ssrs work (purity) ref & asymmetry entangle trade off etcetera rq i w nov 07 1 fabio anselmi...

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

)(loGGW

• 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…

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.”

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.

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)

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)

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

NieTˆˆ

dee NiNiUG

20 : )()1( NieTU

Accessible state• the effective state given the undetectable coherences

no referenceG=SO(2)

“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

“crisp“

Accessible state• the effective state given the undetectable coherences

reduced purity

“The Twirl”

Ex 1: rotations are nondetectable without a spatial reference

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

effective state has random phase

dee NiNiUG

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

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

dEPdW 1dE

)]ˆ([log)ˆ( SDTkW B

subtract initial entropy

von Neumannentropy

)ˆ(log)ˆ( SDW

under G-SSR the extractable work is

])ˆ[(log)ˆ( GSDWG

dEPdW 1dE

)]ˆ([log)ˆ( SDTkW B

subtract initial entropy

von Neumannentropy

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

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

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

Properties of Asymmetry:

reference ancillasystem

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(min{)ˆ,ˆ,(

SGRGSRG

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

neR inN

)(110)(1

NNOneneR

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

}{ 20,ˆ:ˆ)1(ˆ

NieTTU

)ˆˆ( 21 NNie

system:

ref. ancilla:

combined (ref. ancilla + system):

Pegg & Barnett (1989).

110022 UG

U nnR N 11)(G

invariant to

group:

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

NN OnenR

state AG

2loglog

)1(log2 N

neR inN

2loglog1)(,; 1

2 loglog1)1(log N

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

)ˆ(log)ˆ( GG SDW G

)ˆ(log)ˆ( SDW

)ˆ()ˆ()ˆ( SSA GG G )ˆ()ˆ()ˆ( GG AWW

asymmetricsymmetric

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

)ˆ()ˆ( GGGG WW G

hghg TTTTGGG

11ˆ][

Bipartite systems & EnganglementBipartite systems & Enganglement

Local action of the group: local G-SSR

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

)ˆ()ˆ()ˆ( GGGG AWW

can act as local & sharedreference

GGGGGG EEE

Super-additivity:

01001 GGE

Accessible entanglementAccessible entanglement

nnnEpE GG

ˆ ;ˆ

projection onto n particles at A

Examples:

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

local extraction of work

)ˆ()ˆ(L QWW

equivalent method

transfer using a classical channel

)ˆ()ˆ()ˆ(L EWW

pure state

dephase in Schmidt basis

equivalent method for pure states

jijic ˆˆˆ , Q

)ˆ()ˆ()ˆ(L EWW

pure state

equivalent method for pure states

jijic ˆˆˆ , Q

)ˆ()ˆ()ˆ( L EWW

dephase in Schmidt basis

ˆˆ GG

Pure, globally symmetric states

)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -

dephase in Schmidt basis for each charge

Extracting local work under local SSRExtracting local work under local SSR

jijic ˆˆˆ , Q

ˆˆ GG

Pure, globally symmetric states

)ˆ()ˆ()ˆ()ˆ(L GGGGGG AEWW -

dephase in Schmidt basis for each charge

Extracting local work under local SSRExtracting local work under local SSR

jijic ˆˆˆ , Q

)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

)(loGGW

symmetry

asymmetry

reference

TradeoffTradeoff

01100110

1 0 1 2 L

GGGGGG AEWW

1 2 4 L

GGGGGG AEWW

Recall examples for U(1)

ability to act as shared reference

super-additivity of accessible entanglement=GGA

01100110

1 0 1 2 L

GGGGGG AEWW

1 2 4 L

GGGGGG AEWW

Recall examples for U(1)

ability to act as shared reference

super-additivity of accessible entanglement=GGA

GGGGGG AEWW L

Optimum shared reference states?

make zero make maximum

RANRE GGGG

047.1)(log~ 221 NRA GG

0 RE GG

NRA GG 2

NRW 2)(

NRW 2)( NRW GG L 0L RW GG

Hierarchy of restrictions-resourcesHierarchy of restrictions-resources

GG AWW

GGGGG AWW

GGGGGG EWW L

LOCC, GG

for globally-symmetric states

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

Etcetera…Etcetera…

),;(AQ

Complete reference frame when

then system is completely “shielded” from G

Normalised synergy of asymmetry:

• Figure of merit - Quality

1system state

in neN

reference:

1Qquality

(M=30)

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

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.

• reference ancilla

• accessible entanglement and work

• tradeoff of resources: reference ability

versus mechanical work

versus logical work

reference f rame

asymmetric system

)(loGGW

triality

SummarySummary

ssrs work (purity) ref & asymmetry entangle trade off etcetera rq i w nov 07 1 fabio anselmi...

reference work

etcetera rq

effect of t i

detectable operations

local particle number

reference g

reference phase y

imposed superselection

Documents

anselmi catalogo 062016 2 -...

etcetera progress report (2009-dec-01)

etcetera columbia

sancti anselmi cantuariensis archiepiscopi opera omnia 3

joe cool uk jewelry timeless keepsakes etcetera

download the etcetera article for background - betsy woodman

etcetera: checkout workflow

anselmi cantuariensis proslogion

etcetera 16

why alliances entangle but seldom entrap states - griffith

eastfield etcetera aug. 31, 2015

anselmi italia

rfoe i (etcetera

gabriela m. anselmi -...

etcetera lifestyle magazine december 2013

quantum entangle and teleportation

sofie anselmi volunteering victoria...

etcetera 'i'c:i'skitr - repository.tcu.edu

a case study in quality improvement - etcetera

etcetera los angeles celebrity and editorial press images