W ! $B[`#+%1+O"]^.7" '`#+oOV&%(,.^%[#+&*H23<%7@A[OV#+'%&!1LM(,6Y%2T#)(rOQ%! r%*H/7ZA!$#+401.%+O2 &Z`#+0<@ &!1&^r#+,6)&@S#% )*#+@!1(% #+0M(*D-/. -/. 8J2;N@ 4 7 7<FHM I @ JL@ >3MN; %&@ &23`#+ )&%)&.1 K%*'0Dw9/D 9/D %(Y]^.7%$#+%^LM@LM(,B%&@ &23`#+ De '%n]Z.7*H^S#+@@AK1#+7`#+A1`#+% |ψ〉 HM w"]^.7w%73%&@ &23%(,De %()(,[b]^.7%n#+%y2%&2T#+%(,[[+O!$#c4!$#+@@A$&^`#+0<@ (,`#+%(Dlj&@ &2a`#+ Y)*#+[7a:#)% &LM(,~'#+ZA1!$#c/!$#+@@A$&Z`#+0<@ (,`#+%<H^7.nOz1 #+M1+O'#+0<.!1&^b$)`^<%1$i &@@w`#+% |Φ+〉 DyK!$#(AF!$#+0<$T#+W]^.7%y+OeP&Z`#+0<@ (,X2T#+:T#(LMb7a(&X%&2T#+`#+%(,-%2T#+S#+@@A7ZA-#1@S#+0Mb,/ `#+)(<Ha%T#+:-W@ 4)*#+ 6b%&,& M(&20[]^.7% /<H 9[#+,6-#+&:@ 4)*#+ 6%()(&LM&" "M(&20[]^.7 1/D

& =& *,3*?+ 2# #&+$#EA4>& #&!* #."+# +% 3*# +# 1 , ( - * 3 *$3 %1*#& #E2. ,, +% )'*4& 1* ##E#&'3 . 2* |ψ〉 #E #. %*,3*?+ *('=# 4& 1* '3 (3#&*,. $++/0 3'.. #$'3 #& +% # 4

km$%%&2[+Or#+@0M!YH# + +,8. %X+. +.4n n2a&oOV!1(,[~]Z.7% /#+,9/Dw5^)(ryOz.'i &@@`#+%(OV!#+!$#+@g`&*H/&A~#+%#b7T# nOz mn+u]^.7Gy@73&b2T#)( H⊗2 D i &@@w!1*#.%&!1&^P P,d(,%Y73[#2%+J()&LM!1*#.%&!1&^Z%P 7T# DeI]Z.L<#+@ &Z@A~ '#8.`#+A[`#+oOV!$#+ OQ%! i &@@l%$P)(!K2.`#+ T#+@I7T# 2D Oz@@ n(,?7ÂY!1*#.%&!1&ZXP)(!K2.`#+ T#+@E BOQ`,M8cKcL\WGMX\ M&N \ BHV T@M&JdOQJaV8i M IKcbT(OSRd\ JdU@M&`QO M8RWO `QV&N@N@OS`S\WO,k'T M[@N@If\:MXR \dR:M&N@JijV&RWg MX\WIKV&N m


7T# bxz)<D OoD%()<DT9/D 9/D0/dD &%b#+%WOz.2a<%%7@ W.%)(!1(y)(%(%2a,/0[%1Wi &@@`#+%(H#+,Oz"*#)`6+O(%nW@ '%(.@0$%`#+%+Ol%,6]Z.7 X*#.%&!1&^y.%)(!1 |Φ+〉 |Φ−〉 |Ψ+〉 |Ψ−〉5^`#+%8+O]Z.7 1 |ψ〉 Z |ψ〉 X |ψ〉 Y |ψ〉

8 '(#E#3. %*,. %*,+ %1*-*,33!*A'.%+%*,!*$,4 , 1, +* %1*1?'.%*0( # # %*,.,4 & 1* +%*A *,+ #&. +* *,+%+*0% ( 4 '+ 1* *$#&3. C$3 3&%#&' 4

:!1*#.%&!1&^".%)(!1('T#(LMr(]^.T#+@32%7T#+7@ (',&2a&,&^+Og:S#+@`#+%<Hr#+, *#)%)*#%XXT#+@`#+%F+OG]Z.7 1 Y#2.%F`#+%?(]Z.T#+@%2.P`#+%1+O"]Z.7>/K.2U%X#)(!87T#+ B+O'2T#%$#+,7 ?28)(,/ T#+@ ~.%)(!1<D5^)(*#)%+Oy[0Z#+%(X,Y,Z

%]^.T#+%~%/<Hn#+@@~T#(LM%?,6%%()(LM& |ψ〉 :%~)(!K2T#+%K.r!1*#.%&!1&ZW.%)(!1G'?`#+7D9/D0/K#+,X#+22@A-#+22%2S#+%60Z#+%%U]^.7M1/D 1!1*#+$T#+1Y%&,&1!b.12T#%[#?)&@S#% )*#+@!1(% #+0M+Olmn17%"%Kr%()(&LM&2D D wT#(LMn%*#)%(,G#'%&!$#+<#+7@ w%(%.@ .#+ÂW,/%()&j^%&`#)& K#+,8`#+ou!K0@A6#1:)&@S#% )*#+@l!1(%`#+0M8Oz.@@j]^.T#+Z.! `#+%8+Ow]^.7 /8T#73(&`#+oOV&%(,%G]Z.7 1/D T#+' !1%<H# @ 0$# r!1*#.%&!1&^y#+,0Z#+%('n23&oOz()&@AK#+,1"2%&2T#+%(,[&^`#+0<@ &!1&Z e%*#+@@AK!$#c/!$#+@vHM I`#+Oz&n e2a&oOV()&*H%&0Z#+%,/@ (%G+O"[%&2T#+`#+ 7a&mn(&B]Z.7%>/#+,1/D6%!_[2a^P+O"L4 & +O)(!K!b. )*#+ 8 8LM&A?2%!K 0 GkO'n$)*#+!$#+T#+0M1%-,/ 7.%[&^`#+0<@ du!1&^b#+!106%&,&%8#+,?%()(&LM&%H&U]^.T#+Z.!/OV!$#+ )*#+F73Kdc/)`T#+0M(,'.'#+ZAKOz.& ((,1OVw2%%()& ~Oz%! `<Dw5^)(:&Z`#+0<@ &!1&^,/ \u7. `#+M( 2@S#)(:#,/L<#+)(r+Og#)&.T#+@0<T#+@@0K)*#+73:#+@@ n(,%b73@ <%%A#+,6!K2a&oOV()&y'.W#+ÂY +Ow@ <0[!K2a`#+^:!1(``#+0M(*H3#@ 0#8T#+@&^`#+0<@ (,`#+%('#+%y0M44,Dl :0M&&`#+ 6#+,~,/ 7. Y+Oj&^`#+0<@ &!1&Z" 2.23<%+OW]Z.T#+^.! %&23*#+%&*D

OSRWO B OM&k>V&U>\ \WG>Oa`QV.N#P2OSN#\WIbV&N \WGM8\ M \dR:M&N>Ji0V&RWg MX\WIbV.N V.N@OlTM8JdIbJ M&N@V8\WG@OSR M&` \WIKP.OQc g M&U>J\WG@O TM&JdIbJ P2OQ` \WV&RWJ V&i \WG@O >RWJ\ V.N\WV\WG>O T@M&JdIbJ P2OQ`S\WV8RWJ V&iH\WG>O JdOQ`QV&Nk m FHG@IbJ gaO,M8N@J \WG@M8\ \WG@O g M8\dRWIRWOQU>RWOSJdOQN\:M8\WIbV.N V&i M J\:M8\WOBIbN^\WG@OLN@O%B TM8JdIbJHIbJ V.T \:M&IbN@OQk^T ^M&U>U@c IKN>e \WG@O O RWgaIb\WOQM&Na`QV&N[>e2MX\WO V&i \WG>O\dR:M&N>Ji0V&RWg MX\WIbV.N g MX\dRWI lIbN\WG>O V.cKk T@M&JdIbJ]\WV \WG@O RWOSU>RWOQJdOQN\:MX\WIKV&N V&i3\WG>O J\:M8\WO IbN \WG>O V.cKk T@M&JdIbJ # A& U mB Xi mE V&\WOH\WGMX\3\WG@O]N>OQO,k ijV&R M `QcKM&JdJdIb`,M8cgaOQJdJWM8e.OLU>RWOQP.OQN\WJi M&J\WO R \WG@M&N cbIbe.G\4JdIbe.N@M&cbcbIKN>e>m i@\WG@O RWOQ`SOQIbP2OSRk>VOQJ]N>V&\ #N>V B \WG@O gaOQM&Jd[>RWOSgaOQN\HV.[ \W`QV.gaOSJQY#G@IbJ]T(OQJ\Lk OQJd`SRWIbU>\WIbV.NV&i D#[>T@If\B IKJLM8NO2D#[@M&cbc BHOQIbe&G#\WOQkgaI \W[ RWOLV&i5\WG>O J\:MX\WOQJ]IbNl\:M8T m m &Y@T>[>\\WGM8\HIKJ\WG@O J\:MX\WO

ρ = 12#N@V=BLN M8JLM g M IKg M8cbc^gaI O,k^J\:MX\WO

`.m i*m4JdOQ`.m B m Y`QV8RdRWOQJdU(V.N@k>IbN@e^\WV N@VÎbN>ijV&RWg MX\WIKV&N \dR:M&N>Ji0OSR M8\BM8cbc m

kmY :)%T#+2%&rnW%y,/ %)&.``#+!K ` -+Ow]^.T#+Z.!/Oz!$#+ ?*LM&#K %A)`T#+&@vH3' )%? :82%7@ &!T#+r]Z.T#+^.! %&2a*#+%&%#+%P,(0<(,X%[%@LM<D &F2%4)(((,?F%()<DS1/D 9[%,(%)&7aPmnY,/0/&%&Z8]^.T#+Z.! %&23*#+%&W2%%4)(@ *H32T#+ )&.@S#+GErqjs'tTuvA423G%&2a*#+%&r' )%F :P!$#+F.7/J()&yOVr :( *D #+@ %G^%^,/.)(8r!1^,&@gT#+'8#+T#+@A4% #+,Y!b.@S#+ +Oe)%T#+2DT=1#+,)`T#+2D|C#+%r7T#%(,6D ($ +<$ L*$ !km#+ZAF%*#+@ % )$2Â4 )*#+@A/%&!.%(,%Y!K2@ &!1&ZK]^.T#+Z.!R)(!K2.`#+ H %'@@/73n2%(`&Z*Deh:+Oa 2a&`#+ e' )`Kn 23&oOz!'@@732a&oOV()& <0Z#+%(I#+,!1*#.%&!1&^%I'@@T#*LM'&%%n#+,G'A/%&!'@@e732a&oOV()&@Ab `@S#+%(,Dl:&)(]^.T#+Z.!)(!K2.`#+ #+,U)(!K!8. )*#+ !8.%7aG!K2@ &!1&^%(,UOv#+.@ouv%@ &`#+Z@A%YOQ.)& $2%(%&)(~+O"(%$!K23&oOV()& *D-s!K!8. )*#+ #)&%<%8 %A)`T#+&@ " ''T#+"8]Z.T#+^.! %&23*#+%&" ",(0<(,%1#)` &LM<Dj"0M&j#+8 ,*#'+O'T#+g./3()&g %e'@@ZT#*LMw)( ,&l#'!K2@ e!14,&@^+O/!K2a&oOV()&23&`#+ *D

A "#G0<@ du]Z.7y2a&`#+ DI 8]Z.7"A/%&! ",&%(,

Q#+,&^L^%!1&^y

ED `#+M./()&+O

A-#+6#+7`#+AY`#+%

ρ%K7a D < 1>=

AρQE = pAideal ρQE + (1 − p)

2⊗ g

Q(ρQE)x 1/D0/<D0/

n ,*#+@423&`#+%Aideal

#)&%@A8Pn]^.7A4%%&!YDp

)*#+P73w.,&%%44,P#j2%7T#+7@mAbOVe'23&`#+ 1%.)()(((,D K2%7T#+7@mA1−p '23&`#+ KOv#+@ *HK' )`[)*#% "T#(LM"^'@ (,/0M+O3'T#+IT#+223&(,K%"]^.7*Dke e2%+J()&%(,%$8`#+%

/2D" y%`#+%W :)*#+@@ (,X#1 ! ' j- ! + OX`#+%<H73()*#+.%P.%)(!1(+O!1*#.%&!1&Z%PF8#+%K!$#c/!$#+@@A-`#+,!YD1T2%+Jo()&LMK!1*#%.%&!1&ZGZ%#+Â0MT#+@j7T# |a〉, |b〉 nWT#(LMOz%!xV9/D0/<D 4^

P (a) = P (b) = 12

DE N D[@M&N\W[@g IbN>ijV&RWg MX\WIKV&Nl\WG>OQV&R \WG@IbJ\ >U(OLV8i5`:GM8N@N@OSc(IKJ #N@V=BLNlM&J]M k>OQU(V&c MXRWIbJdIKN>e `:GM8N@N@OScM&NkIf\WJL\dR:M8N@JdgaIf\d\:M&N>`QO U RWV.U(OSRd\WIbOQJQY@Jd[@`:G M&JL`,M8UM8`QIf\ Y>GM,P2O T(OQOSN J\W[k IbO,k$# % & m

/D

20 3. QUANTUM REPEATERS

:23&`#+ ~ #+22@ (,~%&`#+LM&@AMH#+,~nr.,/AKr./()&+Oj ~]^.7y`#+%<D:q&ρi

,&%bP,&A-!$#+\c-+OIb]^.7#OQ%&i

%&`#+ +OA

H#+,#%.!1y:S#+@3%`#+%y+Og:]^.7n%P73"2.%ρ0 = |ψ〉〈ψ| H/'@ :r`#+%:+O&^L^%!1&^y #+7`#+AMDn22@A40

A)(%GS#+@g`#+%<Hn0M&

Aρ0 ⊗ ρE = p[Aideal ρ0] ⊗ ρE + (1 − p)

2⊗ ρE

=

[

pAideal ρ0 + (1 − p)

2

]

⊗ ρEx 1/D0/<D 9M

= ρ1 ⊗ ρE

'&%nyT#(LMy.%(,$gQ(ρ0 ⊗ρE) = ρE

' )%[OV@@ *"nOQ%!".`#)(2%23&A+O,&A!$#+\cDIkm,/.)&LM&@A~r,6#Oz%&i

%&`#+

ρi = piAiideal ρ0 +

i−1∑

k=0

pk(1 − p)Ak

ideal

2

x 1/D0/<D21MkmY8)*#`W'&%b8]^.7y `#+!K%(,-LM&r#$)%T#+&@vHa ,*#+@2a&`#+ ?)(%(ou23,%G#b23&oOz()&@A$OV#+/Oz.@3`#+oOV&'#+,r!K2@A$T#*LM

Aideal = De%!x 1/D0/<D21M &Y*#A$%G)(!K2.%ryT,&@A+Ogr`#+%!K%(,`#+%:'%(2a()& %P ,*#+@g%`#+% |ψ〉 D

W7`#+

F (ρi, ψ) = 〈ψ|[

piρ0 +

i−1∑

k=0

pk(1 − p)

2

]

|ψ〉

= pi +i−1∑

k=0

pk(1 − p)

2

x 1/D0/<D 4^

= pi +1 − pi

1 − p

1 − p

2=

1 + pi

2'&%bnG.!K!1(,-P0M(!1& )b2%0<%(` ?%$0M&:8%,X(]^.T#+@AMD%! dc/2%(% ?nK`(8T#+'&&LM&K)`T#+&@I :2a&oOV()&*H3vD <DW'&p < 1

H3T,&@mA6,/%2ydc/23&ZS#+@@AY-8^.!873&:+OI%&`#+ *H|%* #+%,1/2

' )%X yT,&@mA+O!$#c/!$#+@@A!K\c/(,`#+%<D !$#(A+Oe#K)(!K!b. )*#+ X)`T#+&@#r!$#,G.2U+O!$#+ZA?!$#+@@I%&0<!1&Z%H*#)%U'?c(,?2%7T#+7@AF+O .)()((%W#rx 1/D0/<D0/ xze(]^.L#+@ &^@Ab')`T#+&@T#I#yc/(,bOv#+@.%n`#+% 23&.e@ &0<|dDk&OV@@ *" T#+ r`#+!K %% [T,&@mA~,/%2'dc/23&ZS#+@@A~':)`T#+&@@ &0<D $Oz*#+.%<Hdc47%(, &%- .!K2@ 6 `6!14,&@vH' [60M&&`#+@:2%7@ &!' )`YT#'%K73*LM&%)(!1WOz"]^.T#+Z.!)(!K!8. )*#+ X%K7a()(!1WL^S#+7@ <Dkm1)*#+73%(&xz<D 0DN7ZA)(!K2.0g(ρ2

i )PT#+[#

i)&%*#%(*H

ρi0<`#,/.T#+@@A,&2T#+%OQ%!WS#+@l2.%P`#+%W%$7a()(!1b!1%P#+,6!1%8!K\c(,D" y@ <%:+O2.mA~ ')*#+@@ (,OZ+,6X$@)3+.%,+268+ZD'kY#G@0<^@A~!1%W0M&&`#+@ %,(%)&2 Y")*#+67a%(&#8K%(.@8+O'&Z`#+0<@ &!1&^G'U1.%.,/0X&ZL4%!1&Z G0Z#+n`#+M8#G%0<@ W]Z.7'S#+@@A-#P2.%8`#+% |ψ〉 = a|0 〉 + b|1 〉 HT#+,S#+@g%`#+%

3.1. THE NOISE PROBLEM 21

+O 1&ZL4%!1&ZP!$#(AU#+@ %Y7aG`#+M&#2.% |ein〉D[ 1]^.7W W23&oOz()&@A %@S#+%(,OQ%! 8&ZL4%!1&Z*Ha7."`#+M&6%0M&&y&A~Oz!#1)&@ <%(,YA/%&! #+,&)(b&LM@LMW.`#+@AMD q&

U,&%W`#+Oz!$#+ X'&6W%A4%&!&LM@LM(OV"%!1r.23()&\T(,Hc(,!1<DI 0M&&`#+@OV! +O&LM@. 6 '&

U |0 〉 ⊗ |ein〉 = |0 〉|e00〉 + |1 〉|e01〉 U |1 〉 ⊗ |ein〉 = |0 〉|e10〉 + |1 〉|e11〉

U |ψ〉 ⊗ |ein〉 = a|0 〉|e00〉 + a|1 〉|e01〉 + b|0 〉|e10〉 + b|1 〉|e11〉x 1/D0/<D =M

:&%Y-&ZL4%!1&Z`#+%( |ejk〉((,N[73!8..T#+@@A 0MT#+@r$!$#+@\u %(,D G%(bT#+rWT#+@e`#+%G+OIG]Z.7r y~@ 0M&:2.%<Dg`#)&0Y*LM&r&^L^%!1&^y0<LM(#P!K\c(,%(,/.)((,-,&A~!$#+\c

ρ = |0 〉〈0 |(

|a|2〈e00|e00〉 + ab∗〈e00|e01〉 + a∗b〈e01|e00〉 + |b|2〈e01|e01〉)

+ |0 〉〈1 |(

|a|2〈e01|e00〉 + ab∗〈e11|e00〉 + a∗b〈e10|e01〉 + |b|2〈e11|e10〉)

+ |1 〉〈0 |(

|a|2〈e00|e01〉 + ab∗〈e10|e01〉 + a∗b〈e00|e11〉 + |b|2〈e10|e11〉)

+ |1 〉〈1 |(

|a|2〈e01|e01〉 + ab∗〈e11|e01〉 + a∗b〈e01|e11〉 + |b|2〈e11|e11〉) x 1/D0/<D CM

Er&23&,/0P~&^L^%!1&^"`#+%(:xQvD <De~y2T#+ )&.@S#+U

T#+@a,&L4S#+ Oz%! |ψ〉 !$#(A~7a:@S#+0M"!$#+@@vDe:&)(rr!K23`#+)(W+Oj %@S#+0K]^.7%'&@@.@ ,Y7aW#+22T#+%&^*D5^`#+0?Oz%!R!1^,&@ K+Oy %<Hw.)`#G~(1#+73*LM<HwG b23<%7@ ~%-,0M&&`#+@^!1&^,Oz,*#+@0y'8&%%*Dlgdc#+!K2@ yx 1/D0/<D =M)*#+87a.,&%%44,8#j#)(!b7T#+ G+O`#+,!7#+,82T#% ?2#)&0:Pw]^.7 xz`( < 9<9@=42D%4+14^dH<#+,#b)(^,r)*#+&7a:,(0<(,' )`'@@3,&%()&"#+,)(%()&'(%?2*D r0M&&`#+@ ,*#K+OlW%+u)*#+@@ (, &%ou)(%()&0)(4,( / '%K&)(4,WW)(!K2.`#+ T#+@j]^.7%(,/.,#+^@A^%F%&LM&`#+@ 2ZA/ )*#+@"]Z.7%K#+,.%~~%(,/.,#+)&A%X2%%()&G`#+%'+O3"&)(4,(,K]Z.7*DyK!K2a`#+^n)&%& $Ozw#+ÂP%)`&!1+O|OV#+.@ouv%@ &`#+^)(!K2.0K T#+ r*LM&*#,~%(%.%)((#+,)(!K2.`#+ Y!1:!b. 0<%dc/2a&^S#+@@AK'$"%*"+O")(!K2.`#+ H/)("&' %"%)%&!1:]^. )%4@A73()(!1(17%@ &%'&B)&%)&.[* K)&%*#%(,DB1]^.T#+Z.! )(!K2.`#+ K.1.$T#+K7ZA!$#+^0F.`+Or&%ou)(%()&0)(^,([#+ZA)(!K2.`#+ )*#+73W23&oOz!1(,-Ov#+.@ouv%@ &`#+Z@AMH2%L^ ,(,XT#+:82%7T#+7@mAX+Ow&%:OVb7T# )0Z#+%('6)&%)&. '73&@ *#G)(&`#+6%(@ , < 1H9<9>=D54YU2)&2@ 1K2%7@ &! +O `K W%@LM(,D[yn&LM&PK%(@ ,F 8@ *WDEr&23&,/0$)(^,<H' A42 )*#+@@A8%,&"+O10−4 u

10−2 OV'@ ^)*#+@23&`#u ''@ rW%@ &`#+7@ W&%'2%7T#+7@AOV'`#+!K % !$#*A[73r0<& < 1/H9<9>=MD ' "#PLM&A~T#+%%(]^.%&!1&Z"%1#P2`#)& )*#+@g!K2@ &!1&Z`#+ D@ %G"%.@ ,7a%(,T#+'8&%"!1^,&@ )( ,&%(,6#+73*LM8#+%r)(!K2@ &%&@A0M&&`#+@vDneZA/ )du#+@@A6#$]Z.7:!$#(A738&)(4,(,6^%~#[A/%&! 'X!1%8T#+X~@ &LM&@ *Ha#+,-&%%; i If\ IbJ^N@V8\,Y B O [>J\aO#\WOQNk V&[>R k OQJd`SRWIbU>\WIbV.N V&i \WG@O OSN P#IfRWV.N>gaOQN\ \WV IbN@`Qcb[@k>O BLG@M8\WOQP.OSR V&\WG>OSRJ J\WOQgaJ]If\LIKJBOQN\:M&N>e.cbO,k BLIb\WGZm BcKJdV Y JdV.gaO U>G >JdIb`,M8cZM&JdJd[@gaU \WIbV.N@Jg [>J\HT(O g M&k>O M&T(V.[ \H\WG@O N@V&IKJdO&Y>U@M8R:M8cKcbOQcZ`QV&gaU@[>\WIbN>e g [@J\ T(OU(V.JdJdIbT@cbO M&N@k M8N@`QIbcbcKM D#[>T@If\WJ]g [>J\]T(O U>RWVXP#IKk>O,k # %& `:GM&UZm % m BNklM&`Q`QV8R:k>IbN@e^\WV U>RWIbP.MX\WO `QV8RdRWOQJdU(V.N@k>OQN>`QO BLIf\WG m< m<8RWOQN@JdOSN m


)(.@ ,B& `#+M6%`#+%Y.% ,-]Z.7$%.7A4%%&!YHn)*#+.0Y]^.7$`#+%%73()(!1G.,d(,DK8dc#+!K2@ KK)(.@ ,F!$#+0<1&)(4,/01]^.7WZ%Y#2%Y' )`6 #+7`7a(,DT(`%*#%H^#+%!1L<#+%(,%8@ 4KOV&']^.T#+Z.! )(!K!b. )*#+ %)`&!1(*D:.T#+%&@A-%&LM&`#+@%)`&!1(y'?@ (%:0M&^:73.,r#+%82%L^ ,(,X7ZA]^.T#+Z.! %&2a*#+%&n2%%^)(@ HZ' )`#+@@ .;$)& &^`#+%!K % [+O/OV!$#+ ~LM&#+6#+7`#+A,/ `#+)('-&%ouv%@ &`#+)((r#+'W@ &LM&@g+Ol23&%)(&Z < 1/HC/H$5>=D L! $6$' $ )+::T#r7a(&F%(%`(,-%&LM&`#+@e!1(W#+@%*#,/AMHb2.23<%G+OnG]^.T#+Z.!%&23*#+%& %G)&%*#+%r&Z`#+0<@ &!1&Zr#+,,/ 7.%:'#+!101,/0/&%&Z'2T#+ (Dek~ '%()& $)( ,&b17T# )[.)&.%$+O'1%&23*#+%&*D654&LM&`#+@wL<#+S#+Z%G+O'1%&23*#+%&2%%4)(@dc4 Uxz<D 0D < 1/H'C/H 5>=8#+, !1%dHr#+, ?!1&Z 7 .?ANU#+22%M#)`^%^,/.)((,0<T#+@'2T#+2a& < 1@=HI'&%[%&! 2345%X+:93+,(:+2%X #1#+@ %)((, D r#+22%M#)%`#+M&[["!$#+~n$+O n( n#nZ%4,/.)((,~ < 5>=H#+,W,(%)&73'Y!1%W,&`#+@vD/. -/.30 J)=?> ,@ I @ IK7M @K5 77 FH7 JL@ 7 I K7T#% )1.)&.%$+O'K%&23*#+%&8 b*'F0D%1/D0/<D~6@ 4)*#+ 8PHif#+%)(()&%(, 7Â # AN]^.T#+Z.! )%T#+&@vD y.0MM#+@: ~%(`#+7@ #!$#c4!$#+@@A&^`#+0<@ (,N2T#+~7a&mn(& #+,NirDwj73Y23()&\T)<H'`#(AT#+~6`#+%6nX#+! %(`#+7@ %X |Ψ+〉 DW K)`T#+&@e ,/L^ ,(,XZ%

2N %&0<!1&Z%8+On@ &0<L0

H3%~T#+$%`#+@n)%T#+&@@ &0< L = 2NL0

D~lT#+@&Z`#+0<@ &!1&ZK b(`#+7@ (,%&`#+LMW%%&2*D

&?E ?% +*# *,*$+A4<& 1*1 *,)#.%# #% . *'!*$. #&.. 1 - 4 & 1* !*$*$%#&! '#&C - .%*#!*$.#&. $'33&* 0+*# *,*$+ . 2%#&' '+ *'4 & 1* 1/.%#$3 .%/. %*, . (*$+3/#& # *$+*$ . *'!*$. '( *,. '+%* # *$%#$'3 4

RR:Q BA DRg 73&0<'HK#+&Z`#+0<@ (,2T#+? ?)&%*#+%(,pOV?*#)%L0

u%&0<!1&^*D5^)(X-2ZA/ )*#+@r]Z.7%~!8.%[^%&`#)&%73()(!1-&^`#+0<@ (,Hy&^`#+/u0<@ &!1&^1)&%*#+ BT#b%XT#+223&@ ^)*#+@@AMDF ~F]Z.7%G+O:[)&%*#+%(,2T#+1#+%%%(,#+K&&$&,B+O:%&0<!1&Z*DB]Z.T#+^.! !1&!1A' )%N]^.7%K!$#(A731%%(,ULM&b!1~#+,F%& &LM(, b#+U^%&0<`#+@n2T#+P+O $%&2a*#+%&G#+,F WLM&A OSRWO If\ JdG@V&[@cKk'U(OSRWGM8U@J^T(OJ\dRWOQJdJdO,k \WG@M8\^\WG>OQJdO Jd`:G@OQgaOSJaM&`:[email protected]>c \WG>IbJ2>

>m4FHG@Ok V^N@V8\LU>RWVXP#I k O [@N>IKP.OSRWJWM&cD[M8N#\W[>g`QV&gaU@[>\:MX\WIbV.NM8\L\WG@O e&IKP.OQN N>V.IbJdO cbOQP2OQcbJQm,F5V \WG@O T(OQJ\BV8i3g #N@V=BLcbO,k e.O.m

3.2. QUANTUM REPEATERS 23

^L^S#+@%:!K2@ &!1&^ ^&)( K')(%.)& $+O|%&23*#+%&%e!8.)`1nP!8.%e7a2."^%G,/0[#G0M44,!1&!1AMDIyn&LM&r#+" '23Z*H((,"A#+73.1,&`#+@ b+O'* 1!1&!1A-Oz.)& *D~y &Z`#+0<@ &!1&^G W)&%*#+%(,,&23&,["2ZA/ )*#+@aT#+.%+Oy]Z.7%*H/#+,$`8#+@ %8,4(n'OV!+O)&%*#+%(,~`#+%#+,,&0<%(W+Oe&Z`#+0<@ &!1&^y'23<%%(`%(*Dlk' ')&.)&S#+@T#+:T,&@AF

WD *D *D|Ψ+〉 )&@ <%%8.mAMDeE:&23&,/0K$:%)%&!1y.%(,H^w n2a<`7@ "%W,~#8 )&@ n&@!K"Oz'S#+@T,&@AMD) *+HR

&K&Z`#+0<@ (,K2T#+%eT#*LM 7a(&K(`#+7@ %(,1#+w#+@@/`&0<!1&Z%*HZ&A#+%y)(()&%(,[%8)&%*#%":&Z`#+0<@ &!1&^",/ `#+)(<De )(()& w@@.`#+%(,0D1/D 9/DYkmb)(% %b+Oy#6]Z.T#+^.!R%&@ &2a`#+ B+O$&,+Oy#2T#+GZ%-Ov#+oum#( #(AG&,P+O3#&0<^7a.02T#+"xz)<D ODj`()<D9/D 9/D21MdD %(.@ e#+K&Z`#+0<@ (,G2T#+'#~&fT,&@AF ′ dc4%&,/0X*LM&Wm' )(11,/ %`#+)(<H

L1 = 2L0D$j&oOV!K0

+.W +. +.4B68$ +,6# ! $ #+l&LM&A8`()(,84,I2%4,/.)((e#0r+O2N−1 %&0<!1&^%*#)`6+Ol@ &0<

L1#+,Y*#)`6^)()&.2 (,Y7ZA2T#+%"+OlT,&@A

F ′ D

1 2 3 4 1 4

F F F'quantum

teleportation

& ''3&*$!*, *, %#&' 4%3# 3&#*,.#(#&,%* *$''3&* #&+.,4 -*$3&3 *A'.+*$!*, #.< *$+" '+ *2 #E2. ( , '( %1* . %* #&. %*$3&*# '+%%*2 %4 -*,,.%*1* ##E#&'3. 2%*A. +*! + #&!'33&/ *,23*2 '( %1*=%*A. ( !*,'.%+*$!*$2. '+%*!' *$+" *, , 1* *,3# /0#. *#+* * #(1*%+$*,..$4

F ′ < F4

T#+@w&^`#+0<@ (,`#+%17a&mn(&#+,Ui b(`#+7@ (,F7ÂX%&`#+0-1)(/u()& N

!1(*Dpkm *#)% %&`#+ -Z.!b7a&Y+O8`&0<!1&Z%~ ~T#+@LM(, #+, &^`#+0<@ &!1&Zr,/ `#+)( ",.7@ (,67ZA)(()&0$2T#+%Ll = 2lL0

Di ()*#+.%r+Og!K23&oOV()& 23&`#+ "#+,~!K2a&oOV()& ~:&^`#+0<@ (,`#+%(*H4T,&@mAN,()&%*#%([ *#)% )(()& %&2D[0M^4,%&23*#+%&$23&oOz!$#+)(xQvD <D73&%&T#+,/%()&`#+!K % |dH: ,()&%*#%?!b.7a?.7/udc/23&ZS#+@W ,/ `#+)(<De r#+22%M#)%%8%&23*#+%& < 1=3#+,[)&@S#%+O%&2a*#+%&% )( ,&%(,[ '(% #+%W,/ 0<. (,Y7ZA*&AY#)` &LM *D/. -/. - 7<FH7 JL@?7<I >A@K5 I > JL@K> MN;kmYr%y2%2a< #+@ +.4W +2'+24B9 % ! 6X( ! $F#y.%(,%1%(%%b#10M^4,T,&@A#Oz%&*#)%?)(()& D' &%b#+%8%&LM&`#+@g #(A/%K2.\OzA `#+%(+Oe@

F%[`#+%(+Og0<

FDe'dc#+!K2@ ry!$#*A[,/ @@#8*#+@A$!$#c/!$#+@@A[&^`#+0<@ (,`#+%yOQ%! #)(@@ ()& Y+Ol@ * u

F`#+%( 7Â~#P%(]^.&)(:+Oj!1*#.%&!1&Z%*H|#+,%&@ ()& 7T#%(,W.%)(!1(DnkOl8S#+@T,&@mAY+Ol2T#+%y #+73*LMP#K)(&`#+6%(@ ,

FminH

FHG>O \WOQcbOQU(V&Rd\:MX\WIbV.N U RWV`QOQJdJ]V&i e>mB#m IbJ]JdV.gaOS\WIbgaOQJH`QM&cbcbO,k OQN\:M&N>e.cbOQgaOQN\]J BHM&U@U>IbN@e bY>T@[ \]IbNl\WG>O`QV.N\WO#\LV8iRWOQU(O,MX\WOSRWJ B O U>RWOSijOSRL\WG@O \WO RWg OSN#\:M8N@e&cKOSgaOQN\L`QV.N>N@OQ`S\WIbV&N bm V&\WO \WGMX\ \WG@O \WOSRWg U>[>RWI`,MX\WIbV.N IbJM&cbJdV [@JdO,k IKN \WG@O0D[M8N#\W[>g IbN>ijV&RWg M8\WIbV&N cbIb\WO R:M8\W[>RWO ijV&R Mg M8\WG>OQg M8\WIb`,M8c \WOS` G>N@IED[@O^\WV M8JdJdV`QIKM8\WO M U>[>RWO^J\:MX\WO V&iBM `QV&gaU(V.JdIf\WO J >J\WOSg BLIb\WG'M gaI O,k k>OQN>JdIf\ V.U(OSR:MX\WV&RBV&i4M Jd[@T@J J\WOQg Y#O&m e m # & m


&FK#(LM&`#+0MKT,&@AF+OnP&f)(@@ ()& +O2T#+%8#OQ%&*#)Ù%&@ ()& U%.,'@@/73 0<&*Dl:&)( %&`#+0 2%4)((,/.%''@@,/LMF

%*#+%,W/xze%*#+%,#+.223&'@!KFmax

dDwj0D:1/D21K@@.`#+%("W)(!b7(,6)(()& ?#+,2.\T)*#+ D 2%4)((,/.%r)*#+[73^%&0<`#+%(,Z%Py%&2a*#+%& 7ZA$T#(L40P2T#+`#+@@ &@)`T#+&@ +O'1mA^23K0D1/D0/<Db*#)`L0

u%&0<!1&ZP&Hg[)&%*#+%~#)(@@ ()& +O @ uF

2T#+%POz%!' )% #X!$#+@@ &K^.!873&$+Oy*#+@A!$#c4!$#+@@A&Z`#+0<@ (,N`#+%(1#+%,/ @@ (,Dw (%r0</uF

