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Nullityofquantum discordoftwo-qubit X-state systems Gao-xiang Lia, Zhen Yia, and Zbigniew Ficekb aDepartment of Physics, Huazhong Normal University, Wuhan 430079, PR China bTheNationalCentreforMathematicsandPhysics,KACST,P.O.Box6086, Riyadh11442, Saudi Arabia (Dated:January27,2011) Theconditionsareestablishedfornullityofquantumdiscordformixedstatesofageneraltwo-qubitsystem whosedynamicalbehaviourisgivenbyanX-stateformdensitymatrix. Itisfoundthatquantumdiscordcan vanishatafinitetimeeveninthepresenceofcorrelationsbetweenqubits.Thedegeneracyofthepopulationsof thecombinedstatesofthequbitsandthemulti-qubitquantumcoherencesareshowntoberesponsibleforvan- ishingofthediscord. Weillustrateourconsiderationsbyexaminingthecoherent-stateTavis-Cummingsmodel inwhichtwoqubitsinteractdispersivelywiththecavityfield. Thedisappearanceofthediscordisshowntobe dependentontheinitialatomicconditionsthatitmayoccuratadiscreteinstanceorperiodicallyatdiscretein- stancesredistributedoverwholerangeoftheevolutiontime.Thenullityofthediscordisinterpretedasresulting fromequalpreparationofthespinanti-correlatedandspincorrelatedproductstatesinsuperpositionstates. 1 PACSnumbers:03.67.Mn,03.65.Yz,03.67.Lx 1 0 2 Nothing in the principles of quantum mechanics prevents Cummingsmodel,weshowhowitispossibletogeneratesuch n a system from possessing quantum correlations. Quantum states in a dynamical process and what their properties are. a or nonclassicalcorrelationsare unavoidablein quantumsys- Zero-discordstatesareparticularlyattractiveregardingthere- J tems [1], and as it has been established recently, almost all cent prove that vanishing discord is necessary and sufficient 6 quantumstateshavenonclassicalcorrelationsor,equivalently, forcompletelypositivemaps[14],andnoconstrainsimposed 2 havepositivediscord[2–6].Quantumdiscord,oneofthemea- onlocalbroadcastingofcorrelatedstates[15]. ] sures of quantumcorrelations, is closely related to entangle- Wechooseatwoqubitsystemwhosedynamicalbehaviour h ment,aquantumfeatureofcompositesystemsthatcannotbe isgivenbyanX-stateformdensitymatrix,andexamineun- p - represented as a product of the states of the individual sys- derwhatcircumstancesthesystemcanhavezerodiscordwith t tems, but recent theoretical investigationsindicated that this nonzerocorrelationsbetweenthequbits.Weillustrateourex- n a may not be true. Only for pure states, quantum discord is aminations on a simple model of two two-level atoms inter- u equivalenttoentanglementandsignificantdeparturesbetween actingdispersivelywithadampedsingle-modecavityfield. q these quantities have been reported for mixed states. For a Considerapairoftwo-levelatoms(qubits),labelledbythe [ largeclassofstates,entanglementiszero,whereasthequan- sufficesAandB,preparedinaquantumstatedeterminedby 1 tum discord is positive for almost all quantum states. Over the X-state form density matrix ρ. The matrix, written in v the last two years, a series of investigations, both theoreti- the space spannedby spin correlated(|1i ≡ |g ,g i,|4i ≡ 3 calandexperimental,hasshownthatthequantumdiscordof A B 8 |eA,eBi) and spin anti-correlated (|2i ≡ |gA,eBi,|3i ≡ open quantum systems vanishes asymptotically in time [7], 9 |eA,gBi)productstatesisoftheform contrarytoentanglementthatcandisappearatfinitetime[8]. 