12
x xx = xc
E(x)
x = xc
T > 0
x x < xc
x > xc x = xc xx
x = xc x
x
x
x
!
|!1,2" = |!1" ! |!2" = |!1"|!2"
|!1,2" = |!1" ! |!2" =!
i1
!
i2
ci1ci2 |"i1" ! |"i2",
|!!" =!
i!
ci! |"i!",
{|"i!"} #{|"i1" ! |"i2"}
H1 H2 H1 !H2
H1 !H2
|!1,2" =!
i1
!
i2
ci1i2 |"i1" ! |"i2"
H1!H2 {|"i1"} H1 {|"i2"}H2
n
|!" =!
i1
. . .!
i2
ci1...in |"i1" ! . . .! |"in"
ci1i2 = ci1ci2
|!S" = |0"A|1"B + |1"A|1"B|!E" = |0"A|1"B + |1"A|0"B
{|0"A, |1"A} {|0"B , |1"B}
|!A,B" = c1Ac1B |0"A|0"B + c1Ac2B |0"A|1"+ c2Ac1B |1"A|0"B + c2Ac2B |1"A|1"B
|!S"c1A = c2A = c!1
2Bc1B = 0 c1A = c2A = c!1
2B= 1
c1B = 0 c1Ac1B = c2Ac2B = 0c1Ac2B = c2Ac1B = 1 ci1i2 = ci1ci2
c1A c2A c1B c2B
ci
ci
HA ! HB HA HB
{|!Ai "} {|!B
i "}
|!" ="!
i
$i|!Ai "|!B
i "
$i
"!
i=1
$2i = 1
|!" {|!n"}
|!" =!
n
cn|!n"
|!" !
n
|cn|2 = 1,
#!|!" = 1
{|!Ai "} {|!B
i "} {|!Ai "|!B
i "}$i %
HA HB
$i %HA HB %
$i
%
NN 1
|"" ="!
!1
$[1]!1|![1]
!1"|![2...N ]
!1"
N
{|i1"}i #1
{|![1]!1"} {|i1"} #1
{|i1"} #[1]i1!1
|"" ="!
i1,!1
#[1]i1!1
$[1]!1|i1"|![2...N ]
!1"
|![2...N ]!1 "
|![2...N ]!1
" ="!
!2
$[2]!2|![2]
!1!2"|![3...N ]
!2"
="!
i2,!2
#[2]i2!1!2
$[2]!2|i2"|![3...N ]
!2"
{|i2"}
|"" ="!
i1,i2,!1,!2
#[1]i1!1
$[1]!1
#[2]i2!1!2
$[2]!2|i1i2"|![3...N ]
!2"
#[2] #1 #2 |![2]!1!2"
#[n] #[1] #[N ] #[n]in!n!1!n
#[n]in!n!1!n
N $ 1N N
{|in"}
|"" =!
i1,i2...iN
ci1i2...iN |i1, i2 . . . iN!1, iN "
ci1i2...iN ="!
!1...!N!1
#[1]i1!1
$[1]!1
#[2]i2!1!2
$[2]!2
. . .#[N!1]iN!1!N!2
$[N!1]!N!1
#[N ]iN!N!1
{|in"}ci1i2...iN
#[n]in!n!1!n $[n]
!n
m% n A
A = UDV †
U V U†U = I V †V = I IU V D
DA D
D U V
m% n A U m%m V n% n D m% n
#[n] #n #n!1 #[1] #1
i1 #[N ] #N!1 iN #n
%n in q in#[n] {|in"} {|in"}
{|in"} qn {|in"} q
{|i1, i2 . . . iN!1, iN "} qN qN
# q%n%n!1 #[1] #[N ]
q%n q%n!1 $n %n
$[n!1] %n!1 %n!1
$[n!1]1 $[n!1]
2 $[n!1]3 $[n]
x = $[n]!n |!n=x #[n]
%n!1 % q % %n
#n!1 (v, 1, 1) = #[n]in!n!1!n |!n!1=v,in=1,!n=1
#n (1, 1, v) = #[n]in!n!1!n |!n!1=1,in=1,!n=v
in
!