2T#+%#+%W&6)(()&%(,-#+,2%4)((%' "%&`#+%(,D

& 0 %*,+%#&' 3 4 & 1* (+* #&.(2'*, +' 4 & 1* (?*,+'( 3& - *,+ $+%)* .%1 - %1* *$3&#E / %*$+ '* +# ?,# + *,23*#!*$ $'*,=# . *# +*,."?*A=#)*$3&/. %#&'. 1* *$3&#E / ?*# '+*C%1*. %* 4

!K2a&oOV()& Uf0Z#+%(#+,f!1*#.%du!1&^%Y,&%&!KFmax, Fmin

#+, X,/%2T,&@mAf7ZA )(()& HY#+, %p )(,/ F ′ > Fmin

c6&%~73.,~#+73*LM-' )%K%&23*#+%&W73()(!1(W23&`#+7@ <D #+&ou%!14,&@Z+OemA^23x 1/D0/<D0/dHi &0M&@H+2] l*T#+w&%n2%7T#+7@ (n$'2a&%)(&ôuv@ &LM&@a)*#+731%@ &`#+%(,D? &A#+@ %X,&!1`#+%~T#+b%(%.%)((:%(]^.%(,YOVy23&`#+ xQvD <D"bZ.!Gu73&e+OT2T#+`#+@@ &@T)%T#+&@ d#+,Gn!1 ((,(,G%(`#+7@ %X&^`#+0<@ &!1&ZW73&(&U #+,-ip0<%*#+e%e2a@A4!KS#+@@AP'1',/ `#+)(<Dy&)([%&2a*#+%&P b2)&2@ 23&`#+7@ [LM&P@ 0,/ `#+)((y#+,Yr2%(%&)(W+O % < 1>=D OV& &n2.\T)*#+ Y%)`&!1( < 9H 3>=an&%)( ,&%(,#+@%*#,/A < 1>=H#+, #Y,&!1/u`#+%(,8T#+jw((,WOVg!$#+ÂW2T#+`#+@@ &@/)%T#+&@ )*#+Y73&@!KT#+%(,D k < C=" 'YT#+2.\uT)*#+ ~)*#+$73#)% &LM(,['WJo.eb]^.7%n#+*#)`Y^,<D W@ *n(%"2a<`7@ L<#+@.+Ol%(%@ ,OV'2.\T)*#+ Fmin

)/9 < 3 =HT#+, 2%L^ ,(w.'[#y7a&)`!$#+bOVeT,&@AK+Oa"&Z`#+0<@ (,1%`#+%(e0M&&`#+%(,7ÂPy]Z.T#+^.!%&23*#+%&*D /9 w"!$#c/!$#+@a#*LM&`#+0M"T,&@mA1T#+w!$#*AK7a'%*#)`(,7ÂY!1*#++On)&@S#% )*#+@)(!K!8. )*#+ Db r!$#(AY73b%*#+@ %(,XX8OV@@ *'0#*AMDkO8#b2a<%%7@ K7ZAU)&@S#%% )*#+@w)(!K!8. )*#+ N#+@ $%Y(`#+7@ ]Z.7W2T#+%b'#+F#(LM&`#+0MPT,&@mA?#+73*LM /9/H3&?7ZA-2.\T)*#+ U+O#~)(@@ ()& U+On.)%?2T#+%: 8#+@ `2a<`7@ G%Y(`#+7@ #UxQ*#+@Ar!$#c4!$#+@@AF&Z`#+0<@ (,2T#+*D$ ^.8!$#c/!$#+@&^`#+0<@ &!1&Z n.@ ,$7a"(%`#+7@ (,173&(&~,/ `#+^nA/%&!1e'. -P]Z.T#+^.!^%&`#)& 73&n(&<%1A/%&!1r'T#+%%4&LM&*D G'@@e%dOz&%Y /9)&@S#% )*#+@@!KY`&LM&`#+@g^)()*#% '@S#+%&"Yr( D/. -/. 9 @CB/FH7 7 FH7 JL@ 7 Ikm < 5>=gEy.T#+Hq.^H|sn`#)W#+,6ta@@ &T#(LM2%23<%(,Y#P%&23*#+%&'2%%^)(@' )`YT#?%&2T#+`#+%2.\T)*#+ %&2Dkm%*#,&^`#+0<@ &!1&Z)(()& G2%7T#+7@\u )<[email protected]<-#G23()&\T)r!K2@ &!1&Z`#+ ?&%W)( ,&%(, < 5@=H/ "( '%&! -&93+ %,+ 93+,(:+.%%dOV&%r%#+ZA6%&23*#+%&r2%%^)(@e'? .)&.%yvD <D


2%7T#+7@ )[)(()& P#+,UY%&2T#+`#+%12.\T)*#+ D 1Erqjs'tTuvmA^231%&23*#+%& '@@.%`#+%(,0D:1/D 4D

connection

failureent. conection

ent. creation

& ? 5 / * +*# *,%*,+,4l *,*,. #&. ' * 3*,)'*,3&.,4 -(%1*

l %1 3&*$)*$3 , *$+

l$'*A=#. , *,21(C%1*"*$''3&*'#+2. ." ! #&. ' *Ll = 2lL0

4pl

#&. %1* + #3&# /7.%, *,.. + . %*#(#(% +% 3&*$)*$3

l−1 3&*$)'*,3

l/ *$''3&*$!*$ $'*A=# 3 '+- +*,%#&'5 4

y*#)%6&^`#+0<@ &!1&Z)(()& H|%.)()((%' ")(,/ (,XYW.%)(!1rOz%!%!1!1*#%.%&!1&Z*D 2%7T#+7@Apl

+Or7`#+0%(]Z.%(,N.%)(!1Y!$#(A,&23&,$1@ &LM&@ <D "2.23<%+O)(,/ 0P I%82%+Jo()&n")&%*#+%(,Ll

u2T#+l%#`#+%n' )%G!K!K %(,/%2G8T,&@mAWOz%!@ &LM&@4%dc/#+,b e #(A1&.%"T#+I",()&%*#%KT,&@A$#e&Z`#+0<@ &!1&Z',/ `#+)(")&%*#%( '.7/udc/23&ZS#+@,/ `#+)(<DIp@ *,/%2T,&@mA[23&")(()& 6%&2!$#(A73K.0<ZK+O#b7a&0-2T#+ ,FOV87ZA#Y0<2%7T#+7@mAFOV8$)(()& %YOV#+@ 1 − pl

D :y#G0M&&`#+@l.@ <HT8@ *&y8%(]Z.%(,-,/%26F

HTW@ n&pl

!b.:7a<D5^)(WOz"W.T#+ ,(,HT A)`T#+&@gT,&@mAY+O`#+%!K`#+)(b,/%2ydc423&ZS#+@@Axz)<D ODY%()<D1/D0/dHj[)*#+U+O"~E:qls'tTuvA42a[%&2a*#+%&K#bT#(L40X%!1$7.@ouv2.\T)*#+ G8n&Z`#+0<@ &!1&Î)(()& < 5>=Dlkmg.@ ,b7ae%(,bT#+*H+b)(^`#%'%n#+22%M#)%H4&%' w((,GOVn#+ZAG/uv@ 4)*#+@|23&`#+ $%73'^LM@LM(,1 )(()& 1%&2Dkmb2T#+ )&.@S#+Il ()((%`#+Ab%y23&oOV!i &@@/!1*#.%&!1&^%y%1!K2@ &!1&ZrP)(hr:p0Z#+%~xQ[i &@@g!1*#.%&!1&^ "((,(,6Ozy2%7T#+7@ )%&@ &23`#+ Y0D$1/D 9MdDe ' "0M^4,7a()*#+.`r!K2@ &!1&Z0$/uv@ 4)*#+@g23&`#+ '0M&&`#+@@AT#+%,D%! 0D:1/D 4P"%.@ ,7ar)&@ *#+ T#+#P0M^4,]Z.T#+^.!!1&!1A[ !K23`#+Z"Oz23&`#+ Y+Ol#PErqjs'tTuvA42ar%&23*#+%&*Dn5^)(#+@@&Z`#+0<@ &!1&^y)(()& 6#+,0M&&ou#+ F%&2W#+%P2%7T#+7@ )<Hj)(()& U:0M&&`#+ UOzr&0<Z73.06%&0<!1&^%)*#+K73[dc/23()&%(,%?.)()(((,B%!8.@`#+(.@AMD &%dOz%&^`#+0<@ (,`#+%(G((,%K73W%%(,Y'@ 8)(%(23,/0~&^`#+0<@ (,62T#+%:#+%8(`#+7@ %(,OV"&0<Z73.0%&0<!1&^%*Dwy@AG&$)*#+1"%`#+%(I73 %& &LM(,~#+,1)(()& ~)*#+173#+%&!K2%(,D:nOz *#+@ &'2%23<`#+@vH/Erqjs'tTuvA423r%&2a*#+%&")*#+73:!$#,r%bT#*LMW.7/udc/2a&^S#+@|LM&*#,[$%(%.%)(".%:#+,1)(!K!8. )*#+ !1<DI5^)(:#+@@T"4,(#+% ,&^ )*#+@l%A4%&!1H/nW%(WT#+"W%(%.%)(W.% '@*#+yYW)%T#+&@l@ &0<D@ \WG>O U@[ RWI`,MX\WIKV&NU RWV`QO,k [>RWO k OQJd`SRWIbT(O,klIKNlJdOQ`.m*B#m m M&cbJdV^[email protected] U RWV.TM8T@IbcbIbJ\WIK` OScKOSgaOQN\,Y#T(OQ`,M&[>JdO If\[@JdO,k `QV&Nk>If\WIbV.N>IbN@e V.NlgaO,M&Jd[ RWOQgaOQN\HV&[>\W`QV&gaOQJQY#T@[ \HG@O RWO BHO M&JdJdV`QIKM8\WO M U>RWV&TM&T>IbcKIf\ BLIb\WG \WG@O OQN\WIfRWOJ\WOQUi RWV.ghV.N>O cbOQP2OQc(\WV \WG>O N@O#\


j(!$#+%nw)(!K!b. )*#+ K!1ww)( ,&&Z`#+0<@ &!1&Î)(()& P73&(&2T#+% +O@ &0<Ll−1

D &[:)(()& Ov#+@ *H^&^`#+0<@ &!1&Z' @ < [y'@ Ll

u%&0<!1&ZyZLM@LM(,H3#'@@.`#+%(,YY0D$1/D 4HT#+,%&23*#+%&"2%%4)(@T#"%17a%(`#+%(,XOz%!GS#+@I&Z`#+0<@ &!1&^80M&&`#+ ? W%&0<!1&^*Db 1#*LM&`#+0M^.!873&:+Ow#+%&!K2%y7adOV%W)(()& F.)()(((," 1/pl

D 8@ &τ

,&%W8!123&ZG#X0<@ [&^`#+0<@ &!1&Z~)(()& G)&%*#+ N#+%&!K22DZHI#+,tl

,&%8#*LM&`#+0M8!1W((,(,Y%1(`#+7@ %X#+6&^`#+0<@ (,Ll

uv2T#+*DnkOnb#+22%*c4!$#+%b!1b((,(,X%[0M&&`#+%xQX2T#+`#+@@ &@ y[&0<^7a.0Ll−1

uv2T#+%:7ZA6G#*LM&`#+0M!1rOV'0M&&`#+0[%.)%Y2T#+ D,D H/W7`#+-#G%()&.%&)(8%&@S#+ OVtl

tl =1

pl(τ + tl−1)

x 1/D 9/D0/' )`YT#'W%@.

tl = τ(1

pl+ · · · + 1

pl · · · p0) =

τ

pl · · · p0

1 +l−1∑

i=0

i∏

j=0

pj

x 1/D 9/D 9Me.0

l(]Z.T#+@/%

N = log2(L/L0)G ldc/2%(% G0<LM( %`#+@/!1((,(,G%(`#+7@ %&^`#+0<@ &!1&Zy7a&mn(&6p#+,i #"#8OQ.)& Y+Ogr)`T#+&@@ &0<DI 73&T#(L4 .8+O

tN,&2a&,bU123()&\T)K!K2@ &!1&^`#+ %.0<

plD 1%(T#+nOz

tN%bT#(LM:.7/udc/2a&^S#+@,&2a&,&)(r+O:)%T#+&@3@ &0<H/

pl!b.W,/%2@/Bdc/2a&^S#+@AX'

lDK T#%7T#+@S#+)(1'UK)(!K2@ &!1&Z`#+mA73&(&[@ ,/%2~

F#+,$@

pl Z&%y #W`#,du@/673&n(&@ )(!K!b. )*#+ !1#+,B#-0<T#+@"%`#+%1T,&@mAMD `)%&!1' )%N#)` &LM(1.)`7T#+@S#+)( #,&!1`#+%(,67ZAE:.T#+ +2^ I < 5>=dD D:y#1@S#:%&!$#+Y+OI y%()& Ha8!1&^ ?#+&r2.23<%8+OIb!1*#.%du!1&^%'&Z`#+0<@ &!1&^:)(()& X%&2DekY#,,/ Y%G!K!K %01:T,&@A,/%2H^&A$#+%:#+@ `W.%(,$%8)(Z%@3Oz! +O'T#+@3&Z`#+0<@ (,`#+%(n0M&&`#+%(,7ÂU~%&2a*#+%&*DFkO$Oz!+O$T#+@'`#+%(G 84*'Hwb!$#*A73$23<%7@ [%,(0<pxQ2%7T#+7@ % )K]^.T#+Z.! )(!K!b. )*#+ `)%&!1(1' )`N)*#+N23&`#+%Y&LM&'&1'T,&@AK+Oa`#+%(I ew0<DlTwdc#+!K2@ <HZ"&Z`#+0<@ (,~`#+%(e!K0<^`#+MrrOV!

ρ = c|vac〉〈vac| + (1 − c)|Ψ−〉〈Ψ−| x 1/D 9/D21M'&%!$#+0<.,&@A^02ZA/ )*#+@"A/%&!R%?7a~2% )!14,(*DkvO:n!$#+MG#rJZr!1*#.%&!1&^:'U#+&r`#+%G#+,-)(,/ FX82%(%&)(P+On#2%Hl1L<#)&..!_)(!K23&ZG 8dc/2%(% )*#+G)(Z7.%~%[.)du)((%OQ.@3.%)(!1(*DI #(A~ 23<%7@ y%G)(.)&#b2%7T#+7@ )%&@ &23`#+ E \WG>O \WIbgaO JdU(OQN\LV.N `Sc M8JdJdIb`,M&c`SV.gag [>N@Ib`,MX\WIKV&NT(O \5BHOSOQN \WG@O N@V#k>OSJBg M lM8cKJdVaT(O IbN>`Qcb[k>OQkIbN

τmEdE:FHG>IKJ IKJ M&N [@Nk OSRWOQJ\WIbg M8\WIbV&N m FHG>O^M,P2OSR:M8e.O \WIKgaO i0V8R \5B VU@M&IfRWJ \WV Jd[@`Q`SOQO,k IKJ [email protected]>OSR \WGM8N i0V&R [email protected] UM8IfR,mHFHG>O OSRdRWV&R IbN \WG@O @NM8c `QV&gag [>N@Ib`,M8\WIbV&N\WIbgaO k>[>O \WV \WG@IbJBM&U>U>RWV Ibg M8\WIbV.NIKJLi0V&[@N@k BLG@OQN

BHO `QV.gaU>[>\WO `QV.gag [@N>IK`QM8\WIbV.N R:M8\WOSJ i0V8R JdU(OQ`QI` RWOQU(O,M8\WO RWJ IKN `:GM&UZm m \ IbJ M&U>U>RWV Ibg M8\WOQc Mai M8`S\WV&R#m #mE ; V&\WO \WGMX\ IbN # & \WG@O M&[ \WG@V8RWJ `QV&N@JdIKk>O R V.N>c\WG>O k V.gaIbNM8N\B\WO RWg V&i B m #m m FHG>IbJ IbJ IbN M&k@k Ib\WIbV&N\WV \WG>O M&U@U RWV >Ibg MX\WIKV&N g M&k>O IbN V&T>\:M8IKN>IbN@e \WG@M8\Li0V8RWg [@cKM#m


2%%4)(@' )%Y4"'6#+@!1<'2a&oOV()& T,&@A < 5>=De r2%7T#+7@mAOz r2%+u%4)(@n%X.)()(((, G,&%&!K(,7ÂU)(^.;$)& &^cD ~%&.7 .?A%6 b%()<D=/D 4D

/. -/. M 7 3>3; !$#Jw2T#+ +O3n12%(`&Z%(,[$ n( wT#w7a(&[%b#+T#+@A/%"2a&oOV!Gu#+)(~+O,/0/3&%&^KE:qls'tTuvA42a~%&23*#+%&%*D-j-,- Pn[((,#-23()&\T)$!14,&@nOz:2ZA/ )*#+@A/%&!1' )%!$#+M:.2:%&2a*#+%&*H4vD <Dr]Z.T#+^.!!1&!1 (*H/!$#+&:+Oe&^`#+0<@ &!1&ZW)&%*#+ Ha&%)<D yy.!14,&@nP)%4<%8#$%&+OeT#+!1 )<%)&@@S#+%%D

& + #.%$#3&3%+ *,3 4& - .%$#3&3%+ *A. +* '.. #* - #E1(*A'21(+%*# *,%*$+ . *'!*$,4 < #%#&'?3! *A. / * #&)'3&)'*2"#& - +#E* # !+%*A ' +' !*$!'+"#&*,. ( # *$''3&*$!*,='*,*$+2# ( *, %#&' 4 *A. #)'3)* #& $'*A=# +*#&( #$%* / +*74

Te#y0<LM&K%&0<!1&^*HMn'#%`^)&S#+%'#!1^,'K&&I&,D@@/2a&`#+ ' )`#+% 2T#+w+O3'%&23*#+%&I2%%4)(@|#+%#%.!1(,G%W7a',(%)&73(,G7ZAKi [email protected]*LG`#+ouOV!$#+ n+O!1^,y2a&`#+%%wOzn(%#+,~#+./c4@S#+AK!14,(*H/#+,17ZAK2%+J()&LM!1*#.%&!1&^%WZ.!b7a&:`#+%r7T# 8xV9/D0/<D0/5MdD 8!1^,&@vHl[)(!K2.`#+ T#+@7T#% OV8.b]^.7%8 80<LM&7ÂF1@ n("^.!873&y`#+%( |0 〉qubit = |0〉 |1 〉qubit = |1〉 x 1/D 9/D 4^

'4,+Oe]Z.7%' "..T#+@@A~)(% ,&%(,"%G73W#PLM&A[0M4^,&)(4,/0H73()*#+.%%!1b0<@ du]^.7y23&`#+ H|.)`X#yW7l?260Z#+%X

H|#+%8,/0;$)&.@:%$,D :&)(^,/0X

%(]^.%(r#1)%T#+0MP+OI8dc)&`#+ XZ.!873&Kxz<D 0D 2%-^.!873&OVW@0<Z dHlT#+b *Hj#+#)&LM1`#+oOV!$#+ Dk 82%./3&`#+7@ [%Y&)(4,(,#]^.7^%!1^,(W'#[c/(,Fdc)&`#+ UZ.!b7a&b#%)*#+@@ (,F,/.T#+@\uv`#+@n&)(4,/0D:*&LM&*H3#:,&!1`#+%(,X < 5>= H|&^`#+0<@ (,?`#+%(y+Oe8Oz!Rx 1/D 9/D =M'73&@ *!$#(A73'.%(,$%W%&@ &23n,/.T#+@\uv`#+@3&)(^,(,[]Z.7% #+,1&A1)*#+#+@ %73".%(,K~]Z.T#+^.!)&A42%0<`#+2ZA6#+22@ )*#+ *DkO'n[,&%$[!1^,[2a&`#+%%8OVP#Y`&0<!1&ZP7ZAaL, aR

Hl&#Y!$#c4!$#+@@A&^`#+0<@ (,6`#+%8#)&%<`'T#+%&0<!1&^" '0<LM&Y7Â |Ψ+〉 =

1√2(|01 〉 + |10 〉) =

1√2(a†L + a†R)|vac〉 x 1/D 9/D =M

P&Z`#+0<@ &!1&^8)&%*#+ U%&2F#+!1y%[0M&&`#+%G`#+%(:T#+W#+%b)&@ <%P%[ rdcû2%(%% Dl y2%4)((%I #Wi [email protected]`#+oOV!$#+ ~+OaL, aR

#+@ 08'#c4@@S#+A


!14,(lOz@@ n(,K7Âb!1*#%.%&!1&Z%euOVedc#+!K2@ &Z`#+0<@ &!1&Zn!$#(AP7a 0M&&`#+%(,KOz2% )8!1^,(' )%X#+%W&Y@ M#,(,Y^%[#+%!K )W!1&!1A!14,(*D 5^!K@S#+@AMHT&/u`#+0<@ &!1&^)(()& Y "#Pi [email protected]~`#+Oz!$#+ Y+O<D 0DaL

#+,a′R

OQ%! &0<^7a.0G%&0<!1&Z)(!K2@ &!1&Z%(,~7ZA$#+./c/@S#+AK!14,(yxz%( 0D 1/D =MdDk1 n)*#%r!1&!1 (y!$#(A73r%*#,Y."Z%$dc4`#K!1^,(*H' )%X#+%r&6!$#+2.@S#+%(,#+,K!1*#%.%(,DI "%(.@w n#+[&^`#+0<@ (,`#+%"%%(,$$!14,(a′L, aR

' )%1OV!$!1^,(P+O"$& %&0<!1&ZYxz)<D OD0D1/D 9MdD- $!14,(aL, aR

)*#+7a[%`#+0M!14,(n+Og]^.T#+Z.! !1&!1 ( @ 4)*#+%(,~#+n4,( +O%&23*#+%&*H47.n&A1,bT#*LM%-73<Dy@%&T#+LM&@AB)*#+BL^ &)(!K2@ &%Y`#+%~`#+oOV&6xQ'%duv%*#,4u. nOV!$#+@@AY#'2T#++O&Z`#+0<@ &!1&^r)(()& Dn#)%&Z`#+0<@ &!1&^-)(()& uG!$#+%&X,/ `#+)(?@ &LM&@yuG .223<%(,%7a62a&oOV!1(,N 6 #+!16!$#+& :T#+~ *H YOV! +OW?#+./c/@S#+AN`#+%(^LM@LM(,#+,~i [email protected]*L`#+oOV!$#+ #+%#+@#(A/K`#+!1<DqM&' %&^`#+0<@ &!1&ZI)&%*#+ G2%^)((,/.% #+@ %yn`#+!1P#+@@/%&0<!1&^%e+On)`T#+&@vDek2)&2@ <H^'&[yi [email protected]`#+oOz!$#+ '#+,$!1*#.%&!1&Z% +O"2%%4)(@#+%nc(,H<e &.0<1%r2a()&\OzA82.I`#+%OV'&Z`#+0<@ &!1&^n)&%*#+ De `#+%r+O&Z`#+0<@ (,2T#+%#+"#+Â[0<&'@ &LM&@)*#+&73r)*#+@ )&.@S#+%(,xz)(,/ T#+@g)(()& %1.)()(((,[email protected]<$#+%0<@S#+)( ')% )(+O|T#+!1 )'<%)&@@S#+%%w#I#:!14,&@/!$#(AG%(&!%( )&LM<HjW UOV#)&8]^.%K0M&&`#+@vD~X#+ÂF]Z.T#+^.!_A/%&!1!$#*AX731,(%)&73(,7Â-W#+22%(c/!$#+%(,F7ZA-T#+!1 )K<%)&@@S#+%%*DWTWdc#+!K2@ PnK%T#+@@)(% ,&:@0<^^%&`#)&0-'2`#+%(b+O"#+%!K )[&%&!87@ (Dy&Pdc#+!K2@ (b!K0<Zb)&@.,&@ ()&%j823%&ZS#+@4n&@@ gL47`#+ T#+@4!1^,(l+O`#+22a(,8 u3A4%%&!1' )%K#+%)( ,&%(,UOz8!K2@ &!1&Z`#+ +O"]Z.T#+^.! )(!K2.`#+ D6 $i [email protected]`#+ouOV!$#+ $#+@ %:T#' ,"#+22@ )*#+7@AMDIkm)&@.,(*HM7. e%( )&%(,G% #+@@@*#+2 )*#+@n`#+oOz!$#+ xz<D 0D~7a*#+!_2@%&% dHj0M&&`#+ +O"%]^.(&*(,U`#+%(*H#+,@0<ôum#+%! Z%&`#)& *D:*n&LM&KK)*#+K0M&&`#+@@A7a[.%(,%X%*#+1&LM@. .,&r#+!K@%S#+T#+8#+%PW]Z.T#,/`#+ )G?P!1^,G23&`#+%%*DPyKdc#+!K2@ +Oj#82&!1&Y.n!14,&@a n%.;$)& &Z'%b%*#+n :2 )*#+@& ./()&*H^' )` #2%~Z.!873&',&2a&,&^ 2T#%\Ozndc/23& &)((,17ZA~#+[2 )*#+@a!1^,2T#%0%.0<#/uv@*#+8!1(,/.!YD$ 12T#%1\OzW)*#+U2%4,/.)($#+./()&LMK2%/u2%B^%&`#)& H"#+,PT#17a(&B2%23<%(,%?.%~ GOVG!K2@ &!1&Z`#+ +O/uv@ 4)*#+@w]^.T#+Z.!0Z#+%(*DK $:#+!K@%S#+U%&@ &L<#+Z8Oz&b./3()& rOV.%,&"W!1^,23&`#+%%'OV'@0<Z"T&@ , < /*9H/64(=Dy.`#+%&0<AOVl#+T#+@A/0:%&2a*#+%&%l'@@/73I%:,&LM&@ 2P^.!1& )*#+@#+,P#+T#+@A4 )*#+@%4@ eOVw,*#+@0P'$0M&&`#+@[email protected]*L$`#+oOz!$#+ '#+,$2%+Jo()&LMy!1*#.%du!1&^%*D $T#+@@I#+^8%!1%P1,&mA?!$#+\cFOz[&Z`#+0<@ (,`#+%(8#+*#)`Y,/ `#+)(@ &LM&@vH/%K.,/A~ ',&2a&,&)(8r2T#+`#+!1&%&%+Oji [email protected]*L`#+oOV!$#+ H12%7T#+7@ (bOV8!1*#.%&!1&^P)(,/ P%67a1OQ.@\@@ (,He#+,WT,&@A6WD *D *D!$#c4!$#+@@A-&Z`#+0<@ (,?`#+%(*Dy:#*L^0~.)%X%^@ r#+r.:,/ 23<`#+@vH:)*#+~&%& :#+22%2S#+%W2T#+`#+!1&%&%OzL#+ .'%2a()&\T)<H42ZA/ )*#+@3!K2@ du!1&^`#+ y+OeW%&23*#+%&:#+,67`#+6%(.@%"Oz"&y2a&oOV!$#+)(<D ydc#+!K2@ b)*#+?ZLM(%0Z#+%bG,/ `#+)(GLM&:' )Ù#10<XT,&@AX)*#+-73b!$#+Z`#+(,U#+,W,&23&,&)(P+O :6!1&!1A2T#+`#+!1&%&%D'g10M&r#"%*#+@ )8!1^,&@ r#"23<ou


7@ <H4:%.@ ,$!$#+Mr.%yT#+ y%4@ ,&LM&@ 2a(,Y#+@@ OV &%%'.)%#@ <% +Odc)&`#+ Y-#P!14,$xz<D 0D2%6@ <` '!K2a&oOV()&,&%()&%%*D


N

l &,P+O|w2%()((,/0W)`T#+2%&enZ%4,/.)((,1#"!14,&@4Oz0<@ du%&2G%&23*#+%&%7T#%(,66T#+!1 )P<%)&@@S#+%%*D k6W2%(%&^y)%T#+2%&:nP,&LM&@ 2X#K!$#+&!$#+ )*#+@%4@<OVj,0y)*#+@ )&.@S#+ P.)%G#"A/%&! A #'0M&&`#+0yOQ.)& [email protected]*L`#+oOV!$#+ *D &-2%^)(((,6%1dc42@S#+-*W0M&&`#+0[OQ.)& - ".%(,%B#+T#+@A/%??2a&oOV!$#+)(U+OP%&2a*#+%&%*Hr#+,M&%)`* !K2@ &!1&^%(, *)+.'( ! 6XX%&7a4?2%0<`#+!1*Dg$7a(%8+O!bAF^'@ (,/0M<Hl1 ,*#+O'0M&&`#+06OQ.)& U,(%)&73(,F73&@ *f 8,/.G%bDg5aDg54%&%&H KDX@!1&b#+,F>D5^&%%D ($E$'$ LH * &N * y.17T#% )2%7@ &! KOz@@ '0 6[Z%&`#)& 2%^)((%`(KZ%&%(0%F.*H<D 0De^%&`#)& -73&(&6@0<^:#+,6#+%!1*Hnb#+%8+OQ%&6#+7@ 8%K%@LMW(]^.T#+ +OI!1 X6Py& `&Z73&0[2 )&.%<H3#+,6&)(8,?#1i [email protected]`#+oOV!$#+ ' )`,(%)&73(W~&LM@. +O $!1^,[2a&`#+%%Dyn&LM&Gn$n.@ ,@M1%,#+@ %Py&LM@. Y+Ojy`#+%r,&mA$!$#+\c$54)%4,/0M&'2 )&.%<DI 0M&&`#+0$OQ.)& Y y#P%^@%1,G D&&`#+@@AMH0<LM&-#+6#+7`#[email protected]*L`#+oOV!$#+ 66#G%&"+Ol!1^,("n)*#+B,d#Y0M&&`#+0?Oz.)& OQ%!R' )`)(^7. %X#+Â&@ &!1&^1+O[.2.P,&mAF!$#+\cUOz%! #+ZAU&@ &!1&^P+O"$2.P,&%AF!$#+\c)*#+7a)*#+@ )&.@S#+%(,[7ZA1,/0/&%&ZS#+ DI 0M&&`#+0POQ.)& [!1&^,$ w&!K2@ *AM(,~731."^.!1& )*#+@j!b.@S#+ :#+,Y6,&L^0#+T#+@A^ )*#+@jdc42%(%% 'OV'W&LM@. +O,&A23&`#+%bxz)<D ODI)%T#+2DT=MdD #%%.!1yT#+".'A/%&!

S)*#+7a:,(%)&73(,7ZA~#P%&'+OgT#+!1 )8<%)&@@S#+%!14,( ai

Dej1)(.)&r0M&&`#+0$OQ.)& HTn73&0<7ZA0

|n〉i =1√n!

(a†i )n|vac〉 =

[

1√n!

∂n

∂αneαa†

i |vac〉]

α=0

x4D0/<D0/'&%K$2T#+`#+!1&%&

α b#+7`#+AxV#+,U^.8)*#+7aK)`<%&%*#+@ dD~hyfn~#+@ %

1A/

32 4. METHODS FOR ANALYSIS

T#*LM@D eαa†

i |vac〉 = eαa†i e−α∗ai |vac〉 = eαa†

i−α∗aie12[αa†

i ,−α∗ai]|vac〉= e

12|α|2Di(α)|vac〉 x4D0/<D 9M

^.|n〉i =

[

1√n!

∂n

∂αne

12|α|2Di(α)|vac〉

]

α=0

x4D0/<D21M e #r.%dOz.@%(.@I73()*#+.%'n#+@@ "w.I%W&L#[email protected]#+%y)(Z7. nOz%!f'2.%.2.n,&mAK!$#+\cK%!K2@AK#wL#)&..! dc423()&`#+ $L<#+@.(w+O,/ 2@S#)(&!1&^23&`#+%%*HT#''@@%(73&@ *WD "@ &

O,&%.7%A4%&!fOzI' )`$y#+%"^%&%(%(,[1.2.`#+%<H#+,

R,&%X6%&!$#+0!14,(

S = ORD.&!1%?n?#%.!1-T#+#i [email protected]*L`#+oOV!$#+

U

S ^'D W%(,/.)((,6,&%A!$#+\cOV".2."!14,(' '&

ρoutO =

R(Uρin

S U†)

x4D0/<D 4^km?8OV@@ *'06n1#%.!1bOV!K2@ )&A?T#+

S)( %+OgJo.r!14,(*H#+,T#+e#+%'Z%&%(%%(,K1'.2.w`#+%'+O3!14,9KxQ8OVIdc#+!K2@ "+O3 @0<^!14,GZ%&`#)&0-'1#+%!K )K!14,dDKk#+22Din10<LMK0M&&`#+@wdc/2%(% L<#+@ ,OVl#+7`#+Ab!1^,wZ.!873&%*D )(Z7. GOz%! |nm〉12〈n′m′| &@ &!1&^+Or2.`#+%%G |l〉2〈l′|

&@ &!1&Z:+OlW.2.y`#+%r (ρin

S )nmn′m′ · 2〈l|g

1

[

U |nm〉12〈n′m′|U †]

|l′〉2x4D0/<D =M

Erd0[#P0M&&`#+0$OQ.)& F (α1, α2, β1, β2, γ2, δ2)

x4D0/<D CM=

1

[

2〈0|eδ2 a2Ueβ1a†1eβ2a†

2 |vac〉〈vac|eα1 a1eα2a2U †eγ2a†2 |0〉2

]

r&0<ZyOV#)&%")*#+Y73r'%& 2〈l|

1

[

U |nm〉12〈n′m′|U †]

|l′〉2x4D0/<D 3<

=

[

1√n′!m′!n!m!l′!l!

