Bipartite quantum coherence in noninertial frames

3.7 MB·

by  Xu Chen

Bipartitequantumcoherenceinnoninertialframes
Xu Chen,1 Chunfeng Wu,2 Hong-Yi Su,1,3 Changliang Ren,4 and Jing-Ling Chen1,5
1Theoretical Physics Division, Chern Institute of Mathematics,
Nankai University, Tianjin 300071, People’s Republic of China
2PillarofEngineeringProductDevelopment,SingaporeUniversityofTechnologyandDesign,8SomapahRoad,Singapore487372
3DepartmentofPhysicsEducation,ChonnamNationalUniversity,Gwangju500-757,RepublicofKorea
4Center of Quantum information, Chongqing Institute of Green and Intelligent Technology,
Chinese Academy of Sciences, People’s Republic of China
5CentreforQuantumTechnologies,NationalUniversityofSingapore,3ScienceDrive2,Singapore117543
(Dated:January13,2016)
PACSnumbers:03.65.Ud,03.67.Mn,42.50.Xa
6
1
Quantumcoherenceasthefundamentalcharacteristicofquantumphysics,providesthevaluableresourceforquantum
0
computationinexceedingthepowerofclassicalalgorithms.Theexplorationofquantumcoherenceinrelativisticsystems
2
is of significance from both the fundamental points of view and practical applications. We investigate the quantum
n
coherence of two free modes of scalar and Dirac fields as detected by two relatively accelerated observers by resorting
a
J to the relative entropy of coherence. We show that the relative entropy of coherence monotonically decreases when
2 accele√ration goes up, as a consequence of the Unruh effect. Specifically, the initial states with parameters α = b and
1 α = 1−b2 have the same initial relative entropy coherence at a = 0 (with a the acceleration), but degrade along
two different trajectories. The relative entropy of coherence reaches vanishing value in the scalar field in the infinite

] acceleration limit, but non-vanishing value in the Dirac field. This suggests that in the Dirac field, the bipartite state
h
p possesses quantum coherence to some extent with the variation of the relative acceleration, and may lead to potential
- applicationsinquantumcomputationperformedbyobserversinmotionrelatively.
t
n
a Quantumcoherencearisingfromthequantumsuperpositionprincipleisafundamentalaspectofquantumphysicsandplays
u acentralroleinquantuminformationandcomputationprocessing[1–5],leadingtoavarietyofspeed-upquantumalgorithms.
q
Quantum coherence is shown to be related to other significant resources like quantum discord, entanglement, steering and
[
Bell-nonlocalityinthefollowinghierarchydiagram[6], asshowninFig.1. Comparedwiththeotherresources, however, the

1 quantification of the quantum coherence has not been well-developed. The framework of quantifying the quantum coherence
v hasonlybeenmethodicallyinvestigatedveryrecently. ThefirstattempttoaddresstheproblemisbyT.Baumgratzetal. based
1
ondistancemeasuresinfinitedimensionalsettingin2014, bypresentingthebasicnotionsofincoherentstates/operationsand
4
the necessary conditions that any measure of coherence should satisfy [7]. In the reference, several candidates for measuring
7
2 thequantumcoherencehavebeenproposed,suchasdistancemeasures,relativeentropyofcoherenceandlp-norms. Followthe
0 seminalwork,thestudyofthequantumcoherencehasbeenfurtherputforward[6,8–16]. Refs.[8–13]hasfoundthedifferent
1. measures of the quantum coherence. Yao et al. explored the quantum decoherence in multipartite systems and presented the
0 hierarchical diagram [6]. In the literatures [14–16], the protection and transformation of the quantum coherence have been
6 investigated,andthestudiesareofpracticalimportance.
