A new measure of nonclassical distance

HoriaScutaru

arXiv:quant-ph/99080v1 31 Aug 1999DepartmentofTheoreticalPhysics,InstituteofAtomicPhysics,POBOXMG-6,R-76900Bucharest-Magurele,RomaniaE-mail:scutaru@theor1.theory.nipne.ro

PACSnumbers:03.65.Bz;03.65.Fd;42.50.Dv.;.70.+c

Abstract.Inthepresentpaperweshallproposeanewmeasureofthenonclassicaldistance[1].Theproposedmodificationisbasedonthefollowingconsiderations.Ifρ1andρ2aredensityoperators,andF(ρ1,ρ2)isthecorrespondingfidelity,thenfromtheinequalities[2]󰀈2

2(1−

itisevidentthatthequantity

φ(ρ)=supF(ρcl,ρ)

ρcl

canbeusedinthesameextentasameasureofthedistanceofthestateρtothesetofclassicalstatesρclastheHillerymeasureδ(ρ)=supρcl||ρ1−ρ2||1[1].φ(ρcl)=1foranyclassicalstateandφ(Γ(ρ))≥φ(ρ)ifthemapΓistheGaussiannoisemap[3,4].Shorttitle:AnewmeasureofnonclassicaldistanceFebruary1,2008

21.

Introduction

In[1],Hillerygaveadefinitionofthenonclassicaldistanceofradiationintermsofthetracenormas

δ(ρ)=inf||ρ−ρcl||,

ρcl

(1)

whereρisthedensitymatrixofthenonclassicalradiationandρclisthatofanarbitraryclassicalfield,while||A||1isthetracenormoftheoperatorA.TheHillerynoncassicaldistancehasthefollowingproperties:(i)δ(V(u)ρV(u)†)=δ(ρ);(ii)0≤δ(ρ)≤2;

(iii)δ(θρ1+(1−θ)ρ2)≤θδ(ρ1)+(1−θ)δ(ρ2).

wheretheunitaryoperatorsV(u)arethewellknownWeyloperatorsgivingaprojectiveunitaryrepresentationofthevectorgroupR2n(seethenextsection).Thismeasureseemstobequiteuniversal.Unfortunately,itisnoteasytousethismeasureinactualcalculations.Inpractice,onecanonlyprovidetheupperboundandlowerboundofthemeasure.

Itispossibletodefinethedistancebetweentwoquantumstatesdescribedbydensityoperatorsinmanyways[5,6,7,8].Ifthedistanceissmallthesetwodensityoperatorscanbeconsideredverysimilartoeachother.Ontheotherhandalargedistancemeansverydifferentdensityoperators.

Another,morephysicalpointofwiewisthatpresentedin[2].Accordingtothispointofview”theonlyphysicalmeansavailablewithwhichtodistinguishtwoquantumstatesisthatspecifiedbythegeneralnotionofquantummechanicalmeasurement”.Insteadofametricalpointofviewastatisticalpointofviewistakenintoaccount.Ameasurementbeingnecessarilyindeterministicandstatisticthemorephysicalmeasuresofdistancebetweentwoquantumstatesarethosewhicharebasedonthestatistical-hypothesistestingprocedures.

Inthepresentpaperweshallproposeanewmeasureofthenonclassicaldistance[1].Theproposedmodificationisbasedonthefollowingconsiderations.Ifρ1andρ2aredensityoperators,andF(ρ1,ρ2)isthecorrespondingfidelity,thenfromtheinequalities[2]

󰀈

22(1−(2)itisevidentthatthequantity

ρcl

φ(ρ)=supF(ρcl,ρ)

(3)

canbeusedinthesameextentasameasureofthedistanceofthestateρtotheset

ofclassicalstatesρclasHillery’smeasureδ(ρ).