∂n′

∂αn′1

∂m′

∂αm′2

∂n

∂βn1

∂m

∂βm2

∂l′

∂γl′2

∂l

∂δl2

F (α1, α2, β1, β2, γ2, δ2)

]

2T#+`#+!1&%&%αi, βi

`&@ ()&"2.!$#+\cY&@ &!1&^'@ γ2, δ2

%&@ ()&&@ &!1&^%+OPF.2.!$#+\c G#+@@*&%!1*#+T#+?7`#)%M&-.@ , 7aF&L<#[email protected]#+%(,#+α1, α2, β1, β2, γ2, δ2 = 0

D b%(bT#+:\OwWOz.)& F

:4*'XnG)*#+X)(!K2.%r.2.`#+%yOV#+ZA~0<LM&2."`#+%<Dekm '`&%:T#+F

0M&&`#+%('i [email protected]*L`#+oOV!$#+ U

DE O e&OS\ \WG>IbJ T [>JdIbN@e^\WG@O > i

[A, [A,B]] = [B, [A,B]] = 0i0V&R MaUM&IfR V&i

V.U(OSR:MX\WV&RWJA,B

Y \WG@OQNeA+B = eAeBe−

1

2[A,B]

# &5UZm ! m

4.1. THE GENERATING FUNCTION 33

:nn`#+,y0M&&`#+0bOz.)& Fx4D0/<D CMw)*#+ 73,/%()&@A$&L<#[email protected]#+%(,H/)(!1< +Oz%&~r^@A$:`#+oOV!$#+ Y+Og:!1^,r23&`#+%%ai

7. !$#+\c%&2%(%&^`#+ 8+OU

De &%dOz%Wn%&'%F

6#G!1%.%dOz.@3OV!YDl%*H/7Â!$#+^01.`+O'x4D0/<D 9M'#+,YW,d 6+O`#)(n<HT#(LM

F (α1, α2, β1, β2, γ2, δ2)x4D0/<D25M

= 〈vac|eα1 a1eα2a2U †eγ2a†2 |0〉2

[ ∞∑

i=0

|i〉1〈i|]

2〈0|eδ2 a2Ueβ1a†1eβ2a†

2 |vac〉

= e12(α2

1+α22+β2

1+β22+γ2

2+δ22)

× 〈vac|D1(−α1)D2(−α2)U†D2(γ2)|0〉2〈0|D2(−δ2)UD1(β1)D2(β2)|vac〉

'&%T#(LMW`#+M&Y2T#+`#+!1&%&%:+OF

%G73r%*#+@vDh:dc/*HeN#+22Dei 2%*LMT#+K2%+J()& 23&`#+%KB!K ,,/@ +Oy#+73*LMWdc/2%(% Y)*#+67ar'%&

|0〉2〈0| =

∫

dpdx

2πe−(x2+p2)/4D2(

x+ ip√2

)x4D0/<D2DM

km%&0K "^%6x4D0/<D25M'#+,2.@@0[."Z%&0<`#+ H|nW0M&

F (α1, α2, β1, β2, γ2, δ2)x4D0/<D0/*;M

= e12(α2

1+α22+β2

1+β22+γ2

2+δ22)

×∫

dpdx

2πe−(x2+p2)/4

× 〈vac|D1(−α1)D2(−α2)U†D2(γ2)D2(

x+ ip√2

)D2(−δ2)UD1(β1)D2(β2)|vac〉

r0M&&`#+@!$#+ÂZuv!14,Wdc/2%(% )(%(23,/0K%G 0<LM&xzi:D 3<dD Ty,&#i [email protected]*LK`#+oOV!$#+ #,/ 2@S#)(&!1&Z'23&`#+%w`#+oOV!1I^%P#2%^,/.)&+O,/ 2@S#)(&!1&^w23&`#+%%*H^#+,G%1x4D0/<D0/*;Mldc/2%(%%(F

#I#+GZ%&0<`#+@a*LM&e L<#)&..!dc/2a()&`#+ L<#+@.+O#P2%4,/.)&"+Oe,/ 2@S#)(&!1&Z23&`#+%%*DIT#+ZA~0<LM&U

#bT#+@dc/2%(% OVF

'7`#+(,Y6%(W%&2 1`#+Oz!1(,,/ 2@S#)(&!1&ZK23&`#+%%b#+%$)*#+@ )&.@S#+%(,Dec42@ )&@AMHl.,&r0M&&`#+@[email protected]*L`#+oOV!$#+

U †ajU =∑

i

bji ai + cji a†i

x4D0/<D0//

;QFHG>O e.OQN>OSR:M8\WIbN@e i0[@N>`S\WIbV.N gaO \WG@V#kIbJLV.N>O-B]M l\WV^V.T \:M&IbN \WG>IKJ]RWOQU RWOQJdOQN\:M8\WIbV&N m


,&mA2a&`#+%Dj(β)

`#+oOV!1"%

U †Dj(β)U =c/2 [

β∑

i

(b∗jia†i + c∗jiai) − β∗

∑

i

(bjiai + cjia†i )

]

=∏

i

c/2 [(βb∗ji − β∗cji)a

†i + (βc∗ji − β∗bji)ai

]

=∏

i

Di(βb∗ji − β∗cji)

x4D0/<D0/*9M GL#)&..!dc/2a()&`#+ U+O G,/ 2@S#)(&!1&ZP2a&`#+%%W &L#[email protected]#+%(,DP W,1%W7ÂX%(,/.)&0-12%4,/.)&W%6#%0<@ $23&`#+%Ozb*#)%!14,KL4S# ,&Âx < /*9>=g2DI=<;M

D(α)D(β) = ei

(αβ∗)D(α+ β)x4D0/<D0/1M

#+,%()(,Y7ZA`#+^0K8dc423()&`#+ YL<#+@.(+Oj(`W2a&`#+%% 〈vac|D(α)|vac〉 = e−

12|α|2 x4D0/<D0/64^

7 -%(%.@0dc/2%(% Hy' )% #]^.T#,/`#+ )Fdc42%(%% ?.2.!14,W^%&0<`#+ XL<#+S#+7@ (x, p

H| ^%&0<`#+%(,?#+@ 0~'XW2%dOv#)&%%y2%(`&Z:x4D0/<D0/*;MdDUOz%&P^%&0<`#+ n~T#*LMY#+B#+T#+@A4 )dc42%(` OzF

' )%BZLM@LM(@A~α, β, γ, δ

uvL<#+S#+7@ (r#+,Y2T#+`#+!1&%&%+OU

D%!

F7`#+(,K7Â80M0%.0<1%&2"x /ax 1M#+73*LM ')*#+1)*#+@ )&.@S#+% `#+oOV!$#+ Y+O#+Â[2."%`#+%

ρinS

.,&U

De:*&LM&*H'&6#+T#+@A4%0K2ZA/ )*#+@%&.2[nXT#+@@r#+@ %U #+Z%)&@.,-6./()&+OW!1*#.%&!1&Z%*H' )% )*#+73W,(%)&73(,Y7Â6#K.`#+A`#+oOV!$#+ D"km62T#+ )&.@S#+*H|nb#+Z:%$)(!K2.%b`#+%+OT.2.e!1^,(O

)(,/ (,KK#)(&`#+K.%)(!1Oz%!#!1*#.%&!1&Z`!11+O1%&!$#+0-!14,(R

D$.T#+%&@AF10M&&`#+06Oz.)& )*#+73dc/%&,(,%1#+@@ *pOz' D 8)( ,&"!1*#.%&!1&^y%G`#+MW2@S#)(b#Oz%&'`#+oOV!$#+ U

DI%!xV9/D0/<D 4^W.!$#+@ %(,6)(,/ T#[email protected].`#+%r "%(&6%G7aρout

O =

R(PUρinS U

†P †)x4D0/<D0/*=M

'&%'2%+Jo()& ~2a&`#+%P

#)&%w$".7%&w+O!14,(R

' )`[ e!1*#%.%(,D dc/2%(% P%&2@S#)(("x4D0/<D 4^j#+,bnOz!b.@S# ,&L^0:Oz%!l!8.73w!14,/\T(,#)()(%,/0<@AMDk6+uv!1^,Pdc#+!K2@ P#+7aLMnb)*#+Y`#+MP = P1

%$#)&Y!14,/1xz<D 0Dnn!1*#%.%W@0<^!14,8#OQ%&Z%&`#)& F#+,,YW)(,/ T#+@l`#+%+OIG#+%!1 dD &Ha.0

P †P = P 2 = Pxz)(

P r#$2%+Jo()& ?2a&`#+% ,T#+"W@S#'%&! x4D0/<D25M "%&2@S#)((,Y7ZA

〈vac|D1(−α1)D2(−α2)U†D2(γ2)|0〉2〈0|P1D2(−δ2)UD1(β1)D2(β2)|vac〉

x4D0/<D0/*CM ~#+%1Z%&%(%%(,)*#%(b'&%$$!1*#.%(,!14,(G#+%12% )<HI#+,1%du]^.%(,G.%)(!1(w#+%'&&I')&@ )%/$#2%4,&%()&%*DeI,&%()&%%e' )`

4.2. USING THE GENERATING FUNCTION 35

#+%G23&oOz()&87.W,%(%@LMKK^.!873&b+O 2%[xQvD <DG&AF,/ 0<. @A73&(&62%(%&)(8#+,-#+7%&)(W+O@0<Z dH23<%7@ r2%+J()& 623&`#+%%y#+%Pdark = |0〉〈0| Plight =

− |0〉〈0| x4D0/<D0/3<%!x4D0/<D2DM73$+O(%"!$#*AG7a''%&~1%&!1n+O,/ %2@S#)(&!1&Z'2a&`#+%%H^#+,%Pr&0M&&`#+0KOz.)& Y)&@.,/01!1*#.%&!1&Z%y "#+0Z#+Ydc/2%(%%(,Y#L<#)&..! dc/2a()&`#+ [L#+@.(n+O,/ %2@S#)(&!1&Z 2a&`#+%%*De !$#+ZAûv!1^,0M&&`#+0Oz.)& Y)&@.,/0$!1*#%.%&!1&Z%y '0<LM&YxzirD2DMdD

r#(L40(`#+7@ (,XT#+r80M&&`#+0Oz.)& F :0M&&`#+@e&.0<X%.r.2.23<%(*H/!1&Z -n~#,,/ T#+@g.%dOz.@3OV*#+.%("+Ol "Oz!b.@S#+ Dl%@AMH+%wT#+j73()*#+.%In`#+%yx4D0/<D0/*=M jg!$#+@ %(,Hw`#)(ng(ρout

O )(]^.T#+@ wy2%7T#+7@A+Og7`#+0G!1*#%.%&!1&Z".%)(!1r)(%(23,/0P%P

D:&)('7ZA1,d0b"0M&&`#+08OQ.)& #n#+73*LM"7`#+[7a1y)(,/ T#[email protected].%`#+%(#+,&"2%7T#+7@ (yYW)*#+@ )&.@S#+ D54()(,/@AMHgnK%(PT#+W%]^.(&(0+On2.W`#+%(W!$#(A673P%*#+%(,U*#@AMD E()*#+@@T#+b$%]^.(&*(,L<#)&..!_%`#+%1 S(ξ)|0〉 '$`]Z.(&(0X2a&`#+%P0<LM&7ÂxV9/D0/<D 91MdD `#+%IP2)&2@ )(Z`#+l% )(Z7. lOz%! #+@@^&LM&b^.!Gu73&%`#+%(Wx < /*9@=a2D /*CA/dDwin. \Og:n&%r%G23()&\OQA1r)(%(23,/0K2.',&A!$#+\cK-x4D0/<D0/*=MdH^n'.@ ,$T#*LM'%!$#+M:#)&.%@/Y)('n)*#+w`#+M#+7`#+@A0<1,&L<#+LM(I1.e2%0<`#+!1:xzI.! #+7`#+@AG!$#+ZAG%&!1 dDyn&LM&nn')*#+#*LM ,? ()((%mA-7Â6)(23`#+0S(ξ)

^%Ki [email protected]`#+Oz!$#+ U#+,[Z.Z%by0M&&`#+0GOQ.)& D

U w%&2@S#)((,~7Â

US(ξ)#+,[y2.`#+%OVb%]^.(&*(,!1^,~ PL#)&..!YDF ~%(%&2P%?7`#+F

#+%[.)%T#+0M(, x /8,/ 2@S#)(&!1&Z[2a&`#+%%K`#+Oz! %G.,&U

#+,&.,&S(ξ).0

S†(ξ)D(β)S(ξ)x4D0/<D0/5M

=ec42 [

β(

cosh(r)a† − sinh(r)a)

− β∗(

cosh(r)a− sinh(r)a†)]

= D(β cosh(r) + β∗ sinh(r)) = D(E

(β)er + ikm!

(β)e−r)

'&%yy`]Z.(&(0P w~X

u]Z.T#,/`#+.%<Dn6.@\uv!14,:%]Z.(&(0G)*#+~7ay%*#+%(,Y!K@S#+"OV# D + * L( $ $' $ ) * & &6^LM(0Z#+0W23&oOz!$#+)(P+OIL#+ .:Erqjs'tTuvA4238%&23*#+%&%WOV4)&.mnY)&.)&S#+@e2%23& (*HT#+!1&@AXPT,&@A?' )%U)*#+F73P!$#+Z`#+(,*LM&8#~0<LM&,/ `#+)(<Hl#+,F1)(!K!b. )*#+ `#+%1' )%)*#+7a1#)` &LM(,D[ &%K P#!K2@ `#+%&0<A~%$#)()((%"(%2%23& (*H.0K0M&&`#+0$OQ.)&

<#IKN>`QO-B O G@M,P2OA>S†(ξ)aS(ξ) = cosh(r)a− sinh(r)a†

O.m e>m # &5UZm B% m


l%b.@ ,X)( ,&b&Z`#+0<@ &!1&Z0M&&`#+ F%&2D km-%!1b)*#%( %&2~ !K2@A~#%%.!1(,$%P7323&oOz()&'#+,~:0M&&`#+%(,`#+%('#+%y23&oOV()&i &@@`#+%(Dlkm[&)*#%(nn:)*#+~7`#+#+~#+22%2S#+%ri [email protected]`#+oOV!$#+ ~Ozr0M&&`#+ HT#+,&6.%:r0M&&`#+0KOQ.)& 6%K)(!K2.%.2."`#+%+O n`#+oOz!$#+ `#+08OQ%! #,/ %&Z`#+0<@ (,~2.*Deh:%"T#+.)()((%oOz.@T&/u`#+0<@ &!1&^"0M&&`#+ OV'#+Â[%&0<!1&^+Ogr%&23*#+%&'%(.@% `#+!1r,&A!$#+\cHT#+,&)(((,Y@A2a&oOV!W)(!K2.`#+ X)(<D h:dc/n:&Z`#+0<@ &!1&^)(()& Y%&2~ )(% ,&%(,DI%! yi [email protected]*L`#+oOV!$#+ ~Oz )(()& Y:)(%(23,/0K0M&&`#+0GOQ.)& )(!K2.%(,H#+,X.0G&Z`#+0<@ (,`#+%(:OQ%!G0M&&`#+ %%&2F#r2.*H3K%`#+%G#Oz%&.)()((òOQ.@)(()& 8OV.,D EdOV&0-%60D1/D 4H P0<LM(P.b[&^`#+0<@ (,`#+%#+G

l = 1@ &LM&@vD `#+%(1#+K.7%(]^.&Z1,/ `#+)(@ &LM&@ 1#+%~OV.,7ZA%&`#+0Y r2%4)((,/.%K.0K.2.b`#+%(rOz%!K)(()& #2.%rdc/*D

7 lT#+@@AMHOz%! @ "+OI,&mA!$#+ )(("OV.,xV9MdH8)*#+Y0M&yrT,&@\umA?#+,FG2%7T#+7@mAU+O %.)()((%oOz.@&^`#+0<@ &!1&ZG)(()& #+W*#)`@ &LM&@vDK T,&@mA[WD *D *Dei &@@`#+%1x 1/D 9/D =M '*#@A[)(!K2.%(,.%0K,d xV9/D0/<D2DMdD 2%7T#+7@mAOz"&^`#+0<@ &!1&Z)(()& 6%K%.)()(((, !K2@A~`#)(W+O)(!K2.%(,K.!$#+@ `(,1.2.I`#+%<HZ#!1&^ (,1K%()<D 4D0/<D r*'0pl

#+*#)`@ &LM&@vH8)*#+,68#(LM&`#+0M8!1823&Zy6)&%*#+0[rT#+@l&Z`#+0<@ (,-2T#+y#+,&)(bG)(!K!8. )*#+ U`#+%<H3&&:OQ%!G#+22%(c/!$#+%Kdc42%(%% x 1/D 9/D 9My7Â[^.!1& )*#+@j!b.@S#+ D X`#+%&0<AB!$#*A7a6.%(,73 Oz[Z.!1& )*#+@!8.@S#+ #+,NOz~7`#+0#+T#+@A4 )*#+@%(.@%*Dl e e!K2@AG#y]^.( 1+Oa'&&en'23()&\OQA8Z.!1& )*#+@TL<#+@.(OV"W2T#+`#+!1&%&%[email protected]*L6`#+oOz!$#+ D"k-b#+T#+@A4 )*#+@l)*#`<H)(!K2.%&ouv0M&&`#+%(,dc/2%(% OVI",&mAK!$#+\c$#+,K'T,&@mA1)*#+17a LM&A)(!K2@ dcH#+,6y y&X()((%`#+A6%1!$#+MG%!1G#+22%(c/!$#+ X%~!K2@\OzAY&!YH<D 0D7Âdc/2T#+,/016%!$#+@@2T#+`#+!1&%&%*D

8#+,GOV@@ *'0-)`T#+2%&b)(^`#+K!$#+U%(%.@%8+O Kn?73&,U ( D kmF2%(%&^6)%T#+2%&6nU,&LMF0M&&`#+@P#+T#+@A^ )*#+@Pdc42%(%% *H' )`'@@#+22@A%F#+ZAE:qls'tTuvA42a%&23*#+%&1.0?`#+!1&^`#+0<@ &!1&Z)(()& %&.2#10<T#+@I2%23<`#+@ < 5>=#+,FOQ.@\@@0-#Oz& #,,/ T#+@ #%.!K2 *D[kXOV@@ *'0 )%T#+2%&Y#+T#+@A4`F%(?2T#+ )&.@S#+6%&23*#+%&6A/%&!1*H:)(!K2T#+0#+T#+@A4 )*#+@2%(,/ )& y'YZ.!1& )*#+@l!8.@S#+ D * :* ($ $ < L L*s ,&'r!1<%'0M&&`#+@gi [email protected]~`#+Oz!$#+ 6,(`)&70[#P)(!K2@ &%W`#+%`#+oOV&8xQvD <De'%duv?#+,.7%(]^.&Z"%*#,4u. Oz#K]^.T#+Z.! !1&!1A D

a′1 = b1a1 + c1a†1 +

∑

i

biâi + ciâ†i

xV=/D0/<D0/

'=3&* *". *%+2. *$+A4 & 1* . 2%* *

a1

#. . '+* #&%1*?% !*$!'+/ '(3&%*$+ +* %+#&*$)'*2 # %1* *

a′1

4

:&%a1

rG!14,K2a&`#+%WOzG2.!14,1)(/u`#+018`#+%%K73W%%(,W!1&!1AMH|#+,a′1

b.2.!14,WZ%[' )%X8%`#+%W "%*#,X.*D' %&!$#+0623&`#+%%8,(`)&7a1#+./c/@S#+A-!14,(^LM@LM(, X'%duvp#+,N%*#,4u.2%4)((%%(*D b

#+,c)(4.;$)& &Z%:#+%0M&&`#+@j)(!K2@ dcD5^`#+0OQ%! $(]^.T#+ NnY'@@y,&LM6#+T#+@A4ou )*#+@'%(.@%KOzG73&T#(L4 .$+OrErqjs'tTuvmA^23%&23*#+%&%.0G#2T#+ )&.@S#+'&Z`#+0<@ &!1&^")(()& %)%&!1#+,

4-6]Z.T#+^.!!1&!1AMD $, W7ZAX23&.7T#+ 82T#+`#+!1&%&%:+OI8!1&!1A6i [email protected]`#+oOV!$#u #*#(A?Oz%! K)*#%K+O 2a&oOV()&W`#+%G`#+oOV&*DK &^`#+0<@ &!1&Zr)(()& -%)`&!1W @@.`#+%(,60DT=/D 9E V8R RWO,M&k>OSRWJ BLG>V M8RWOBHV&RdRWIbO,k \WGMX\ \WG@O k>IbJd`SRWOS\WO^Jd[>g IbJ N@V8\ \WG@O gaV.J\ e&OQN@O R:M&c3i0V8RWg Y \WG>O gaV#k ORWO,k>[>`S\WIbV.NV&i\WG>O N@O#\LJdOQ` \WIKV&N `,M8N T(O M&U>U@cbIbO,k M8cbJdV BLG@OQNl\WG@O Jd[>gIbJLRWOSU@cKM&`QOQk T lM&N IbN\WOQe&R:M8c VXP2O RBM`QV.N\WIbN#[@V&[@J]IbNk O m

1H3

38 5. A PERTURBATIVE APPROACH

& ( #%#&''3 *,('3&*$!*$ $'*,=# 4 & - *$''3&* ?#&+. , . '+* #& ! *A.a1out, a11

(a2out, a21

'+%* *, %*274& 1* ! *,.a11

'(a21

'+%* - +%# *$ #& ( %1*,+*,* +' - ' *,!'+#*A.$4& 1*0 ""! *,."'+%* !# + * ' *,."3&#E%%*$+ (!!*,. +%*274.& 1*+ #&3#+/ ! *,.

a12, a13, a22, a23

#)'3)* #& 1* . %* +'. *,+- #&33 * *+3#&*2 #& %1* *#+ . *A=%#&' 4

[email protected]<Fb0<.%K,&2 )&%!K\c/0YF#[73*#+!2@%&W#:Ozr@0<^W!1^,(H3nM(&2[$!K,1T#+n7a*#+! 2@%&Z%&`#)& )*#+$73"%*#+@ %(,~#+@ %OVw&2ZA/ )*#+@A/%&!1"#+,&)(WW)(()& -,4(''T#(LM8%K2a&`#+%8Y@0<Z < 9A/ZH94=D5^.)()((òOQ.@a)(()& n)(,/ (,#W0<@ y)&@ )%1,&%()&%'@AMDe 2%4)((,/.%6)*#+ 7a6.,&%%^4,^.LM&@A #KOV@@ *" Xn#)% +Or6&^`#+0<@ (,2T#+%' )%Y#+%y7a&0G)(()&%(, +Og"OV! x 1/D 9/D =Mn#+,)(^`#+'ydc)&`#+ Dkm[)*#%[+O" ,*#+@!1&!1 (K#+,,&%()&%%H&%~#+%[%($23<%7@ (8OV8%*#,4u. l<H4"I73$+Oaydc/)&`#+ #+%"%*#,1.*DkOa8dc/)&`#+ [ I%*#,.*Hj&%$)*#+7aK-)&@ )%#+,$)(()& OV#+@ *DkO73dc)&`#+ G#+%1%*#,.*H4&%r'@@37a:mnK)&@ )%/"#+,~r)(()& OV#+@ DlkOJo. dc/)&`#+ %*#,.*Hl173*#+!_%2@%&8'@@ &`#%[#+ÂF/Oz!$#+ B#+7a.b' )%2T#+8P0<T#+%(,Oz%!YH|#+,6&)(8bA4%%&! @ dOz"?#1.23&2a<% -+OIdc/)&`#+ X6!14,(a1out

a2out' )%- "dc#)&@A~W,(%(,6`#+%<D "Z.LMrdc42@S#+T#+ ~)*#+$73'!$#,"0M%.n7ZAK)( ,&0P./()&w+O73*#+! %2@%& ~r`#+% |Ψ+

1out,11′〉 ⊗ |Ψ+2out,21′〉

#+,.7%(]^.&Z 2%+J()&LM:!1*#ou.%&!1&^%*D G,&`#+@ W#+%8%dOV&%(,X%#+22Ds:D &-P!1&!1 (W#+,?,&%()&%%#+%123&oOV()&*Hw12.$%`#+% K1!$#c/!$#+@@A&^`#+0<@ (,HT,&@A'@@73&0<1%8,&0<`#,y$)(()& #+,1"0MM#+@|+O3"Oz@@ '0G#+T#+@A/ w e%W%&@@!b.)%X#+,6'T#+"!$#+&" "T#+223&*DEydOz&0:7T#)%b%y0D1/D 4HM.e#+! l%",K#+Kdc42%(` 8OVl,&%A8!$#+\c+Oa'&Z`#+0<@ (,K2T#+%w#+I*#)`$,/ `#+)( @ &LM&@vD Z.!b7a&+O|!14,(6xV=/D0/<D0/l c(,6#+,!$#*AY2)&2@ W73@S#+0M<D %%%&2Y%* #+%,y y&,'@@g&%dOV%73r%G%(,/.)(!14,Z.!b7a&*HT%GW)(ZLM& &^:%*+Ol%(<D

#%$ $ &N *km' "#+@#*A4y2a<`7@ r%K,d&WH,&23&,&Z:!1^,23&`#+%%

a2, a37Â

b2a2 ≡∑

i

biâi c2a†2 + c3a

†3 ≡

∑

i

ciâ†i

xV=/D 9/D0/

5.3. PERTURBATION 39

'&%b2, c2, c3

#+%W)(!K2@ dcY)(^.;$)& &^%*D W)*#+6&Y%&'%$xV=/D0/<D0/ a′1 = b1a1 + c1a

†1 + b2a2 + c2a

†2 + c3a

†3

xV=/D 9/D 9M .n&i [email protected]*L$`#+Oz!$#+ H/' )`* ZLM@LM('@A[%(y!14,(*D !1^,(

a2, a3#+%ydc#)&@AGr#+./c4@S#+AK!14,(w+O0D4=/D 9/Dlkm$"0<.%<HZ%,dc6%dOV&%r%[b2T#+1xQ@ dOz:r0<^ dH3'@ Gb%()(,X,dc?)(%(2a,r%[,dcxV=/D 9/D 9MdD

b, c)(4.;$)& &Z%r.@ ,73W7`#+(,6OQ%!_xV=/D0/<D0/#+,Y&%8 %!1:Oz%((,! Y)` )(8+O2T#%(*D LM&`#+@@2T#%r+Oj.2.'!14,r '!K23`#+Z W#+,&%dOV%rnr!$#(A#+@ #(A/b#%.!1GT#+

b1 %*#+@#+,F&0Z#+LM<D$%!Kdc42%(%% FOz

b2a2n%(:T#+#b23<%7@ r)(!K2@ dc~2T#`

b2)*#+73r#+7`7a(,~Z%1:,d -+Oj!14,823&`#+%

a2#+,X%$nP)*#+?#+@ %~#%.!18T#+

b2 y%*#+@ <D"qM&' `b82T#%+O

c3)*#+7aW#+7%73(,

a3De" !1^,rT#(LMc∗3a3 =

∑

i

c∗i âi −c∗2b2biâi

xV=/D 9/D21M#+,Y`GnW)`^<%

c3%*#+@vDI:&)(0M&&`#+@jnT#*LMb1, b2, c3 ∈ R 0 ≤ b1 c1, c2 ∈ C

xV=/D 9/D 4^ 1L#+@.(b+O

b1, c1#+%10<LM&FOz%! xV=/D0/<D0/dD~WK%&!$#+0?)(4.;$)& &Z%bnT#*LM

|b2|2 = [b2a2, b∗2a

†2] =

∑

i

|bi|2xV=/D 9/D =<#M

c2 = [a2, c2a†2 + c3a

†3] =

∑

i

bicib2

xV=/D 9/D =+7||c3|2 = [c∗2a2 + c∗3a3, c2a

†2 + c3a

†3] − |c2|2 =

∑

i

|ci|2 − |c2|2xV=/D 9/D =)

y,7a()*#+.à′

!8.%"2%(%&LMW)*#+ )*#+@l)(!K!8.`#+ -%&@S#+ b21 + b22 − |c1|2 − |c2|2 − |c3|2 = 1

xV=/D 9/D CMh:*WHl'[#+73*LM~dc42%(` WOV8$!1&!1Ai [email protected]`#+oOV!$#+ #+,[r)(^.;$)& &^% ZLM@LM(,H/n#+%y%*#,/A1%P)( ,& %&2a*#+%&'23&`#+ ~23&.7/u#+LM&@AMD

:$ " * j)(!K2.%''0M&&`#+0Oz.)& KOV&^`#+0<@ &!1&Z )(()& HZ ((,K%r4*K)(%(23,/0Y`#+oOV!$#+ FG!14,(r' )%#+%G!1*#%.%(,Hj#+,FF

; O `,M&N ` G>VV.JdO M&N lP.M8cb[@O ijV&RBIf\,YT lM U@G@M&JdO JdG>Ibi \BV.N \WG>O V.[ \WU@[>\LgaV#k O.m O g M M8cbJdV^\:M2O

b2\WV^T(O N>V.N@N>OQe2MX\WIbP2O T@[ \]\WG@IbJLM&JdJd[>gaU>\WIbV.NIKJBN@V8\LN@OQ`SOQJdJWM8Rm


!14,(1' )` )(^`#+ ^%&%(0.2.~`#+% A vD <D !1^,(a1out

Ha2out

0D3=/D 9/D b((,X:^b`#+Oz!$#+ F?P#+./c/@S#+A6!14,(:' )%U#+%`#)((,1.:xz%('%()<D4D0/"#+,6xzirD2DMdDw5^)('"nb.2.I!14,(w#+%'I^LM@LM(,1n)(()& b2%4)((,/.%<HI`#+oOV!$#+ G8(%w!1^,(j g!K2@Aw ,&ZmAMD%! 0DT=/D 9PnW7`#+6Oz'r!1*#%.%(,Y!14,( a′′11 =

1√2

(

a′11 + a′21)

a′′21 =1√2

(

a′11 − a′21) xV=/D21/D0/

'?2!1(,?2a&`#+%%r0<LM&?7ÂxV=/D 9/D 9MdDkm?#,,/ F%~b!1^,P`#+oOV!$#+ H~#+@ %-((,$!1*#.%&!1&^12a&`#+%%G)(%(23,/0?%Y~,(%(,.%)(!1(OV$.)()((%OQ.@:)(()& Dpi ()*#+.%X.;$)& &Z~2%/uv%(%@L^0B,&%()&%%#+%YLM&A,/0;$)&.@'%P!K2@ &!1&Z~2`#)& )(<H/)%4<%:%Pn%@ &@A$',&%()&%%'T#+',e)(.Ze ^.!873&I+Oadc)&`#+ *De/u)(.Z0P,&%()&%%*H<!1*#.%&!1&^23&`#+%%#+%0<LM&Y7ÂUx4D0/<D0/3<dDr#(L40)(!K2.%(,I0M&&`#+0"Oz.)& b#)()(%,/0% e%(w%&2+O%()<D64D0/<H8)*#+Y,-8.2.r,&mA!$#+\c-+Oe!1^,(a1out

#+,a2out

OVy#+ZA2.r&/u`#+0<@ (,-`#+%<D5^)(P#+@@j%&0<!1&Z%r+OeW%&23*#+%&r#+%8 ,&Z )*#+@vHaW2.:`#+%(:+Or0<^:#+,@ dOz'2T#+y#+%rW`#+!1<D P,&LM1#+Fdc/2%(% -OzrG,&mA6!$#+\cX+OnP&^`#+0<@ (,F2T#+%W##$Oz.)& +OnK,/ `#+)(1@ &LM&@vDPkm?G ,*#+@w)*#%G+O"#$23&oOV()&W]Z.T#+^.!!1&!1AMHb1 = 1

?xV=/D 9/D 9Me'@ %&!$#+0G)(4.;$)& &Z%L#+ %D ,W23&.7T#+ Y#( #(AOz%! )*#%"~*#)%+O)(4.;$)& &Z%'%&2T#+`#+%&@AMH^M(&20G&% #+w*&%D '@@23&oOz! ,&L#+ 1K 73(ou)*#%'`)(&T#+ W+O#:23&oOV()&I&Z`#+0<@ &!1&Z0M&/u&`#+ %&2H^vD <Djy2.n%`#+%%8'%&Z`#+0<@ &!1&Z")(()& n!$#c4!$#+@@A&^`#+0<@ (,DN K!1*#+KT#+[.$,&L#+ N0<LM(K73(K2a<`7@ ~2a&oOV!$#+)(OV #P%&23*#+%& .0$#P]Z.T#+^.!!1&!1A~,(`)&7a(,~7ZAFxV=/D0/<D0/ #+,~r)(,/ 0%)`&!1W+Ol0DT=/D 9/D X`#+%&0<A6.`6 [%2T#+`#+!1& `-X,&mA!$#+\c #?OV! ' )` K2%(`&LM(,7ZAB)(()& %F@ *([%,&[23&.7T#+ D iA2a&oOV!K0)(()& ?'X y2T#+`#+!1& `(,-!$#+\c?#*HT2.rb7`#+F)(.2@ (,X%()&.%&)(%&@S#+ 8Ozb12T#+`#+!1&%&%P$!$#+\c3He#+,%@L40Y(%11,#+dc/2%(ou OzP[!$#+\c#P#YOz.)& +O"[Z.!b7a&K+Oy&^`#+0<@ &!1&Z1)(()& K(]^.L#+@ &^@A~#'#WOQ.)& Y+Ogr,/ `#+)(:@ &LM&@vD :2%(%&^&%:r,&L<#+ Oz23&.7T#+ ?b2

Drj&.7T#+ -G&)(^.;$)& &^%r :%*#+%(,X%!K@S#+@AX#+,~%()&.%&)(%&@S#+ $#+,dc/2%(% bOzG,&mA!$#+\cB)*#+73$Oz.,#+22DEGD [`#+P7ZAU)(!K2.0?$0M&&`#+0-Oz.)& OV8~)(()& HOz%!i [email protected]*L`#+oOV!$#+ xV=/D21/D0/dD b#%.!18WS#+@j2.:`#+%8%173 |Ψ+〉 HvD <DGUP%%8%&2Fn$)(()&8Yi &@@w`#+%(*DK 12.8`#+%1+O'#+@@n#+./c4@S#+A!14,(w .223<%(,1%b73"L<#)&..!YDek%&0P 2. #+,[y0M&&`#+0bOz.)& OVwy)(()& ~^%G. (*)3+.'( ! 68<uv2%0<`#+!YHy7`#+#+dc42%(%% 1OVw.2.P,&AF!$#+\cHl' )%n[!$#(AUdc/2T#+,Ub2

H#%.!K06T#+b2 1

D

5.3. PERTURBATION 41

!$#+M:OV@@ *'0~.7. b1 →

√

1 − b22 c1, c2, c3 → 0xV=/D21/D 9M

'&%GGT#(LMG.%(,BxV=/D 9/D CMdDGc/2T#+,/0Y*#)%&ÂX+OwG!$#+\c?%~@ *(8%,&&Y0<LM(bxQYZ.!b7a&y`#+%7T#

ρ1 =

1/3 + 2 b22/9 0 0 00 1/3 − b22/9 1/3 − b22/9 00 1/3 − b22/9 1/3 − b22/9 00 0 0 0

xV=/D21/D21M

Ey&2a*#+0P:2%^)((,/.%W#WOz& !1(':& ,&%A$!$#+\c#2.!1Lû#+%('."%KA[:OV@@ *'0[2T#+`#+!1& `#+

ρl =

1 − 2fl + 2glb22 0 0 0

0 fl − glb22 fl − glb

22 0

0 fl − glb22 fl − glb

22 0

0 0 0 0

xV=/D21/D 4^

'&%fl, gl

#+%:Oz.)& "+OlD r*23&oOz!r&^`#+0<@ &!1&Zr)(()& Y'y2T#+`#+!1& %(,

ρl#n2. #+,#Oz%&n%!1:!K2@\T)*#+ n,[T#+:.2.`#+M('Oz!

