Fabio Anselmi

1U HMilano 05

Fabio AnselmiFabio Anselmi

University of Hertfordshire Hatfield, UK

John VaccaroJohn VaccaroHoward WisemanHoward WisemanKurt JacobsKurt Jacobs

Griffith University Brisbane, Australia

quant-ph/0501121

2U HMilano 05

W

GGA

)(loGGW

GGE

This talkThis talk•Superselection Rules (SSRs)

– conservation of local particle number– general symmetry groups

•Reference frames– asymmetry: ability to act as a reference

•Extracting mechanical work•Bipartite systems under SSR•Accessible entanglement•Extracting local work•Hierarchy of restrictions/resources• Complementarity

S

g

h

3U HMilano 05

Superselection RulesSuperselection Rules

all physical operations conserve particle number

cannot observe coherence between subspaces of different particle number

effectively a superselection rule.

n

1n

2n

1n

1n

ie

Conservation of particle number

4U HMilano 05

Symmetry group: }{gG unitary representation: GggT )(

Operationally accessible states

N

dee NiNi

ˆ

2ˆ)ˆ(

2

ˆˆ

G

Expressed as symmetry group U(1)

General symmetry group

Operationally accessible states

Gg

gTgTG

)(ˆ)(][

1ˆ 1G

projective measurement of

Ssystem under SO(2)

} { ˆ NieG

reduced coherence

S

“crisp“

5U HMilano 05

Reference systems (frames)Reference systems (frames)

Measure of asymmetry ˆˆ)ˆ( SSAG G

Reference frames break the symmetry preserve coherence of system

R

reference system

asymmetric system

S

von Neumann entropy induced by G

Any system with asymmetry can act as a reference system (frame) for G

A symmetric system cannot act as a reference.

Gg

gTgTG

)(ˆ)(][

1ˆ 1G

6U HMilano 05

iff is symmetric:

0)ˆ( GA

Properties of asymmetry

0)ˆ( GA ˆˆ G)ˆ(GA does not increase for G-SSR operations Q

GggTgTgTgT )(]ˆ[)()](ˆ)([ 11 QQ

Synergy of is given by)ˆ(GW)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGG WWWW

})ˆ(),ˆ(min{ 21 GG AA

ˆˆ)ˆ( SSAG G

Gg

gTgTG

)(ˆ)(][

1ˆ 1Gi)

ii)

iii)

iv)

7U HMilano 05

Example of Abelian case (particle number)

N

n

inNR

ne0

11)(

10 S

N

n

iinNRS

nene1

11 110)(

-invariant to -coherence is preserved

22

222

2 loglog11 N

Niegˆ

)(

Nie ˆ

22

222

2 loglog)( SGA

system:

asymmetry

reference:

)1(log)( 2 NA RG asymmetry

reduction in asymmetry (synergy):

R S

combined:

Pegg & Barnett (1989).

8U HMilano 05

Example of Abelian case (particle number)

N

n

inNR

ne0

11)(

10 S

N

n

iinNRS

nene1

11 110)(

-invariant to -coherence is preserved

22

222

2 loglog11 N

Niegˆ

)(

Nie ˆ

22

222

2 loglog)( SGA

system:

asymmetry

reference:

)1(log)( 2 NA RG asymmetry

reduction in asymmetry (synergy):

R S

combined:

Pegg & Barnett (1989).

9U HMilano 05

Extracting mechanical workExtracting mechanical work

)]ˆ([log)ˆ( SDTkW B

)ˆ(log)ˆ( SDW

1

under SSR)ˆ(log)ˆ( GSDWG

10U HMilano 05

)ˆ(log)ˆ( GSDWG

)ˆ(log)ˆ( SDW

)ˆ()ˆ()ˆ( SSAG G)ˆ()ˆ()ˆ( GG AWW

GA

)ˆ(W

resource?

GW

11U HMilano 05

acting separately

acting as single system

Upper bound

asymmetry is a resource

S R

gG fG

advantage of acting as a composite system

Synergy

R S

gG

)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( SGRGSRGSRG WWWW

)ˆ( )}ˆ(),ˆ(min{)ˆ,ˆ,(

RG

SGRGSRG

AAAW

12U HMilano 05

Bipartite systems under SSRBipartite systems under SSR

)ˆ]([)ˆ( BAGG WW GG

Ghg

BA hTgThTgTG ,

11 )]()([ˆ)]()([][

1ˆ][ GG

Local action of the group: local G-SSR

][ BA GG g

h

13U HMilano 05

iff is locally symmetric:

0)ˆ( GGA

Local asymmetry

0)ˆ( GGA ˆˆ]1[ˆ]1[ BABA GG

)ˆ(GGA does not increase for locally G-SSR operations QSynergy of is given by)ˆ(GGW

)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGGGGGG WWWW

})ˆ(),ˆ(min{ 21 GGGG AA

ˆˆ][)ˆ( SSA BAGG GG

i)

ii)

iii)

iv)

)ˆ()ˆ()ˆ( GGGG AWW

)ˆ(GGW

GGA

)ˆ(W

can act as local & sharedreference

GGW

g

h

14U HMilano 05

fixed total number of N particles

N

nnNnnf

0,

+ +

N

n N-n

Accessible entanglement under SSRAccessible entanglement under SSR

f n

nNnE ,

A B

nNn

N

nnGG EfE

,

0

2

Wiseman and Vaccaro, PRL 91, 097902 (2003).