1
The general theory of relativity and quantum mechanics are the foundation of modern theoretical physics. The integration
:
v ofquantuminformationandthegeneralrelativitygivesbirthtothetheoryoftherelativisticquantuminformation[17],helping
i toexploretheapplicationsofquantuminformationandcomputationprotocolsperformedbyobserversinrelativemotion. The
X
effects of observing from accelerated observers on quantum entanglement, quantum discord and Bell nonlocality have been
r
a studiedinRefs.[18–23].Therelativityeffectinfluencesthequantumresources,andhencethequantuminformationandcompu-
tationprotocols[24–27]. Actually,theaccelerationofobserverscanbetreatedascertain“environmentaldecoherence”andthe
effectwillinfluencetheperformanceofquantumprotocolsinanegativeway[18]. Quantumcoherenceisaquantumresource
differentfromtheabovementionedones. Theflourishinginvestigationsinthequantumcoherencemakethestudyofthequan-
tumdecoherenceinrelativisticquantumsystemspossible. Suchstudieswilldeepenourunderstandingofthequantumfeatures
ofthesystemsandmayresultinpotentialapplicationsofdifferentsystemsinquantuminformationandcomputation.
InthispaperweinvestigatethebipartitequantumcoherenceofscalarandDiracfieldsinrelativeconstantacceleration. We
showthevarianceofthequantumcoherenceaccordingtotheFermi-DiracandBose-Einsteindistributions,basedontherelative
entropy of coherence. It is found that the relative entropy of coherence decreases monotonically with increasing acceleration,
in both Dirac field and scalar field. In the scalar field, the coherence approaches zero in the infinite acceleration limit. While
in the fermionic case the coherence will never be completely destroyed by the infinite growth of acceleration. This indicates
thatthefermionicsystemismoreusefulasaresourceforcertainquantuminformationandcomputationprotocolsperformedby
observersinnoninertialframes.
2
Results
Consider a quantum system shared by two observers, Alice who is in inertial frame and Rob who is uniformly accelerated.
FromtheperspectiveofAlice,theMinkowskicoordinatesareutilizedtodescribethesystem,whiletheRindlercoordinatesare
utilizedfromtheviewpointofRob.TheworldlinesofuniformlyacceleratedobserversintheMinkowskicoordinatescorrespond
tothehyperbolasinregionI(lefthandsideoftheorigin)andregionII(righthandsideoftheorigin)intheRindlercoordinates,
andregionsIandIIarecausallydisconnectfromeachother[18]. Inourcase,RobtravelsalongthehyperbolainregionI.The
stateofthesystemisassumedtobe
(cid:112)
|Ψ(cid:105)=α|0 (cid:105)M|0 (cid:105)M + 1−α2|1 (cid:105)M|1 (cid:105)M, (1)
s k s k
oftwoMinkowskimodessandk,whereαistheparameter. Weexplorethequantumcoherenceofthesysteminthescalarfield
andtheDiracfieldbyusingtherelativeentropyofcoherence(seetheMethodsection).
Quantumcoherenceinthescalarfield–. Wefirstconsidertheeffectofthescalarfieldonthequantumcoherence. WhenRob
moveswithuniformaccelerationwithrespecttoAlice,theMinkowskivacuumcanbeexpressedasatwo-modesqueezedstate
oftheRindlervacuuminthescalarfield[21],
∞
1 (cid:88)
|0 (cid:105)M = tanhnr|n (cid:105) |n (cid:105) , (2)
k k 1 k 2
coshr
n=0
withtanhr = e−π|k|c/a ≡ t, k isthewavevector, aistheacceleration, r istheaccelerationparameter, cisthespeedoflight
invacuum. |n(cid:105) and|n(cid:105) indicatetheRindlermodesinRegionIandII,respectively. ThefirstexcitedstateintheMinkowski
1 2
modecanbeeasilyrepresentedas
1 (cid:88)∞ √
|1 (cid:105)M = tanhnr n+1|(n+1) (cid:105) |n (cid:105) . (3)
k cosh2r k 1 k 2
n=0
Duetothecausalitycondition,RobisconstrainedtoregionIandhencewetraceoverthestatesinregionIItofindthemixed
densitymatrixbetweenAliceanRob
∞
1 (cid:88)
ρs = tanh2nrρ , (4)
AR cosh2r n
n=0
where
√
(cid:112)
α (1−α2) n+1
ρ = α2|0,n(cid:105)(cid:104)0,n|+ |0,n(cid:105)(cid:104)1,n+1|
n
coshr
√
(cid:112)
α (1−α2) n+1 (1−α2)(n+1)
+ |1,n+1(cid:105)(cid:104)0,n|+ |1,n+1(cid:105)(cid:104)1,n+1|, (5)
coshr cosh2r
where|m,n(cid:105)≡|m (cid:105)M|m (cid:105) . Theeigenvaluesofthedensitymatrixρs isgivenby
s k 1 AR
tanh2nr(cid:20) (1−α2)(n+1)(cid:21)
λs = α2+ . (6)
n cosh2r cosh2r
Itisclearthattheλs isdependentonαandhyperbolicfunctionofr.