ρl+1 =

1 − 2 fl2−fl

+ 2f2

l +2gl

(fl−2)2b22 0 0 0

0 fl2−fl

− f2l +2gl

(fl−2)2b22

fl2−fl

− f2l +2gl

(fl−2)2b22 0

0 fl2−fl

− f2l +2gl

(fl−2)2b22

fl2−fl

− f2l +2gl

(fl−2)2b22 0

0 0 0 0

xV=/D21/D =M r%T#+rOz! +O!$#+\c '2%(%&LM(,Y\OlW%&

fl+1 =fl

2 − fl

#+,gl+1 =

f2l + 2gl

(fl − 2)2xV=/D21/D CM

P)(/!1b.G)% )(~+O'2T#+`#+!1& `#+ He\O'(%$%()&.%&)([%&@S#+ PT#*LM~#. ]^.W%@. D '@@l%(T#+"&A,H|#+,6nW)*#+X7`#+6rOz.)& T#+@OV!+Oρl

7Â~`@L^01&!YDj%@LMY%()&.%&)(YOVfl

Hnn-,d-#?&L#+S#+7@ fl ≡ f−1

l

H'#+,B%()&.%&)(8%&@S#+ Y7a()(!1( fl+1 = 2fl − 1

xV=/D21/D 3< ' y#G@*#+"!10M&(.:,/0/3&%&)(b(]Z.T#+ Y'-)(`#+Z)(4.;$)& &Z%H|#+,T#w#. ]Z."`@. De%!xV=/D21/D21M"S#+@|)(,/ $

f1 = f−11 = 3

DkmI I*#A%)`()%XT#+fl = 2l + 1

W#`@. Hg#+,?&)(GnKT#*LMfl = 1/(2l + 1)

DGhy


fl)*#+-738&@!KT#+%(,XOz%! b%()&.%&)(P%&@S#+ -OV

gl7ZA%&0P%@. D r0M&

gl+1 =2(2l + 1)2gl + 1

(2l+1 + 1)2xV=/D21/D25M

y0Z#+b l(]^.T#+ b!$#*A873w!K2@\T(,P7ZA8#y)%T#+0M +OL<#+S#+7@ <D gl ≡ (2l+1)2glr,

gl+1 = 2gl + 1xV=/D21/D2DM

KS#+@n)(,/ g1 = 9g1 = 1

#+,F1`@. U W%(&U%Y7agl = 2l − 1

D ^."r%&@S#+ bxV=/D21/D CM''YS#+@g)(,/ 0<LM&Y7ZAUxV=/D21/D21M'#+%W`@LM(,7ZA fl =

1

2l + 1

#+,gl =

2l − 1

(2l + 1)2xV=/D21/D0/*;M

h:%8T#+rP%@. ? :L#+@ ,?#+@ %$Ozl = 0

H#+,XT#+:G#+22%*c4!$#+ NxV=/D21/D 4^%GW,&%A!$#+\c~ "L#+@ ,6#"@ 0[#'r%&!glb

22

"!$#+@@vDj0M&&bxV=/D21/D 4^ #+,UxV=/D21/D0/*;M,&%&!Kr,&mA[!$#+\c~Oz'&^`#+0<@ (,2T#+%#+$#+Â,/ `#+)(@ &LM&@+O%&23*#+%&KOV1' )% 6#+22%(c/!$#+ 1L#+@ ,H'#+,&T#+7@ .w%b)(!K2.%2Â4 )*#+@@AK^%&%(0K]Z.T#+^ (*Dekm$"dc/n`()& $ny@ 4#+'rT,&@mAMDkmX#+22D3E nb0<LMbP,&mAY!$#+ )((*H3%()&.%&)(P%&@S#+ W#+,-`@. :Oz23&.7T#+ --b&:)(^.;$)& &^%c1, c2, c3

D' b%@. -'&X!1%WT#+?+O8?)(4.;$)& &Z%Y#+%X*&% 7`#+(,!K2@A7Â #,,/0B*#)%23&.7T#+ %&2T#+`#+%&@Ab%r %@. GOz23&oOz()&e`#+% `#+Oz&"xV#0<LM&K7ZAxV=/D21/D 4ê#+,xV=/D21/D0/*;M'&b2 = 0

dDlTl l%:73wL<#+@ ,823&.7T#+LM'dc42T#+% b!b.)(^`#+$#+ZA)&%<%j%&!1e+OT`#+!1%,&e#ln@ *(I%,&%&!OVl#+Âb0<@ )(4.;$)& &Z*D T#*LMy)`()%M(, n7ZA123&oOz!K0P:dc42T#+ ~~#+@@3)(4.;$)& &Z% !8.@`#+(.%@A$ *)+.'( ! 6X#+,&@!KT#+0$0<&y%,&%&!1*D * $ *jK@ *(%,&"

b2:T,&@mAY+O

ρlWD *D *D |Ψ+〉

Fl = F (ρl,Ψ+) =

2

2l + 1− 2

2l − 1

(2l + 1)2b22

xV=/D 4D0/ w%l0: )(wOz%! ldc/2%(% P jT#+&LM&GPw ,*#+@)*#%+O

b2 = 0"T,&@A1 n,()&%*#0bLM&AGOv#*DOQ%&w@AKmnG)(()& %&2F = 2/5

' )% b7a&@ [)&@S#%% )*#+@ /9uv@!K*D? ,()&%*#%[ P%.7/udc423&ZS#+@ Ll/L0 = 2l H7.I\O T,&@mAK I73&@ *K/9W#OQ%&wnP)(()& &LM&1OVw#+$ ,*#+@!1&!1AMH^ ':+Oe!8.)%?&)(.`#+0M&!1&Z*Dk"@ 4/'@M8%!1&0[ "'%0['-.%)`&!1 n&LM&W.@ ,Y)`()%Ydc#)&@A['T#+2%7@ &! !K0<Zy7a<DkmOV#)&K#+dc/23&ZS#+@',()&%*#%~

l.@ ,P)(!1#K#Y0<%*#+K.2 `<DXkm G,/.~%X$Ov#)&GT#+Kn#+%~.0?/u)(.Z0,&%()&%%*D Ey()*#+@@T#+G)(,/ T#+@'&Z`#+0<@ &!1&^1)(()& B+O'0Dj=/D 96n~)*#+F^.LM&@A+O"%(

5.4. CONDITIONAL FIDELITY 43

23<%7@ ( 6%*#,N.$+O<H 6$7a +O:-dc/)&`#+ ~)(Z`#+(, mnF&Z`#+0<@ (,2T#+%DFhy~@ *#,%6?)&@ )àHw#+,~@ *#,P%F#60<@ )&@ )`#+,.)()((òOQ.@I)(()& DkmF$)*#%G'&%$6dc)&`#+ P#+%K%*#,.*Hg1&LM&^.@ ,87aI% Jo()&%(,P#j#Ov#+@ (,W)(()& De:*n&LM&l73()*#+.%I+O7.)%0#+ge73*#+!2@%& HI7aBdc)&`#+ G'@@ &^%&K`#+!1,&%()&%*Hn#+,'&B,&%()&%%#+%~/u)(.Z0 G'@@2%4,/.)(@A#-0<@ )&@ )%3DU:&)([ K&LM&^K'@@ 7a#)()(&2%(,P#g# .)()((%oOz.@M)(()& H+7.I)(,/ T#+@4`#+%I+O/I.2.g!14,( L<#)&..!YD IL<#)&..!)*#+12%2T#+0Z#+%%W.7%(]^.&Z,/ `#+)('@ &LM&@ w+O'%&23*#+%&*H)(-)(()& +Ob#UL#)&..! `#+%Y'#+ &^`#+0<@ (, `#+%6)*#+ #+@ %U@ *#, %#,&%()&%1)&@ )`B#+,B#XL<#)&..! .2.*D%! xV=/D21/D 4^Pn`([T#+KL<#)&..!)(!K23&Z+OW?,&AN!$#+\c #+[l @ &LM&@r,&L^S#+%([Oz%! /-7ZA

2flH#+,Oz%!r%@. UxV=/D21/D0/*;Mn ,&L4S#+ Y,()&%*#%('dc423&ZS#+@@AMDwk y0<%'0L<#)&..! )(!K23&Z' )`Y@ *#,"%G8,()*#(A+OlT,&@AMDk yy&U#$2a&@ (%:`#%[7.@ ,X8%&23*#+%&r.0~/u)(.Z06,&%()&%%h:H/nT#*LMrOV.,T#+"r2%7@ &! "'YrL#)&..!)(!K2a&^y+OW`#+%(*H#+, N )*#%6&%- #F#+%.,DkmNX0<T#+@r2%2a<`#+@ < 5@=y7ÂNErqjs't#~%&@ &23`#+ 2%%4)(@eOVW]Z.7%8U,/.T#+@w`#+@w&)(4,/0Y #8,(%)&73(,HgF' )`-L#)&..! )(!K23&Z%+O8X&Z`#+0<@ (,`#+%([.%(, ,)(Z7.%?%T,&@mA+O'%&@ &2a`#+ DY [2%%4)(@w bb`#+0<ôOz #+%,%6!K2@ &!1&Z w8 2%7T#+7@ ):#+,KZLM@LM( #:23<%& w)(/!$#+ +O3.)()((òOQ.@23&`#+ ~#Oz%&e)(,/ 0W%&2Dli.j lT#j:?.&)(P2a&`#+ K+OTn%&2a*#+%&*H' )`!K2@A1,&@LM&%wT#+@3&Z`#+0<@ (,~2T#+%wT#+'#+%".`(,$["%&@ &23`#+ ~2%%4)(@|Oz]^.T#+Z.! )(!K!b. )*#+ Dnkm]Z.T#+^.! )&A^2%0<`#+2Â&^`#+0<@ (,Y`#+%( '6#b@S#+0ML<#)&..! )(!K23&Z#+%-#+@ %2a%&^S#+@@A.%dOz.@vD 6eM&~2%%^)(@yOV$T#+0#-%()&%&GM&AU73&n(&Bmn?2T#+ (P%(]^.%[~2T#+ (G%XT#+%#Y@S#+0M~^.!873&1+O&^`#+0<@ (,62T#+%y+O]^.7%*Dwi X]Z.7%y+O*#)`62T#+ "!1*#.%(,H|#+,62%(%&)(+OW#+ dc)&`#+ N'@@"^.$736#+.%!$#+ )*#+@@AN)(/!1(, < //2=DN:&)(Y2T#+%$' #L<#)&..! )(!K23&Z !K0<^n&@@a73".`(,1[%.)%#W2%%4)(@vDw5^)(y]Z.T#+@mA$+O%()&%&M&A8nT,&@AG+OT%&@ &23`#+ K.0r E:qls't[2%2a<`#+@/ l?.&)((,7ÂKL<#)&..!)(!K23&Z'+Oy0M&&`#+%(,[&^`#+0<@ (,~%`#+%(*HZ((,#& 0<.%+Or!1&KOVKY]Z.T#+@mAB+O:(%6`#+%(*DBkm%*#,+Or~T,&@AMH nY'@@".%

6X$( O ! ! $ OZ+ ! -[,d(,6# Fl ≡ F (ρl/

(ρl),Ψ

+)'&%

ρl ≡ ρl − |vac〉〈vac|ρl|vac〉〈vac|xV=/D 4D 9M

b)(,/ T#+@jT,&@mA y%!K2@A~8T,&@A'&XWL<#)&..!)(!K23&Z+Oe,&mA!$#+\cY %&"%1*&%Dn 8%&@ &2a`#+ -2%2a<%(,67ZAErqjs't?4y23&oOz()&'&Y)(,/ T#+@jT,&@A+OW&Z`#+0<@ (,-`#+%('.%(, )/<Dh:*nP)*#+6%&L4 &p82a&.7T#+ Xb2

D'%! xV=/D21/D 4^" T#+%,6%K,T#+ Fl = 1

xV=/D 4D21M G>OQN\ BHVÛ>G@V8\WV.N@JLOQN\WOSR M ^T(OQM&g JdU@cbIf\d\WOSRLijRWV&ghV.U@U(V&JdIf\WO JdIKk OQJQY@\WG>O BLIKcbc5M&c B]M J O >If\BV.N\WG@O JWM8gaO JdI k O.m]FHG>IbJ k>O RWIKP.OQJ]ijRWV.g \WG>O U(V.J\W[@cKMX\WO \WG@M8\B\WG@O BHMXP.O i0[@N>`S\WIbV.N V8i T(V.JdV&N@Ib` U@M8Rd\WIb`QcbOQJBg [>J\M&c B]M JLT(O J gagaOS\dRWIb`.m FHG@O O OQ`S\LIbJL`,M&cbcbO,k T@[>N@`:G@IbN@e Km


y2T#+ OQ%! #P0<%'0$L<#)&..! )(!K2a&^*H&Z`#+0<@ (,Y2T#+%"+Olr%&23*#+%&#+%23&oOz()&'&c1, c2, c3 = 0

D1 W W7a()*#+.%1K`#+oOV!$#+ xV=/D 9/D 9M: 2T#`LM G)*#%<HevD <D?G)(Z`#+1@A#+@S#+ 2a&`#+%%*Hn#+,&%dOV%-dc/`#dc)&`#+ )*#+-#+ `W,/.01W`#+%W`#+oOz&'X#+,Y.+O!1&!1AMDny@@T#+)*#+T#+223&~ @ <% +Ojdc)&`#+ '^%b#+./c/@S#+A$!14,a2

H4' )%)*#+@ *#,&&%Y)(()& Ov#+@.%18L<#)&..!_.2.*DW23&.7T#+ c1, c2, c3

K`#+ouOV!$#+ - y:2T#%LM<H3#+,-)(%(]^.&Z@A-b)(,/ T#+@lT,&@mA-,()&%*#%(:')&%*#0[,/ `#+)(@ &LM&@vDkm~Oz@@ '0%()& *Hnn'@@")(% ,&1%(2T#+ )&.@S#+12ZA/ )*#+@"%&23*#+%&A/%&!1*D P'@@l)(!K2T#+%Gb%(.@%r+OwZ.!1& )*#+@I%!8.@S#+ XOz:G)(,/ T#+@T,&@mA'P#+T#+@A4 )*#+@Z2%(,/ )& gOQ%!e2a&.7T#+LM #+22%M#)%HM#+,Wnw%T#+@@^%(T#+n&% nLM&A10M4^,$#+0<%(&!1&^''[`#+0Mr+O3L<#+@ ,/A[+O23&.7T#+ D

km1 w)`T#+2%&n&!K2@ A1'0M&&`#+0bOQ.)& ~!1&^,1%,12a&oOV!$#+)(+O'%(~,/0/3&%&^b2%23<`#+@ OV8%&23*#+%&b%A4%&!1Hj7`#+0-OzP*#)%+O"&!T,&@mA[' )`6)*#+7a:!$#+^`#+(,6*LM&:#80<LM&6Z.!873&+O)(()& 6%&2"#+,)(%(23,/0-)(!K!b. )*#+ `#+%<D$w#)`%&23*#+%&8 W,(%)&73(,F,&`#+@ #+,F%(.@%nOz%! Z.!1& )*#+@!b.@S#+ 6#+%y2%(`&Z%(,Y#+,)(!K2T#+%(,'r#+T#+@A4 )*#+@2%(,/ )& D

! ! $' L$'+

y@@%(G%&2a*#+%&%W#+%G7T#%(,?#+%!K )K&%&!b7@ 1]^.T#+Z.! !1&!1 (^%&`#)&0'U@0<^*D5^.)`#!1&!1AF)( %%W+O#+&%&!87@ [+O'#+%!1b)(Z`#+(,'#)(&@@' )%#+@@ *"8#)()((%r7Â6@0<^7a*#+!1*H<D 0DWOzrP!1&!1A?+O%()<DC/D 42T#+`#>;1/u)(M#+%(,0<@S#%')(&@@ "#+%r.`(,DeinA[2T#%01#b@0<Z'2.@ %r%.0<)(&@@vH/ 23<%7@ %~`#+oOV&yG]^.T#+Z.!`#+%G+O#[@0<^:!14,P%~b^%&T#+@I`#+%G+OwK#+%!1*H#+,@S#+%&K%X%& &LMG7ZAB#+&K2.@ %<DU @0<Zoum#+%! Z%&`#)& GG`#+!1:OV'%(W`&.2*H/7."' ")(!K!1Y%1#+@@j+Ol&! T#+' ')(@@ ()&LM#+%!K )b!1^,8' )%? y.%(,OVy%`#+0M<Dyk,/L4 ,/.T#+@e#+%!1:#+%b:#,,/%(%`(,D %(K%&.2b#+@ %~.%GK`#+!1G&Z`#+0<@ &!1&^P)(()& `)%&!1<HT#+!1&@AXK0<LM&0D=/D 9/D? K)` )([ #G!1L#+%(,7Â%G!K2@ )&A#+,7ZAU~S#+@ ,*#K+Oe 2%o()& n%K%&yWduv2T#%y!1&!1Axz%(W%()<DTC/D 4^'Z%$W0<T#+@Erqjs'tTuv%&23*#+%&2%%4)(@ < 5>=l#+,~,W2a&oOV!$#+)(<Dsn%@ 0< )*#+@@AMHjGErqjs'tTuv%&2a*#+%&b :K@ ,(2%2a< #+@'xV9<;<;A/OV@@ *(,F7ZAGduv2T#%r!1&!1AxV9<;<;4^y#+,?Pmn+uv2T#%r@S#+!12%()((%%0~!1&!1ABxV9<;<;<=MdD 8%*#+b@S#"A4%%&!1"-223<%b%,&*HT73()*#+.%W !$#+M(yOzy#+X*# &%*#,D

4Z=

46 6. THREE REPEATER SYSTEMS

($ )$6$'H $ $%G%&2a*#+%&G2%%4)(@ n~@ ^#+b P0<T#+@'E:qls'tTuv2%%^)(@ D 2%(%&^%(, < 5>=D km X%%Y%()& 73&@ * n,(%)&7a??2Â4% )*#+@W%&.2#+, 0<LM?i [email protected]*L`#+oOV!$#+ e#+22@ )*#+7@ n%y&Z`#+0<@ &!1&ê0M&&`#+ G#+,P)(()& H' )`Y 'T#+"nr((,%G4*p%1#+22@A[r0M&&`#+0KOz.)& DIkm:Oz@@ '0%()& W2%(%&^"%(.@%+O.%!8.@S#+ -+Or2%%4)(@vD/. -/.30 7 @ F @ &LM&@y.)&.%Y+Or6#+%!11%&@ &L<#+Z$OzK@0<ôuv!$#+%&[^%&`#)& %U7a.%(,N *' N0D C/D0/*#/D ?#+%!1#+%?#%.!1(, %736,(`)&7a(, 7ZA #.λ

uA4%%&! D mnF@ n&$%`#+%(K!$#(AOoD cD)(%(2a,%?Â^23&oY.7@ &LM&@ $+O#+#+@#+@I!1&`#+@ < 5>=Dy@@n#+%!1P#+%$S#+@@AU2%&2T#+%(,10<%.,`#+% |1〉 #+,dc)&`#+ ")*#+&73%%(,7ZA~,/.)&0K`#+ |1〉 → |2〉 D Ey*#,4u.)*#+7a#)` &LM(,?7ZAYPZLM&%`b`#+% D P*Z%4,/.)(K]Z.T#+^.!2a&`#+%%:%~0<LM#bOV!$#+@g,(%)&2 6+OWA/%&! +O@0<Zy#+,-#+%!1*D

&?E 3*,!*$. 4 '21 %1*?'% !*$!'+#*A. .%#&. .7 *$.%*$ 3&*%' . - #E1 %1* 3&*$)*$3. %+ %+* .%1 - 4;& 1*+'. #%#&'?. #&)'3&)'*2 # *$''3&*$!*$C*$*,+%#&''( $'*A=# +* #&."3 /'* 4% , *A.%. 3 *$''3&*$!*, '*,*$+2# +*#&+%*A.( 3&#& #&* * %*A='+ '3/4 & 1* #&$'!#( *,' . +* 3%*,+%*2 - / , 01*1% '*A. 3&# '1 '.. #*- #%1 '!#*#+ ##.-#&. *A'.%+*74

W@0<Z"T&@ ,:#+%WT#+.`#+@@A,(%)&73(,Y7Â~!14,W23&`#+%%aL

H'&%b,/0/3&%&^!14,(y#+%W,/ %0<. (,62T#+S#+@@AY7ZA23@S#+ `#+ -" #(LM&@ &0<D 8)*#+6!$#+M#P!K@S#+",(%)&2 OV'#+%!1*D ,dr)(@@ ()&LMr0<%.,xz L<#)&..! `#+%"Oz #+~#+%!K ):&%&!87@ y%W73 ⊗j |1〉j

'j

.0K*LM&'#+@@T:#+%!1*De E VaN>V&\ gaIbJ\:M2O \WG@O \WOSRWgaJ RWO(O,MX\WOSR M&N@k \ (O RWO(O,MX\WOSR bm FHG>O >RWJ\ RWO i0OSRWJ V&N@c\WV \WG>O V8RWIKe&IbNM&c>RWV(V.JWM&c # &BLG@IbcbO \WG>O JdOQ`QV&Nk RWOSi0OSRWJH\WV M T RWV2M.k OSRL`QcKM&JdJLV8i>RWV8\WV`QV.cbJ `.m i*m4JdOQ`&m B#m #m B! ; V8RM&N RWOQM&c MX\WV.g \WG@O M8`S\W[M8c cbOQP2OSc J\dRW[@` \W[>RWO IbJgaV8RWO `QV&g@cbIb`,M8\WO,k \WG@M8\ \WG@IbJQYBM8Nk If\IKJM&NM>RWV IKg MX\WIbV.N\WVa[>JdO \WG@O

λ J J\WOQg k OQJd`SRWI>\WIbV.NZm ]`Q`QV8R:k>IbN@e \WV>RWIbP&M8\WO `QV8RdRWOQJ(V.N@k@M&N>`QO BLIf\WG m< m

<[email protected]#k O (O RWIKgaOSN#\:M8cZM RWV >Ibg MX\WIKV&N@JH\WV \WG>Oλ J J\WOQg`,M8N T(O V.T \:M&IbN@OQkZY>M&cf\WG@V&[@e&G If\ IbJLN@V8\O,M&J m

6.2. THE DLCZ-REPEATER 47

&%&!b7@ '&6,(`)&7a(,7ÂY#PT#+!1 )8<%)&@@S#+%"'-)&%*#+ 62a&`#+% a†A =

1√NA

∑

j

|2〉j〈1|xVC/D 9/D0/

'&%NA

w^.!873&w+Oa#+%!1*D "!#$%$&!' )($*+, |3〉 !-!/.01!2'+!%%3425!2%768!%$*9$0'%29% #2#:$#2%7;<' =$1+,%?>=.

aA:/!$@! .=A #6B2+,1 0$;CD

2$E>FG%2$H!I142% &!J% #K&!ML%G%=1>4#&N!2 O62%.!GL%J% I>F#9)(P!2%I!#J!O-Q$RS2#TJ =$

aA;9UV/-+,#9J#2%$

!K425!2%H-<<!-!/.0J>4G$2A%!%=/6Q!%$WK2+,X 0$K>F!/+, YZ,1!&!# 2%I5!2#" 0$;&?[2#T% !% %=',%#!2%\VO-<%\%?L;F]P;^>7;N_% !% %=9

,%#!$X>4D-%MD-QJ<$P#. P-QT$I%2,%#!'%=!%$X :!%$`2#A>a=% .,;bCc%HS!2JMdTe8f<ghi#202 j!2A $0J%2&3!3[<a !%= 2#.,;k*#)hi%`S2#!#>#!#.`!l:= =$Nm%2A %=$n;mUV/-h+,#/6=m%2Q%!#.JS2#Q#4/!#24#!2%76!2m2%J!2Q%=!% % ">4D-%%0$T/!%*>F&!>$n;IlX#/!A% !% %=&!K-Q/!Z\2#% 9Jm=-<R!G$%%Ko#2p |1〉 → |3〉 #!%2%\:!2$3#2R/!2Rq1/;kr?!JG!n#2>!>D.M1.=A #'2+,1 =$''%I>'-<7%>4Q)($n6,!%$A!% !205!$As=2Z,22%J-<P>4QA$ ,;/!#$ s=2Z,='TL#$\o#2tJ%'=/67Au($*2%E!`>F/!v#1!%$R /!2#$n;Jw%'ZX%?2%9$2#:2%.`<#a=#$MS2#"Y[n2%%2%7;I% #!22%*>4D-%R1=T!%$R!2 T/!%@>49%$#00$@./n+,.@!2

!`c-Qhi 0$Ga=x%@!2%@2%*G2+, 0$aA!%$RA /!2#$* 0$

aL<Ks=2Z,1 /;GG$!1< $#+!2%yG!.R>4J[2%$y% <z = ;G

/!%J!AM!3!|[#J!2!,MVK%A= !\a=x/$!XNS2#~SP;^;,&-<)(P>A4#Y[A2##5!2%>4D-%!2A3)(!2%G!%$2Vs=2Z,:22%/;mV/--Q1/!%@T!:2#:%=!% % T,%#!2%?:+,#.#A%%=KT32%%2%/-<%%bL;QP;P;MD-W!2A@A5!.b#2JN2J 0$ !%$W/!2:&%=!%$b-<!\s=2Z,J=& 0$;\/!2#%\G=I 0$1/!2$&\!%H%=!%$!G"X!2AKO;?G!G!/!$!N!#'+,%3%?!7;FfV;8E>4@!>@H15!@?,%#!2%76"-Q@%$\2,2>FO+#!%Yh

S2#G!2%H[2#& =$aA!%$

aL;R#2`#&+,%H% z| 6-M/!%EL%$H

)(P#2%7

a′L = aL

√

1

1 − pc+ a†A

√

pc

1 − pca′A = aA

√

1

1 − pc+ a†L

√

pc

1 − pc

~S]P;P;,

|B" i| Oi P |4 i R" O |8 S Sij |1〉 → |2〉 i j¡ Y i |i Sij¢¤£¥ i ¦ £ § i£|Y¨4©B£|8ª£ i£ H< ¢D ¢ ¥¥:i [ Pi Y Y¦ ¥ « ¤ jSS

Pi q¬ i |i ni < 7S£ )Si O¢Y i ¡Oj®/j¡|£ ¥ i Y¨ ¯ S Y i ¥[ < ¡O ¦2 i¢ ¥ i 4i O¢Y ¡9¥ ® ¥

|3〉i ¡9i ¢cij

j¥i V ¡:i Y S)¥£|i Pi Y/¦ ª/£ i n ij °7i [i /¡ Y¨


-<#@# X$%2?\2F#!2#X!|[#K@% #!22% !%$pc$#A%`

#2>!>c.3K#/!J!%\!2A&)(!2%7;"@2#I#.,6B9#2>!>c.@m#)h!%9!/!22%)(P!2%

(pc+p2c)/(1−pc)

62-<A#$Ppc-<%

pc 1;

Cc%M# ")(04# % !.32%=#25!>1!#! #/6pc<+,%3>=.F

pc = 1 − e−κ2 ~S]P;P; -<#

κ9A4+, 2%@#%*S2#'J Yhc!2 %=#!22%76l-<

κ2

#242#2%!T<%=1>4#Q7!2 Q!%$A%=1>4#Qn22%q%A"%q z 6n^6 ;#2 /!5!2%G%A!7; fE8[2%$J!#2>!>c.1S2#% !% %

2%%2%MJ$`pc!%$K!T$+05!2%3lT#/!$M!o#2 |Ψ+〉!&l2#$#

O(pc);UN%V#N!9#!2$)hD@>4D-%M:c-QJn/;w

L$c.`#a=#"O-pc6P>"<-<8$#/!2'92AI%/!2%@#!;

dT%2%\K2J24#!2#1>=.E$2>G# O68G[m#!%YS2#G!2%H[2#%=!% % V,%#!2%?%$P%GT>4/! #:>F2 /

a′′L1 =1√2

(

a′L1 + a′L2

)

a′′L2 =1√2

(

a′L1 − a′L2

)

a′′A1 = a′A1 a′′A2 = a′A2

~S]P;P; =-<3# $?2F#!2#:+,%3> .*~S]P;P;,);&\S2#V%=!% % 12%%2%RTA5!#T L;4]P;^>76n)(N!T

%2A%W `3G%/-t!*#2%W2#%=K3#2%!%=`2% |2〉 → |3〉#!%2%7;MI 2%=+,#&G)(!2%I2#$E%E`!2 1%=?=I2% |3〉 → |1〉 #!%2%y!%$@/!+,TJ!2 N%\&2+,J#2%$\!;NG%. 72% +,#2%M/!%K>F:G!2$"+,#. :K^9 6=!%$K:2##42%$P%2,2>4/+X#!%YS2#G!2%?<#)S2#

a′′L1 =1√2

(aA1 + aA2) a′′L2 =1√2

(aA1 − aA2)~S]P;P;,

D-Q?#!%YS2#G!2%Wa=!2%G~S]P;P; =9!%$~S]P;P;,9!#G!q!9-A%$*Z0%/- M!.@&,%#!%X[%2%* =$n;I#2t&#!%YS2#G!2%R[2#'%Ph!% %=q,%#!2%A-Q<2A%5!% !%$G!B<!2A<2#!, 0$'~oi;;T!'!M+,nlT#4/!#);qlIL%$XT!<!<#+,:"<%@G% : Z=%O- 12,2>4/+M#!%YS2#G!2%3S2#<#/!2$0hD2/;CD%L;mP;\-@!#?!O-Q$*%ZV 0$

a1out, a11, a21, a2out!2 2#!,M 0$/6

#!#N!%@%N =$:[2#N9 2#/;:7-QÌ =$V!#9#/!2$\2V!%$ /!2#$n60!%$J<#%9!:B<#G!%%9c-Q1 =$/6,-< m!%)(P+,n8V#4/!#/6P<2A$n;q'$Pn#%T>Fc-Q%M-!/.`/!5!2%76!%$bM2%XA$>=.EL;qP;\AT%RA42#!%;CDJAA.!Ra 2%-<#IJL%!m% !%$H!A,%#!$E> .RA#4/!#1-<q>4 2%=!%$y%9!2AI2#!,T =$N2#<%?2 '2#I~o= 0$;

i # & i [) 4[ ,i 7 /¡ « 4 ¢¤i √1 − pc

P

np

n/2

cn!

(a†La†A)n|vac〉

¡Fu¥¥j§74i Fi |i ¦¦ j¥ O¢Y i

(pc + p2c)(1− pNa

c )/(1− pc)§7

Na

i l £ ¦ i Y¨ ¯ |j ® 7i | SSj i / |¢ |n§7 Y pc

l® :¢Y¥) i )¨


/. -/. - 7 =A@K=V%\M2,2>4/+W#!%YS2#G!2% 2>!%$b%bK#+02 2%b-K!/+,I5!$?1dVef<ghi#F/!#VS2#"+2!#.0% +!:9!#! #

pc!%$3-Q9!/+,

S2%$EGL$D.b!&/!2$P!%`+,i;\2#J!@#!%b+2!`pc-Q`!+,M!

2>!%$HX2A1%/!2%#!;*Cc%HK 0$<:XdTe8f<ghi#F/!#A#202#%=$\%?I#+=2T2%76F&% !% % 92%%2%*\V% 5!.!2 $?K>F1F#YS 6B-<I

pchD$4%$%&%=#T%@&% !% %=9,%Ph

#!2%X7;V!%!.0/!F =$Fl;=9-!2Q>!2$K2%KN24V!2A2%!%$W$=I#)[2#K%2&!.E%E #% J/!2;3UV/-Q+,#A2 `!%!.=/!#)h$P2%IS2#1`2A1%/!2%#!K-Q#K+,%b!#/!2$P.W% ;N%.*+,#.y#2G!K'XL$D.-#3+,%#6"!%$-3#%=`!2#!?+2!J[#2%= #/!I5!2%y!%$?,T!%@)(P5!%!2%?S2#"I2>#+,$?>4!+02#/;ky>4%M> .M$P.0% VL$D.?!%$M>a %=.X=2ZM!"'2A1%/!2%R#!;V2 !J)(1S2#1`2A1%/!2%#!K-<b1[2%$E> .W!?=!2

0$n6P-Q'$J%2%$P/!1!%=.K%#!%="%M2#<% #/!7#/;'##2#%39% #/!2"[#2 K-# ∼10−16 ~oi;;:^/A$P:#2%B)6-< !A/!2M^R2#$# NG!%$?J!%MG!A%=1>4#A%2##O62##F2%$P%K !J#5!+,9##2#: ∼10−10 ;ky!/+,<15!$J#4/!#%T%2%PhD2% %920$2#/6 !%$ !2#)h

$P$&% ;2P; 'QL$D.J$P#2 .0F#>42/!.1-<A2A1%/!2%`$P!%Ll~oI#$P2%!J!?-<%H%=!% % 2%%2%H14#Y[/6!%$W

2#T!/+=%G!%MA4#YS<,%#!2%?3$0%2<B);qw:%M)(!AT<2$%XL;]P;

.! "$#%'&)(+* *,% -(/.10 &324* 56**7&)(6*85 - &19:96 ;"<#%&=(6*2>&=( pc = 10−30 @?=(+*A(624&=(

(+2*CB)&=%#*89 D?1%;%? - (+2*45E9/(3(6*856"F?<3 0 40 25GB1*,5+.IH J,*,%.K0 ¯ ¡5)£O4 F£ 6L ¢Y Y ¢"¨ B )SS ¦ ¥ iQ Si/¡|£|¢ l 7+L ¢Y Y ¢u4i |i¢ SY¨

§7¥¥0¢¤ <¦ ¢ "iQin§7 Y § ¡|S¢Y£|SlS£|i¢ l Si¨


ky"%2"!!#/!2$P.G!|[#qc-Q&2%%2%`mL$D.G!2m$P#24$ >4/-35!2/!.2>!%!>3+!3

1/2;:/![#2 ;mP; E! ?$P#2K!

2%a=%9q!%M%#/!2%G+!2 2A42%% /;UV/-+,#/6u-Q92%$#:$P#>2%?% !% %=NS2#"#.02#! .M2#")h

42#!2%X+=5!IV#2=2n,$K%* 60%XNL#V% #T2%$Puh2%!=L$D.,;Cc%9L;2]P; -/-HQ2%$P2%!PL$c.I2>!%$I[#2 I5!2%'dTe8f<ghi#F/!#:[2#"L+,'$P4#%=:+!"

pc;

! "$#4%&=(6* J?KH -(+ ;?1H4&=% *8%; ;(/.K0(E&=56(6 H D5+?K" (62*$(6?(+2*$B)&=%#*79 ?pc

D?15 (624*4B1*%;?K(+9A&K56*

10−10 5+* 10−3 ?K5E&=HK* 10−2 ?K% BK* 10−1 156*8*,H ∼ 0.5 %#*

wV )(PF$MGL$`!#@2>!%$-<%pc G!T!%$#

2#$#An&!2A$2>X)(P!2% !#X#$n;RkyK%2`!J`2!#VYhD!4$n

F#G!%"! 2%!% S2#!1S-r>4)[2#T$P#2%G!>#.

>4/- '5!2/!7A1/2;

CcV%y!#!TM!K2%!% 9+!;90%32%R&L#T!#'Q&2/6'5!/!\!%$3$P#276:/!%?>F'%@2%3#"%42%W~[; ;q>=.X2%G"2%!I%/!#:/!OQJ!/+,''!4T!I[%2%3VS2#

1 − β(Ll/L0)α ;ky'/!%

%$#!%$3<%3TS2/-<% G!%%#/;kyVZ0%/- ~[; ;q!7;f<m!"##2#<2#$#

O(pc)%`S2# $2>')(!2%

%=#,%#!$J% !%$A!/;8Cc%IS!2=G%=qQ$2>)(!2%# 1\,%#!$!@

pc/2!%$rR2%$P2%!TL$D.r1R!\

F = 1−pc/2;V/-962uB!<!#2% !%%Tc-QT)(P!2%2%%$&-<

Ψ+〉 % )¥ Q J ¡|® ¢ i ¡! YY ¢Y¥)£| ¦ ¥ ¡I ¡ iY¨ ¯ " £O S£¥j¡ ¦ ¡ ¦ ¥

°i i ¥ YY ¡ ® ¤ ¢Y¥j£|il¢ | ¦ ¡|[i £|S ¡,¨


G?KHH*7J (+ ;HI& ?1# %. *J, -(+* 4&K ;570 H (62* 4&K ;5 ?=(+2 9+&=" %*89@&=5+* *J ;(6* 0! H4J,*G5+*8&?1#( '9 *85 D*7J ( (624*G*J, -(E&)(+ ;?1H$ H$" ?* '9 &K% 3&.9 5+*8&C?K#(80 (62* *,(6*7J (6?15+9 J?1#H(24?=(6?1H49C(+2* 5+*8&?1#( D5+?K" ">?* "$#9/( (+2*,5+*D?156* * B)&1J##" D?15$9 #J,J*7969 D#% J?KH4H*8J(6 ?KH&KH " ?*89 &K56* (62*8H H (62* 9 (+&)(+* |11〉 0 2?K(6?KH9<&K56* H4?=(IJ?K#4H1(+* 5+*8&?K#($? A(62* *J ;(+&=(6 ?KH &K%96? J?KH(+56 #(+*89<(6? 96#4J,J,*89+9 D#%AJ,?KHH*7J (+ ;?1H&KH (+2* ?K#( #( '9

(|01〉〈01| +|11〉〈11|)/2 0

22% 2% %$2#/6<%@!?2%$P2%$ 2%YoV2%%2%! !%b!2&D-\)(!2%3~[GL;]P; =);*I /!%&!J&D.0F`N##2#J-<#2! !MR#K$P!%M+, TM#4/!#/;M4 4# !\[2#%2%PhD2%=%3$2#/64-<R!K0G%

1/√

2[2#NJ$2>I)(!2%R#

X2/;s=%XK%=1>4#GN%5!"%=!%$!#J-<2% #>M*!%%=!%$?!#N!:+,

l

2l !%$39##2#"#2>!>NS2#N!l !#9a=!i6-A)(04' $2>)hD)(!2%W##2#'%yA%=!%$E!6l!%$y%J%

F6n

%#/!2J)(PF2%%=5!.R-<lF ≈ 1 − βal 64-<# β 2$\>FJ2#$# pc

;9V/-l = log2(Ll/L0)

!%$?