Letρ1andρ2betwodensityoperatorswhichdescribetwomixedstates.ThetransitionprobabilityP(ρ1,ρ2)hastosatisfythefollowingnaturalaxioms:(i)P(ρ1,ρ2)≤1andP(ρ1,ρ2)=1ifandonlyifρ1=ρ2;(ii)P(ρ1,ρ2)=P(ρ2,ρ1);

3

(iii)Ifρ1isapurestate,ρ1=|ψ1><ψ1P(ρ1,ρ2)=<ψ1|ρ2|ψ1>;

|then

(iv)P(ρ1,ρ2)isinvariantunderunitarytransformationsonthestatespace;(v)P(ρ1|A,ρ2|A)≥P(ρ1,ρ2)foranycompletesubalgebraofobservablesA;(vi)P(ρ1⊗σ1,ρ2⊗σ2)=P(ρ1,ρ2)P(σ1,σ2).

(vii)P(µ1ρ1+µ2ρ2,σ)≥µ1P(ρ1,σ)+µ2P(ρ2,σ)when0≤µ1,µ2≤1,µ1+µ2=1.

Uhlmann’stransitionprobabilityformixedstates[9,10,11,12]

P(ρ1,ρ2)=󰀏Tr(√ρ1)

1/2

󰀓2(4)

satisfiesproperties1–7.ThefidelityisdefinedbyF(ρ1,ρ2)=P(ρ1,ρ2).Adetailed

analysisforthestructureofthetransitionprobabilitywashamperedbythefactorscontainingsquareroots.Duetotechnicaldifficultiesinthecomputationoffidelities,fewconcreteexamplesofanalyticcalculationsareknown.Thefirstresultsinaninfinite-dimensionalHilbertspacewererecentlyobtainedbyTwamley[13]forthefidelityoftwothermalsqueezedstatesandbyParaoanuandScutaru[14]forthecaseoftwodisplacedthermalstates.In[15]Scutaruhasdevelopedanothercalculationmethodwhichallowedgettingtheresultforthecaseoftwodisplacedthermalsqueezedstatesinacoordinate-independentform.AgeneralformulaforthefidelityofanytwomixedGaussianstates(i.e.multimodedisplacedthermalstates[16,17,18,19,20,21],fromwhichthepreviousresultscanbeobtainedasparticularcases,hasbeenobtainedrecently[22].

Anothermodificationisbasedonthefollowingconsiderations[23].Ifρ1andρ2aredensityoperators,thenitiseasytoestablish[23]theinequalities

2(1−Tr

√ρ12)≤||ρ1−ρ2||1≤2[1−(Tr√

ρ2)2]ρcl

√

ρ1

√

4

continuousfamilyofunitaryoperators{V(u),u∈E}onHwhichsatisfytheWeylrelations[19,21]:

V(u)V(v)=exp

i

5

(i)CF0(O)=TrO;

󰀏󰀓

†(ii)CFuV(v)OV(v)=CFu[Oexpiσ(v,u)];(iii)CFu(O1O2)=

1

2σ(v,u)dv;

(iv)CFSu(O)=CFu(U(S)OU(S)†).

Themultimodethermalsqueezedstatesaredefinedbythedensityoperatorsρ

whosecharacteristicfunctionsareGaussians[15,19,21]

󰀇

1

CFu(ρ)=exp−

4

󰀊

󰀄󰀆uTA2−(A2−iJ)(A1+A2)−1(A2+iJ)u.

2

󰀑−1

6

Whenρ1=ρ2wehave

CFu(ρ)=(detA)

2

−

1

4

u

T

󰀍

A−JA−1J

>A.HenceforanyGaussian

statethecorrelationmatrixAmustsatisfythefollovingrestriction[21]

2

A≤−JA−1J.

3.

TheclassicalGaussianstates

(19)

AmultimodesqueezedthermalstateisaclassicalstatewhenithasaP-representation.TheP-distributiononthephasespacewhichdescribessuchastateisthesymplecticFouriertransformofthenormalorderedcharacteristicfunction

NCFu(ρ)=exp{−

1

det(A−I))−1exp{

1

detGexp{−vTGv}

(23)

(Gisapositivedefinite2n×2nmatrix)andV(u)aretheWeyloperators.Itiseasy

toseethatinthecasewhenρisaquasifreestatewiththecharacteristicfunction

CFu(ρ)=exp{−

uTAu

7

thecharacteristicfunctionofthestateΓ(ρ)isgivenby

CFu(Γ(ρ))=exp{−

uT(A−JG−1J)u

TrA=d(m2+

1

m

󰀑

.Thenitisevidentthat

m

)=2

TrA

detA

(27)