n
Wefindtherelativeentropyofcoherenceofthestate(4)isasfolloes:
C (ρs )=S(ρs )−S(ρs ), (7)
rel.ent. AR ARdiag AR
with
1 (cid:88)∞ (1−α2)(n+1)
ρs = tanh2nr(α2|0,n(cid:105)(cid:104)0,n|+ |1,n+1(cid:105)(cid:104)1,n+1|), (8)
ARdiag cosh2r cosh2r
n=0
and
∞
(cid:88)
S(ρs )=− λslog λs. (9)
AR n 2 n
n=0
ThevariationoftherelativeentropyofcoherenceasafunctionofrwithdifferentvaluesofαisplottedinFig.2. Thefollowing
observationsarefound:
3
(i) When r = 0, the relative entropy of coherence C (ρs ) = −α2logα2 −(1−α2)log(1−α2). For the case that
rel.ent. AR
0<α≤ √1 ,thelargerthevalueofα,themoretherelativeentropyofcoherenceis;butinthecasethat √1 ≤α<1we
2 2
haveconverseresult,namelythelargerthevalueofα,thesmallertherelativeentropyofcoherenceis.
(ii) The monotonic decrease of the relative entropy of coherence with increasing r for different α is obtained. Where r is
between 2 and 4, the coherence C (ρs ) reduces rapidly, and it approaches zero when r → ∞. The result means
rel.ent. AR
thatthestateinscalarfieldgeneratedbytheUnruheffect,containsnocoherenceinthelimitofinfiniteacceleration.
√
(iii) Oneinterestingresultfromthenumericalresultsisthattheinitialstateswithparametersα=bandα= 1−b2havethe
sameinitialrelativeentropycoherenceatr =0,butdegradealongtwodifferentcurvesexceptforthemaximallyentangled
state, i.e., α = √1 . Thisphenomenon, duetothedependenceoftheeigenvaluesonαandthehyperbolicfunctionofr,
2
showstheinequivalenceofthequantizationforascalarfieldintheMinkowskiandRindlercoordinates.
QuantumcoherenceintheDiracfield–. WenextinvestigatetheeffectoftheDiracfieldonthequantumcoherence. Inthe
case of the Dirac field, the quantum system will be expressed in another way. With the single-mode approximation [18], the
MinkowskivacuumintheDiracfieldisgivenby
|0 (cid:105)M =cosθ|0 (cid:105) |0 (cid:105) +sinθ|1 (cid:105) |1 (cid:105) , (10)
k k 1 k 2 k 1 k 2
andthefirstexcitationintheMinkowskimodecanbewrittenas
|1 (cid:105)M =|1 (cid:105) |0 (cid:105) . (11)
k k 1 k 2
withcosθ = (1+e−2πωc/a)−1/2, adenotesRob’sacceleration, ω isfrequencyoftheDiracparticle, cisthespeedoflightin
vacuum. Theaccelerationparameterθsatisfies0≤θ < π for0≤a<∞,|0 (cid:105) and|1 (cid:105) indicateRindlermodesinRegionI
4 k 1 k 2
andII,respectively.