F ≈ 1 − βalog2(Ll/L0) = 1 − β(Ll/L0)log2(a) = 1 − β(Ll/L0)

α ~S]P;P;],1D.0FA[%2%E$09%Ri!2'L9 I5!$y#9[2#

Fa A-,;

;]P;&/-"N!IL<IT2S2#pc = 10−3 ;q2#<T2<lL;]P; -Q'L%$ α %T#!%,

1

2!%$

β ∼ 0.5pc;

s=%&-&L%$@,00$\!# %=9>Fc-Q%*JA& 0$<~S]P;P;],T!%$\JJh5!2%/6B"N#/!22%!>IG4'!N::$2>)hD)(!2%T%?1,%#!$%=!%$G!#m-<X/!<:2%$P2%!BL$D.A'S!;:dTe8f<ghi#4/!#4#Y[2# &-<%W# A

F6q!2I2%E!2&%R$2>)hD)(!2% % #AX.P36

>"!2<02%X!2"!&$2>')(!2%3!#T#% /6F$P#2/;' T>#/!Z0$O-<%

-<%+,#J#Kj!?22%b=?12 -<#;\K#2>!>D.HN!%H)(P#!22% #O-"q%/!#.J-< %=1>4#Q4#F/!#m % /6 i;;8-<G"$P!%Ll60!%$`/!2$J!942.=%2A5!n/!%&S2#QT##2#%

F;:+!T

α%2

)(!2.y^'>4/!9&##2#"[#2 1%5!!:$ %2VA.?!2$$n;Ns=o2%%2%\+,% :2A% o#2 2%%2%\c-Q l!"!+,1!&+!2 2Jh42%% 9!%$*!#1#)S2#&!#5!.\#$@%@J2%$P2%!L$D.,;1QV2&[#22%%2%T!?$2>)hD)(!2%-<!?<!K!2&%@+!22A42%% T!%$M%2:#$n;V2J!1!`42.=%2A5!h:2#9!'/!29>PhD)(P42%% 5!h:/!%XS2#'3~[2%$Puh

2%! NL$c.*92% %=5!q#a=# % I[2#1!`#4/!#I.0t3>F/! ! ,i Sil

F§ SS£ < ¡<i ¦ ¥ ¡N Y ¢ )£O /¡ ¥, S¥ 7 |i® Y<¤ ® niY¨


;( ? (62*D#H4J(6 ?KH 0 0 (+? F ? (+&= H* D56?1" 96 "$#%'&)(+ ;?1H A -(+2pc = 10−3

032*( 2&K9

α = 1.8&KH

β = 0.4pc

0

'5!2/!7!%%i6B!%$X-T!+,9%M!"'dTe8f<ghi#F/!#V$=%Xi!2< "$G!%$n;V/- [2#1K2AI%/!2%H#!;Xkr%E-K15!K #4/!#1-Q`2>!%

!1/!2W+,l %%2#G!$H$%c.yG!#u(46!%$W> .*!Z0%? #!2K<

-'L%$31"#2>!>c.pl[2#"I% !% %=92%%2%@/!2$P%KA

+,i;q#2 T"#2>!>:-9/!%315!992A1%/!2%\#!;q!,2# !M-R\EL%$ *2AI%/!2%#!RX)(P5!%$%!7;"%32%=)(0`T!%%:Au(P%y#4/!#/;CcAA>!2$2%=!2M#0~oi;;J9I!`Z=%$*<@2%=)h¤f!#@15!2%B)68>F/! /!2# % I%ya !%= !%"!#N%#%=.G#!%$23;qdVN1V%!#T7'!,2# N#%= 2A)(PD.K

O(2(L/L0)) = O(2N+1)6P-<#

NV% I>F#"82%%2%@

#a=#$JT,%#!"!NL%!% !%$G!#m!%$L2!% F#4/!#/;

J%=1>4#1#%9%!#.3S2#',0=$R!I!X%#/!2'%/!#.R-<*

tN#a=#$y,%#!@!@L%!N%=!%$!#/;#2 3G!E~ ;P;,

-?/!%M!GG`! /!2K)(PF2%%=5!N%N (= p0)

6%pl < pc

S2#K!#Q+,/;N!,2# -!2m-<#%K% 7 ,6 -<Gm%2!T#/!q02S2#I%= #/6Q!%$H&#!#A/-9;*#)S2#`X$P!%`+, !%$W+!J

pc!T-QA/!%R#/!2*-<%* =$#!J#%= 1!#A2%$#!>.?##$n; N%*2#!#$P-!#6P> .A#%%%J%G!#! ∼20

27N#2#! QF>"&2>!%O+,#% '!&,=0$MG!TS2#"T#!'-<3#/!22%!>1!"-<%

N = 3!%$

pc = 10−3 ;lwV!N2%a=%QA!2%O68!2%2>F%&4>QNI5!&2A1%/!2%*#!IS2#:G!% .\$P4#%=T+2!T

pc;VII5!2%T!T-Q


!/+,<F#YS2# $K!#:AG!#$A% !>7;=]P;^<>FO-96,2,#Q-< "2##42%$P%!%!.0/!7+!o#2 ~ ;P;,);

Cc%?!7/!2N = 3 pc = 10−3 pc = 10−2 pc = 10−1

s=I5!$3#![10−5 · τ−1] 6.6 ± 0.6 73 ± 2 880 ± 8w:%!.0/!7#![10−5 · τ−1] 15 150 1700%!l2%$P2%!lL$D.

FP; z P;] P;

41G?K" "<#H 'J,&)(+ ;?1H 5+&=(6* D?K5A(+2* 56**7&)(+*,570 32* 4H4&K% B)&=%#*893? FA2 'JE2 "I&.

*C56*7& ? 0 0 &=5+* &K%96? K BK*,H 0

9#!V!#'+,%@%3%:10−5τ−1 6-<# τ <9 I4% :2%@! %%=!% % A,%#!2%2#&2%%2%!A/;\!I

τIK Yhc!2%Ph

#!22%@ 9"1 94% V2%3 /!2# % '!%$35!2/!82AI%/!2%>4D-%@$Pn#% : %=NT#4/!#/;kyQ%2Q!8!%!.=/!)(P#2%M~ ;P;,7O+,#G!Q#!m>=.I!i!22#

!#/(0G!.MP;P;$PNI'!#(PG!2%<G!2$V%KT$#+!2%MI)(0#2%76F!2N+,%?%R; ;P; ;:wNV!A027S2#N2#$#YhDohiG!%$AG!/62#K% #/!9I5!2%`2%PL# @+2!$Pc.G~ ;P;,K!%$%\!E92##42%$P%I)(P#2%G$A%? 6 -<GQA.&:$2A%!% Q# q~ ;P;,);9$2A%!%=:# "5!#,#"!%M9 q!792#"# "> .3!IS!22#"q!/!2

O(p0) = O(pc);

I#!T2>!%$\#J!#I%2T$P#.?2A!#!>J-<@J2AI%/!2% Q-<X-Q#:S2%$G>=.GdTe8f<gM%\ 6,>4/!N. 2%$#2%. 22%`2%=Yh%y$2#G!%$2A4%!M>=.E%$P%W!%%G%.HS2#AX% !% % 2%%2%r7;Cc%M!4#? G 5! $!K% #0$P%H!%%G%.S2#"92%%2%R3Na +!% NA%G%2%PhD2%=%`$2#O6B!%$@!%!.0Yh/!)(0#2%I[2#I`2AI%/!2% M!#K$#+,$b2%HI>!2/;@UV/-+,#-1!/+,&Z,$*!V&)(0#2%N+,%R#1[2#V&2%%2%*#2>!>~o-<y/!2$*XA)(P#2%RS2#V OT!2!.\A.\22%E2%=%; 2#!A!%%8%

L ≈ 24L0!%$@!%?G%.\ [2#N9% !% %=92%%2%

dTe8f<g@#$P:!I#!'O(10−6τ−1)

6P-<3-<%32%'2#$#"8G!%$V[#2TO-QN#!'%M!>7;]P;^;G#!1+,%W%y!>7;8]P;^`!# O- [#2 +0-<42% I<#!2/!a !%=

2AI%/!2%7;qlV,q!:[%VS2#8% I>F#O6-Q</!%J!Z,τ ≈ 1

/;Q$@%RJ)(04# % T-<RJ2%)hi!2T 2#.?!/+,J'2#$#'mG!%$^/ 6!%$X-QT/!%XV&,<!%X$/!&8V#!;kr

τ = 1 TO-Q<#!T

!>7;]P;^T>42 O(10

−1)2##42%$P%AA!>F2 P;&a=> |A% ;Q2#"2Jh

!#2%764%y!Ka !%=tZ,.@$P#>2%y)(P4# % '4#Y[2# $\%N%%!K%y 6&Z,.?x/&-!2 ∼80, 000

>K 67!%$\6B-<*JdTe8f<ghi#4/!#'#!6n&Z,.-2$M!Z,9!>F2 J 2%=3A%$n;qCD

τ/!%M>FV#$P$M !I%!%2%$3/-Q+,#/6

JZ,.RG!.\>4A% 9% ∼15A% O6l-<EA 1>FAG%=9S2#9)(04# % !

+,#uL/!2%\T#202i;q2#"1dVef<ghD.0 <G!.X>4TF>TJ'2#


;_q+,%y2yJ#!TS2#'AdTe8f<ghi#F/!#J!#A/-96l.R!#A>!% 5!.

#T!%\[2#T$P#92A1%/!2%E>Fc-Q%y%$#9!%$\#+,#/;'ky&/!%\+0-$P#12AI%/!2%!2I!X!2%W%y-<W %0$9<A#F/!#J!# 2%Ph%$ya %=5!.?[#2 2%A%$*A!%%mO-!#$IA2#/67%/!2$EQ%!#!i;9C);;'% !% %=12%%2%yV4#Y[2# $@L#9!V%=$AP6F%E!N%0$ ;A%=qA5!21%=$J'#/!2$n;G_%=!% % J,%#!2%E9F#YS2# $*%2%W!#!i6>&2%` % !/!2$H"G2%%2%7;R`2A1%/!2% %#/!2)(PF2%%=5!.K-<`T!%%n%

L/L0;Ci7-N

pc = 10−3 !%$`"! :m#2>!>D.AS2#m%=!% % "2%%2%M!2q-!2Q$A%`2>!%%!>7;0]P;^6,%G:#!<S2#m$P#2A1%/!2%M/+,#!9!%%4n%

L = 23L0 ∼10−26τ−1 ;: 1021 "<!%3T#4/!#:#!;Cc%@AG!#.,64-Q&!+,IS2%$?!VJdVef<ghi#F/!#924#Y[2# T$P#'2Jh

I%/!2%M>F2K%K# nL$D.X!%$K2AI%/!2%3#!;kyN!+,N[2%$K!T'$2>A)(!2%9-<y/!JA2%$P2%!L$D.\M$#!2$;&kyI!/+,! [2%$@!"S2#V :N&2A1%/!2%*#!I:0 O-S2#:#!2/!q!Ph/!2% FQ#4/!#/6 !%$J!S2#)(P4# % V>4Q+05!>2%%2% τ%$V>F#$P$AT>!%=5!.I>4/-2#$#qFG!%$6 ; ;V%+,i;R!% .`2#<l%2V#%=%3!I#/!T.P !+,T>4%`%2#$`%M2##/! %=/6P!%$G"n5!2/!2A1%/!2%M!2m>F%K2A. %$n;'##2#N2#"-<n>4'$P$3%M92=2ZX;0;P;

! "#%$&#(')*+,!!-/.012$&34

Cc%X"2%3-92%$#:!&#4/!#">!2$M2%3'a=!% 2#.X#24$X%P^ ;RfQ#%K@ZE"KR!|(0h655!%Z=hDCD%I87#:9N!%=%2ZF6<;T#G!% .!2IZ=%$P.W!#$E#&#A!%$W#/+0$$bK2,2>4/+E#!%[2#G!2%HS2#&![l!1#!%Y[#N% `!%$?2N1 2#.,6%$P%`%29# :[#2p!2A$2#%V!%$A#>=2% O;q /!%q!m4<#".P -<#/!q%O6=T#[2##4/!#:!#NZ,.`&+,TV <#/!V#q!2!74#Y[2#G!%;

?A@?6B C,DFE2GCc%32% #!2NA9 2#.X$3%?9dTe8f<ghi#4/!#/69#2F!QP^ :!&[uh=$Ha=!% 2#.E-<b>42W-<#)hi%!%$W#/!2$0hD2/;y;]P;]>H/-"J#0$P#M-<A3! `S2#G2#!,@!%$##+2!i;wp#2%W2#% G=>4/! F25!#$3!2%A(`G!Z,"D-QG!2#2@9!2A9%1>;s=!<>4G-<#%b%=@` 2#.b!#K%0$$H%b!@-Q/!ZE. hi425!#$WL$bF#4$2%XT2#% ">4/!36!%$`.K/!%M>4T2#$`%M!%$X##+,$`[#2 T2+,!2A %76l-<E1!@!2 $*M>4 #2%.R425!#$E!2%?(4; K!H ¬ ¢ 4 " ¡ ¥ ª£|

κi ¦ " ¥¥ §7¥ ui JI S <Y κ ¡ S ¦ ¥ K ¢ ¥j¥,i

κjOi Si 2 ¥ iQi i ¥ |i V £ Q¦ Lc¨

6.3. REPEATER WITH TWO-PASS LARMOR PRECESSING MEMORY 55

!#V##+,$X%X. hi425!#!2%7;q2#"2A%<-QT %=2%M!<u82%'0"!$P#2%&S2#82%$I!28Bm=Q24Q:2%Q/-<%J%1L;2]P;]>~o>422 N2B)62%A2%2>!%! QD.0Fc-Qhi 0$<a x%T%=#!22%>4D-%E!2 1!%$\ 1!2T%RJ% !% % I,%#!2%EmJdTe8f<ghD ~[;]P;P;Ô);UV#?O-Q+,#/6:-QR!#@2%#%$r-<r!E#4/!#K%W@/-<% 2#.X#202n[2#<-<?T%=#!22%@!Z,:!A%-[2#3; !2A +,q##J#+2!%='S2#'J% #!22%E%=+,2+,'S2#'>+,

!2Q/-<%`% L;0]P;]!P;ky:a=!% ::.P !2%1(G!%$ Va !%= %=1>4#m#)S#JV(=hD2A42%% N8T2!l!%5!#" 2 %= !%3!23;qT42Ph

5!2%RmI#2%$\! |1〉 T$2A%!%=/6n !T&2+,&%@NF25!#$!2%X(2;I&(0hiF25!#$*2#% 9I!#A$%$\o#2tJ#!%2%R[#2#2%$H\)(P$H+,"!%$W2J@#!%2%J>4D-%>+,ANa=!

m-<'9. hi425!#$`L$32:J#!%2%"-<@!%,m;

4 4 ? 4&1969 %&K56" ?15 5+*8J,*89+9 H ">*8" ?K5+.K0 *,B1*,% 9/(+56#J (6#456*>?G(62*I&)(+?K"I9 ;H (624**(+*,5+H4&=%3"I&1H*(+ J4*8% 0 324* J,?K%%;*7J (+ ;B1* &=(6?K" 'J 9 H 9I&19696#" * (6? * ?1%&K56 '9 * A ;(62&?1" ;H4&KH( ? 4#%&=(6 ?KH ? (624*

m = 1/2K5+?K#H>9 (+&)(+*K0 !*(+# D?K5A5+ -(+* ;H ?15 56*7& ?K#(7032*

%*,HK(62 ? (+2* % K2( #%'9 * 9<&K9+9 #4">* (6? * "<#4JE2 %;?1HK*85C(624&KH (62* 4&)(+2 %;*8H=(+2 D56?1"5+9 (*8H1(+56. (+? 96*8J,?KH * ;( ? (+2* *,H496*," %;* &=H( ? 4&K5 (E9 ? (+2* #%'9 *>(+2*,5+*D?156*< ;H(6*85+&1J ( A -(+2(+2* &)(+?K"I9A9 "$#4%-(E&=H*8?K#496%;.10

2#=qa !%= %P[2#G!2%G%0$$J%AG!$+=5!2%F2!0425!#Yh!2%X[#2 T(0hD$P#2%76!%$MA5!#.`S2#:!2 <<"%0$$X%3$+=5!2%N!%5!#l 2 % [#2 (0hD$P#2%7;UN%mq#+!% a !%= +!#5!>S2#"$#>% T 2#.3!#''425!#!2%@!%$3!%5!#: 2 %=3;mdVMK#2%M(0hiF25!#!2%E11F>J3G!Z,`!XD-hD$P %2%!<$#2%-<3-<702ZK9! V[2#<=V!%$?!2 9^/6P^6P6 ;;,]P; V#!qS2# /;Nf!#5!%K!|(Pm2##F2%$ '2AF2%%=

7N :s=2Z,+,2#9~[">4/-N)6!%$`+,#. 42% 2%KV#Yi!2N7V###%=V!&4>&s=2Z,"+,2#N425!#$M ;m2#T!J#2%.`(=hi425!#$ ¬ £¢D [ Pi Y ¤¦ ¥ 7S < i < n¢ ¥¥ ¡ ¢Y ¤ 7| [ K ¬|¬ L 5¨P i7 Sj¡ i ¢YS ¡|ni¥ 7§7i £ )¥ ij ¡V¢Y Y ¨


4 4@?K%'9/(+*, H 56 "I&K? &5+? ;"I&=(6 ?KH 0 32 '9 K#456* &193 ;H9 5+* . 0

>4/! "!m!#"=/!$A.A!#2%$ S1hiF260!%$J<G!%$:7!% .

$+05!2%y[#2 GF2A11bG!#1!%EG /!%WS1;XCD%H!M &#2%

!#2%$3423@#/!N,2 #.G!.>43#5!2$> .!*c-QhD$P %2%!5!%!#:2%;s=!#%\-<EG /6#/!mo#2 5!2/!Q2I!1GF25!#!2%H:!

=N>4/!pV$#A%$\> .3#As=2Z,V!#! #T-<y!#II$Pn#%T%=%&>Fc-Q%HF25!#!2%!2%R2#2,2%!!|(9(n6 .,68>Fc-Q%!|(P1#2!$+45

!%$ −45o#2 (y!%$E>Fc-Q%H# Yh9!%$y)[YhD#5!#I=X~[G; ;*^/] );

'a=!% !%!2/6/!$ ! E,6F!#9$)L%$M> .4

S1 ≡ 1

2(a†xax − a†yay)

S2 ≡ 1

2(a†+45a+45 − a†−45a−45) =

1

2(a†xay + a†yax)

~S]P; ;Ô

S3 ≡ 1

2(a†l al − a†rar) =

1

2i(a†xay − a†yax)

1s=2Z,N2F#!2#N2>F.`1!%5!#" 2 %= 2AI!2%?#5!2%7

[Sα, Sβ] = iεαβγSγ~S]P; ;,

Cc%X9/!2T-<#92%'425!#!2%3:$2A%!% /6"#5!2%@/!%3>4V#5!2$3> .M!UV% >4#2hiky.0m~oF2%Phi 2 %=K<2AI!2#/;m)(!2.K'!2%S2#"I-<#)hi%R!%$?#/!2$0hD2N=/<% ⟨

a†xax

⟩

⟨

a†yay

⟩ 94>I22 l /!2# % N

S1!#V2%%=#!$3=.X!#2%$M2 VL($`+2!

s1 = 〈S1〉6

!2I#!$H%EL;]P; 0;\#)[2#`-QX/!%H#5!2`K24#!2#S1> .

s1 6!%$$)L%'%-+2!#5!>/

XL ≡ S2√s1

PL ≡ S3√s1

[XL, PL] = i~S]P; ;


-<#:N>#LmS2# ;N425!#!2%K7: m%O-2A.

$#>4$y>=.E!M!# 2%X5!2#&-<W#/!2%24#!2#a†L = (XL − iPL)/

√2~o-<?<n+,.

a†y ;

kyG/!%y2>!%W!MA5!#1$#2%*[2#'K!2AG!%5!#1 2 %=3;G2+,R!%5!#K 2 %= 2F#!2#G[2#K?%I>\!#@$)L%$A5!#G~S]P;P;Ô)

Jα =1√NA

∑

i

jα,i α = x, y, z~S]P; ; =

-<#jα,i

&`2F#!2#&S2#1i !23;@X!%5!#J 2 % 24#!2#

2>4.MI2AI!2%*#5!2%b~S]P; ;,)64i;;[Jα, Jβ ] = iεαβγ

!%$?-<%JxV$2Jh

%!%=/6nA!%5!#V 2 %=vG!.@>FJ$#>4$\> .\2F#!2#XA, PA

67!%!2,2XL, PL

S2#<' /

XA =J2

√

〈J1〉PA =

J3√

〈J1〉~S]P; ;,

krK:+!#5!>+,#.=%J-QV%$ÌZ=%O- !>42N$P.0%!Al:=Yh!2 %=#!22%@/!%3>4')(P#$n;A$#+!2%yQA2,2>4/+R#!%YS2#G!2%R[2#'-<#)hi%E!%$R#/!2$0hD2I

J 2#.*/!%*>4&[2%$R%P^ 68!%$*-QJ-<q%2'##0$PA'#;Jky A.2%'9%#$P% N%M9 ;G% #!22%bUV!A2%5!%WS2#9 I!2%3#2!%W!2AG!A Z,

N2%<$ #:-!2$#+,$ %\^ 6=!%$G!2q<S2#XLXA

2#PLPA

$4%$P%2%@,2 #.,;N_m!2*q1%=#!22%9!2%I$K%2V/!2$?`! 2#.3#202i;s02 %\ 2#GI%$$n6q!%$y%WG#% J/!2 I1#O+=$$W> .yK2%$!2!,9' J~o%@;]P; ;^9-Q1!l1!%2#T!#,!2B);<'%=#!22%$P#%b@L#X!2!,\

PLPA; Ra !##Yhi-!+,*5!6VO-<%r%rL;<]P;]>76

%=#!%, #5!#A!%$E%/!#&425!#!2%b!%$W%K% #!%,XL

!%$PL!%$3!"'! ' T',2 #.M12%$X!2!,'A"!"1!2A

+!#5!>"!#9!J% #!%,$n;QT%=#!22%3%MT2%$X!2!,T#)S2#XLXA

;21>%!2%GQc-QT%=#!22%m$08/!2$J'!: 2#.I#202i;-!2</-<%M%* ;UV/-Q+,#:N#202n#%=$X#T%2.=A #V%X!%$

P$PI J!2.0A #.?qI?L;4]P;]>7;:I5!2V%#$P% 'q

#%=9 67IG!%&L$n67N%=#=$P$y`2A4%!J9!2.0A #.,;Cc%XT#%'!%3)(P#%!nG!%TL$3!2%AV(0hc!|(0O6PT.,6 xhD2AF2%%=l'!2A'!%5!#< 2 % -<4)(P4#%Ve8!# 2#<#2%7;e7

τ>FN

G%H!%$tG K"2% !2!, #2W 02WL;8]P;]>7;Xkr%

\el!# 2#X[#a %.ΩKL(P$rr!

t Ω−1 τ6<i;; r!X

!%5!#& 2 % 1o#2x/%b$P#%R% #!22%-<b2%KG:K & ¯ |i¢ ¡|£O § ® ¡ S¢¤i ¦ ¡" u OiO " i | i £|¥ <¤ i£ E ¥ )Y¦ ¦

i 8 Y/¦ IJ ¥ ¥ Y¦ ¥i£ Bª£ i i)£| j§ ¥j¥ [ ¦ ¥jS ¡ i ¥ i£| ¨ 5S < i < P¢ ¥¥ ¡mi ¥[ I i " OiO " i ¨ ¬ < ¢ £Oi S|)£|¥«¡¦ Y §7i'i |i ¢D ¡ ¤ [S ¡T i m m" i ¤" i¢ ¥¥ ¢ °£¥PS <Y ¨


>'#2!9O+,#9A2!6l%*G!2.0A #.@T-!2$y2/;AwV9!`#X2,2>FO+W#!%[2#G!2%$#>%*` 2#.WA2A.H.0A #6!%$\%\J!>%&%2I&G%.R!K!&#!%YS#T$4%$'2%.?2%R#%@T%=#!22%@>4D-%\!2 N!%$X=/;&#!%YS2#G!2%*S2#T&o!J#!%YS#T%R !>%AQ%2&/6F[#2

P^

X ′L = −(1 − e−κ2

)XL − e−κ2/2√

1 − e−κ2XA + e−κ2/2XL~S]P; ;]!,

P ′L = −(1 − e−κ2

)PL − e−κ2/2√

1 − e−κ2PA + e−κ2/2PL

~S]P; ;]>B

UV#K# I$%2KG2& /6XL, PL

!.*? -<#)hi%b6-<XL, PL

!#II%V24#!2#V[2#N1##+!;κNJ2%M#%

!%$W&!2 $E@>4G`! G[2#&-<#)hi%!%$E#/!2$0hD2/;κ2 I#2F2#2%!

V% I>F#"!2 "!%$K22%'^60 2;8A2ANV,%#!%A[%2%76-X#)[#JR#-<#XMa=!2%G!2 !\2,2>4/+b#!%YS2#G!2%NX 0$24#!2#/;q<:/!2.,6%'.M!#9.0A #'%

X!%$

P

a′L = −(1 − e−κ2)aL − e−κ2/2

√

1 − e−κ2 aA + e−κ2/2âL~S]P; ;

V2 ! 2#. 4#Y[/6 -<%κ 1

;

?19+9 *79,0

Cc% 2#yI5!2% -Q$%2E!2!. b%2)h`#!%YS2#G!2% ~S]P; ; ); kyR!Z,R% !22%=3#>=)h2%RT-<%*J='!2T#2*A-<*!2 /6!%$E$2#%X<K!2A`%H! >=.RF2%Ph!%2IA2%7;XdT2#%K<G!2A`!%5!#I h %= !4%-<%J!2 /!#22%% V2# 0$:!%\'S2#-!#$\ 0$;:J!%5!#N 2 %=vG!2 I9!%,$E>=.\K/!#%;`ky 2%$#&2%.#!%+,# $2#%6-<# A.=hN!%$*xhD2A42%% JJ!%,6=>q%q2%$P%!F$2#%6=%A-<

Jx"!%,$n;Q2,2>FO+'#!%Y[2#G!2%&%$P%V%2m2%=!%"%$F%$%=8 0$/6

!%$M-<8>F12A$3# ;Qk%@-Q92AI',%#!%Ko%2%?[2#:%=!%Ph %="2%%2%3-QVL#"!.`V =$T#$P2%3;P;P;qV!#! #lV#!%YS2#G!2%?!#

κ6PV#>=2%3=G%=

r~oT! NS2#!nn-!

!%$KV!2AT$2#%T#!η/τ;.K!#V%$P/!$X2%KL;P]P;P;kr%

r → 0!%$η → 0

T%2.G#!%YS2#G!2%M#$P<3~S]P; ; );qV2V!"2#"2#"22%3<>Fc-Q%@TD-QG!2"2#<>4)S2#9$2%?/!%@!J>4T%$$3%

r;

¯ ¢Y)£¥ ¢¤ [ ) j ni < 7°7i I ¨ § ® i ¦ i |iSi ¥ i|i V ¡ i £ ¦ iY¨ & ¥jS ¦ ¢ £| F|i[i £|S¢c| ¡O¥ S ¡I £ £ ¦ ¥jS ¡T S£¥i ¡ Y ¢ 7§4£¥j¡ ¦ " Si i Oi/¡|£¢ i Y ¨


N)[##%R?L;P;?K[Q#!%[2#G!2%bS2#&%=!% % `2%%2%>F)h2 /

a′′L1 =1√2

(

a′L1 + a′L2

)

a′′L2 =1√2

(

a′L1 − a′L2

) ~S]P; ;,

-<9# $I2F#!2#l+,%9> . ~S]P; ; m~[2#7q2##42%$P%:%2.V#!%YS2#G!2%B);

! 41' 4 4:H(E&=H1%;*" *,H( K*8H*,5E&)(+ ;?1H 0

s09S!#m-Q"!/+,N!$A%2%J!>42Q%=!% % ,%#!|h2%7;ms=%:"#%=Q#4/!#mm%Ia=!% 2#S2# /6=<-!/.J',%#!:%=!% % -Q2$ >49#)h!K% !%$!&" A!%$W2# >a=% .,;I/!%y>F !2+,$y; (n;A> .R!#! #`$/-<%W2%=+,#2%7% !N%2%Phi%/!#m#.P!2%22%G/!%J>F2% +,#$G% Tc-Q22%QB/-#q[#a %.,60!%$A<2q=m% %G!c-Qhi 0$9a=x/$M! ~SP;^;,)6i;;<<% !%$n;QCi-Q2%$P2%`2%`!'%:ZA%G:KO-<%G%AL;=]P; z 6 c-QAX!/!%X>4V$G&,%#! |Ψ+〉 ;CcQ4>J%$9?%=!% % V,%#!2%?%32#:I5!2%~o\a=x%b24#!2#XG!.>F@%$$r% !W,%#!%[%2%B); UNO-Q+,#/6T-QR-<1!2 *A.r!H4#YSX%=!% % \,%#!2%7;J'9!`,00$\!#%3F2%='M)(!A% J>F!/+=2#1J#F/!#9[2#+!#.=%A+!"' 2#.X!#! #/;s=AA%G760[2#<I5!2%?-QTT'2,2>4/+`#!%YS2#G!2%MS2#<Vo

!Q#!%YS#%$P%'%26,!%$&-Q"!2 Q%5!!% =$>4TF#YS.M% !%$?!# |Ψ+〉 ;m'%5!!9q!l2#" 0$":!2 $J>4T+2!23;


?A@? D Eky:!+,NF#YS2# $G%= #/!n15!2%M7ND-Qhi!2Q#2024$#>F$X!>4/+,!%$`S2%$`'2%$P2%!nL$D.3!%$M2A1%/!2%@#!;q2#<<#4/!#<#h024!4N#<8!7;PJ!.,;/- -'2A!#N'!%!.0/!4#$P2%[#2 F##>!2%@2#.X-<3T#:T% #/!I5!2%7;#A!#&#&!#! #/6

κ6r6η6F%R2#T 2#.@ =$i; :m

κ!%$

η#%E2I%29M>4 #5!qS2#9K2%$P2%!mL$c.,68-<rA1 %=1>4#

<2%%2%H'S2#'-<!X,=0$\L$D.y/!%W>FJG!%=!%$n;κ1AF2#!%=

S2#QT#2>!>D.X8T% !% %=N2%%2%?!%$`%V[2#T2JhI%/!2%\#!;VT!K$4%$T2%

r6F!2R-/!Z,#/;NI+2!&

ηN%2

#5!S2#JXF#YS2#G!%MN`#202i6!%$HS2# !?#/!X+2!η ≈ 0.1

4#Y[2# :% 5!.X'! I!2[2#

η ≈ 0;q;]P;^JO-""9$4%$%9

2%$P2%!lL$D.M2%κ!%$

η;

4! ;"<#%'&)(6*$J,?KH ;(6 ?KH&=% *8%; ;(/.K0 @**8H*8H4J*3?KHη032* %?=(E9 962?

F4%;?K( (6*

D?15η = 0 56* η = 0.01

η = 0.1

η = 0.2 K5+*,*8H 0 * *,H*8H4J*$?KH κ 0 F '9 %?=(6(6* D?15

κ = 0.1κ = 1.0 5+* κ = 2.0

κ = 3.0 K5+*,*,H 0 4?K5 ?=(+2 &! &KH r = 0.01

0 ?K( &=%%%?=(E9A&=5+* B '96 %* #* (+?>?)B1*,5+%& 0Cc%KL;P]P;^^V-V$P5!.

FS2#Q+!#.0%I+!<lV#>=2%3=G%=/;kyV,

T! TYhD!4$K2"!2mS2#TdVef<ghi#F/!#/6%$P/!%A!<V !%/!%M&L$c.@M$P#2*G!/.@>4AA5!#/;

FT2$RS2#

κ = 2!%$

η = 0.1;

w<2#$P%KA#+2!I2##42%$%9-<RfV;w&;@Z3!%$;Bs=#2%@1!##/!@% I>F#M?W?AM9-<!M/!%r>F\!2+,$)(PF# %=!.r!#%=/;V2!2%$P2%!L$D.J$F%$#2%.&2%

r;8CD$#/!>VZ,

r!29O- !294>68-<%G2%G2$y>4A+,#.*/!#)o!>F21G9!J!#

$I%J)(04# % /;#2%'$F%$2%rA 2+!N$P4%-<

/6!%$HGF2%=5!uhi#!F$!2A`2$J%/!2$n6q!22#J#2> G!/.K>4'% +0!,$3%3?.P /;kyI-Q2$@Z,IK2A!#&&15!2%R#N-<@J!%!.0/! 0$

!7;mP;#)S2#M-QM!/+,?!$ 0$X#$P2%r!2G$#>F$%;mP;\


4 ! "$#4%&=(6* J?1H ;(6 ?KH4&K% *,% -(/.10 ?K5A&K%;% %?=(+9κ = 2

&KHη = 0.1

*J* (G(+2* (6? D?15 A24 JE2

η = 00 5+?K" (+? (6? ?K( (+?K" (+2* B&K%;#4*89

r&=5+*

r = 0 56* r = 0.01 ?K5E&=HK*r = 0.05 156*8*,H r = 0.1 %;#4* 0

[!"#!%Y[#Q2,2>4/+A#!%[2#G!2%7;qm#!%YS2#G!2%K2% !%%$4%$% V =$&~oi;;

i%W~SP;^;Ô"#%<[#2 ^9GPÔ:!%$M10G% V

0$J2F#!2#9!#&2A/!$y)(P#2%V%κ, r

!%$η6B>T%+,#9#

!#A2 &%&S/!#/T#'A=G%=c1 = 0

;Js02%$n67[2#1! 0$9#bi = 0

2#ci = 0

6P-<@A"!c2 = 0

;q#$n6T#G!%%K0G% N#%2"J>4'#/!i;qUV%'-'%$?2%.X$A4##>!2%?%

b2!%$

c3!%$M-'%$M%2

-2##.@!>F2N9!2O; :/!8!N#&!#1%K#:# T!%$3!V-Q&2>!%K1!%2J4##>!2%b%

b2, c3A.*>=.E!2$$P%\K!#!`22%/;

#2 :)(P#2%Q%K;=P; !%$K!7;0d 2%"G!/.AL%$ V$%D.AG!#u(ρl[2#

&% !%$\!#T!Nl @+,qI#4/!#/64!%$?[#2

ρl-&/!%\2A

F!%$

F;kyTL%$n

F =2

2l + 1− 2

2l − 1

(2l + 1)2b22 +

1

3

(2l − 1)(2l − 6)

2l + 1c23

~S]P; ; z

!%$MJO-QV2#$#"%b2!%$

c3

F = 1 − 1

22l(2l − 1)c23

~S]P; ;^,

V2m!q2%.c3!4/!#%&2%$P2%!PL$c.,;ky/!%J%$#!%$J%J

S2/-<%'-!.,; :)S##%T>!2ZIJ~SP;P;,lS2##$P$A2,2>4/+&#!%YS2#G!2%


lV!V#!%YS#Q%760!"!9%2%Phix/#c3!%,V#!%YS2#G!2%K[#2

!2+,IM!2+,;&N2%Phix/#b2/!%%21!2$$y!% .\)(0#!`)(!2%'% XA.P36

>N%2%Phix/#c3/!2$NG$2>&)(P!2%O6F!%$n6F!2:-9!+,&%769/!2$@G!