Evidently,tothecorrelationmatrixΓ(A)therecorrespondsanewsqueezingparameterΓ(m)givenbytheanalogousequationwithAreplacedwithΓ(A).Fromtheformula(26)itfollowsthat

−1T−1T

detΓ(A)=det(g−1I+d(OG)SSOG)

(28)

and

TrΓ(A)=TrG−1+TrA

Fromtheformula(28)weobtainthat

detΓ(A)=detG

−1

(29)

+

TrG−1TrA

Γ(m)2

=

TrG−1+TrA

+detA2

(31)

Withtheaboveparametrizationwehave

󰀂󰀕

󰀕g+dm22

Γ(m)=󰀒d

g+

(1

g+

d

8and

g

−1

=

d[(m2+

1

(m2−

1

d

)2

m2−2

1

Γ(d)

1+(

m2−

1

2Γ(d)

m=Γ(m),¸siΓ(d=1)=2¯n+1andm=exp(r).

Thenthefirsttwoequations(A6)fromtheAppendixAofthepaper[4]become:

m2−

and

1

4(m2+

1

4

(Γ(m)2+

1

1

Γ(m)2

)

(36)

2¯n,

′′

1+gm2

m2

−

󰀎

m2

m2)

(1+gm2)(1+

󰀂

󰀕󰀕g++󰀒

1

g

g

+m)+((1

21

m2)

11

m2

1

g

+

1

g

+

d

(40)

4

whereA=2ΣandΣisthecorrelationmatrixofthestate.Wehaveforanytwodensityoperatorsρ1andρ2:

󰀉

ivTJu−nuCFu(ρ1ρ2)=(2π)CFv+−v(ρ2)exp{2

uTAu}

9

whereJ=

󰀍0−I

󰀑

√I

.Thisbecomesintheparticularcasewhenρ1=ρ2=0

+v(2

√

−v(2

√

2

}dv(42)

Ifwesupposethat

√

CFu(

thenitfollowsthat

CFu(ρ)=K2(detφ(A))−

1

4

uTφ(A)u}

(43)

8

uT(φ(A)−Jφ(A)−1J)u}

(44)

1

InorderthatthisequalitybevalidforallvaluesofuwemusthaveK=(detφ(A))

I+(JA)−2)

Indeed,wehave

A=STDS

(47)

(48)

withSTJS=J(i.e.Sisasymplecticmatrix)andD≥Iisadiagonalmatrix.ItiswellknownthatifSisasymplecticmatrixthenSTandS−1arealsosymplecticmatrices.Fromthisfactweobtainthat

(JA)−2=−S−1D−2S

Alsowehave

Jφ(A)JA=−S−1D(D+

󰀈

D2−I)2S

(51)(49)

√

Because(D+D2−I)thedesiredresultfollows.Thefollowingformofthefunctionφ(A)isalsouseful:

󰀈

−1

φ(A)=−S(D+

ρ1√

10

itfollowsthat

Tr√

ρ2=(2π)−n

󰀉

√CFu(

ρ2)du()

Itiseasytocomputethisintegral.Theresultis

󰀂󰀕

󰀕detφ(A1)detφ(A2)√

ρ2=󰀒Tr

)2

(55)

󰀍

󰀑2

M0T

Whenρ2isasqueezedstate(i.e.whenA2=O2O2)itisplaussible

0M−2

tosupposethatthemaximumvalueofthisquantityisobtainedforD1=I,andO1=O2.Inthiscase

χ(ρ)=

1

2

)

(56)

(wherewehavedenotedthedensitymatrixρ2byρ).Itisclearfromthisformulathatthenonclassicityofthestateρisentirelyduetothesqueezing.WhenM=Iwehavenosqueezingandχ(ρ)=1.5.1.

Theone-modecase

Intheone-modecasethegeneralclassicalstateisgivenbyacharacteristic󰀍most2

d1m10T

functionwithA1=O1d1

0

2

d2k−1)

fork=1,2.Ifρ2isalsoaclassicalstatethen

√

ρ2=1supTr

ρ1

(57)

󰀍d2m22

0

0

isobtainedforρ1=ρ2.