!2
$[1]!1
#[2]i2!1!2
$[2]!2
#[3]i3!2!3
$[3]!3
#
& #1 i2 #[2]
#3 i3 #[3] q%1 q%3 #[2]
$[2] #[3] q%1 q%1 q%1 q%3 q%3 q%3 ##[2] #[3] #[2] #[3]
$[1] $[3]
#2 q%1 q%3
%2 $[2]
%2 & $[2]
#2 %2
#
e!iHt/!
qN
qN qN !H e!iHt
e!iHt
#
H =N!1!
l=1
H [l,l!1]
= H [1,2] + H [2,3] + H [3,4] + H [4,5] + H [5,6] + . . .
H = H [odd] + H [even]
H [odd] = H [1,2] + H [3,4] + H [5,6] + . . .
H [even] = H [2,3] + H [4,5] + H [6,7] + . . .
e!iHt
H [odd] H [even]
e!iHt H
[A, B] &= 0 ' eA+B &= eAeB
e(A+B) ="e(A+B)/n
#n= lim
n"#
"e
An e
Bn
#n
n n = 1/'
lim#"0
"e(A+B)#
#1/#= lim
#"0
$e#Ae#B + O('2)
%1/#
' ( 1 ' ) 0
e#(A+B) * e#Ae#B + O('2)
O('2)'2 ' |'|( 1
O('2)' = it A + B = $H
T = 1n
e!iHT =$e!iH#
%T/# *$e#F e#G
%T"
iH [odd] iH [even] F G
e!iHT *$e#F e#G
%T" =
"e!#H[odd]
e!#H[even]#$
( = T#
H [odd] H [even]
G[n,n+1]
q %
H
e!#H[n,n+1]G[n,n+1]
#[1]
qN qN
#
{|in"} n
{|in"}
$[n!1] #[n] $[n] #[n+1] $[n+1]
& G[n,n+1]
& & G[n,n+1] &$
& #[n] $[n] #[n+1]
$[n!1] #[n] $[n] #[n+1] $[n+1]
#$[n] $$[n] #$[n+1] #[n]
$[n!1] #n!1 #[n+1] $[n+1] #n+1 $
#n %n q%n!1 q%n+1
#
O(q!t) %n * %n+1 q
N $ 1$[n] qn q(N n) %n q n = 1
qN/2 n = N/2 %n = q n = N
%n
%n % # % q %$ % % $ %
# %
%n
$[n] %
%%
%
%q$ (
%%
O(qN )
ci1i2...iN qN N q %N % q %
N $ 1 $ % O(Nq%2)+ Nq%2 + N%$ %
NN N
$
O(q%3) %
%
% $
#
%
$
S()) = $Tr() log ))
= $!