$#/!2NF;:#%:7$2>N)(P!2%/60-<%

c3q%2%Phix/#I/!%G>4"%

$P#.@o#2tA)(P#2%RS2#'A$%c.\G!#u(R+,%y%E!7;ndJ;J#2 ~[dJ; =-'!+,

l#(|11〉〈11]ρl) = 2

2l − 1

2l + 1c23

~S]P; ;^Ô

-< A3!M*#2>!>c. [2#?!b$2>E)(P!2% =#M3$P#.#242#2%!7

c23!%$?!#,!2

c23-<M%#/!2%K+,i;

4 H&=%.1(+ J J,#56B1*$&=H 96 ;"<#%'&)(6* ?=(+9 J,?KH -(+ ;?1H4&=% *8%; ;(/.K0 32* 4&K5+&K" *(6*85B)&=%#*79A&=5+*

κ = 2r = 0.01

η = 0.1

032* &19 2* BK*,56(6 'J,&K%%; H*C H 'J,&)(+*89 (+2* 5+* 'J (+* %;*8BK*8%&=( A24 JE2

F56?49A& &. D5+?K" 10

Cc%*L;]P;^/M-G2F2>!%$Eo#2 4#>!+,X!#,!276q!%$EK2#Yh

#42%$P%E+!A[#2 I5!2%S2# 32#!%,\#+,3TL;Q]P;^^; V2M!~S]P; ;^,'9 O-QJ2#$#&!#/(0G!2%b?A[m)(0#2%W2>!%$E%E4#>!2%76!%$G%'!#-<=V2<2%.AS2#G!

l;kyN:!#V

,0=$M!# %=:% l'$+=5!2%3Fo#2v^T>42 <5!#,6Pi;;%=7'##2#

%F>42 "%uL/!%=/;s=A5!#N!# %=:S2%$X[2#<2#"+!"'!#!|h

#o##<2N/!%3>FVS2%$X%@!7;_:;ky'/!%3 ~S]P; ;^,<J+,1!J#2G!'T+,l!<-<?9##2#<%32%$P2%!7L$c.`>42 "%uL/!%=/62#1a=+!% .*A+,!1-<W,=0$*#4/!#14#Y[2#G!%G>#/!Z0I$/-<%7;X


>#/!ZP$/-<%\=#<-<%?'G!%$1c3hi#p!#,!2&^6i;;-<%7

1

22l(2l − 1) = c−2

3

~S]P; ;^/,

2#V!% .M2I!#! #κ, r, η

'+!9c3"2A$\!%$M:a=!2%

"2+,$MJL%$M1+2!9l-<#

F$P#2 ;ky'!/+,9%$P/!$3IG!$

>#/!ZP$/-<%J2%'L;|]P;^/:!%$9-Qm%2q!7#m7,0=$I!# % -<1q%= #/!I5!2%3~[!V!7;2_Q);+2!8S2#8Q=/!2%A>#/!ZP$/-<%A2>!%$[#2 ~S]P; ;^/,A2$2%.b>FX!Z,% !2A#2G!K%6Q-<%

c3hi#

!#,!21^6PVF##>!2%?&L#<2#$#"%c3-<3/!2$X3~S]P; ;^,/!2

>4:+!$M!%$XT$P#2M!I%2"!I-Qn0/!$M% c.,;qT>#/!ZP$/-<%38#4/!#4#Y[2#G!%I!%$MT>#/!Z0$O-<%@'4##>!+,I!#,!2\2%$;ky\2$%2\!K\)(0#2% ~S]P; ;^,G#I>X@A@ 0$T

~S]P;P;],m>%2Q$%=/!71/;Cc%X;0]P;P;I-V$P$`%2!+,'!%=.K)(!2<!%!.0/! 0$S2#mN+,22%X7##2#<!%$ -NA.G!/-Q$`:FO-Q#

αLl/L0 = 2l

@$+=5! o#2 ^ >4/! -K2>#+,$E!&2 G2% #>2%JF2% !%%

##2#m!#:#$Jq!% 2#/;CD%J<#% m/!2-<!+,N!T 2#"!2#!$#2%RmI$%D.@G!#u(\!%$31[2# q&##2#/;:Cc:G!/.3-Q>49!V!2!S2#

F[2#Q:dTe8f<ghi#4/!#<!9% +,2+,<+,#!# 7$P4#%=

4/-#A%Ll/L0

6Q!%$b>=.EL%*-3/!%bL%$b!AX =$1~S]P;P;],JJ%bS!2G!,0=$!#/(0G!2%r~S]P; ;^,J[2# O- +,/;UNO-Q+,#`3%=5!"S/!#MT! T%X>42X/!2/'##2#%32%$P2%!nL$D.`#/-"<42.0%2A5!.K%X2AI%/!2%r$P!%

Ll/L0 = 2l ;2#4/!#K G24#YS2# $P#2AI%/!2%b~o+05!A!J%2.X!%% :!2<#a #$n;2#lq2AI%/!2%&#!m2>!%$1>=.T15!2%&-Q!/+,m2%.9%%=1>4#/;

<! !%M$P:IV##2%AF$G>=. V#% '82#<!,2#3;ky2>!%J<2A1%/!2%G#![2#2#!%,<#+,<L;]P;^^-<%

N = 3;#

2%%2%ETT!>42'JT-QA/!%*,X-<R 0$#!A#% I!%$R-Q=?@2#!%,\#+,@>F/!\!RL$c.-Q'!>4/+,@?5!2/!J^2*A`G!%=!%$n;:.X%#/!2%

κ2%9G!/.@!A%#/!211#!6>

κ = 2N!#/!2$P.

%I:)(PF# %=!BA!%$G'-QN$9%2m,&!>FO+,"Q+2!;qs=%N% I5!2%yq&#F/!#T-<R$@`9L$VL;n]P;^Î-QA!2 $\4#Y[,%#!2%Xl% !% %=/6i;; |Ψ+〉 6=-QV$9%2Q!+,T!9m#2>!>c.A[2#Q,%#!2%b7;M#)S2# -QK!+,K3G!Z,`!32;3l@/!2G2A!#2%H-<'dTe8f<ghi#4/!#N#/6P-Q9=

p0 = 10−3 ;kyTL%$nwN !%H-QXK!IM!%!.0/!.E/!5!$b#!G&0?H> .b!Mi!22#J

!>42`P;P6<>`X!2#!@-<%r\2#$#`9G!%$; ?#!?S2#K$P#2AI%/!2%@ ∼10−27τ−1 6i;; 1022 "O;#2 \15!$r#!K?4#YS2#G!%@9\dTe8f<ghi#F/!#X` .

>4#1!%W[2#1K.P "&2%7JK#!&!#G-<%Wc-QR!%$!#$$+05!2%/6><VL%!l2%$P2%!lL$D.`S2#<'dVef<ghi#F/!#N9^ z #/;Y¯ S)¥£|i ¢ T¦ °)£ ¡ i £ < i¢ ¥¥ ¦ S)¥®/ |

x(x− 1) = 2c−23

¨ £|Si x = 2l

¥«¡Ol = log2(1 +

q

1 + 8c−23 ) − 1

§7 § ¢D i Si® i8S ¢ l ≥ 0

¨


N = 3s=15!$@#!

[10−5 · τ−1]wN%!.=/!l#!

[10−5 · τ−1]%!

F

κ = 2η = 0.1 5.3 ± 0.7

^/ P; zr = 0.01

G?1">"<#H 'J,&=(6 ?KH 5E&)(+* D?15 56**7&)(6*85 A -(+2 ( ? 4&K9+9 %'&=5+" ?K5 56?K(+&=(6 H " *8">?156. &KH4H&=% J?1H ;(6 ?KH4&K% *8%; ;(/.K0

τ 9A(62* (+ ;" *D?K5 & 96 ;HK%* J,?KHH4*8J (+ ;?1H ?15 K*8H*,5E&)(+ ;?1H &)(6(6*8" (80

UV/-+,#/6%yGI5!2%KdTe8f<ghi#F/!#&-Q`$P$y%2&2%$#&!%=.*%22#O6-<#-Q!/+,<!Z,%A% 1!22%=q#/!#>=2%K!%$ !2A$2#%;QUN%'-9/!%%2:!/.X!%=.=%`2%+,1!>42"-<?#2=274#YhS2# '>4#/;KCc9A I>F % #%\3)(!A% 2AI%/!2%H#!G%y/!2'%J%26

r = η = 0;

##G2#GT%2X-<-Q#3%2 %$$%2# I5!2%G-<:>F$P$X%M9202Z`; 0;P;

- 3 012$&# 4

X#$!%$5!2A#F/!#A-3!"02ZH!A >!2$2%Ma=!% 2#.#/!$*)(P4# % !.R> .RAa=!% 2'#2H!'A:92#9Cc%~o!#`9>NwVN$ 5 );Q#K#G-#?#42#$% ^/ ; N%.?-<#)hi%K!2>F%`$ 2%#!$X%K)(PF# %="!%$G%X15!2%-V!2 V!9F#YS#/!2$0hD2/;ky"!12%$#Q!V#/!2$0hD2m%A# Q4a %=5!!2q7!T=q6-<3/!2$"J#!%YS2#G!2%@a=!2%:A5!#"J9-<#)hi%7;

?2?6B C,DFE2G'3[2#<-<#)hi%@J" 2#.M/-<%?%XL;]P;^ <+,#.XA5!#"A!N;7]P; ;^6!%$*A+,Q## G!2 9'G! ; wVT[2#' D-Qhi!2 6M% #%Ha !%= !`!#3%0$$%!R-/!ZH.=hi425!#$bL$4#4$?2%*!G#2%G(0hiF25!#$@2#%=V>4/!36n!%$?.\!#1-<#%\% `!2A"!%5!# 2 %=3;q<.P a=!% $ !2%T(J!%$JF25!2%KN#2%$`+,~o-<

m = ±1/2q<!2 $GI>FN$2A%!% /6I!V!%5!#

2 %=pN#2%.XF25!#$?%3'(0hD$P#2%7;UN%11UV2%Ph65#G!Z,!#/(0G!2% ~ L;l]P; T!I! !%76!%$*A &!%$E!2 1!# $#>4$R> .!# 2%\5!2#Ka=!2$P#!#\2F#!2#

XL, PL, XA, PA; wV %=2%$#+0uh

2.E!@%K!2&#2bKI%2&%2b@G!Z,X!@a=!% 2#.,;2#K3c-Qhi!2X ?@2%$!2G-!2K% #0$P$W2>!%@$#$2#!,G<!/;KCc%yG#% & G /!2# % A!%$*S$P>!2Z*1$n;KwV%$P/!$H2%W L#6K24#!2#

XL[2#I JI /!2#$b!|o#I!2!,

#2XVi60!%$`V22 V7N /!2# %=<mS$ >!2ZG2% &T!2A S i 2 ¥ i¢D |)£ ¡ ij ¤ u £ i£ |i¢Y S §n§n§¨ ª£ i ¨ /¦ ¨ ¡

6.4. REPEATER WITH ONE-PASS MEMORY 65

24#!2#PA%A!'G!%V;krGQ[$P>!2Z "-<#)hi%Ma=!2%Q[2#

'a=!2$P#!#124#!2#">42 ^ 06n^/

X ′A = XA + κPL

~S]P; ;^/!,

P ′A = (1 − κg)PA − gXL

~S]P; ;^>B

-<#m# l$%2Q2F#!2#!|o#lq% #!22%A!%$'S$P>!2Z4;κ7m2%

#%E!%$gTJ[$P>!2Z@ !%7;AV2J!'Aa=!2%I!# V.0A #

%X!%$

P>F/!J ,2 #.R *T%21.=A #J%*.*!%$RxG!%$

>4/!'1!>%9q!% .M)(P#%!7G!%TL$n;

44 H* 4&1969$" *," ?K5+.K0 *,BK*8% 9 (65+#4J(6#5+* ?A(+2* &)(6?1"I9 H (62* *,H9 *8" %*K0 32*J,?K%%;*7J (6 BK* &)(+?K" J 9 4 ;H 93&K9+96#" *I(+? * ?1%&K56 '9 * A ;(62 & ?1"> H4&KH1( ? #%'&)(+ ;?1H ? &)(6?1"I9G H&KH *,B1*,H 96# *,5 ?9 ;(6 ?KH ? (+2*

m = ±1/2K5+?K#H 9/(E&)(+*8980 !*(+# D?15 A5+ ;(6* H 0 "I&1H*(+ J

#4%96*CJ?KH -(+ ;?1H* ?1H " *8&19 #456*8">*8H(3?1#(+J,?K" * D*,*9XL

&1J ?KH(6?>(+2* &)(+?K"I9,0k*#)hi% $#>F$ > . ~S]P; ;Ô`!2M>4% $ 2%#!$ % )(PF# %=3-< !

L$c.Kl2#$`!K ∼70%!2Q#42#$K%R^/ ;UV/-+,#%1#/!2$0hD2[#2

JT!2T>4%y$ 2%#!$n6l!%$R%*2#915!2%9-QJ-<q!2 J!T#/!2$0hD2T$\%RI%=!% % &2%%2%yRVA.3F#YS/;'CcN-2$*2#m>Fm#)[#!>Q"!/+,!" 2#q#/!Q 0$,S2#lq#/!2$0hD2/62>!27%2%Q!2>4%\$ 2%#!$y!%$@#&V%X2> +02/64)(04# % !.@S/!2>&#2=2"~[>4/-N8-!Z,m!#,!27;AV%9!VF#YS#/!2$0hD2q /!%!2#qI5!2%-<+,"!%A4#>F2%$A2%J4#Y[2#G!%:4!V#4/!#>!2$A2%A<2%%2% NL;PP;J!%$`T2%)hi!2 2#.,;qwN% .GM#4/!#-<3!I#/!V#/!2$0h2MM>42%$b4#Y[2# -2#*!%2#M$/!I/!2; k32RS2#M#/!2$0hD2/6'o7!9#!%Y[#<%=K!%$32"' 2#.X<+,%3>=.F

X ′L = X ′

A = XA + κPL~S]P; ;!,

P ′L = P ′

A = (1 − κg)PA − gXL~S]P; ;>B

-<?%M# :T 0$92F#!2#:!Z,<TS2#3

a′L = (1 − κg

2)aA +

κg

2a†A − i

2(κ+ g)aL +

i

2(κ− g)a†L

~S]P; ;


kyX%2X!G%Aa !2%76#XA%*23NM!#! #κ, g

S2#-<@#!%YS#GKF#YS/; 7-H# `$P#2 2G-<%

κ = g = 16"!%$

#!%YS#:-2$\>F1F#YS/6@`! !21uo/6Buqa†Ahi# -!2N%2V#;:

!2(−i) 2% aL

-Q /!%yA%! A.?> .*!π/2

!2Au[T2%*J%2A%=/;3l\#I

a†A2% #>2%b3`2&#a #IG%5!!2A

!T >F'a=x/$n;N/![#2 ;0P;^;9!!9a=x/$`!"!9!:[2#m-<X"%#!%=D.

~o+!#5!%O2%989a=!2$P#!#124#!2#<<>4/-'A"%M9UV% >4#%#!%=D.b#5!2%!&X)(PF%`T!?5!#,#J%#!%=D.b%b`2%Y !M2Ph#!2#/;Eky`/!%b,%#!3a x/$b!J>=.H!2%R-<Xa x%y24#!2#~SP;^; )6!%$X!2$P$K!T%$Kl; ;^602F#!2#</!%`>F:%$$K%=@2,2>4/+#!%YS2#G!2%r$H2A@@,%#!%Ho%2%7; /!%<!:-Q92$3#5!2''%N!2A924#!2#

aA>=.R~[ ^/ 7;"^/ )

S†(ξ)aAS(ξ) = cosh(r)aA − eiθ sinh(r)a†A~S]P; ; =

#2~S]P; ;,-1'!N""XAhDa=!2$P#!#9!N-Q9%$? a=x/;QUV%

-Qθ = 0

;2#2%=+,%%<-Q<!V$)L%!"%-b!#! #s-<&[#2 ~SP;^;=

<Vi!22#:a=x% %M9a=!2$P#!#/

s ≡ e2r cosh(r) =s+ 1

2√

2sinh(r) =

s− 1

2√

2

~S]P; ;,

wN.=%NQa x%N:m%!2A2F#!2#!%$&!"!2Qu[l"Q%=/6-QTL%$3!"92,2>FO+X#!%Y[2#G!2%?[2#<1!T#!%YS#">F2 O

a′L =s(1 − κg) + 1

2√s

aA +s(κg − 1) + 1

2√s

a†A +1

2(κ+ g)aL − 1

2(κ− g)a†L

~S]P; ;],

ky&A $P5!.R&!VJ!&#!%YS#TTF#YSV%\JAκ = g = 1

!%$s 1

;8?I5!GG#F/!#I%?G,%#!%?[%2%E 0$n6-QG%$y

#!%YS2#G!2%MS2#'% !% % V2%%2%3L;P;P6%$P% V>4/! Yh#/;92,2>4/+M#!%YS2#G!2%3$3%M915!2%:/

a′′L1 =1√2

(

a′L1 + a′L2

)

a′′L2 =1√2

(

a′L1 − a′L2

) ~S]P; ;

-<A# $A+!#5!>+,% >=.X~S]P; ;],);kr a !2%\~S]P; ; -Q!/+,<%$$Aa=x%'2F#!2#/6,-<A,%#!a=x/$I%q!/62% V!#!%YhS#&2,2>4/+H#!%YS2#G!2%76Q!%$H`%J!`NX!2AX 0$J2$#)S2#V>FV!Z,%X&>FV+!23;qT!<lV%< :!#T4$`&>F4#Y["l! |Ψ+〉 2A% [#2p!%3$/!8% !% % T,%#!2%R7;qwN% .#/!K% !% %=&,%#!2%76H!21!#! #K$O-<%H2%=+,#2%76-<m/!2$ !&-2#94#Y[2#G!%1T#4/!#/;wN>4/+,9-1!$?!:#1"%2V!%\2> +02N)(P4# % !.MS/!2>9#202

S2#J#/!2$0hD2GNM2%)hi!2 2#.,;WkyM/!%O-Q+,#G,G!R X-<


/!2$m9a !2%Qn"! S2# !2:~S]P; ;Ô);; ]P;^ &/-"m<#2F$ 7;I% #I#/!2$0hD2N1G!Z,:c-QXa=% 5!!2:#2@I~oN"%2%=#!2 *ML;q]P;^M-<#MXM uV%M!AO);#!%YS2#G!2%?a=!2%"%3:/!29!#

X ′L = XL + κ2P

′A

~S]P; ;!,

P ′L = (1 − κ1κ2)PL − κ1X

′A

~S]P; ;>B

UN#κ1, κ2

!#Q2%'#%%JQL#!%$&2%$I!28#F+,.,;kyI!

κ12##42%$T-<R1S$P>!2Z\ !%\[2#T-<#)hi%R-<

κ22##42%$

1:2%A#%7;"# 2%KN!2AV2F#!2#%$P/!N!!'=!T!2N!#/!2$P.`>4%?2#$3%M $P#% -<#)hi%7;

4 A*8&?K#( D56?1" ?KH* &K9+9 " *," ?K5+.K0 (+*,5 (624* 5+9 ( 4&K9+96& K*8(+2* *,H(6 56*3% 12( #4%96* '9 J,?K#%* ;H(6?<& *8%&.<%; H* -(3 '9:H?K(G" *8&19 #5+* &=HI(62*8H "I&K*79:&$96*8J,?KH &K9+96& K*3(62456?1#K2(+2* J*8%;%0

&@L;n]P;^ KV%2V+,#.3#/!&-<\##%=9)(04# % !%22.,;s0a=% 5!F!2m#a #V!'$5!.A%6P!2q#!$n60-<X!%K2/!%`#/!#!%W #/!2$0hD2&A%7;MCc%^/ K$P#!2%H<-<#)hi%H9I2%E2#$#'Q T-<* /!%1!>4/+,X^Z=v$5!.\%;1wTJ##% 1)(PF# %=!!,?` A4> ;UV/-+,#X-Q\2$%2X)(P$@@F>D.![#1)(PF# %=VA T>F12%#$\-<\2##NI%N2#T2#-<6 !O- !J$5!.X%6!%$3!"'$P.=%!AN<~S]P; ;,A :>4T#/!$ ;:#G!#.A#/!22%G!-Q:!+,N % 2%$`"#/!2$0hD2 N!>4/+,Nq!

V15!2%"-QN!+,T4#Y[2# $`/!%X>FN%$#0=$X!#%!+,.X!2I5!2%q!&#F/!#:% <#/!2$0hD2V '2,#N-<@!&4#Y["-<#)hi%@E6~S!224$JT4#Y[#/!2$0hD2Q!%$J-<#)hi%G$#>F$J>=.X~S]P; ;Ô);krAF#YS-<#)h

¢Y¢YD¡| i|i® ¢YS ¡ Y ¢ §7i|¨ ¬ iS ¨ l ¦ ®/)£| ¥ 2 ij® iV ª£ Y i ¥ S Qq Y ¡ ¦2 ¢ iVi ¥ Q ¢¤ ¥ "2 ¥j

§7iVi §ni JI ¨ § ® ni|l l¥ [ 8¡O L ¢Y£¥ni Y ¥ m i <¤ ¥¥ ¨


%76P'#!%[2#G!2%?[2#<'!9#!%Y[#<>42 /

X ′L = XL + κ2P

′A = XL + κ2P

inL

~S]P; ; z !,

P ′L = (1 − κ1κ2)PL − κ1X

′A = (1 − κ1κ2)PL − κ1X

inL

~S]P; ; z >B

-<#Xin

L , PinL

!.`J9%" "-<?"2#$M%M1!2 :!%$XL, PL!.3Ì#/!2$0hD2'I>F)S2#&% #!22%7;9IS2# T~S]P; ; z N!%$~S]P; ;,:

)(!2.3J! J)(PV!'I%2.?# V`>F&a=x/$\%/- 2 Io#2 #/!2$0hD29=9 !%$R%2'[#2tJ%5!Q!2A !;&UV%A2#1I5!2%#<!.à !. -n&!I#4/!#-<`V#/!2$0hD2L;]P;^ ;qV.=A #.>4D-%*Ic-QX 0$'2 No#2 IS!2T!T&$% c.@#!%YS2#G!2%-<b$#>F&`F#YSAH%W>42b#2=2 2A1A-<!%=.W2##!%YS2#G!2%7;

?2? D Eky:-<B#0$K!2%&N! "%!2m%`;0]P; ;P;N #/!415!2%72%)hi!2 2#.`#4/!#N!#T2A!#$M-<?!%!.=/!l#o#2 T4##>Ph!+,J!#,!2R%R!7;4P6F!%$@-QJ!GL%$?I2A1%/!2%y#!1[2#T! 4uL29!#! #O;Cc%2#` 0$V9@2%)hi!2` 2#.#*!#3#\!#! #

κ6g!%$

s6l!%$*%32#I$2#%G!+,G>F%y%$$n;GkyA!+,G%2$E!#/!2$P.R%`$#2%N2#J =$!JKa x%

s&#5![2#Ià=!D.HN!

!`#!%YS#A% y!%$2 VX 2#.!%$W%M[2#J#4/!#AF#YS2#G!%;Cc%ML;]P;^/ -Q9!/+,12$?12%$P2%!8L$D.XS2%$M[#2 I5!2%?S2#<L+,$Pn#% '+!'

s;1J+!9

κ6g!#A%*`>4A2G!-<*J+,%

a=x%; N%TQ,%#!%'[%2%&-!+,4#Y[2# $&QL#% !% % 2%%2%!%!.=/!.E% E!%$E2>!%$b!%W)(P#2%E%

s, κ, gS2#

JL$c.y!|o#12%%2%7;`2#'L($*+!1sI)(0#2%*-!212A$

-9; #/; /;κ!%$

g!%$M1#%G+!"-#9%3$3%3%= #/!I5!2%\

I#G!%%32%%2%* 6 ;VI+!T!#I+,%R%\9L#J/!2%y!%$-9%2I!:.?!#I!-!.PT9@^6-<\N12G!2I-<%

s 1;

'#+,V! !%3!Z,1!%?YhD!F6!2<-<?>F23.P <#+02.M!%!.0$n;\0$:#$P2%MlV2,2>FO+G#!%YS2#G!2%E~S]P; ;],q[2#N!:#!%YS#

</!2.,6%T<"!#/!2$P.`2%MT#$P$XS2# ~SP;P;,)6!%$X-9/!%3A.G#/!2$?'0G% /

b1 =1

2(κ+ g) b2 =

s(1 − κg) + 1

2√s

~S]P; ;^,

c1 =1

2(κ− g) c2 =

s(κg − 1) + 1

2√s

c3 = 0

H 5 SS ¦ ¥ S ¥ i9 ¦ `K ¢¤ 2 ¥ i¢ O SS u<i ¡ Y S A" Si ¢D&¥ ® ¥B®/ i )Y| i | u£ ¢¤i ¦ ¢ £ i ¢Y ¥ O ni | SSj )¤ ¡ ¦ ¢¤ : ¥Sj® ¥ §7iAi ¥ ® ¥ ¡JS9¡O Qi i < Q qi )¤ i | SS Y¨


4 ! "$#%'&)(+* J?1H ;(6 ?KH4&K% *8%; ;(/.K0 5+?K" (+? (6? ?=(6(6?1" (624* B)&=%#*89 ? [s, κ, g]&K56*

[109, 1.00, 1.00] 5+* [106, 1.00, 1.00] ?K5E&=H1* [103, 1.00, 1.00] ?K% BK* [10, 1.01, 0.96] K5+*,*,H [1, 1.17, 0.83] %;#4* 0

w<$$P%"m22%nS2#74##>!2%1%b2, c1, c2

o#2;OP; !%$I!7;/dH-m2>!%M$%c.WG!#u(

ρl%XF##>!+,\!#(PG!2%r!%$bM2##42%$P%

L$c.M!%$?2%$P2%!7L$D.4

F =2

2l + 1− 2

2l − 1

(2l + 1)2b22 +

1

3

2 · 23l − 6 · 22l − 5 · 2l + 9

(2l + 1)2c21 +

1

3

(2l − 1)(2l − 6)

2l + 1c22~S]P; ;^Ô

!%$

F = 1 − 1

2(2l − 1)c21 −

1

22l(2l − 1)c22

~S]P; ;^/,

V2q!Q0G% c2%M~S]P; ;^/,7l! !2l

c3hD=G%=%X~S]P; ;^,);

<">F/!T4##>!2%?%c2!%$

c3/!2$MJ1! T!#! #!2%@

$%c.XG!#u(*~[I!7;d9);wN !%76

c16c22%1&!2+,I %=TI!9#!%[2#G!2%76F-<

b2$="%2"2% #>9J12%$P2%!nL$D.,;Q#2 ~[dJ;Ô<!%$y~[dJ; =Q'#2>Ph

!>c.KS2#:!A$2>9)(!2%3%ρl/

7#(|11〉〈11|ρl) = (1 − 2

2l + 1)c21 + 2

2l − 1

2l + 1c22

~S]P; ;^

wN%$&o#2 ~S]P; ;^/,-QG!.AG!"+,!-< #4/!#q4#Y[2#G!%<>#/!ZP


$O-<%3> .M2+0%A9a=!2%

1

2(2l − 1)c21 +

1

22l(2l − 1)c22 = 1

~S]P; ;^ =

4 @H4&=%.(6 'J &=H 96 ;"<#%'&)(6* J?KH -(+ ;?1H4&=% *,% ;(/.K0 32* 4&K5+&K">*,(6*85 B&K%;#4*89:&K56*s = 10

κ = 1.01

g = 0.96

0 32* &19 24* B1*,56(6 'J,&=%G%; H* ;H J8&)(+*89$(62* 56* 'J (6* %*,B1*,%G&=( A24 JE2F5+? 9 & 3&. D5+?K" 10

G=/!2%I<>#/!Z0$O-<%H2>!%$W9-!.E!# %$P/!$H2%*L;]P;^/]?!%$VL#:%@!7;_";qwVS2#<'#4/!#N;]P; -Q'L%$M,=0$3!# % V-<%= #/!8I5!2%7;V2N!QV)(0#2%R~S]P; ;^/,m42.=%2A5!4%`N2AI%/!2%?$P!%

Ll/L0 = 2l 6!2[2#<>42?T#F/!#N N$P$M#+02.,;wV'S2#9G#F/!#&-<EGD-Qhi!2I5!# 2#&#2!%\ 2#.,6-Q !/+, Y

!R%3% 1>4# S2#A32AI%/!2%#!3'X#4/!#G%*?2%)hi!2 2#.,;kyN!+,T2A$`:#!:S2#Q#V2%%2%37:#%M#+,%HL;]P;^/P;ykyX0XA#+,6Q%

N = 3JKA% 1>4# V

-:/!%K15!N-<G 0$#!:#%= !%$n6P!22#$P%&9#+!:2##42%$%-< ;ns=#2%y!%$R2#/67!Ga x%M

s = 10NIG!|(P1t-<y/!%R>4

!2+,$R-<\#% T%22. ;T2#TJ% !% %=9,%#!2%76n-<*-!2!2 $y3>4A4#Y[/6-G0X!MI#2>!>D.W

p0 = 10−3 ;M2#I!#! #/6-Q'!/+,'S2%$n & ® S¥® ¡Ni ª£ ij T £ < i¢ ¥¥ ¨ V ¢Y ® Y i ¥ i <Y ¥[n iª£ ¤ ¡ ¦


N = 3s=15!$?#!

[10−5 · τ−1]wN%!.=/!7#!

[10−5 · τ−1]%!

F

s = 10κ = 1.01 6.8 ± 0.7

^/ P; 2g = 0.96

4 4G?K" "<#H 'J,&)(+ ;?1H5+&=(6* D?K5@5+* *8&=(6*85 A ;(62 ?1H* 4&K9+9A" *," ?156. &=H 4H4&=% J?1H ;(6 ?KH4&K%*,% -(/.10

τ '93(624* (6 ">*D?15 & 9 HK%*CJ?1HH*8J(6 ?KH ?K5 1*,H*85+&=(6 ?KH &=( (+*," (80

wNJS2#:N.P &!%!.=/!8#!1NR>=.?!Ji!22#Nq/!P;"P;A>T!#-<MTI5!$M#!'2%X'2#$#"8G!%$6!%$X'$P#"2AI%/!2%#!G ∼10−26τ−1 ;`k%b!`S- !%$!#$W$+=5!2%I #!`!#9-<E>42"dTe8f<ghi#4/!#<!%$ <#4/!#m-<X!TD-hi!2m5!# 2##2!%1 2#.,6P!%$-V/!%X+,V%J2%2%?!2&-<M#202F4#YS2# >F#/;kyV%260276!Q%9##2#Q2#m-#"%$$K%KI5!2%Qn"#% V!%$G+,%%$#"T$/!l2%$P2%V!&@a x%G<#a #$M[2#<#4/!#:24#!2%7;


!"$#

Cc%X<!#<-Q'AG!#9!%$X2%$'V#-<3!/+,T>F%X#%=$!%$J$P$J%J!7;,T!%$ ]P;8ky<!T2%[#-Q2#Z48)(P%$P%9I5!2%J%$1%29!%$M22%?2%=%;

% '& (3-+*),+/ 3-#&dTe8f<ghiD.0FI#4/!#N!/+,1>4%@$P$?%@:O;"_m!2Rq I!$Pn#% q!2A<a=!% 2#.,6>!Pm%=!% % Q2%%2% &2#%!QdVef<gW#2F!"~ L;7P;,N-<*%2%PhD2%=%?20$2#/; q#.$/!$@ 0$N!/+,&>F%R2%$#$@[2#VI 2#/;TCD%*!8#&/!2T#-<#)hi%M2#m#/!2$0hD2-!2!2 $GI>4"4#Y[/6!%$`2%.A[2#Q:c-Qhi!2 2#.;]P; -!2<%29$P#% 2#.M24#!2%@2%$#$n;8?G!Z,K`15!2%&4>K! 2#&-Q#`#/!$b!21!# 2%XYh

5!2#T.P /6!%$R!A#!%,IC7 X#2#! V-#I$+,24$n6>!2$?2%X,%#!%Eo%2% 0$'!7; ;X#2#! +,K2%uhA!K>!% 5!!#'mJ2T[#2 IG!2#VN# /;9.@!O-I5!2%H:!%=.ydVef<ghiD.0FK#F/!#&%R!X 2#.y-<HG!.*>4 $#>4$>=.@!# 2% 5!2#/;ACD%$$R.*!/- 'M2AA 29!&[#2!%=.G2,2>FO+ #!%YS2#G!2%76P+,%KN%/;.G/!%`>F:$ &2>!%K>42%= #/!!%$3!%!.0/!l#/6!2@2 T% #/!8!2Z0N!#'2+,$M#!#%G%=.,6,$Pq<qS!28! 7 V7%2!<+,#.Ti!275!%!,m[2#7%= #Yh/;1/- %= #/!!,2# V2$M>4TF2#!>9JS!2#<5!%!,/;m<!2$2%K"!# 2%V5!2#Q 0$n!%$ ",%#!%&[%2%76 -V!1$+,24$`!4##>!+,q 0$=!<dTe8f<ghiD.04m#4/!#7%T!,%#! !#! #$Ia !%= 2#.R!%$@4#YSV%=!% % 1,%#!2%7;1T =$-!29!/!>JKD-2"'#9.P "$P$n;2#'A#J#F/!#1-<y-# $P$RA#+!% 1 /!2# QAa=!uh

c.`,%#!$?%=!% % V-!2"[2%$MJ>4T92%$P2%!lL$D.F6P>4/!

%2%PhD2%=%J$2#/!2$KI!9#!$P. %#/!2%&+2!2 2AF2%%=<7",%Ph#!$@!/;<s=!"-<?,00$32%$P2%!7L$c.3G!.X>4'$XS2#:a=!% Z,.

74 7. CONCLUSION AND OUTLOOK

$P#>2%M2#%K#2>!>TF2#!2%X-<XF#2#2%PL#G!2%38)(!|h2%/6!2)(05!%$X%R ;T2%$P2%!4L$D.`$#/!2F2.0%2A5!. -<X2AI%/!2%E$P!%&S2#T!#A.P /6n!2V#a #$\`>4/!T$P#92AIPh%/!2%760!%$J-<% +2!$Pc.I#!%,:B<4##>!2% #<,0=$A!# %=>4D-%

F2A$*[#2t% #/!15!2%W!%$

F2A$R[#2v2#I!%!|h

.0/! =$i;*` 0$+,J,00$WG!ANK+,9~oi;;*`%=1>4# 2%%2%1!|o#&-<

F$!#1[#2 ^6!2IJ2%$&-<WK+,

-<#'T4##>!2%?>#/!ZP:$O-<%7;CcX-!23%r!M*2>#+,$$#/!2*%rL$D./!%>4\)(P5!%$ > .r

0##%1)(0#! )(P!2%V#/!$3$P#%G=Yhc!2p%=#!22%7;9$2>)(!2%/!% /!$#/!2i!2Yhi!%Phi%/!#q%J<2A1%/!2%K%

L6,!%$

-Q>4+,8>4/!!-<2q)(!2%q!##$I 2#Q%F;AN#

!%!.0/! =$#/+0$')(P#2%N[2#TI#2>!>D.*Q$2>)hD)(!2%'0#<%3# "'4##>!2%?!#! #/;2#:#/!I%,:9 2#.X!#! #/6B12AI%/!2%*$P!%

S2#-<K!V,0=$J2%$P2%!L$c.A/!%A>FG!% !%$`!##!#/-9;<.P8;P]P; 6!I#>=2%30G% :

r = 0.01#!%%AT##% N)(04# % !