ThequantumstatedescribingthesystemintheDiracfieldcansimilarlybeobtainedbylookingatRegionIonly,
(cid:112)
ρD =α2(cos2θ|00(cid:105)(cid:104)00|+sin2θ|01(cid:105)(cid:104)01|)+α 1−α2 cosθ (|00(cid:105)(cid:104)11|+|11(cid:105)(cid:104)00|)+(1−α2)(|11(cid:105)(cid:104)11|), (12)
AR
where|mn(cid:105)≡|m (cid:105)M|m (cid:105) . WethengettheeigenvaluesofdensitymatrixρD asfollows,
s k 1 AR
λD =1−α2sin2θ, λD =α2sin2θ. (13)
1 2
FortheDiracfield,theeigenvaluesdependonαandthetrigonometricfunctionofθ. Similarly,onehas
ρD =α2(cos2θ|00(cid:105)(cid:104)00|+sin2θ|01(cid:105)(cid:104)01|)+(1−α2)(|11(cid:105)(cid:104)11|). (14)
ARdiag
AccordingtoEq. (17),therelativeentropyofcoherenceofstate(12)isexpressedas
C (ρD ) = S(ρD )−S(ρD )
rel.ent. AR ARdiag AR
= −α2cos2θlog(α2cos2θ)−(1−α2)log(1−α2)+(1−α2sin2θ)log(1−α2sin2θ). (15)
ThebehavioroftherelativeentropyofcoherenceasafunctionofθfordifferentvaluesofαisshowninFig. 3. Wefindthat
(i) Atθ =0,therelativeentropyofcoherenceCrel.ent.(ρDAR)=−α2logα2−(1−α2)log(1−α2). When0<α≤ √12,the
coherence is increasing when α goes up; but in the range √1 ≤ α < 1, the coherence is decreasing with increasing α.
2
Theresultsarethesameasthoseobtainedinthescalarfield.
(ii) Atfiniteacceleration,wefindthemonotonicdecreaseoftherelativeentropyofcoherencewithincreasingθfordifferent
valuesofα. Howeverunlikethescalarfieldcase,thecoherenceC (ρD )doesnotundergosharpchangebutshows
rel.ent. AR
a slow variation. In the limit of infinite acceleration, state (12) still bears quantum coherence to some degree when
θ → π/4 or a → ∞. Analytically, we find the relative entropy of coherence in the limit to be C (θ = π/4) =
rel.ent.
−(1 − α2)log(1 − α2) − α2logα2 + 2−α2log2−α2. The expression means that the state always possesses quantum
2 2 2 2
(cid:113) √
coherence to some extent. The maximal relative entropy of coherence happens at α = (5− 5)/5 and the maximal
(cid:113) √
value is 0.694 when a → ∞. Thus we know that in the infinite limit when 0 < α ≤ (5− 5)/5, the coherence
(cid:113) √
increaseswithαgoingup;butintherange (5− 5)/5 ≤ α < 1,thecoherencevariesconverselywiththechangeof
α.
4
√
(iii) WecanalsofindthatinFig. 3,thestatewithα=bandtheonewithα= 1−b2havethesamecoherenceatθ =0,but
degradealongtwodifferenttrajectoriestotwononvanishingminimumvaluesintheinfinite-accelerationlimitexceptfor
thecasethatα= √1 .
2
(iv) Letusdefinethelossoftherelativeentropyofcoherenceofeachfixedvalueofαas
π α2 α2 2−α2 2−α2
∆=C (ρD )(θ =0)−C (ρD )(θ = )=−α2logα2+ log − log . (16)
rel.ent. AR rel.ent. AR 4 2 2 2 2
By resorting to ∆, we can study the which gives us the of the relative entropy of coherence in noninertial frame in a
(cid:113)
qualitative way. In Fig. 4, it is shown that the maximal value of loss is given when α = 2, where ∆ = 0.322.
5 Max
(cid:113)
Thus we know when 0 < α ≤ 2, the coherence increases with the growth of α going up; conversely in the range
5
(cid:113)
2 ≤ α < 1, the coherence reduces with the growth of α. Interestingly, we can easily find that, the state with the
5
maximalquantumcoherenceininertialframe(α = √1 ),thestatewhichcanpossessesthemaximalquantumcoherence
2
(cid:113) √
(α = (5− 5)/5), andthestatesufferingthemaximalcoherencelossfrominertialframetononinertialframe(α =
(cid:113)
2),aretotalydifferent.
