SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston

11

SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera

RQRQ II WW Nov 07 Nov 07

Fabio AnselmiVenetian Inst. Mol. Med. Padova

Kurt JacobsU. Mass, Boston

Graham White Howard WisemanJoan VGriffith Uni.

arXive:quant-ph/0501121v2

Quantum Reference FramesQuantum Reference Framessuperselection rules, reference ancilla & superselection rules, reference ancilla &

entanglemententanglement

22



W

GGA

)(loGGW

GGE

• Superselection Rules (SSRs)– restricted operations– general symmetry groups

• Reference & Asymmetry– asymmetry: ability to act as a reference

• Work - a measure of purity

• Entanglement - limited by SSR

• Trade off between resources• Etcetera…

S

g

h

OverviewOverview

33RQRQ II WW Nov 07 Nov 07

SSRs SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera

Wick, Wightman & Wigner, Phys. Rev. 80, 101 (1952).

“We shall say that a superselection rule operates between subspaces …

• if a selection rule operates between them… and if, …

• there are no measurable quantities with finite matrix elements between their state vectors.”

n

1n

2n

1n

1n

ie

Superselection Rules (SSRs)Superselection Rules (SSRs)Selection rules forbid transitions of a given kind – m = 2 not allowed for optical dipole transitions etc but not transitions of any kind – e.g. electron collisions etc.



n

1n

2n

1n

1n

ie

10 ie

impose the rule: physical operations conserve local particle number

then coherence between subspaces of different particle number are nondetectable

an imposed superselection rule.

E.g. optics: the phase in

is unobservable ….

Example: local conservation of particle number

Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967).A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004).S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)



impose the rule: physical operations conserve local particle number

then coherence between subspaces of different particle number are nondetectable

an imposed superselection rule.

E.g. optics: the phase in

is unobservable …. except relativeto a local oscillator (a reference phase)

n

1n

2n

1n

1n

ie

Y.Aharonov and L.Susskind, Phys. Rev. 155, 1428 (1967).A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004).S.D. Bartlett, T. Rudolph, R.W. Spekkens, Rev. Mod. Phys. 79, 555 (2007)

10 ie

Example: local conservation of particle number

reference

Pegg… PRL 81 1604 (1998)quantum scissors, phase shift



Consider set of unitary operators whose effect is not physically detectable: G = {T1, T2, T3, …}

Non-detectable operations form a group

GTGT ii 1 then , if• the effect of a product of two such

operators is also non-detectable, thus

GTTGTGT jiji then , and if• clearly the identity operator is in G

Thus G = {T1, T2, T3, … }

is a group which expresses the symmetry of the system

• if effect of Ti is not detectable then

effect of time-reversed operator Ti1

is also not detectable, i.e. ie

n 1n2n

3n

ie

NieTˆˆ



GT

ggGg

TTG

1ˆ][

1ˆ G

10

1100

222

ˆˆ

21

dee NiNiUG

20 : )()1( NieTU

Accessible state• the effective state given the undetectable coherences

S

no referenceG=SO(2)

S

“crisp“

Bartlett and Wiseman, PRL 91, 097903 (2003).

Ex 2: optical phase shifts are non-detectable (without a reference)

reduced purity

“The Twirl”

Ex 1: rotations are nondetectable without a spatial reference



S

“crisp“

Accessible state• the effective state given the undetectable coherences

GT

ggGg

TTG

1ˆ][

1ˆ G

reduced purity

“The Twirl”

Ex 1: rotations are nondetectable without a spatial reference

10

20 : )()1( NieTU

Ex 2: optical phase shifts are non-detectable (without a reference)

Bartlett and Wiseman, PRL 91, 097903 (2003).

noreference

equally likely to be any value of

UG

effective state has random phase

1100

222

ˆˆ

21

dee NiNiUG


SSRs Work (purity) Work (purity) Ref & Asymmetry Entangle Trade off Etcetera

T

Z

eP

kTE

E

/

0 1

Extracting work (purity measure)Extracting work (purity measure)



2log

0

1

Tk

dEPW

B

kTE

kTE

e

eP

/

/

1 1

dEPdW 1dE

)]ˆ([log)ˆ( SDTkW B

subtract initial entropy

T

Z

eP

kTE

E

/

0 1

1


von Neumannentropy

dim.