i
$2i log $2
i
$2i )
$i
$%
%# $
$
# $#
|1|# # $
# $
%
O('2)'
$t$t $t
$t'
''
$t
k k + NN
NO(qN )
k kk
q2k%k k NNk q2k%k
N/k & k # (k + 1) $N/k &
& k # (k + 1) $ %3&k
i=2 qi + 2q%
$ %&k
i=2 qi!1 + %3&k
i=2 qi k = 2 %&k
i=2 qi!1 = q%
%3&k
i=2 qi = q2%3 k = 2N/k
kk i = 2 k N/k
kk
%&k
i=2 qi!1 %3&k
i=2 qi i = 2 kk = 1 #$# k = 1
H N
k = 2
NN
'
tf $ ti = i,
tsE(ts)
ts i,ts
ts
tits
ti tsts
''
''
' '
' ( 1
'b
' = 'b
' = 'b
' 'a 'b 'c
'a 'c
' = 'c ' = 'a
'a 'b ' = 'b 'b 'c
' = 'c
' t!#
t!# '
'
#"|H|"" - E0
E0
|""
O(Nq%2)
#[1]i1!1 $[1]
!1#[2]i2!1!2$
[2]!2 . . .#[N!1]iN!1
!N!2 $[N!1]!N!1 #[N ]iN
!N!1
#[1]i1!1 $[1]
!1#[2]i2!1!2$
[2]!2 . . .#[N!1]iN!1
!N!2 $[N!1]!N!1 #[N ]iN
!N!1
|"" #"|
#
$$% $
ci1i2...iN ci1i2...iN
qN q N {|in"}in q in|in"
in #
#A[2]" = #"|A[2]|""
A
A|in"
A n
n
#$ n # 5(n$1)$2
5(N $ n)$ 2
A
An in n
#n n#n %
n
A{|in"} A A
A
#A[n]" =#"|A[n]|""#"|""
A
B
%n b 4(N $ 1) + N + b
b in in+1 # $B
B
B
O(qn%3)O(q%3)
O(q2%3) NO(qN%3) N
Mx(h)
Mx(h) =1N
N!
i=1
#"|*[i]x |""
#"|""
$%
$ #
# $
$
$#
$
$
S($[n]) = $!
i
$2i log $2
i
$i i $[n] $[n]
S()))
$[n] S()) - 0 S()) = 0
E1($[n]) = $ log $2max
= $2 log $max
$max $[n] eln(x) . loge(x) log(x) . loge(x)
log 1 = 0
- 0
0 + $i < 1 /i log $i < 0 /i| log $i| -| log $max| /i
1 + | log $i|| log $max|
=log $i
log $max
log $i < 0 /i
S($[n])E1($[n])
=$
&i $2
i log $2i
$ log $2max
=$2
&i $2
i log $i
$2 log $max
=&
i $2i log $i
log $max
=&
i $2i log $i&
i $2i log $max
$i
| log $i| -| log $max|
S($[n])E1($[n])
- 1
S($[n]) - E1($[n]) S($[n]) = E1($[n]) $i = $max
/i $i $i
$i = 1&"/i
S($[n]) = E1($[n])0 + $i < 1 /i $i = 1
0$i = 1 0$i = 1$max = 1 $i = 0 i
S($[n]) = $!
$2i log $2
i
= $2 log $max
= E1($[n])
S($[n]) = E1($[n]) = 0 S($[n]) > E1($[n])S($[n]) = E1($[n]) = 0 S($[n]) = E1($[n]) &= 0
S())N N N * ,N ),
S - E1
SE1
SE1
= 2
S = $k!
i
Pi log Pi
Pi i kPi k = kB
)
S()) = $Tr() log ))
)
) =!
i
pi|!i"#!i|
pi |!i")
) =!
k
Pk|k"#k|
S()) = $!
i
#i|) log())|i"
= $!
i
#i|!
k
Pk|k"#k| log
'!
k
Pk|k"#k|(|i"
{|k"}
S()) = $!
i
Pi log Pi
|!" {|n"}
|!" =!
n
cn|n"
) =!
n,m
c%ncm|n"#m|
#!|A|!" =!
m
!
n
c%mcn#m|A|n"
#m|)|n" = cmc%n
#!|A|!" =!
m
!
n
#n|)|m"#m|A|n"
=!
n
#n|)A|n"
= Tr()A)
$2i )
S()) = $!
i
$2i log $2
i
$ log Pmax
$
% qN h
h
12
12
H =!