5
(v) InFig. 5,wealsoshowthetransitionofthestateswiththemaximalquantumcoherencefrominertialframetononinertial
frame. Thepointintheredlinerepresentthestatewhichpossessesthemaximalquantumcoherencewithafixedvalueof
α. Theredlineisjustthetrajectoryshowingtheevolutionofthestatepossessingthemaximalquantumcoherencewith
thechangeofacceleration.
BasedontheresultsforthescalarfieldandtheDiracfield, wefindthatthequantumcoherenceoftherelativisticsystemis
related to the acceleration qualitatively. The decoherence due to the acceleration is manifested by the numerical results, and
differentfieldsinfluencethecoherenceindifferentways. Itisclearlyshownthat,inthelimitofinfiniteacceleration,bipartite
coherencevanishesinthescalarfield,butneverintheDiracfield. ThedissimilaritybetweenthescalarfieldandDiracfieldis
causedbythedifferencesbetweenFermi-DiracandBose-Einsteinstatistics. Fermionsonlyhaveaccesstotwoquantumlevels
whilethereareinfiniteexcitationsavailabletobosonsduetothePauliexclusionprinciple.
Comparewiththestudiesofentanglementintherelativisticsystems[24],wefindthatquantumcoherenceandentanglement
are both affected by the acceleration in negative ways, however, the quantum coherence is more resistant to the decoherence
effectthantheentanglement. TheresultisofimportancesinceitisshowninRef.[28]thatanynonzerocoherencewithrespect
tosomereferencebasiscanbeconvertedtoanequalamountofentanglementviaincoherentoperations.
Discussion
Inthiswork,wehaveexploredthebipartitequantumcoherenceasobservedinnoninertialframesbasedonthequantumsystem
oftwofreemodesofscalarandDiracfieldsasdetectedbyAlicewhoisinertial,andRobwhoundergoesaconstantacceleration.
Decoherenceeffectduetotheaccelerationisfoundnumerically,andthatistherelativeentropyofcoherenceismonotonically
decreasingwithincreasingaccelerationfordifferentvalueofα,asaconsequenceoftheUnruheffectinboththescalarfieldand
√
theDiracfield. Inbothfields,wehavefoundthatthestatewithα =bandtheonewithα = 1−b2 havethesamecoherence
√
at a = 0, but degrade along two different paths except for the maximally entangled state(α = 1/ 2). The results exhibit
the inequivalence of the quantization for a free field in the Minkowski and Rindler coordinates. But the two fields influence
the behaviors of the degradation of the coherence quite differently. In particular in the infinite acceleration limit, the state in
thescalarfieldhasnononzerorelativeentropyofcoherence,howeverthestateintheDiracfieldalwaysremainsnonvanishing
relativeentropyofcoherencedependentonα. ForthecaseoftheDiracfield,wehaveshownthatinthelimit,maximalrelative
(cid:113) √
entropy of coherence can be achieved when α = (5− 5)/5. This nonzero coherence in the fermionic system makes the
system a more desirable candidate for implementing quantum information and computation protocols performed by relatively
acceleratingobservers.
Method
Themeasureofquantumcoherenceutilizedintheworkistherelativeentropyofcoherencewhichisexpressedas
C (ρ)=minS(ρ||δ)=S(ρ )−S(ρ), (17)
rel.ent. diag
δ⊂I
whereI denotesthewholesetofincoherentstates,S isthevonNeumannentropyandρ isthediagonalversionofρ,which
diag
onlycontainsthediagonalelementsofρ.AccordingtoRef.[7],therelativeentropyofcoherencemeasuresatisfiesthefollowing
necessaryconditions,
5
(C1) C(ρ)=0iffρ⊂I.