Whenρ2isasqueezedstate(i.e.whenA2=

ρclass

√

󰀋

TO2

d2

(58)

φ(d1)φ(d2)

+F(∆θ,m1,m2)

wherewehavedenotedwithFthefollowingfunction

F(∆θ,m1,m2)=

2

2+sin(∆θ)2(m21m2+

1

m2

)2+(

m2

φ(d1)φ(d2)

+F(∆θ,m1,m2)(60)

11

ItisevidentthattheminimumvalueofHisattainedford1=d2andforthosevaluesof∆θ,m1andm2whichminmizethefunctionF.TheminimumvalueofthefunctionFisequalwith2andisattainedeitherfor∆θ=0,andm1=m2orfor∆θ=π

HencewehaveobtainedthatthevalueofFisinbothcasesequalwith(m2+

nonclassicaldistanceforathermalsqueezedstateρ:

χ(ρ)=

2

d

√

d2√

1

>d2

m22

).d2

m2itfollowsthattheminimum

√

Γ(m)

+

Γ(m)

Γ(d)

(62)

whenΓ(d)>Γ(m)2orbyχ(ρ)=1whenΓ(d)<Γ(m)2.TheintuitivefactaccordingtowhichΓ(ρ)isclosertoaclassicalstatethanρisreflectedquantitativelyintheinequality

χ(Γ(ρ))≥χ(ρ).

Hencewemustprovethat

󰀈Γ(m)

√m

orinamoreconvenientform

Γ(m)2Γ(d)

m2

(63)

d+m2

()

m2

+1)2

(65)

whichbecomes

(

m2

d

g)1

m2

m2

++1)2

(66)

d

Fromthisitfollowsthattheinequalityχ(Γ(ρ))≥χ(ρ)isvalidonlyfor1d−

gintroducedbytheGaussian

noisemapisinacceptable.

6.Thefidelitydistance

ThefidelityF(ρ1,ρ2)fortwodensityoperatorsρ1andρ2isdefinedby

󰀍󰀋

√ρ1ρ2F(ρ1,ρ2)=Tr

12

SincethecharacteristicfunctionofaproductofoperatorswhosecharacteristicfunctionsareGaussiansisalsoaGaussianandthecharacteristicfunctionofthesquarerootofaGaussiandensityoperatorisaGaussianwecanfindasimpleformulaforthe

√

characteristicfunctionoftheoperatorρ1:

󰀊

√√

ρ1)=CFz(zTOz,(68)

4where

L−1=detΦ(A1)−1det

2

󰀍

Φ(A1)+A2

󰀑

(69)

whereU=(A2−iJ)(Φ(A1)+A2)−1(A2+iJ),and

O=Φ(A1)−(Φ(A1)−iJ)[A2+Φ(A1)−

(A2−iJ)(Φ(A1)+A2)−1(A2+iJ)]−1(Φ(A1)+iJ).

Thenapplyingofthepreceedingsectionwecanobtainthecharacteristic󰀈theresult√functionofρ1ρ2

󰀑

√

ρ1=

󰀊

1

[LdetΦ(O)]zTΦ(O)z.

4

(70)Fromthisformulaandtheproperty1ofthecharacteristicfunctionweobtain

󰀈

F(ρ1,ρ2)=

I+

(JO)−2

󰀓.

(72)

Inordertosimplifytheformulaforthefidelityweobservethat

󰀍

Ai+Aj

tijk=Trρiρjρk=det

󰀁,

2

andthatt123=t231=t312.IfwetakeinthislastidentityΦ(A1)insteadofA1weobtain

󰀔

Φ(A1)+A2

det

󰀁

=det

󰀍2A1+A2

Henceweget

L=󰀔

det

󰀍A1+A2

1−

1

detO+

√

󰀈

detO−

√

det(A1+A2)

,whichgivestheresultof[15]

F(ρ2

1,ρ2)=

det(A1+A2)+P−

√

󰀈

(d21−1)(d22−1)

.