17

6 11

EE GG

15U HMilano 05

Super-additivity:

01001 GGE

-releases “latent” entanglement-a kind of distillation

cross terms represent 1 particle at each site – no particle entropy

A B

A B

1,1

2,0

0,2

+

Examples

GGGGGG EEE

16U HMilano 05

Extracting Extracting locallocal work work Oppenheim et al PRL 89, 180402 (2002)

)ˆ()( loW

)ˆ()( loW

17U HMilano 05

)()(, ˆˆˆ B

jA

ijic QQ

LOC

C

local extraction of work)ˆ()ˆ()( QWW lo

classically-correlated state with min entropy

equivalent method

classical channel

18U HMilano 05

classical channel

)()(, ˆˆˆ B

jA

ijic QQ

LOC

C

local extraction of work)ˆ()ˆ()ˆ()( EWW lo

classically-correlated state with min entropy

pure state

dephase in Schmidt basis

equivalent method for pure states

)ˆ()ˆ()ˆ( )( EWW lo

19U HMilano 05

Extracting Extracting locallocal work under local SSR work under local SSR )ˆ()( loGGW

)ˆ()( loGGW

][ BA GG g

h

20U HMilano 05

classical channel

ˆˆ GFor pure, globally-symmetric states

)()(, ˆˆˆ B

jA

ijic QQ

LOC

C

local extraction of work)ˆ()ˆ()ˆ()ˆ()( GGGG

loGG AEWW

classically-correlated state with min entropy

dephase in Schmidt basis

][ BA GG g

h

)ˆ()ˆ()ˆ()ˆ( )( GGGGlo

GG AEWW

21U HMilano 05

work

local symmetry

local asymmetry

)ˆ()ˆ()ˆ()ˆ( )( GGGGlo

GG AEWW

W

GGlo

GG EW )( GGA

can act as a local reference

22U HMilano 05

mechanical

logical

local asymmetry

can act as a local reference

)ˆ()ˆ()ˆ()ˆ( )( GGGGlo

GG AEWW

W

GGA

)(loGGW

GGE

local symmetry

23U HMilano 05

A B

A B

0110

S R

R

ability to act as shared reference

super-additivity of accessible entanglement=

01100110

1 0 1 2

)(

GGGG

loGG AEWW

1 1 2

)(

EWW lo

23

21

)(

2 4

GGGGlo

GG AEWW

2 2 4

)(

EWW lo

GGA

Recall examples for U(1)

24U HMilano 05

A B

A B

0110

S R

R

ability to act as shared reference

super-additivity of accessible entanglement=

01100110

1 0 1 2

)(

GGGG

loGG AEWW

1 1 2

)(

EWW lo

23

21

)(

2 4

GGGGlo

GG AEWW

2 2 4

)(

EWW lo

GGA

Recall examples for U(1)LOCC LOCC+LocalG

25U HMilano 05

Symmetry group: }{gG Unitary representation: GggT )(

Locally accessible states

Details of general group caseDetails of general group case

Gg

gTgTG

)(ˆ)(][

1)ˆ( 1G Globally-symmetric states

ˆˆ G

Ghg

BA hTgThTgTG ,

11 )]()([ˆ)]()([][

1ˆ][ GG

26U HMilano 05

Elemental globally symmetric state

,

,, ,

, mmmm

mmd

)()()( , gTgTgT

where and are conjugate w.r.t. :

,,

,, )()( mmmm ggT

i.e.:

General globally symmetric state

ji

jimm jmim

DC

,

,,, ,,,,

multiplicity “flavour”

1 dim irrep

“charge”

“colour”

27U HMilano 05

PPdEPE mmGG 2, log)(

,

,, ,

, mmmm

mmd

entanglement due to multiplicity indices mm ,

reduced by entropy associated with

DPPPA GG

1log2log 22

ji

jimm jmim

DC

,

,,, ,,,,

entropy associated with mixing of i & j indices

entropy associated with charge fluctuations

28U HMilano 05

W

ability to act as a reference frame

RFW

asymmetry

Complementarity of resourcesComplementarity of resources

symmetry

asymsym

)ˆ()ˆ()ˆ()ˆ( )( GGGGlo

GG AEWW

)(loGGW

GGE

M L

GGA

ability to perform work

29U HMilano 05

Hierarchy of Hierarchy of restrictions-resourcesrestrictions-resources

GG AWW

GGGG AWW

GGGGlo

GG AEWW )(

EWW lo )(LOCC

G

GG

LOCC, GG

WW -

for globally-symmetric states

g

h

g

h

30U HMilano 05

• reference frames

• accessible entanglement and work

• complementarity of resources: reference ability

versus mechanical work

versus logical work

SummarySummary

R

reference f rame

asymmetric system

S

1,1

2,0

0,2

+

W

GGA

)(loGGW

GGE

triality

31U HMilano 05