(C2a) Monotonicity under non-selective incoherent completely positive and trace preserving (ICPTP) maps, i.e., C(ρ) ≥
C(Φ (ρ)), where Φ (ρ) = (cid:80) K†ρK and K is a set of Kraus operators that satisfies (cid:80)K†K = I and
ICPTP ICPTP n n n n n n
K IK ⊂I.
n n
(cid:80)
(C2b) Monotonicity for average coherence under subselection based on measurement outcomes, i.e., C(ρ) ≥ p C(ρ ),
n n n
whereρ =K†ρK /p andp =Tr(K†ρK )forallK with(cid:80)K†K =I andK IK ⊂I.
n n n n n n n n n n n n
(cid:80) (cid:80)
(C3) Convexity,i.e., p C(ρ )≥C( p ρ )foranysetofstatesρ andanyprobabilitydistributionp .
n n n n n n n n
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Acknowledgements
J.L.C. is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and
the Natural Science Foundation of China (Grant Nos. 11175089 and 11475089). C.R. is supported by Youth Innovation Pro-
motion Association (CAS) No. 2015317. H.Y.S. acknowledges the support by Institute for Information and Communications
TechnologyPromotion(IITP),Daejeon,RepublicofKorea.
AuthorcontributionsX.C.andJ.L.C.initiatedtheidea. X.C.,C.W.C.R.andH.Y.S.wrotethemainmanuscripttext. X.C.
andC.R.plottedthefigures. Allauthorsreviewedthemanuscript.
Additionalinformation
6
Competingfinancialinterests: Theauthorsdeclarenocompetingfinancialinterests.
Correspondence and requests for materials should be addressed to C.W. (chunfeng [email protected]), H.Y. S.
([email protected])orJ.L.C.([email protected]).
7
FIG.1:Venndiagramofdifferentresourcespresentedincompositequantumstatesforquantuminformationandcomputation[6].
1.0
0.8
1(cid:2) 2
C 0.6 1(cid:2)2
3(cid:2)4
0.4
1(cid:2)8
0.2 7(cid:2)8
0.0
0 2 4 r 6 8 10
FIG.2:√Relativeentropyof√coherenceofthebosonicmodesversusrfordifferentα.Theblack,red,blue,green,orangecurvescorrespondto
(cid:112) (cid:112)
α=1/ 2,1/2, 3/4,1/ 8, 7/8,respectively.
1.0
0.9
0.8 1(cid:2) 2
C 0.7 1(cid:2)2
3(cid:2)4
0.6 1(cid:2)8
7(cid:2)8
0.5
0.4
0.0 0.2 0.4 0.6 0.8
θ
FIG.3: Re√lativeentropyofco√herenceofthefermionicmodesversusθfordifferentα. Theblack,red,blue,green,orangecurvescorrespond
(cid:112) (cid:112)
toα=1/ 2,1/2, 3/4,1/ 8, 7/8,respectively.
8
1.0
C(cid:2)Θ(cid:3)0(cid:3)
C(cid:2)Θ(cid:3)Π(cid:3)
0.8 4
(cid:5)
0.6
C
0.4
0.2
0.0
0.0 0.2 0.4 α 0.6 0.8 1.0
FIG.4: Relativeentropyofcoherenceofthefermionicmodesfordifferentαwhenθ = 0(Red)andθ = π (Green), andlossofrelative
4
entropyofcoherence∆(Blue).
(cid:19)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:21)
θ
(cid:19)(cid:17)(cid:23) (cid:20)(cid:17)(cid:19)
(cid:19)(cid:17)(cid:25)
C
(cid:19)(cid:17)(cid:24)
(cid:19)(cid:17)(cid:27)
(cid:19)(cid:17)(cid:19)
α(cid:19)(cid:17)(cid:24) (cid:19)(cid:17)(cid:19)
(cid:20)(cid:17)(cid:19)
FIG.5:Transitionofthestateswiththemaximalquantumcoherencefrominertialframetononinertialframe.Thethree-dimensionalgraphic
isplottedbythefunctiongiveninEq.(15).Thepointintheredlinerepresentthestatewhichpossessesthemaximalquantumcoherencewith
afixedvalueofα.