A/;rkrG+2!\!*2%$P2%!VL$D.-Q'!>4/+,3@5!2/!&^2*AJG!%=!%$2%.W[2#J#32%%2%/;bs=A5!#.,6-<!\a=x%s = 10

S2#[email protected] 9;m]P; W-Q?/!%r$y#@2%%2%K>4)S2#F$P#2

>4/- ^2P; 2#M*dTe8f<ghi#4/!#?\AF2#!%=3!#! #3MR)(!2%#2>!>c.

pc6,!%$J&L$m!#<2>!%$J-<%

pc 1-<AG!.&>F<!2+,$

>=.*%@=JIN/- 22%H%=1>4#3~oi;;?-/!Zy);@2#IJ.P!>42Nu(?N/!%\>F1#/!2$\-<\,=0$

F-<R!

pca=!GI% !% %

,%#!2%E'#2>!>D.RS2#'A2#9c-Q3.P /;JUV/-+,#F'Z,.\

$P#2?S!2#:uq#>=2%@N!%$?2#N%212#V!#1%$$n; $P%G[#2#%=m)(04# % q-< % !%$ 22%O6< % m%

L0B#4/!#

G!/.M>FIm2#$#&^)h^ Z0 6B 6B] ;"#I2%%2%*N%R2##F2%$N!J2!2A1%/!2%*%\Q!:

L = 23L0 = 800Z03;"UV%9-9L%$?!

+,%`-<%M2#Q$/!$` 0$/60:%=#YhD2% #.M2A1%/!2%?A$K>=. c.K%! <%XL; ;^TS!#[#2 !&#/!9%!#&S2#<'.P <2%$#$n;QCD/!%%2<>4')(P$$X!N2%'8 G!/.K>4V$`[2#")(P4# % !7+,#uL/!2%\T#4/!#"#202i;f2AI%/!2%X#!q-Q#V2>!%$AS2#q#:!#! #%,Qn"dTe8f<gh

#4/!#1!%$RS2#92%.\2%A!#! #I%MS2#9/!2EJ#G!%%?#4/!#/;X#!K2%PL# ! M#4/!# 24#Y[2# $P#`2A1%/!2% %uL)h!%=.FK-<%bM2A1%/!2%$P!%

L = 23L0!%$H`J#2>!>D.

S2# % !% %=`,%#!2%p0 = 10−3 6!:#M#F/!#G+,M#! '2#$#

10−5τ−1 6n-<AJ#!A[2#9$P#12AI%/!2%W ∼10−26 ;AUV# τ TA 4% N2%\!A%I2%%2%\2#N,%#!2%R!A/;<!Z=%τ = 1

O6-<3[#2 |8 4§FS i 4i ¢¤ni 4i ¢Y < £ j¢ i ¢¤¥j i ª/£ ¥ N Si ¢¤

i i£ i < ¥j i £|S £¥ i ¨ § ® i| SO¨ J ® ¢c|) Y ¥j[i¢® ¥£ 7i q<Y < i ¡N°)£ ¡Vi l°li Si | Y¨ & ® ¡o£[ ¡i < i4i " | i ¨

7.2. OUTLOOK 75

^/ &!3#/!G+!6 >Yhi#!`>42 ∼0.5a=> |A% ;\w: !%HI

IG0T/-[2##!2/!4!/!2%QK!2qa=!% Z,.A$P#>2%76 >qG!.!`>4J%G% 9[2#9)(PF# %=!q+,#uL/!2%EJ#2=2O67%J%y)(PF#uh %=",=0$M!"<#a #$M[2#:2%+,'#/;/!*2#M 0$?!#R+,#.$/!$MM$PG3!/.r!%=.=%!>42

-<M#F/!#<-2$M>FT/!2#JA %="%?!%M)(04# % !l%62#-<#4/!#K-<V4#Y[2# >F#`%$#X#/!\2%$P2%/; w:V#@.0 K!!/+,?>4%r$P$)(P>K?AF2#!%=`[/!#XI!ya=!% #4/!#/y>Ph)(PF2%%=5!=/!%VN~[2%$P2%! nL$D.1!%$12AI%/!2%J#!l-0!>4/+,I#!I[2#T$P#T#!%A2%7;'UV%I/!2ym G!.?%R#%&>41$@%)(PF# %=!q+,#uL/!2%W<dTe8f<ghiD.04 a !%= #4/!#/;A dTe8f<ghi#4/!# J\G!%=!%>F#JL$c.b! !\2A1%/!2%#!`VX! X2#$# G!%$T!2q<[2%$ [2#qN2#mc-Q9#202/6=>mm#"G!. !%,-<%b#>=2%J!%$b2#A##2#J2# !#G%$$n;y2#&`#4/!#A;]P; 6N#>=2%M!X\$M1RT2%=!%%*!2A*!A-!21GG! Y2#&2#K<%26!%$E A42#!% J!#! #1S2#1K2%$P2%!L$c.I-!2#>=2%G=G%=

r;/! 2#$&%AD-92#

#202m2%.&#a #:2%<!2q4 Q6,-"$T%2m)(PF#>=2%K`>4&!2'+,#IS2#TJ.0 /6F>V.R!#AZ,.@`5!.R!K#2;12#T#4/!#:%K12%)hi!2N 2#.@! @a=x%GN#a #$3[2#NF#YS2#G!%2A!#!>1A!N;]P; %$#N$/!l2%$P2%/6F!%$MuTL$D.M[##$#!2$$*>=.@#>=2%E9!%$*2#9%2J9#F/!#'-<%294#YS2# -i;UV%-Q<,! 2#<)(PF$P%=TA % m#<dTe8f<ghi#4/!#2#<'#4/!#"%GTD-Qhi!2: 2#.X%3)(PF# %=/;

% + ) &-Q2#Z@#% $\%RVVV+,#.@24%\%$$\%RI%1!TIy/!%>4'$2%'JA#/+,&!%$3)(P%$M'#/;#/689I$#!>GA.*3F#YS2# 2#K15!2%I!%E- !+,`$2%6

%&2 J!24NmI#4/!#N-#&%2V$P$\+,#.32#2.\$P1K5!2Zn N94#Y[2# "#a=#$ #2#! #%/;5!#5!#.G[2#mN2AI%/!2%#!2$G>4:%=#%JI2>!%X 2#V$!!I9L%$`; ;/- N#!N$4%$2%32AI%/!2%r$P!%

L!%$M2%W!#! #

κ;wVG!\!#G

/6:A=N>41!2$P+!% !,2V 42#:&!,2#p$? I5!11#!9!@5!%!,M-<!%$PA% #AS!2J!% 7 ;b2# )(!A<;#2 @F#>!+,\ 0$N-\G!.!H2>!% !%!.=/!')(P#2%KS2#GT#2>!>'S2#9% !% % J2%%2%768!%$*o#2 J#2>!>c.*-QG!/. ,!%M!%!.0/!4)(P#2%KS2#mV2A1%/!2%3#!60-<M2$`%`>F2A!#$?-<315!2%7;s02%$n6%?!l2#"-2#Z`-'!+,I2%$#$32%.X%2%PhD2% %`20$2#/6

>4/!JG% '22%y2%=%3$2#1!#&+,#.@$PG'X2%#9)(PF#uh %=!.,;UNO-Q+,#/6=m/!%%2q>4<)(P$$A!m!V22%`2% %&$2#m!>


S2#TJ#F/!#9 9$P$R#A2$R>4J$+,2F$E!%$RTT%=#%3)(!A%GO- K#202I-2$HF#YS2# -<bb!@$2#/;(5q22%b2%=Yh%AG!/.`/!2.`>4N#/!$3-<MT,%#!% [%2%M =$?!%$X%Xi!2<$P#%V-Q2#ZX2%M# /6-QT!/+,9!#/!2$P.KA %=$322%?2%=% !#5!.,;5#2#! A2A%*`,%#!%R[%2%H[2#J% !% %=G2%%2%%22%\2% %X$2#:!/+,1>4%3-<#%@[2#N!l#9 2#/64!%$3'4#Yh#>!2%3%M!7;&!2<>4%`!#5!.M2+,$`S2#"!&#F/!#<%A"2%%2%#0$P#;Xs=I5!2%&/!2$P%?M+!1AL$c.E!%$y2A1%/!2%b#!n#G!%?A>F'/!##$?2/;#$n68!%$R 'AF2#!%=/68!`2'##2#I2#T!+,A>4%R%$*%*

I5!2%l!!+,m>F%&$2%m"S!#/;8q 8AF2#!%=Q##2#!# ,;2#Q!N20$2#qG!.&!4%A!<$2#mZPq+,%A2A%22%X#% /;s=?!%M+,% </!$M!&$!#ZK2% /;mdT!#ZK2% :!#'!I#2c.=4JQ##2#'>F/!I.R/!%R/!2$RX$2>A)(P!2%'%RA%=!%$y!'3#4/!#/6:!%$-Q@!+,\%!`$2>@)(!2%X/!$\2>#+,$$#/!2J%y2%$P2%!mL$D.,;ACD!X$!#ZR2% 10#1!'J A<2%%2%D-Q?% !%$E!/64-QAG!/.@,'A!2%y/-<%y%\L; 0;^;JV2%AQ)(!2%J%bKc-Q*%=!%$b!# !#K#/!2$b2/6>JX$2# ZP !%$%J2%%2%R'!2 $@K>4&Y[i;NI#%M!Jm 0$K^J!%$A:!J$2>1)(P!2%7;

! 41' 41 &=5 J,?K#H(A*,5+56?153%;*7& H>(6?I& ?K# %;*C*J ;(+&)(+ ;?1H 0

w/-$!#Z12% #!^2UNxQ&^ 2U:x 62-<J[2#!%J% !% % 2%%2% N

τ = 1 Q2##42%$I!I$!#Z 2% #2>!>D.`$P#%&2%%2%Mm^:

^ ;VV#2>!>D.?N%2V%>67!%$?%J$!#Z?2%=NIT>FI)(04$@!+,`%uL/!% AA!2A2%H#F/!#&4#Y[2#G!%;RCc1&F> @#/!&$!#Z2%=9-<%E ,%#!%?[%2%y 0$*> .*2%$#%R!#!.RAD-!2%OÔ%1$!#ZA2% 0#/60,m!'$!#Z 2% 0#/;2#/!2K!2%`,%#!%V[%2%J/!%J>4Q2A$3~o#/!P! /!2# %=m!#m%$$J%&,%#!%M[%2%B)6!%$y!9/!2y WJ#4/!#9-QG/!%y%y2A'!I!|o#"%=!% % '2%%2%\!2

ρ = (1 − pdc)ρ1 + pdcρ2~ 0;P;Ô

-<#ρ16ρ22 q[#2 % !% % Q2%%2%&-<&D-N,%#!%V[%2%

2##42%$P%A3~YÔ!%$y~S,);CD%2%3$!#ZK2%="%X'15!2%"<8#2#c.`%3o#T-2#ZF;

¬/Y ¨ ¨BS §n§n§¨ ¡| [¨ ¢Y ¡ ¢¤iiY¨

7.2. OUTLOOK 77

Cc%!2$$P2%\$!#ZH2% A#M!#X2#J .P/!"2#AN##2#J#%=/N>=2%18!%$I!2Am$2#%-#m2%$#$9[2#7m#F/!#;O]P; !%$R2$*!`>4&2%$#$?S2#T&D-QM2#9.0 O;'dV2#'%G%.\a=+2!%=l<22%18!%$9/!%1>F%$$1> .9!%9%#/!2mP#>=2%&=G%=/;<!2Z0#2%$`22%%IV$2#!+,V:! Vn!2$!#ZG2% <!%$2$3>FT25!$@! !%/;2#`!%$A5!2/60!2q% :2#%!FdTe8f<g`#24! 6,"Qn5!2/!2Jh

I%/!2%A!28>4%A2A.1%$J%J2#15!2%2AI%/!2%#!;Cc%2q2>=+=2q! <F%=q2%A5!2/!%PS2#G!2%G)(P!%,<%2h>A2A!#$RX& J%$$\S2#T%=!% % &,%#!2%E!%$*2%%2%!A/6P!%$X!'2#2M!%!.0Q7:)(!2! 2%=l5!2/!B2AI%/!2%%$$XS2#<#4/!#:2F#!2%@-Q2$M>4'$#!>;Cc%MAG!#.,6I?/!%34>4T$2%T%M# <%2V%2%MJA#O+,92#

#'TF#YS2#G!%'T#T#F/!#:.0 2%$#$3%M</;N=2%$$KI$&J!#:5!#,.G%K5!2V%X2#$7 AA %=!2%J,%#!%Mo%2%E =$n;J\TAF2#!%=I!# $!#Z@2%=/67-<yG!.>4T%$$M> .3!J=" 0$PuL/!2%@'#2#! "-'!+,I$+,2F$n;


$

!"$#%#%&')(*+',-#%.#/!0123#4'56#2(*7879:/;#%5<8+#/=>?'??ldTe8f<gh@BAC9D92FE)AG753#C1HJI2#%#/=892K#%#C+#%?L#%M8E8>%7?(*?@NO@PQRP'81981)3#/K84)AC2?S'56?53)>#%LNT53?'AU>8?56JVH',8P*wK/W7X8X8+++:/WX?Y564Z[?'AC2?Y>#%?'X#%LNT[?',*#%'5\/W#%)X532?9'5"X?]#%G !%S'56?53-+1^' 7 7#_I88148`+#%565"X2#%>RPbaY812Z/W#%F53#C1HJI2#%081M8538#C1MX8+1M/W#%K'5"X?]#%0+)+1'M8D>+#%8FE9?#c DP

41 &=5E&=%%;*8% (/. * 56* *8&=(6*,5 JE2&=HH*8%980 32* (+*,5+" 9 H?*&KH 9 (+&=(6 ?KH 2&BK* *,5+*,H(A" *8&=H4 ;H9&193 ;%%;#9/(+5+&=(6* 0

dH/W#%X?'AM'53e%K8#%d?'?#c S+1fZ8#%Yh#%X,Rg-X532?Z?'AC#/7 Fh+#%565"X2|h#%i>+^2iE$0%>(*+1i F'i8&53#C1HJI+1#%#)+#%?@P8#%Y4%+X>d?'AHO,*#C#C13>Yh+"4a-X'+1N15<81H',j5<-AUX8"#/K#%X?',*#%'5k/W#%=2%7>53>',8Pwd?'8#%X,R S^+(*Ê$+l2m/nX?KE$#%X78^+#%56?mVH'BAo#/86?',*#%'5p'd%D<X-'536#%j8Z#%1_#/q G8#%X7#%U#%Xe5<%>'8+ /;#%D53#C1><+#%5Jh5eX2#%h1H>8++F8142?'d(%?'X8+Y#/L8+#%8+#%r#%EE'?''+cPMwde53-#%8+1s'.8

(C#%X8S>2NC'T S#%X?1<+(*dE9+4m/W>E?d#_X8t_)X532?'?'A6/;%b#%,R>5Jh56',h?],RX,*4X8o% k#%e98 Gu_vwxudyKP4k.62(*3X8+1.8<?',*#%'5#.#%E>'z8+#%565"X2#%i>+3,R'(*z'LPY|.81[2(*0?>#f5<%1&f/W '5"X?]#%8`#/>?'??>8?x56JVH',6/W#%F8_AC53`1HX8+1G8RP

~'& H/d+2?'?9F8_+#%565"X2#%>t/;#%K")X-X592FY8t-X5"E9d#/xI8?)>,R?+1.>+D+>E?'8+1j9_X'_'53RPZB/

m1-2?T$2e6#%$>+1

'81$81-?'AG'M>?'??@NH8h8+#%5eE'8+1&+#%565eX2#%&> F'?'?E9m'53+

¯ © Y i£ o" ¢D n i © F¢Y £| 4i ¨%

80 8. PARALLEL CHANNEL MIXING

8<>3/;#%7&',R?4922P.< Y-Z#h')(*+',-RNO8#2 5"X8o8<+#%5"E'8+1>_2hE$'56#2(*+1h/W#%`>?'??@BAC9e92KE-AG56JVC',48?2P

d+2?'?8/W#%5 +RP P HP !->,R?53)KE9 S+^8#)1+!#/x>#%L0

S']?'?'A72+1<81783mVC81+16#c(*b'82%',D1H>8++OE)A"X8+++>'(*F+#%8+#%_(*A&>++#%81&8#C1RP.Z+#%81eÂH5 #/

m>?'??@ A)$Z$22N

810 S6X9#R7_8#C1+'f1H$)D92 F>r6_8Z>5361H>8+/W#%5 8781H$#%'-77?#)2+1f?#R7#G2%r#%8D]?'?'A'f>#%8e%_8#2 F'UI8,8PHP q P@8#%?'?#2 F',f8@BAC94$27#%#C+#%?@NO2%o->,R?53)<+#%8+#%l30#%EE'?'(*-6 F[/;'?Z F'0#%EE'?'' Ao1$81H',.#%l81H>8+h?(*?WRP /P[I8,8P P mP [8h2%#/t/@'?'X)>,R?53-G3?#R<#%[8 F8#%?M,R53)<')(*#%?'(*+1l'[8M+#%8+#%LP[B/t+#%8+#% !%<X8+++:/WX?S/;#%688',R)E9#%X',<>,R53-`#%M8_ 53?(*?@NH8+'F F'?'?8ME$d1H?DX-'?x78 56_X8++++12Pz'&>?'??$2>8?2N$'t5<+A8m/;#%"9j !_+(*_8e'X#%8#c Fh'MI8,8PHP ;mP

H ? JE2&=HH*8%980 9 ;(6#4&=(6 ?KH A24*,5+* *,H(+&KHK%*," *8H1( J?KH4H*8J(6 ?KH49 ?KH J?1565+*89 ?KH ;H 96* 1" *,H(+9 24&B1* &= %;*> ;HI?1H* JE24&=H4H*,%4&KH 9 #J,J*8**> ;H (+2*A?=(+2*,570 32*A*8H4J 5EJ%*H?*79 &K56*$%;?J,&=(6*?KH (62*>96&K">*>9 &)(6 '&=% %?!J8&)(+ ;?1H 0 * J,&=H (62*856* D?K5+*$J?1HH*8J( (+2*$*8H(+&=HK%*4&K ;5E9 ;H *7&KJE2 JE24&=HH4*,% !(6?IJ5+*8&=(6*C&>H* 4&= 5A? ?K# %;*C%*,H=(62 0

82%1s#/`?2+(C',j84>,R53-Z/;#%e F>[+#%8+#%l%7X8++++1+1r1H?RNb S2s+#%5eE'84'D/W#%5 >?'??Y8?6%D'?'?'X8>+1r'fI8,8PHP !nEmPMaYD Y+A8_+#%565"X2#%0>5<2AGE9'56#c(*+1P

Kd8#%d?26#%AM8#2 k?],*"8D'56#2(*53)_'8"+#%565eX2#%f> F'?'?dE9RP d 8&#%881 8)X5eE$M#/D9#R'E? Q#%X+ S4?2%1 #r.I8?)>,R?+1h'"b@PQRPS8_)X5eE$d#/b+#%5eE'#%8#/8#)1+` F0e+#%8++1h#268<'`'82%+"1H>5<2?'?'A. F'.86)X5eE$7#/Y>?'??S>8?"/W#%?],*

LPr@/` !G>$#*h%++#%X-3#/t8^/;%Z7IVC',.#%8^#%XMmV?'X81+3?'?K#%8

#%X+Y F>M+(*_->,R?53)`+#%8+#%8K#%G8#C1+F+1^ F'^8dI8F#%8_)AD?(*?@N%8Z8S)X5eE$O#/ 1H$) !+AH#dY'">?'??

mI8?H->,R?+1

'= ∼ (m!)2L/L0PAdZ8S#%8O81N%8!#%EE'?''BA"/;#%)A"#%8Y#/88+!#%X+

#G#C+X_5eX8D1$81f#%j86+#%8+#%.#%EE'?''+plN81&'_D)D

's842%>< F8pl = 1

/;#%Z?'?!?(*?lN=8^ F'?'?!E948#'56#c(*53-<#c(*Z8

'81$81-K>?'??L>853_E$+2X88Y8(*F8++1^/;#%F)A^)>,R?+1M'F#E$1H?RP%z8M8

pl1F/W#%5 q !DmVC$+F8_'56#c(*53-t#6'82%RP

a#E9YE?Y#t5<$#*Kt-X)'>'(*`?'ACO#/98?56JVC', SY8++1Z#$+J/WA8S1>'?=#/8S92AH5 !T?#C#&#e2PxaY8T/W2X+ S!8#C#RT/W#%#%X=AH5

8.1. IDEA 81

D%Y/;#%?'?#2 KO• aY8"

m>?'??x8?t#/?,R

L = 2NL0P

• _2%>f'53+j8"5< VH'5<?O-X5"E9D#/!->,R?53-7+#%8+#%8e6 56+1&K2%&1H>8+D?(*?'h>?'??@P

• 8K8#C1+`t/W++1G/W#%FX8>e /nKX8+++:/WX?L+#%8+#%82P

• !#%8+#%8DZ?' !2AC_(*A++#%81&>#%0#%f<,R'(*&?(*?@P 81#/O7'K#<+#%8+KYIV+1P

• aY8D8#)1+K#/=I8?x)>,R?+1h'dt/W++1G/;#%FX8e /WK#%8_'53+LP.3%>X53Z8Z92_#M#%$>6'.1HZ'53+8D#/Y1HX>#%

τ F>

5<2A&E$7>$#*r%8Z'533$-D#%U^',R?6)>,R?53-Z2#%U#%D+#%8+#%562P 2%>f'53+U81j2%.1H>8+Z?(*?7 !68+ #&/;#%)>,R?+1j''r8',R-E$#%X',j>,R53-7 Fs5<+A.E$3+#%8++1NT81s?'?S$#R'E?<+#%8+#%8h56+1N!J/d8+++>>ArE)Ar56JVH',.8?^%Z'oI8,8PHP HPoaY653286U)>,R?+1.'"'r^,R'(*r,R53)e F'?'?b8(*"E9Z1H?RNX?+_8<8',R-E$#%X',,R53)K%F/; Sd#%F8#6'F#3+#%8+K F'LP

4 4 G24&=HH4*,% " H 5+* *8&=(6*8580 (96?K" * * (+ ;" *79/(+* (624* K#456*79 9 24? *7&KJE2 '9 (E&=H4J,* %;*8BK*8% D?K5 & ?19+9 %* 9/(E&)(+* ? & 96 " 4%;*JE24&KHH*8% " H 56**7&)(+*,5 A ;(62

N = 3&=H

2*8H4J*L = 23L0

0

=',8P HP 8#c K_6$#R'E?e>"#/S$2` F'>8?x56JVH',8P d2%?(*?!8<)>,R?+1r'Z81r8<+#%8+#%8FF F'?'?E$_56+1&t'1H2+1Pd#%478<$27#%$ +T'6>?'??#%4?'??(*? @?(*? ,&53)b#/$t8? F>4 /n+ @PQRP/;#%T F>48`8#)1+YK8#%S!#/x-A)>,R?+1 ' M F'?'?6'5653+1H]?'A E9X8+17/;#%O8 )>,R?53-Y2#%LPOBX?]2Nd F88(*hU+#%8+#% 56ZX8++++1e#%oM?(*?S',R8eq N!8 @?(*?`>,R53-^81H',U#%[8

8#C1+e')(*#%?'(*+1s'rZ+#%8+#% n@PQRP8#R48'?+1s'UI8,8P HP !; "3/W++1Pt#)1+D_8<8?T81_ F>./;#%5 4I8?b->,R?+1U'2N3%X53+1.#GE9X8+1./W#%"+#%565eX2#%l'.8<'53+U/;#%?'?#2 F',08'"2#%LPaY8AU38/W++1^/W#%FX8>RP

FIV+14?(*? 8t>#%8F?#%,"8d$2S F8t->,R?53)`+#%8+#%5<2AE$"$:/;#%53+1j7IV+10#Ê97(*A0>++#%810>#%j#%f?(*?@P d8"56',R)'5<,R'8D'X#%^ F84->,R?+1^'Y+#%X?1<E$KmVH81+1<E)A<+#%8+',7'!8_81^ FY8#%`K#%8t#/8+>#%82NHEXF'MK2%>d !_8#C#R>8F#?22(*D8D'`1H?RP!aY8e2%#%M/;#%K8#%t?'?#c F',4`BAC9e#/O+#%8+#%02N5<'?'AD'e S#%X?1r'56?'Afh',RJI2-7+#%56?'2#%s#/F83'5"X?]#%o?',*#%'5 t81?#6#/8_$2K#%#C+#%?L'?J/P

D¨ q ¢DV¥ ® ¥= " Ö¨


~ ) .3#*:$t#2 DNC+(C',Z1+1+1^#%^8dX?]F#/x8`$2!#e !#% #< F'LNC S`2%1HA/;#%7'5eX?]#%LPt< !G#%X?''848Ê%^X8XG#/`8M?',*#%'5 X8+1PjaY8%X?9+#)1FT8#%T'8?'X81+1<'<8F8+cP= T24E9Y/;#%X81<#%?''8RN)m/;b#"LP=0/W#%86m/;8+RP t#%ZDE$+2X8683)>,R?53-Z2#%s81.+#%8+#%8Z#%EE'?'d(*-2N)8K'5"X?]#%<T8+++>'?'A3_#C%K#)++cNR@PQRPb0#%-Y?#.'5eX?]#%LPgaY8M'X<#.8?',*#%'5

mNNNFj?'<#/8X8+++

#%EE'?''+pi/W#%<)>,R?53-M+#%8+#%z81,*8>#%z81l8h-X5eE$G#/

'53+8F/W#%K F>h8_$2`8#%X?1ME$XLN,R'(*hE)AT/τ

P 8"53#Rt1H'+#*%>U0?',*#%'5p2 #*+0>% #&#/T(*A',R?D8#C1

'48K$22NC#%',"'H/;#%5<#%GE$#%XT F88Y8`8#)1`S)>,R?+1G814 F' F=#%8x8#C1RP x2e8"?#)#%"#%X,RZ?'?)8#)1+x#`>8+ #D F88=->,R?53-+#%8+#%<#%b2#%426E9K56+1<81Z$:/;#%5 8K56b F8F$#R'E?RP X8>63#*%>6 S#%X?1Z8++17#_?'?#)2F6>+AZ#/?,R

m2N n8!#%>?C)X5eE$#/78#)1+ G81 8jX-'53U+#%56?mVH'BAz/;#%o',R?f'53+ S#%X?1gE$.M?2%O(m2N )

P .D2&1#3>#%53 FdE$K` F'0Z?+>`1H'+t#*%>LPFaY8?',*#%'5 !DX8`#%X?''8+1M'MI8,8PHPHP

82%1o#/#*+',f>% #s#/t'81H'()1HX?F8#C1+2NT SM#%^8G$#%X?]#%[#/t>,R?+10't/;#%2%>f,R53-D#/T87$2e_2%j?(*?@P dj?(*?

l86

2N−l ,R53)T8e1H)>,R?+10?(*?x% 2N F81#6 !D8++1&#3#%70>+A#/

2N +N∑

l=0

2N−l = 2N+1 + 2N − 1 = 3 · 2N − 1 HP HP q

'),*2P aY4O(3 · 2N )

'81$81-M#/m81^z'56#2(*53)h#2(*^8

#*%>jE$#2(*e F8m > 3

P#%d2%&'53+&#%02%0?(*?N > l ≥ 0

!D8?#C#%<#%X,RM?'?'!#/L,R53-!E$B S+G F>^+#%8+#%^!?'?#c S+1PbaY82N−l−1 ,R53)D'D#%.?(*? l P6aY-X82Nx'8?'X81H', 2N '>#%8_/;#%D->,R?53-2#%^#%<8K1H)>,R?+1<?(*?$81<#%8`'>#%3/;#%O/n+',781 @9#%')S'cN*8#%>?LX)'53e+#%56?mVH'BAG$F'53+hE$++#%53+

2N + 1 +

N−1∑

l=0

2N−l−1 = 2N + 1 + 2N − 1 = 2N+1 HP HP

F><O(2 ·2N )

P ,-'<Tb'81$81-F#/ m 813!<'56#c(*53-F#2(*!81H'+d#*%> F8

m > 2P

aY8KX9#RF#/$8F?#C#%3#2(*!,R53-b'TO#$:/W#%5 )>,R?53-Y+#%8+ #%82PaYTT1#%8d%b/W#%?'?#c K2P.`>TE-AZI881H',e8K?#2 S+S$#%X?]#%<)X5eE$

n'h8"B !#4>,R53-2PFaYdK8e-X5eE$_#/b+#%8+#%8d F>02&E9756+1PB/'SS8#%2#NH St1H>+ _)X5eE$KS>81#%5 %++#%1H',Z#7DE'8#%56]?1H'EX#%b(n, pl+1)

N F8pl+1

d8D#%EE'?'' Ah/;#%d+#%8+#%f#/OB S#^?(*?l'`#4X8

+++1PDaY>81#%5p)X5eE$Dt87-X5eE$e#/SX8+++:/WX?+#%8+#%U56 L8+/@'?@P<aY8<X8+++:/WX?b+#%8+#%86%11f#M86$#%X?]#%r#/Y83+#%+$#%81H',

8.3. PRELIMINARY RESULTS 83

,R53)b#%Z?(*?l+1

PaY8Y#%8+O FZ/@'?8%11Z#t8T/W+!'b#%Z8F1H->,R?+1?(*?@P='?'?'A

nTXE>%+16/W#%5 8F$#%X?]#%8T#/8F S#e>,R53-b'<8F'2P

/WKX',<87?',*#%'5/;#%K8D$+JI+1h)X5eE$t#/O'53+8` SDI881h8>e'hX'`#/

τ−1 E)AG1H'(C1H',<8e#%>?x)X5eE$t#/b->,R?+10>e,*8>+1#%?(*?NE-A^8_-X5"E9t#/='53+8

T/τP

~ ')*) $& -# 4 +) a#h?'AC786+_#/!8?T56JVC',8N !68#%X?10Xf83?',*#%'5 /W#%_(R#%X8',*F#/=8>53K'G853#C1?@NH@PQRPO8_8?L?,Rh156'8+1hE-A

NN

8e-X5eE$m#/b>?'??O8?2N81&8"+#%8+#%.8102#%&#%EE'?''+

plPOaY8d+#%565eX2#%M>+!#%E>'8+1<S !2A38#%X?148<E$`+#%56+14 F'<8

+#%+$#%81H',>+D/W#%_'81$81-Z8?" Fs2.E$7/W#%X81fE)Af2?'',h8>/;#%X81G F8

m = 1 F'

mP

.`+(*`X8+138`?',*#%'5hNC F'm = 1

N)#e+#%56X`8`+#%565"X2#%^>+,R'(*0'0>+#%8|HP HP HNL|HP P ^81f|HP 8P HP .72(*7?#45<%1Z#%8"'5eX?]#%.#/b8mB1$818+7#/b8">" F'j81& F'8#%X_8?56JVC',8PtaY87+X?'`8#c F

'0I8,8P HP 8PeaY8Z?#% !%_#%E>'8+1. F'j+#%8+#%U#%EE'?''+D#/S?J/!8"(R?'X8#/b8#R"+#%+$#%81H',^#<8e>e/;#%`8"92t F' S#$@%?]53#%t#%>',5353#%A n>ELP|HP mP

4 &)(+*89 A -(+2 &KH A -(+2?K#(CJE24&KHH*,% " ! H0 G 5+J,%;*79 962? (+2* 5+&=(6* A -(+2 " ! H(+56 '&=H1%;*79 A ;(624?K#( " H40 ?=(6(6* % ;H*79 ?1#H(+2* *,5+56?15@ H % ;H4*8&=5 *!(+5+& ?1%&=(6 ?KH D5+?K" (624*96 ;"<#%'&)(6*

m = 15+&=(6*10

.Z#%E8(*60'82%7'f+#%565eX2#%.>Z#/!E$#%XD6/@%#%B S#4/;#%_2%m > 1

P_aY87>7 F'056JVH',h92t#^+(*6^?'',R)?'A&X$?''82_2?'',G F'


mNREXS8K1()]#%4b8#%!',RJI2-?'A3#%X1`8K#%2POa#_+b/;#%bX8>^_1(C

#%38Y>`8#%X?16E$Y?#%+1Z/W#%Tt?],*T>,*K#/m@(R?'X8+2P 88S#%O8#%b8

>_ F'&8?x56JVH',3F2?'?'AMX$?''822N8_'56#c(*53-_#2(*d'81981)>?'??=8?tK8#%dE',8PFB#%Xt2?X?]#%8t SD2(*e+#%56??'Ah8,R?++1&8'82%7'f?]%2?+#%565eX2#%r Fft8+++>Ah/W#%_8?O56JVH',8P'8#%D',R?M>8?`2%8#)1G8++1l#%?'Ao+#%565eX2 F'lB S#U#%838#)1+332%>'53+LP '.>8?T56JVH',2%r8#)1Z8++1e#M+#%565eX24 F'

2m#%8

8#C1+2P d8+h8h53#%X-4#/_?]%2?F'H/;#%5<#%mV,*8++1+1o/W#%3>8?56JVH',e2?+T?''82?'A7'

mNC816J/$8F'56#2(*53)!'<>`#2(*T'81981)!>

8?Fd?#R_#3+#%8>-;%`X,R,*++1hE)AÎ8,8PHP !8h8_,-'&'h>_E)A^56JVH',56',R)D S?'?SE9628+?'?+1UE-Aj86'533$-"#%.?]%2?T+#%565"X2#% )P 53#%8#%#%X,R?'AHY#/8+Y#/?]%2?+#%565eX2#%&#%^8>t!8++1+1G#156'8_8#c 5"X8LN8J/b-AC',8N2ME$,-'8+1hE)A4>?'??=8?56JVC',8P

K6+#%8?'X8#%&#%h8?''56'AM+X?'È$#2(*RN S_5<$#*_ S#6)A)$#%8++/q 8h56JVH',<>8?d%++#%1H',<#Z8D53#)1?L1+'E$+1M'>+RPHP q8D2?'',3#/8D+#%565eX2#%0>_ F'

mKX$?''822P

z'j353#)1>7)X5eE$#/S>?'??>8?m ∼ O(10)

8#^',RI2-`,-'M'&+#%565eX2#%0>_K#%E>'8+1hE)A^>8?x56JVH',8P

8 i|)£ Ii ¥ Y m ¢¤ i i ¥ £ ¦ 4¢Y¥ SS¢ ¥ < S q Y m¡| ¢ S i¥ ¥ N ¥ Y l ¢¤ ¢Y < £ ¢ i i < ¨

8.3. PRELIMINARY RESULTS 85

4 4 &.K?K#( ? &K% 1?K5+ -(+2"D?K5 96 ;"<#%'&)(6 ?KH ? &IJE24&=H4H*,% " ! H 5+* *8&)(+*,570


$

$ ! !