4 . Understandingandmanipulatingdynamicsofthesequantum 1 4 featuresareofgreatimportanceforbothfundamentalphysics 0 ρ = ρ |jihj|+(ρ |1ih4|+ρ |2ih3|+H.c.), (1) 1 andnewemergingquantumtechnologies. AB Xj=1 jj 14 23 1 In principle, it is possible to construct multi-qubit states : v withzerodiscord. Anexplicitconstructionofsuchstateshas where|g iand|e i(j = A,B)representthegroundandex- j j i X recently been investigated in many publications [9–13]. In citedstatesofthequbits. Thestate (1)isoftheformassoci- theseinvestigationsconditionsfornullityofquantumdiscord atedwithquantumcoherencesexistingbetweenstatesofboth r a havebeenestablished. However,thezero-discordstatessuch qubits,notbetweenthestatesofindividualqubits. Theseco- constructedsuffer froma commonhandicap: theypossesno herencestellusaboutthecorrelationsbetweenthequbits. anycorrelationsandcoherences. Therefore,itisnotsurpris- We now determine for which states of a bipartite system, ingthatthediscordiszero. specifiedbythedensitymatrixofanX-stateform,thequan- Somewhatlessfamiliarisaconstructionofstatesthatcould tum discord can reach zero even in the presence of internal havezerodiscordatafinitetimeeveninthepresenceofcorre- coherences. Wedefinethequantumdiscordbyrelationtothe lationsbetweenthequbits. Thisisthepurposeofthepresent quantummutualinformationandadopta procedurewhichis Lettertoshowthatitisreallypossibletoprepareatwo-qubit bothanalyticandsimple,andpointsouttheunusualbehaviour systeminaquantumstatethatcanexhibitavanishingdiscord ofthediscordinastraightforwardandtransparentfashion. ata finitetimewithnon-zeroquantumcoherencespresentin Thetotalcorrelations(quantumandclassical)inabipartite the system. Ina simple exampleof the coherent-stateTavis- quantum system are measured by the total quantum mutual 2 informationI(ρ )definedas Accordingly, C results from the projective measure- AB m2 mentsonB,withθ =π/4andφ=(φ −φ )/2: 1 2 I(ρ ) = S(ρ )+S(ρ )−S(ρ ) AB A B AB = D(ρAB)+C(ρAB), (2) Cm2 =S(χm2)−S(TrA(χm2)) (1+Υ) 1+Υ (1−Υ) 1−Υ whereρ andρ arethereduceddensitymatricesofthe =− log − log , (6) A(B) AB 2 2(cid:18) 2 (cid:19) 2 2(cid:18) 2 (cid:19) subsystem A(B) and the total system, respectively; S(ρ) = −Tr{ρlog2ρ} is the von Neumann entropy, D(ρAB) is the whereΥ= (ρ +ρ −ρ −ρ )2+4(|ρ |+|ρ |)2 21, quantumdiscordwhichprovidesinformationaboutthequan- 11 22 33 44 41 32 tum nature of the correlations, and C(ρ ) describes the φ1 =arg(ρ1(cid:2)4),φ2 =arg(ρ23)andthezero-discordstate (cid:3) AB classical correlations, which can be obtained by use of a 1 measurement-basedconditionaldensity operator[2–5]. Nat- χ = {(ρ +ρ )(|1ih1|+|2ih2|) m2 11 22 2 urally, if the mutual information is larger than the classical +(ρ +ρ )(|3ih3|+|4ih4|) correlations, the quantum discord is positive, different from 33 44 zero. Clearly,thequantumdiscordisobtainedbysubtracting +(|ρ |+|ρ |) |1ih4|eiφ1+|2ih3|eiφ2 +H.c. . (7) 14 23 theclassicalcorrelationsfromthetotalmutualinformation. (cid:2)(cid:0) (cid:1) (cid:3)(cid:9) The mechanism for a positive discord is a quantum cor- Evidently,thestatesχm1andχm2whichposseszeroquan- relation between the A and B subsystems. The subsystems tum discord have the same X-state form as the state ρAB. are correlated if the measurementof an observable of the A From this observation, it follows that for a two qubit sys- system projects the B system into a new state, and vice temdeterminedbythedensitymatrixofanX-stateform,the versa. Here, we limit ourselves to projective measurements quantumdiscordvanisheswhen performed locally only on the subsystem B described by a ρ =ρ =0, (8) complete set of orthonormal projectors {Π } corresponding 14 23 k totheoutcomesk.TheclassicalcorrelationsC(ρ )arethen AB orwhen definedas ρ =ρ , ρ =ρ , and |ρ |=|ρ |. (9) C(ρ )=max[S(ρ )−S(ρ |{Π }], (3) 11 22 33 44 23 14 AB A AB k {Πk} Thus,wehaveextractedtwodistinctconditionsfornullityof where the maximum is taken over the set of projective the quantum discord. The former, Eq. (8), is the condition measurement {Π }, and S(ρ |{Π }) = p S(ρ ) is k AB k k k k involvingonlythediagonaldensitymatrixelements(popula- the conditional entropy of the subsystem AP, with ρk = tions)withallquantumcoherencesequaltozero. Thisisthe Tr (Π ρ Π )/p andp =Tr (ρ Π ). B k AB k k k AB AB k trivialcasewhichsuggeststhatthezerodiscordoccurswhen WenowlinktheclassicalcorrelationsasdefinedinEq.