T

Z

eP

kTE

E

/

)ˆ(log)ˆ( SDW

1

under G-SSR the extractable work is

])ˆ[(log)ˆ( GSDWG

0 1


2log

0

1

Tk

dEPW

B

kTE

kTE

e

eP

/

/

1 1

dEPdW 1dE

)]ˆ([log)ˆ( SDTkW B

subtract initial entropy

von Neumannentropy

dim.


SSRs Work (purity) Ref & Ref & Asymmetry Asymmetry Entangle Trade off Etcetera

Reference ancilla (frame)Reference ancilla (frame)• the SSR imposes a symmetry which

reduces the purity• we need to break the symmetry

& preserve the coherence• this requires an asymmetric ancilla

• define symmetric state as one for which

ˆˆ G• define asymmetric state as one for which

ˆˆ G

GT

ggGg

TTG

1ˆ][

1ˆ G

The Twirl

• use loss of purity to measure asymmetry

ˆˆ)ˆ( SSAG G

von Neumann entropyAsymmetry



iff is symmetric:

0)ˆ( GA

Asymmetry (reference ability)Asymmetry (reference ability)

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

GggTgTgTgT )(]ˆ[)()](ˆ)([ 11 QQSynergy of is given by)ˆ(GW

)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGG WWWW

ˆˆ)ˆ( SSAG G

i)

ii)

iii)

iv)

• any ancilla with asymmetry can act as a reference to (partially) break the SSR

Properties of Asymmetry:

R

reference ancillasystem

S



SR

gGfG

acting separately

acting as single system

Upper bound

asymmetry is a resource

advantage of acting as a composite system

Synergy Synergy

)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( SGRGSRGSRG WWWW

)ˆ(

)}ˆ(),ˆ(min{)ˆ,ˆ,(

RG

SGRGSRG

A

AAW

S

gG

R



Example: local conservation of particles [U(1)]

N

n

neR inN

01

1)(

10

)(110)(1

11

1

1

NNOneneR

N

n

iin

• decoherent free subspaces (superselection sectors)• coherence is preserved

}{ 20,ˆ:ˆ)1(ˆ

NieTTU

)ˆˆ( 21 NNie

system:

ref. ancilla:

R S

combined (ref. ancilla + system):

Pegg & Barnett (1989).

110022 UG

n

U nnR N 11)(G

invariant to

group:



combined: )(....conj herm....110)( 1

11

1

1

NN OnenR

N

n

iU G

state AG

2

2

22

2

2loglog

)1(log2 N

10

N

n

neR inN

01

1)(

)(R

2

2

22

2

2loglog1)(,; 1

NRAG

2

2

22

2

2

2 loglog1)1(log N

N)( R

Synergy of AG: the reduction in entropy due to combined action

R S

S

R



)ˆ(log)ˆ( GG SDW G

)ˆ(log)ˆ( SDW

)ˆ()ˆ()ˆ( SSA GG G )ˆ()ˆ()ˆ( GG AWW

GA

)ˆ(W

GW

asymmetricsymmetric


SSRs Work (purity) Ref & Asymmetry EntangleEntangle Trade off Etcetera

)ˆ()ˆ( GGGG WW G

Gg Gh

hghg TTTTGGG

11ˆ][

1ˆ

2G

GGG

Bipartite systems & EnganglementBipartite systems & Enganglement

Local action of the group: local G-SSR

g

h



iff is locally symmetric:

0)ˆ( GGA

Local asymmetryLocal asymmetry

0)ˆ( GGA ˆˆˆ 11 GG GG

)ˆ(GGA does not increase for locally G-SSR operations Q

Synergy of is given by)ˆ(GGW

)]ˆ()ˆ([)ˆˆ()ˆ,ˆ,( 212121 GGGGGGGG WWWW })ˆ(),ˆ(min{ 21 GGGG AA

ˆˆ)ˆ( SSA GGGG G

i)

ii)

iii)

iv)

)ˆ()ˆ()ˆ( GGGG AWW

GGA

)ˆ(W

can act as local & sharedreference

GGW

g

hg

h



GGGGGG EEE

Super-additivity:

01001 GGE

Accessible entanglementAccessible entanglement

N

nnnEpE GG

0

n

nnnn

pp

ˆ ;ˆ

projection onto n particles at A

Examples:

A B

A B

1,1

2,0

0,2

+

N particles shared between A and BWiseman and Vaccaro, PRL 91, 097902 (2003).