'i,j(
J [i,j]*[i]x *[j]
x + h!
i
K [i]*[i]z
*z *x
*x =)
0 11 0
**y =
)0 $ii 0
**z =
)1 00 $1
*
q = 2
{|in"} =+)
10
*,
)01
*,
*z
#i, j"i j j
j = i+1
#i, j"
J [i,j]
J [i,j] = J J J
J
J < 0 J > 0
J = $1
J [i,j] K [i] J [i,j]
K [i] = K*x *z K = $1 K = J = $1
H = $!
i
*[i]x *[i+1]
x $ h
2
!
i
"*[i]
z + *[i+1]z
#
H [i,i+1] = $*[i]x *[i+1]
x $ h
2
"*[i]
z + *[i+1]z
#
h&
i *[i]z = h
2
&i
"*[i]
z + *[i+1]z
#
!
i
*[i]z =
"*[1]
z + *[2]z + *[3]
z + . . .#
=12
"*[1]
z +"*[1]
z + *[2]z
#+
"*[2]
z + *[3]z
#+ *[3]
z + . . .#
=12
N!1!
i=1
"*[i]
z + *[i+1]z
#+
12*[1]
z +12*[N ]
z
*[1]z *[N ]
z
N ),
H [1,2] =-$*[i]
x *[i+1]x $ h
2
"*[i]
z + *[i+1]z
#.
i=1
+h
2*[1]
z
= $*[1]x *[2]
x $ h
2
"*[1]
z + *[2]z
#+
h
2*[1]
z
= $*[1]x *[2]
x $ h
)*[1]
z +12*[2]
z
*
H [N!1,N ] = $*[N!1]x *[N ]
x $ h
)12*[N!1]
z + *[N ]z
*
*x
*z hh h = 0 h = 2
h = 1 h = 1h = 1
J < 0 h = 0
h = 0
h < hc h > hc
% = 10 % = 20 % = 30 % = 4% - 10
N %
N # % q % N $ 1 %#
i
% % $# #
$
h = 1
h = 1 % = 30
% = 30 %
% = 30h = 1
#$
%% = 20 % = 10
h = 1 % = 10
%
'b
'c 'a 'b 'a
'b 'c
h = 0.7 % = 30
h = 0.7 % = 30
h = 0 h = 1
h = 0.7 h = 1.4
h = 1.4 % = 30
% = 30 h = 0.7 h = 1.5h = 0.9
h = 0.7
h = 1h = 1
h = 1h = 1 h = 0
h = 0 |#*x"| = 1h = 0
|#*x"|
|#*x"| % = 20
% % = 10
h = 1%
hh = 1
h h = 1 h = 0.9 h = 0.8 |#*x"||#*x"| = 1
h = 1h = 1 h = 1
#*x" = 0
% = 30 h = 0
h = 0 h = 0|#*x"| = 1 h = 0.5
h = 0.7
Mx(h) =1N
N!
i=1
#"|*[i]x |""
#"|""
h ) 1
n = 50 h > 1#*x" h > 1
#*x"
h < hc h > hc
h = 1 h#*x"
h < 1 #*x" h > 1#*x"
h > 1h < 1
h < 1
G G|!0" =ei%|!0" [H,G] = 0G|!0" &= ei%|!0"G|!0(h)" = |!0(h)" h > 1 G =
&i *[i]
z
#*x" &= 0G G
G
hh h > hc
h hh h > hc h < hc
h h
% = 20
h = 1 h = 1.1% %
#*x"h < 1
h = 1h > 1
h = 0J J J
% = 10 h = 1
hh
h = 1 h = 1h = 1 h = 0.9 h = 1
h % = 30
h = 0.9 h = 1
h
% = 30 h h = 0 h = 2
h h = 1h = 0.9
h = 1.9 h = 2
h = 0 h = 2 h = 1 h = 0.7
ht($'1) t($'2)
t($'1) hh = 1.1 h = 2 t($'1) h = 1
t($'1)t($'2)
t($'2) h ) 1h ) 1
h ) 1
h h = 1
% = 30h = 0 h = 0.9 h = 1.1 h = 2
h ) 1 h = 1
h = 1
h < 1 h ) 1
h ) 1n = 1 n = N
n = 1 n = Nh
h - 1h > 1
h > 1h
% = 30 h h = 0h = 2
h
h % = 30
% = 30h = 0 h = 0.9 h = 1.1 h = 2
h
h h
SE1
h E1S
h
% = 30
h = 1SE1
(h)
E1S (h)
% = 30
hh = 0
E1S h
S E1
h = 1
% = 30
h = 0 h = 0.9 h = 1.1 h = 2
E1S h = 1
SE1* 2
SE1
= 2 N = 100SE1
E1S
%
E1S h
h = 1E1S
|1" =
/
0100
1
2 |2" =
/
0010
1
2 |3" =
/
0001
1
2
H =!