GK981HJVG S :X8J/WA38+X?'F8#c Fh'G>ELPHP q P25O?2%m/;F#Z+RPHP HP /;#%F8DXLP

xD86'']?T><#/Y)XE' q E$Z,R'(*.E-A |ψ〉 = a|0 〉 + b|1 〉 F8 a, b ∈ C81 |a|2 + |b|2 = 1POaY8 #%'-Y'']?9>` |ψ〉1 ⊗ |Φ+〉23

POax#eI881^8d>t#/L8)XE't /W`8"532%X53)t !e2(*e#^%t#%&d>e F'

UBell/W#%?'?#c S+10E-A

#%8_#/8_# +#%&#%$>#%

|00 〉12〈00 | |01 〉12〈01 | |10 〉12〈10 | |11 〉12〈11 | P q

.e F'?'?1#68"2?X?]#%&/;#%K8"++#%81#/O8+RN +#%+9#%81H',^#38e532%X 53)`#%X+#%53 q P

'8+UBell

38>8:/W#%5<#% /W#%5 8?Y?'?`#.8+#%56X>#%?E%2N%',&#h83?m/WD'"X8"+#%56X>#%?YE%">+D')#@Y?'?S>+2P d8+< SI881n

[

|01 〉12〈01 |]

UBell

[

|ψ〉1 ⊗ |Φ+〉23] P

=

[

|01 〉12〈Φ−|][

|ψ〉1 ⊗ |Φ+〉23]

=|00 〉12〈00 | − |00 〉12〈11 |√

2

[

a|00 〉12 + b|10 〉12√2

⊗ |0 〉3 +a|01 〉12 + b|11 〉12√

2⊗ |1 〉3

]

=1

2|00 〉12 ⊗

[

a|0 〉3 − b|1 〉3]

8#%5 3#%XX6 SM+G68^#%EE'?'' A[#/t#%E>'',.8G532%X53)#%X+#%53 6 1

4

'81$81-_#/b8e+#C<-a, b

'8+Dd`8"-X"#/O88#%5<?'>#%0/@%#%K/;#%d8"> n S7+#%X?10?#<+(*"/;#%X81htE-AM87>8181#%X!#/8>$#)',t8S>%+!#/88S# +#%Z#%$>#%= F'"8S+1HX8++1Z18'BAD5<JV/;#%F8_532%X+1hAC5 mP

zbO53#%RN- SK>+YT8!I8?>`#/$)XE' b+>?'AZ|ψ〉 %b?]'53+1'7>ELPHP q PaY8Y2?X?]#%8/;#%=8S+F#%8532%X53)T#%X+#%53+bY'56'?]2P

88 A. QUANTUM TELEPORTATION

$

k! ! !

Md$81HJVh SD>DmVC+>#%8F/;#%K8_,*8>',</nX8#%0#/b!#%,*#%?''XE$#2(>8:/;#%5<#%r#%srE'>A W+#%X->E? D-X5"E9Z#/!53#C1+2P .6+(*<EX'?'e#%81lmVH81+1[l12j1HX8G# P P 8LPYm/;#%4 !M#C+++1Nb SG#c(*M8/;#%?'?#2 F',

|vac〉〈vac| =

∫

dpdx

2πe−(x2+p2)/4D(

x+ ip√2

)P q

F8 |vac〉 F8_(R%XX5>t/;#%K853#C1D#%M FD%2P

aY8)X%1H>Xt#%$>#%K1mI88+1Ê-A

X = (a† + a)/√

281

P = i(a† − a)/√

2P

aY8K',*8>+S#/X/;#%5 _+#%56?tS814`18#%+13E)A |x〉 N- F8 x ∈ R

nX#Dd/@%#%2N

Xb8Y9#R'#%4#%9>#%b#/$8K'56?Y53#%d#R'?'?]#% mP '8+F8

+#%565"X>#%_[X,P ] = i

N98">8?]#%j#%$>#%t/W#%t8XB',*8>+t,R'(*

E)Ae−iP∆x W+_RP ,8P LLP !#" mP ._5<+AG8m/;#%_ F'!$

|x〉〈x′| = |x〉〈x|eiP (x′−x) = |x〉〈x|e−(a†−a) x′−x√2 = |x〉〈x|D(

x− x′√2

)P

t#2 g/;#%`-Ax′ ∈ R

!_+(*!$∫

dp

2πeip(X−x)|x′〉 =

∫

dp

2πeip(x′−x)|x′〉 = δ(x′ − x)|x′〉

= δ(x′ − x)|x〉 = |x〉〈x|x′〉 P&%R

90 B. MANY-MODE GENERATING FUNCTION

81M88+D8D#%9>#% |x〉〈x| +)X?K8_'),R>?mVH+#%#%8_?m/n2PTB8',F')#P ! Se#%E>' $

|x〉〈x′| =

∫

dp

2πeip(X−x)D(

x− x′√2

) =

∫

dp

2πeip( a†+a√

2−x)

ea† x−x′√

2−a x−x′√

2

=

∫

dp

2πea† x−x′+ip√

2−a x−x′−ip√

2 e−ip(x+x′)/2

=

∫

dp

2πD(

x− x′ + ip√2

)e−ip(x+x′)/2 P F84841H>->,R?'',f8+#%5\ !%7X8+1U'r8^++#%81r?''8RPz'r848?'o#/P 81 8P q P q t !62.mVC+8Z# +#%U#%f8Z(R%XX5p'f87/W#%?'?#c F',5<8 n8#%DK8_(R%XX5>F8#%K8_ 53e%

x = 0 $

|vac〉〈vac| =

∫

dxdx′|x〉〈x|vac〉〈vac|x′〉〈x′|

=

∫

dxdx′|x〉〈x′|〈vac|x′〉〈x|vac〉

=

∫

dxdx′dpdp′

(2π)2D(

x− x′ + ip√2

)e−ip(x+x′)/2e− 1

2

˛

˛

˛

x′−x+ip′√2

˛

˛

˛

2

e−ip′(x+x′)/2

=

∫

dxdx′dpdp′

(2π)2D(

x− x′ + ip√2

)e−i(p+p′)(x+x′)/2e−14 [(x−x′)2+p′2]

=

∫

dydy′dpdp′

2(2π)2D(

y + ip√2

)e−i(p+p′)y′/2e−(y2+p′2)/4

=

∫

dydpdp′

2(2π)D(

y + ip√2

)δ(−p− p′

2)e−(y2+p′2)/4

=

∫

dydp

2πe−(y2+p2)/4D(

y + ip√2

)P

?]E$?'?'',38_(R]E? y →

x8+#%8?'X81+K8D#)#/O#/=8?565<

t#2 r ST,*8>?'T8T,*8>',t/nX8#%7#/>+RP 8P q #`5<-AD53#C1+2P x'e+#%j !Z+(*<^+#%?'?+#%U#/T53#%<#R'?'?]#%_53#)1+ ai81j Y#%,*#%?''XE$#2(

>8:/;#%5<#%f,R'(*jE-A 8P q P qRq mP dI8 S6+#%818Z2%7 F'8#%X_532%X 53)2P .3?"'81mVoXs#c(*"84#%XX"53#C1+2N F'?

iX87#2(*6?'?T53#C1+2P

aY8,*8>',</nX8#%h`1mI88+1hE-A $

F ≡ axi\o

[

out〈vac|(

∏

o

eδoao

)

U

(

∏

i

eβia†i

)

|vac〉〈vac|(

∏

i

eαiai

)

U †(

∏

o

eγoa†o

)

|vac〉out

]

= 〈vac|(

∏

i

eαiai

)

U †(

∏

o

eγoa†o |0〉o〈0|eδoao

)

U

(

∏

i

eβia†i

)

|vac〉

= e12Σ × 〈vac|

(

∏

i

Di(−αi)

)

U †(

∏

o

Do(γo)|0〉o〈0|Do(−δo))

U

(

∏

i

Di(βi)

)

|vac〉 P |

91

F8Σ ≡ ∑i(α

2i + β2

i ) +∑

o(γ2o + δ2o)

P ?'?=8">53D#/ F e2? 1H " )]#%0 DP 2P 2P

αi, βi156'8e8"'Xd5<JV0?53-2N9 F'?

γo, δo156'8

8_#%XX2P.F1mI88

wo ≡ (xo + ipo)/√

2NdW ≡∏o

d2wo√2π

=∏

odpodxo

2π

81Σo ≡∑o |wo|2

Pj$#)',`X8T#/8SE$#2(*S?565<`81e8b>8:/;#%5<#%7X? 8P q P q L Sb8Z#%E>'8t/;#%5#/

F FK'56?53-+1h'&#%XK#%,R>53 $

F = e12Σ ×

∫

dWe−12ΣoK

P F'h8#*8? $

K = 〈vac|(

∏

i

Di(−αi)

)

U †∏

o

Do(γo)Do(wo)Do(−δo)U(

∏

i

Di(βi)

)

|vac〉

= 〈vac|(

∏

i

Di(−αi)

)

∏

o,i

Di(γob∗oi − γocoi)Di(wob

∗oi − w∗

ocoi)Di(−δob∗oi + δocoi)

(

∏

i

Di(βi)

)

|vac〉 tP t#2 k S78#%X?1&?' #*e#^mVH81`/W#%5"X?]4#4?#4'8?'X817532%X53-2P.

+#%81FD>X4'^ F^8`532%X+1453#C1+Yd1H'8S/W#%5 8t#%XX!53#C1+81f F8383#%?'A&532%X53-Z#%X+#%53+D S<+#%81H'#%r#%r68<#%8+D+#%+ $#%81H',j# 8P q P q mNO@PQRPj84+8+G#%6E88+G#/t?' #U's8G1+#%2P ?'?532%X53)!F#%^',R?F53#C1RP 0#%Y,*8>? 2%>+b26E9Y2+1< F'3,*8 ',/WX8#%82NREXO F'?'?X +!/;#%O#%XOX9#R+cP -#C1HX8',_B !#_53#%Y'81H++l, d

#'81mVZ+9+'(*?'A78FXE8T#/ 53#C1+/W#%O F>3532%X53)TAC?1b?' #81^8tXE8Y#/53#C1+S/W#%! FM532%X53)FA)?1Y8#7?' #$N P TS53#C1HJI+1#ZE$++#%53 P mP

92 B. MANY-MODE GENERATING FUNCTION

F = 〈vac|[

∏

i

Di(−αi)

]

∏

o,i

Di(γob∗oi − γocoi)

∏

o,i

∫

d2wo√2πe−

12|wo|2Di(wob

∗oi − w∗

ocoi)

∏

d,i

∫

d2wd√2π

e−12|wd|2Di(wdb

∗di − w∗

dcdi)

[

∏

l

−∫

d2wl√2πe−

12|wl|2

∏

i

Di(wlb∗li − w∗

l cli)

]

∏

o,i

Di(−δob∗oi + δocoi)

[

∏

i

Di(βi)

]

|vac〉 tP aY_D83mVC+>#%f Fr%_E9+U'56?53-+1r'f86#%,R>532P3 f86'5 ?53)>#%l84mVH+#%oZ?' !+AH6mVH81+1s#%X2PjaY84'-,R>?F'o2%o5X?'?+1.#%X"#GD8Z(R%XX5 mVC$+>#%.D(R?'X+1jI8e81.86+X?'',)X%1H>/WX8#%hF'-,R>+10 /n Y12N %`1+'E9+1h'h8 q %6#/O+RP 8P q P

$

! !

r ! r Uk!

tD !",R'(*78#%1'(%#%8t#4Xt#%&I85 ,R#%X81&8"#C++1HXD/;#%t)>,R? 53)`+#%8+#%&#/O+RP HP q 81M8e->,R?53-t,*8>#%&#/O+RP8|HP HP q P

D E .t+#%81!8dXG#/LI8,8P tP q 814+)X'e',R??' #3'G#%8t1+#%Y#%?'A3/W#%X8+++>:/nX?L+#%8+#%LP

! ""#$&%'$&)(+*$,(+.-*"/*"0'(+0!$, 1!-2*435* $76 8 9 "0.$:--;!- "#2#<=$& > 8#2$?3

@#%e'56?''BAj?DX8"%X536D83?m/nD1+#%7?' #H2P3aY86# +#%U#%>#%7+#%+9#%81H',0#h7#%X+#%534

a†2′ |vac〉〈vac|a†2′P t#2 8<E925 ?''

>$#*+7(R%XX5 #&(%%XX5 n'Ze%>'(* e81r>8:/;#%53a2 → 1√

2(a2′ + a3′)

81a3 → 1√

2(a2′−a3′)

Pxa=$#)',t8b')(*S#/x>8:/W#%5<#%LN% !S>+O#%X532% X53)`K+)X'(%?)K#<7# +#%#%-#<8_> 1√

2(a†2 + a†3)|vac〉 = |Ψ+〉23

PB/L8t'']?>d#/53#)1+ q !%7 |Ψ+〉12 ⊗ |Ψ+〉34

!t8G#%E>'G8tX8#% 5<?'+1MI8?x>t/W#%5 $

!%

94 C. ENTANGLEMENT CONNECTION AND CREATION WITH BEAM SPLITTERS

23〈Ψ+|[

|Ψ+〉12 ⊗ |Ψ+〉34]

=1

2√

223〈vac|

[

(a2 + a3)(a†1 + a†2)(a

†3 + a†4)

]

|vac〉all

=1

2√

2(a†1 + a†4)|vac〉

tP q 8#%5<?'',8NK>_FmV8%?'A |Ψ+〉14

%K1+'+1P 8"2%" F8e8"',R-1+#%_?' #H2N8"#%XX_>e#/b53#C1+ q N ^ F'?'?

E$ |Ψ−〉 F&2E9>8:/;#%53+1M# |Ψ+〉 E)AG6?#)2?%eJ/nK#/53#C1 q P D D E t#2 +#%8138h->,R?53)G,*8>#% X['oI8,8PS|HP q ELP /n6')>%#%E$B !+l8<'X6X?G81s8M#%532NO8 :#%')Z>^#/#%56^8>5eE?h81 #&#*+S#%XXT?'',R-SS S#$@53#C1d>-X8+ 2+14>RP 838dmVH'>#%4#%EE'?''BAK5<?'?

pc 18_>_ $

|ξ〉 = (√

1 − pc +√pca

†Aa

†L)|vac〉 tP

aY8Y+#%81H'#%',">853Y8F>53F%'ZI8,8P tP q F'653#)1+ q N _E$',_8F#%5653#C1+e81f53#C1+ HN&%GE$',M8 #&#*+D?'',R)D53#C1+2P t8+RN%_E$m/W#%RN=G?' #j'8"?m/nt1+#%_+#%+9#%81t#4# :+#%f#%)# |Ψ+〉23

Pdax#<I88108"#%XX>#/Y8<#%56353#C1+2Nx S3$2D832?X?]#% tP q F'.53#C1+ q N 81%HN h'88 'Xd>_,R'(*hE)A tP mP%.D,* $

23〈Ψ+|[

|ξ〉12 ⊗ |ξ〉34]

=1√2

23〈vac|[

(a2 + a3)(√

1 − pc +√pca

†1a

†2)(√

1 − pc +√pca

†3a

†4)]

|vac〉all

=

√pc√2

(a†1 + a†4)|vac〉 tP&%

#Z?#c S+t#%1K'pcP `8#%5<?'',3,R'(*+ |Ψ+〉14

%K1+'+1Pt#%F/W#%5 tP&%bT8d#%EE'?''BA3/W#%S8`?m/WS1+#%!#e?' #3

pc/2POaY)X8

84#%>?S#%EE'?'' Ar#&,*6',R?^?' # 81r8m/;#%</;#%78<,*8>#%o#X8++++1

pcPdr84#%8781N S45<%1484%X56#%o

pc 1Ph@/

_+#%81H'#%ft?] V+1NL#%_#/!#%1O(pc)

F'?'?T$2'086#%XXD>7#/53#C1+ q N "8168`>Y F'?'?88#_?#%,*SE9Kt$:/W+ !?'? >RPBZ/@%2N*J/$ SKmVH818Y,*8>+14->,R?+1<> /W#%X81ZE)AZ8F,*8>',D/WX8#%3538#C1=#?#2 !+#%1K'

pcNH SD#%E>' $

0 0 0 00 1

2 − pc

412 − pc

4 00 1

2 − pc

412 − pc

4 00 0 0 pc

2

tP

$

" ! !

dm/W',4#4+RP HP&%6'0t$81HJVh S",R'(*"8e>53>#%8_#/O8718'BA5<JVr72%l1H>8+3?(*?S/;#%e$XE#%r'r84+#C -

c1, c281

c3#/

HP HP mP.T>T8T+X8+S?]#%8/W#%2%7>5381",R'(*b8S#%?'X#%82P 98Tb#%?'A"8Y%>K#/

c1-T'-#_8F>#%?'X#%82PB F'?'?8E$F8#%X,R

/;#%D#%XDX$#R+D# #C8#2 83#%?'X#%. F8.e%62#N81f88+6 !31#8#%YI881M8D,*8>?x#%?'X#%M/;#%dE'>AG%D#/

c1P

5b#)+++1H',G?#%,38?''8+`#/O+RP HP&%Z !I881$ D E E

c1

X56',Z c1 1NH F8h8% 2#N8 StI881M8t/;#%?'?#2 F',3>53

#%#/8_18' A^5<JV$

ρi =

1 − 2fi + (2fi − 1 + 2gi)c21 0 0 (1 − 2fi)c1

0 fi − gic21 fi − gic

21 0

0 fi − gic21 fi − gic

21 0

(1 − 2fi)c1 0 0 (1 − 2fi)c21

ZP q

aY8">53fi1#)+d8#%d+#%53e/n#%5 8e$XE#%LPtBKd+)dE$+2X8

!"DX8',<8#%B+#%X-',M1+#%2N$%`1H>X8+1M'0+RP HP 8N$81#%E$AC`8D>53+X8+7?]#%j,-1H?+#/T Ff+#C - S6"9:/;#%56',G8e$XE#%h'LPOaY8D+X8+_/;#%

fi !%d>+1h81h#%?'(*+1G'&+RP HP&% H/;#%K+#%56?8+

!_,R'(*_8_#%?'X#%&,-'E9?#c DP@8#%K8>53gi !I881$

gi+1 =4fi(4 + gi) + 11f3

i − 20f2i − 4

2fi(fi − 2)2

=4(1 + 21+i + 22i)gi − 22+3i + 22+2i + 3

2(21+i + 1)2 ZP

96 D. SOLUTIONS FOR PERTURBATION

/n7M,*^#/F(%]E?4# gi ≡ 2(1 + 2i)2gie+X8+<5<2AfE93>#%?'(*+1.'

RN,R'(*o8<'']?F+#%81H'#%8n !^>e/W#%5\h9:/;+Z->,R?53-,*8>#%0LNH@PQRP/W#%5 8Y?'?L> |Ψ+〉 mPbaY8D#%?'X#%8F/W#% fi, gi

!$

fi =1

2i + 1

ZP&%R

gi =−21+3i + 3 · 21+2i + 5 · 2i − 9

3 · 2(2i + 1)2 ZP&%E

D E E c2

X56',6 c2 1N8>53 #%& $

ρi =

1 − 2fi + (2gi + ki)c22 0 0 0

0 fi − gic22 fi − hic

22 0

0 fi − hic22 fi − gic

22 0

0 0 0 −kic22

ZP

F'fi%FE$m/W#%e81G8e#%8K+X8+D?]#%8 $

ki+1 =2(ki − fi)

2 − fi=

2(2i + 1)ki − 2

2i+1 + 1

ZPR

gi+1 =−2f3

i + fi(−2 + 2gi − 5ki) + 2ki + 3f2i (2 + ki)

fi(fi − 2)2

=2(22i + 21+i + 1)gi − 22+3i + 22+2i + 2

(21+i + 1)2 ZPE

hi+1 =2(1 + f2

i − gi + 2hi − ki + fi(−2 + gi − hi + ki))

(fi − 2)2

=2 · 3(2i + 1)hi + 21+2i

3(2i+1 + 1)

ZP% aY8+X8+&?]#%8M&#%?'(*+1' /n^,R',o(%]E?+4#ki ≡ (2i + 1)ki

Ngi ≡ (2i + 1)2gi

81hi ≡ 3 · (2i + 1)hi

POaY8D#%?'X#%8t!$

fi =1

2i + 1

ZP |R

gi = −2

3

23i − 3 · 22i − 2i + 3

(2i + 1)2 ZP |E

hi =2i+1(2i − 1)

3(2i+1 + 2)

ZP |%

ki = −22i − 1

2i + 1

ZP |%1

97

D E E c3aY8!>53>#%Z/;#%$XE#%6'

c3=1-2?#t8!>53>#%6 F8

mVH81H',3'c2P

D D D E @8#%G?'?`8+X#%?]#%8ME$#2(*h8&#%?'X#%8GhX)X8'8+RN! F8z8A F'& F'h8,R'(*h(R]E?+`,*+`8AGE9++#%53_?''82`'8#%53#%,*8+#%X81H "8+M+-X#%83 F'[+#%8>-<+#C -^81[&/W#%5 #/t8G'8#%53#%,*8'BA F>hA)?1`X-X8_>#%?'X#%82P @8#%`mV856?/;#%F9XE#%&'

c2$

gi+1 = 2gi − 22+3i + 22+2i + 2 ZP

aY8_t6#%8_#3#%8e+#%+9#%818+"E$B S+&8D#%',R'?=818_8 (R]E?+2N81h>#Z8'']?L+X8+D?]#%8d+(*_X)X8D#%?'X#%8cP

?'8#%X,Rg Sj2(*f#%?'(*+1z8j+X8++M' *Nt>#%?'X#%8M2gE9#%E>'8+1ME)AG81ME-A^8_538#C1h#/=X8156'8+10+#C -2P

98 D. SOLUTIONS FOR PERTURBATION

$

U

M`$81HJVG !D+#%?'?+FI8,RX+`'M F8_+#%81H'#%?I1?''BAM#%E>'8+1G/W#%5'5"X?]#%hF+#%56+1M F'M8_+#%81H'#%?I1?'' A^/n#%5 #%XF$XE'(*_2?X?] #%82P

aY8FI8O/;#%XOI8,RX+S8#2 '5"X?]#%8S#/8F S#$@%!#%#C+#%?#/L+RP-|HP&%HPaY8>532NL

κ = 2Nη = 0.1

WmV+t/W#%87#%&?m/Wη = 0

mNL81&/W#%5 #%0?m/W#ZE$#%#%5 ',R- $

r = 0Nr = 0.01

Nr = 0.05

Nr = 0.1

P

R

100 E. ADDITIONAL FIGURES

aY8S8mVHxI8(*TI8,RX+8#2 o'5eX?]#%8O#/88!#%8 @%=#%#C+#%?C#/ >+RP |HP 8P @#%5#%[?m/W<#UE9#%#%5 8M>53

[s, κ, g]!$

[109, 1.00, 1.00]N

[106, 1.00, 1.00]N

[103, 1.00, 1.00]N[10, 1.01, 0.96]

N[1, 1.17, 0.83]

P

$

! !

Zb981HJVZ SY?'8 D#%,R>53T= !FX8+17'6#%E>'',D8?'AC2?$814-X532?+X?'T+)+1^'48d8+2PaY8d#%>?$53#%X)Y#/x+#C1+#%81>E?RN=81&8m/;#%Z S61#^8#%_#^'-_'2P ''j S#M53#%)j/n#%5 8XE56>#%.186#/S8+cN86#%X+Z F'?'?OE$Z2(%'?]E?<#%?''8RN/;#%t8ZE$8mI8#/8_mV56'8cNK- $ F F DP /WAH2P #)XLPQ1# #%$+]?CP

.!,R'(*Kd?'O#/ 8 e8#%E9#C#&#DI8?+O+#%)>'',D8Y53#R'569#%>)#C++1HX+D81j2?X?]#%f Fj SZX8+1&'f#%E>'',M8e+X?'+-+10'< S#% #$P d?''8RNY8hI8?+4&#%,-+1['l/;#%?1G%++#%1H',r#f882%1H?''8+E$?#2 DP

!#C1D8#%X?10?' !2ACdE9 !?'?=+#%5653)+1P `8+#%5653+10+#)1DK?],*?'AG'?'?,R'E?#7-A*#%8EX!8dX8#%2N8814 F'?'? E$`#7?#e/W#%S8`#%,R>5653K /W!8#%!'53RP.2(*"56+1h#6+#%5653)t?'?x+#C18#%#%X,R?'A*P

$ 5O#2(C1+653#C1HX?+Z#j+#%56XG84,*8>',j/WX8#%/W#%6,R'(* !#%,*#%?''XE$#2('8?'X81H',^532%X53- F'08#%@+#%?'()',G1 +#%K>8:/;#%5<#%LP

$5O#c()1+K53#)1HX?+F#Z+#%56X8_#%XXK18' A45<JV/n#%5 Y#%,*#%?''XE9#c(">8:/;#%5<#%LNR,R'(*68!,*8>',_/WX8#%Z/W#%=8>8:/W#%5<#%j81M8'X`>RP

$5O#c()1+S53#)1HX?+T/;#%T#%E>'',"8`18'BA65<JV4!2%?(*?$#/e$22NH,R'(*48t'']?> /nY->,R?53)K,*8>#%81M8,*8>',3/WX8#%h/;#%F8_->,R?53)+#%8+#%LP

$ 5O#2(C1+G53#C1HX?+</;#%4'5eX?]#% #/e8?56JVH',8NF##%E>'h8D+#%565eX2#%0>RP

"!$#&% ' ( *),+ .- $!#%56X+O8Y,*8>',/WX8#%<816#%XXO5<JVe/W#%5

->,R?53-t,*8>#%LPq/Hq

102 F. LIST OF PROGRAMS

( *) - $ !#%56X+e83,*8>',h/WX8#%f/W#%D)>,R?53-6+#%8+#%LP( *), $ '5"X?]+K8 @$22P( *), $ !#%56X+d+#%565eX2#%j>+Y/;#%F8 @$22P

%,' ' * $%!#%56X+!8`53#)1d+1HX8#%^#/L8K/nX?'?9>

>8:/WY#%,*#%?''XE$#2(M>8:/;#%5<#%LP* - $ !#%56X+<8h,*8>',U/nX8#%/W#%<)>,R? 53-`+#%8+#%LP* $ '5"X?]+`8 S#$@%d$22P* $ !#%56X+_+#%565"X2#%U>+d/W#%8"B !#$@%_

922P* eu $=B56?53)!8K$XE'(*t#*%>4#e8KB !#$

%F$22NX8',38,*8>?=#%?'X#%M/;#%F9XE#%LP %,' '

- $!#%56X+=8T,*8>',d/WX8#%"/W#%=->,R?53-+#%8+#%LP $ '5"X?]+`8_#%8 @%`922P $ !#%56X+Z+#%565"X2#%[>+D/;#%78<#%8 @%7

922P u $T 56?53-t8_$XE'(*7#*%>&#38e#%8

%F$22NX8',38,*8>?=#%?'X#%M/;#%F9XE#%LP

.- $ !#%56X+Z8^,*8>',j/nX8#%s/;#%Z)>,R? 53-j+#%8+#%kX8',8.53#C1.+1HX8++1 ,*8>?"/nX?'?7>.>8:/;!#%,*#%?''XE$#2(G>8:/;#%5<#%LP

$t'(*+"81f#%?'(*+86+X8+6?]#%8_'.2%>#/L8d>53Y'^8d9XE#%LNH88+#%E>'',68`/nX?'?$#%?'X#%#Z8_9XE#%LP

aY8K-X532?9+#%< F>< !%SX8+16'42?X?]#%8T Y% j%'8 O+#%≈ q |61H',R'2P

q P P !?'? $ 92$#RE?T81eX892$#RE?O'e)X-X5 53++T+#%?'?++1e$81+1P'NY5eE1H,* `'(*' A O+ / / 8P

tP 7P Y8 $ OXJI2#%#/8#%A4)>,R?53-t814/@'H/nX??$#%> #%()]78#%AG8?%N O)AC2P `($P x2P %|HN& HN q RR|HP

&% 7P 8P !,*? $dX-X5 $2 $xaY8T#%?S#/8'56$:/;+?#C2?)#%$>#%8'h)X-X5 +#%565eX2#%%N O)AC2P d(9P x2P q N R!% HN q R HP

#P bP S8 $ +#%53K#%E8(R#%Z/W#% !X+I1?'' A"E$B !+3B !#D>+#/bZ)XE'%N O-AH2P d(9P | HN / % / %HN / / HP

=P S8 $ S>% c#%[#/F5< VH'5<?'?'AU->,R?+1s S#$B)XE'Z>+D(C]M8Y?'? k'8+-X?'' A%N8)X- @ / / / / CN / / 8P | TP S'?1H+ $ @X?' @#%?>)M-X)X5 +#%565eX2#%iE%+1#%#%?'1

>8#%#%&56'%N b-AH2P `($P L2PR|HN / / / 8N / / |HP ZP X> $dX)X5 '(R%Ae56?'JI2#%381D8T>+X'BAD#/8-X)X5

AC#%,R>-Ah#c(*K8#%AG8?%N O-AH2P d(9P x2P CN q HN q RR|HP TP rP tX $ L#%,&B1H>8+M)X-X5 +#%565eX2#% F'o#%56G

>5eE?+d81h?''82`#%+%N X q 8N q %HN / /Hq P TP rP tX $ X?53->Ah'H/;#%5<#%M/;#% tX q 8N q %HN / /Hq P q/ TP rP tX $dX)X5 +#%565"X2#%ME9 S+M#%56d8>5eE?+!X8',

+#%8)`?'',R)%N O)AC2P `($P x2P HN R| %HN / / / P qRq P6P%#* $dX-X5 AC#%,R>-AlE%>+1l#% Y?'?Q68+#%5%NO-AH2P d(9P

x2P|CN|R| q N q R q P q tP tP _A*N =P bP',R- $ -#C1HX8#%A.-X)X5 #%+%N%Y5eE1H,* d'(*

'BA O+2N / / HP q % "PP @A)5< $ '5"X?]',U)AC+Z F'[+#%56X%N! -2PPOaY8+#%2P O-AH2P

q N q HPq/ %

104 BIBLIOGRAPHY

q # P ?]8A $ 56?53->#%G#/L-X)X5 %Y,-FE)AZ8K#%2? _ "+%N8)X- @ / / / qRq/ N / / / P

q ZP #%+5<LN mP bP S-X, $ 53#%8>', 8(C]E'?''BA #/ X'(*>?)X-X5 +#%56X>#% X8',p?$#%>#% 81 ',R? B)XE' #%$>#%8%NX / HN %R / %R!%HN q RRHP

q | `P d+>- $ d+ R+1P'N 11H#%.+?A*N / / HP q PRX?,-R1$ b->,R?53-681j-X)X5 '->%#%U F'f5<%#R+#%6,-% 56?+%N OLP ZP8+cN `'(*' AM#/ -X82Nt5< #$N / / %HP

q PRX?,-R1 $ OVC$'53->?6153#%8>#% #/<-X)X5 5353#%A /W#%?'',R)%N X % HN HN / / 8P

q `P'?'? $ >853Z/;#%e -7-X)X5 +#%56X>#%s F'.?''82e#%+%N tX / HN -|HN / /Hq P

/ =P P '?#%@N8P "PbE9?'A $ x%%N '?AM81 #%82N q HP q tP P &X8 # $ )3-X)X5\5353#%Ar81r)>,R?53-3E$B !+

?'',R)"81.#%56385"E?6X8',M5<,R8ZI?1%N O-AH2P d(9P #%HN / | !% RHN / / |HP

sP P `?LNe P bP S)X, $ dX-X5 +#%56X>#% 81 -X)X5 'H/;#%5< #%%N Y5eE1H,* d'(*' A b+2N / / / P

!% P=#%9 $ O>%2?D)X-X5 #*Az1H'EX#%i F'z$#%?] c#% >,R?+1h8#%#%8%N d+OVC+ q HN% R| HN / / 8P

# 7P ZP#%X?LN6P ?'53 $ dX-X5 E$25 ?''`/;#%t#%53%N O)ACcP d(9P | HN / %!%R| q %HN / / HP sP ZP t+AC53 $ $>E'?''BA['#%/W#%3>-X)X5 AH53%N

b-AH2P `($P |CN / q/ 8N / / %HP R| 6PP `+> $ t'EX',->,R?53)Z81f',R?Z8#%#%8_#%X,Rr

')> B'BA*N/n+ B%+_-X)X5 8?%N d+VH+ q %HN / HN / / HP P8P $#)X> $ 0#C1r)X-X5 53++4($Px+1P'N 11H#% .+?A L#%,&5< q R8P P 8#% $ ',R)t)XE'`#%>,*781M(%?X8',35<%#R+#%7#%56

8>5eE?+%N b-AH2P `($P 8N /HqRq / HN / / |HP R =P P 8#% $ =#%?'AC8#%56]?@'53<?',*#%'53/;#%t'53"/@%#% c#%r81j1H

?#%,-'53"#%rG)X-X5 +#%56X%N 8PL#% !#%56X>#%s(*#%?@P R|HNL8#PHN q RCP

&% / "a tP . $ j VC'5<?F->,R?53)<(*X86)#%-As/;#%756JV+1o-X)X5>+%N O-AH2P d(9P |CN / qRq/ N / / %HP

i

Y?'?532%X53)2N q Y?'?L>+2N E''D)>,R?53-2N E'8'LN q Y#%,*#%?''XE$#2(G>8:/;#%5<#%LN q %!X:X5eEX>HN EX8>',8N %+#%8)d>RN qRq+#%56X>#%?E%2N q +#%81H'#%?xI1?''BA*N +#%)#%?'?+1 ta_N q |18 #G+#%X-2N %|1++#%88+RN /18' AG5<JV$N 18' AM#%9>#%2N 1H)>,R?'',<8+#%5hN % 1H?]%+53)d#%$>#%2N qRq@ A)$_$22N1#%XE?DmV'>#%LN / N8| HN |R)>,R?+1>RN)>,R?53-2N

5< VH'5<?@N q/+Y/W#%2N q/

)>,R?53-+#%8+#%LN!%)>,R?53-tXJI2#%LN!%I1?'' A*N CN +#%81H'#%?I1?'' A,-RN q ,*8>',</nX8#%LN% qt#%?'O'5<$#*#"4#2VC'5<#%LN R|?#C2?L#%9>#%LN q

5< VC'5<?'?'A^56JV+1h>RN q 532%X53-2N9|56JVH+1M>RN|53#)1RN q %53#)1_#%$>#%2N q %8#$B?#%',38+#%5hN q |8#%RN q 8#%@?#)2?#%9>#%LN q 8#%B>E?e>RN-X5"E9`>RN qRq%8'LN q #:+'(*D532%X53-2N9|X_>RN|XJI2#%LN [->,R?53)UX

I2#%-X%1H>Xe#%9>#%2N q -X)X5 'H/;#%5<#%LN q -X)X5 8#%RN e8#%-X)X5 $22N -X)X5 ?$#%>#%LN q -XE'2N q +1HX8++1h18' AG#%$>#%2N|922N ")X-X5$2>E?D>RN)X8+ 2+1M>RN q )X8+ +',3#%9>#%2N q >>8:/;2N% #&#*+K#%$>#%2N R|B !#$@53#)1"-X8+ +',8N q %X8+>'- A*N-

-X532?@N-q/

Documents

· 2015-09-01 · ykm . % rov &@ 2/oz. @j,/ %)&. % % y#+ , ... zc c/d 9/d 0/ 54 & . 2 d dwd8d dwd dwd dwdwd d8d dwd dwdwd dwd d8dwd dwd dwd dwd8d 4 zc c/d 9/d 9 e ( . @ % dwd8d dwd