Extracting local workExtracting local work Oppenheim et al PRL 89, 180402 (2002))ˆ(L W

)ˆ(L W



jijic ˆˆˆ , Q

classically-correlated state with min entropy

Q

LOC

C

local extraction of work

)ˆ()ˆ(L QWW

equivalent method

transfer using a classical channel




Q

)ˆ()ˆ()ˆ(L EWW

pure state

dephase in Schmidt basis

equivalent method for pure states

jijic ˆˆˆ , Q

LOC

C






Q

)ˆ()ˆ()ˆ(L EWW

pure state

equivalent method for pure states

jijic ˆˆˆ , Q

LOC

C

)ˆ()ˆ()ˆ( L EWW



dephase in Schmidt basis




ˆˆ GG

Pure, globally symmetric states

Q

LOC

C


)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -


dephase in Schmidt basis for each charge

g

h

Extracting local work under local SSRExtracting local work under local SSR

jijic ˆˆˆ , Q

GGG




ˆˆ GG

Pure, globally symmetric states

Q

LOC

C


)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -


dephase in Schmidt basis for each charge

g

h

Extracting local work under local SSRExtracting local work under local SSR

jijic ˆˆˆ , Q

GGG

)ˆ()ˆ()ˆ()ˆ( L GGGGGG AEWW


SSRs Work (purity) Ref & Asymmetry Entangle Trade off Trade off Etcetera

mechanical worklogical work

)ˆ()ˆ()ˆ()ˆ( L GGGGGG AEWW

W

)(loGGW

GGE

symmetry

GGA

asymmetry

reference

TradeoffTradeoff



0110

01100110

1 0 1 2 L

GGGGGG AEWW

23 2

1 2 4 L

GGGGGG AEWW

Recall examples for U(1)

A B

A B

S R

R

ability to act as shared reference

super-additivity of accessible entanglement=GGA



0110

01100110

1 0 1 2 L

GGGGGG AEWW

23 2

1 2 4 L

GGGGGG AEWW

Recall examples for U(1)

A B

A B

S R

R

ability to act as shared reference

super-additivity of accessible entanglement=GGA



GGGGGG AEWW L

Optimum shared reference states?

make zero make maximum

NR

0110

RANRE GGGG

047.1)(log~ 221 NRA GG

N

N

nnnR

4

1 BA4

0 RE GG

NRA GG 2

NRW 2)(

N4Dim

NRW 2)( NRW GG L 0L RW GG



Hierarchy of restrictions-resourcesHierarchy of restrictions-resources

GG AWW

GGGGG AWW

GGGGGG EWW L

EWW L

LOCC

G

GG

LOCC, GG

WW -

for globally-symmetric states

g

h

g

h


SSRs Work (purity) Ref & Asymmetry Entangle Trade off EtceteraEtcetera

Etcetera…Etcetera…

)(A

),;(AQ

G

G R

Complete reference frame when

then system is completely “shielded” from G

Normalised synergy of asymmetry:

• Figure of merit - Quality

M

m

mM 01

1system state

N

n

in neN

R01

1

reference:

1Qquality

(M=30)

N


SSRs Work (purity) Ref & Asymmetry Entangle Trade off EtceteraEtcetera

repeated use of a reference ancilla with independent systems reduces its reference ability…

• Consumption of reference ability

• Complementarity – generalisation

S1

RS2

R’

The symmetry-asymmetry dichotomy is fundamental to a system. Arises from its “geometry”.

It may help understanding of the fundamental particle-wave duality in terms of a symmetry-asymmetry dichotomy.

3434



• reference ancilla

• accessible entanglement and work

• tradeoff of resources: reference ability

versus mechanical work

versus logical work

R

reference f rame

asymmetric system

S

1,1

2,0

0,2

+

W

GGA

)(loGGW

GGE

triality

SummarySummary

3535



Documents

SSRs Work (purity) Ref & Asymmetry Entangle Trade off Etcetera RQ I W Nov 07 1 Fabio Anselmi Venetian Inst. Mol. Med. Padova Kurt Jacobs U. Mass, Boston