'i,j(
J [i,j]"M [i]
x M [j]2x + M [i]2
x M [j]x
#+ h
!
i
K [i]M [i]z
Mx Mz
Mz =
/
02 0 00 $1 00 0 $1
1
2 Mx =
/
00 1 00 0 11 0 0
1
2 M2x = (Mx)2 =
/
00 0 11 0 00 1 0
1
2
#*x"
Mx + M2x =
/
00 1 11 0 11 1 0
1
2
Mz *z
*x12
12 #Mx+M2
x"
J = $1J = $1
H = $!
i
"M [i]
x M [i+1]2x + M [i]2
x M [i+1]x
#$ h
2
!
i
"M [i]
z + M [i+1]z
#
H [i,i+1] = $"M [i]
x M [i+1]2x + M [i]2
x M [i+1]x
#$ h
2
"M [i]
z + M [i+1]z
#
H [1,2] = $"M [1]
x M [2]2x + M [1]2
x M [2]x
#$ h
2
"2M [1]
z + M [2]z
#
H [N!1,N ] = $"M [N!1]
x M [N ]2x + M [N!1]2
x M [N ]x
#$ h
2
"M [N!1]
z + 2M [N ]z
#
J = $1 h = 1
h h = 1
h = 0.5 h = 0.7 % = 25
h = 0.9 h = 1.1 % = 25
h = 0.5 h = 0.7h = 1 h = 1.1
h = 0.5h = 0.5 h = 0.7
h = 0.7h = 0.7
h = 0.5 h = 0.9h = 1.1
h ) 1h = 1 h = 1
h = 0.5 h = 0.9h = 0.5 h = 0.9
h ) 1,h > 1
12 #Mx + M2
x"
12 #Mx + M2
x" % = 20
h h = 0 h = 2h = 1 h = 2
h = 1
h h = 1h < 1
|1|h h = 1
% = 25
h = 1 h = 1.1
h > 1 h ) 1
% = 25
h = 1h = 0
h = 1h > 1
h
% = 25
h = 1 h = 1.1
h = 0.9 h > 1h > 1
h = 1h ) 2
% = 25 h = 0 h = 0.9 h = 1 h = 2
% = 25
% = 25
% = 25 h = 0 h = 0.9 h = 1 h = 2
S E1
SE1
E1S % = 25
E1S = 2
%
% = 30
h = 1 SE1
= 2
h = 1
% = 25 h = 0 h = 0.9 h = 1 h = 2
h < 1h = 1
h > 1h > 1
i
' = 0.01
h = 2h = 1.8
h > 1 |#*x"|
h = 1.8h = 1.9 % = 20
h = 2
10!8
h = 2
h = 1.8 h = 1.9 % = 20
h = 2 h = 1.8 h = 1.9h = hg h > hg
h < hg
h ( hg
%%
h = 2 h = 1.8 h = 1.9 % = 20
h = 1.8h = 1.9 % = 25
h = 1.8 h = 1.9 % = 25
%%
h
A1
A†AXX = Y X =
1Y
#[1] #1 i1 #[N ] #N!1 iN #[n]
in #n #n!1,#[1] #[N ]
# #[1] #[N ]
#$# &q%n!1 $[n] #
O(x) x x
' = ite!#H ' = t
e!#H
O('2)' O('2)
'
J J