Then

φ(ρ)=supρF(ρcl,ρ)=

.

cl

(d2

−1)2

+d2[

√

2m

+

md

]2−√(d2

−1)

WestressthefactthatfornonclassicalGaussianstateswehave

13

(76)

(78)

(79)

14

GaussianstatesourresultsarecomparablewiththeupperboundsobtainedforHillery’snonclassicaldistance[1],whichisdefinedusingthetracenorm,becausetheseupperboundscontaintheoverlapsbetweenthesqueezedstatesandthecoherentstates.References

[1]HilleryM1987Phys.Rev.A35725

HilleryM19Phys.Rev.A392994

[2]FuchsCAandVanDeGraafJ1999IEEETrans.Info.TheoryIT-451216[3]HallMJW1994Phys.Rev.A503295

[4]MusslimaniZH,BraunsteinSL,MannA,andRevzenM1995Phys.Rev.A514967[5]W¨unscheA1995Appl.Phys.B60S119[6]Kn¨ollLandOrlowskiA1995Phys.Rev.A511622

˙[7]ZyczkowskiKandS󰁍lomcz´ynskiW1998J.Phys.A:Math.Gen.319095[8]DodonovVV,Man’koOV,Man’koVIandW¨unscheA1999Phys.Scripta5981[9]BuresD1969Trans.Am.Math.Soc.135,199(1969).[10]UhlmannA1976Rep.Math.Phys.9273[11]JoszaR1994J.Mod.Opt.412314

[12]BartosewiczA1983Bull.Polish.Acad.Sci.31273[13]TwamleyJ1996J.Phys.A:Math.Gen.293723

[14]ParaoanuGh-SandScutaruH1998Phys.Rev.A58869[15]ScutaruH1998J.Phys.A:Math.Gen.313659[16]FearnJandColletMJ1988J.Mod.Opt.35553

[17]ChaturvediS,SanchyaR,SrinivasanVandSimonR1990Phys.Rev.A413969[18]LoCFandSollieR1993Phys.Rev.A47733

[19]HolevoAS1970ProblemyPeredachiInformacii4

HolevoAS1975IEEETrans.Inform.TheoryIF-21533

HolevoAS,SohmaMandHirotaO1999Phys.Rev.A591820HolevoAS1998quant-ph/9809022

HolevoAS1982ProbabilisticandStatisticalAspectsofQuantumTheory,(NorthHolland,Amsterdam),Chap5

[20]Oz-VogtJ,MannAandRevzenM1991J.Mod.Opt.38,2339[21]ScutaruH19Phys.Lett.A141223

ScutaruH1992Phys.Lett.A167326ScutaruH1995Phys.Lett.A20091ScutaruH1998J.Math.Phys.3903

[22]ParaoanuGh-SandScutaruH1999quant-ph/9907068.[23]HolevoAS1972Theor.Math.Phys.13184[24]GlauberRJ1963Phys.Rev.1312766[25]LachsG1965Phys.Rev.138B1012[26]VourdasA1986Phys.Rev.A343466

[27]VourdasAandWernerRM1987Phys.Rev.A365866[28]FearnHandCollettMJ1988J.Mod.Opt.35553[29]LoudonRandShepherdTJ1984Opt.Acta311243[30]VourdasA1988Phys.Rev.A37,30(1988).

[31]LeeCT1991inWorkshoponSqueezedStatesandUncertaintyRelations(NASAConference

Publications3135),editedbyHanDetal(NASA,WashingtonDC),pp.365-367.

[32]BuzekVandKnightPL1991Opt.Commun.81331

[33]HallMJWandO’RourkeMJ1993QuantumOpt.5161[34]MarianPandMarianTA1993Phys.Rev.A474474

MarianPandMarianTA1993Phys.Rev.A474487

[35]DodonovVV,Man’koOV,Man’koVIandRosaL1994Phys.Lett.A185231

DodonovVV,Man’koOVandMan’koVI1994Phys.Rev.A492993

[36]FollandGB19Harmonicanalysisinphasespace(Ann.ofMath.Studies,PrincetonUniversity

Press)

[37]BalianR,DominicisCandItzyksonC1965Nucl.Phys.67609

[38]EzawaH,MannA,NakamuraKandRevzenM1991Ann.Phys.,NY209216