(3) there are no any quantum coherences in the system. From to the quantumdiscord. We see that a state that couldmini- this it is clear that the quantum discord vanishes. It is also mize the conditionalentropy S(ρ |{Π }) would then cor- AB k clearthatoutsidethiscondition,thediscordisalwaysdiffer- respondtoa zero-discordstate [2]. Followingthis, weintro- entfromzero. Thishasalreadybeenreportedinseveralpub- ducethebasisoforthogonalstatesofthesubsystemB;|+i= lications[9–13]. cosθ|e i+sinθeiφ|g iand|−i=sinθ|e i−cosθeiφ|g i. B B B B Thelattercondition,Eq.(9),ismoreinterestingandinfact Then, we define a general one-qubit projector Π = |kihk| k surprising. Itisageneralresultvalidforanyinitialstateand (k = ±) on the subsystem B. From this, we find that the any dynamical process that involves the X-state form den- minimumvalueoftheconditionalentropyS(ρ |{Π }),that AB k sity matrix. Itshowsthata zero-discordstate doesnotguar- maximizes the classical correlations, can be analytically ex- antee no quantum coherences in the system. The condition pressedas to be satisfied requires the pair degeneracy of the populations of the energy states and equal magnitudes of the one- min{S(ρ |{Π })}=min{C ,C }, (4) AB k m1 m2 {Πk} andtwo-photonquantumcoherences. Itiseasilyverifiedthat the condition is related physically to the preparation of the where C results from the projective measurements on B, m1 spincorrelated(|1i,|4i)andanti-correlated(|2i,|3i)statesin withtheangleθ =0orπ/2: equalsuperpositionstates. Notethatingeneral,thesuperpo- sition statesso obtainedarenon-maximallyentangledstates. C =S(χ )−S(Tr (χ )) m1 m1 A m1 Inotherwords,theydonotcoincidewiththeBellstates. ρ ρ =−ρ log 44 −ρ log 22 Wenowdemonstratetheoccurrenceofthenullityofquan- 44 2(cid:18)ρ22+ρ44(cid:19) 22 2(cid:18)ρ22+ρ44(cid:19) tumdiscordinapracticalphysicalsystem:Thecoherent-state ρ ρ Tavis-Cummingsmodeloftwoatomscoupledtoasinglecav- −ρ log 33 −ρ log 11 , (5) 33 2(cid:18)ρ +ρ (cid:19) 11 2(cid:18)ρ +ρ (cid:19) ity mode [16]. The interatomic separation is assumed to be 11 33 11 33 muchlargerthanthetransitionwavelengthsothatcooperative withthezero-discordstateχ = 4 ρ |jihj|. effectscanbeignored.Furthermore,weconsidertheatomsto m1 j=1 jj P 3 bewelllocalizedandneglecttheirexternalmotion.Ifwespe- TodemonstratethatEqs.(12)canmatchtheconditionsfor cialisetothecaseofahighlydetunedcavitymode,|δ| ≫ g, nullityofthequantumdiscord,wemustcomparethepopula- where g is the coupling constant of the atoms to the cavity tions and the coherences. Since the populations ρ (t) and 11 field, and δ = ω − ω is the detuning between the atomic ρ (t) are constants of motion, the condition of ρ (t) = 0 44 11 transitionfrequencyω andcavityfrequencyω, theeffective ρ (t) and ρ (t) = ρ (t) can be adjusted by a suitable 0 22 33 44 Hamiltonianofthesystemreads choiceoftheinitialatomicconditions.Thus,theproblemsim- plifiestofindif therearediscretetimesorperiodsoftimeat Heff = 21λ (|ejihej|aa†−|gjihgj|a†a) winhgicahreththeecochaeseresnwcehe|ρre23|(ρt2)3|(=t)||ρ=14(|tρ)1|4.(Pta)|rti6=cul0araltytin→tere∞st-, j=XA,B i.e.,ifthecondition(9)canbesatisfiedinthesteadystate. + σAσB +σAσB , (10) Wenowpresentsomenumericalcalculationsthatillustrate − + + − (cid:0) (cid:1)(cid:3) theaboveremarks. Wespecializetothetimeevolutionofthe where σ+j = |ejihgj| and σ−j = |gjihej|(j = A,B) are quantum discord for different initial conditions to show that atomic spin operators, a† and a are respectivelythe creation dependingontheinitialstate,andalsoonthedissipationrate, andannihilationoperatorsofthecavityfield,λ=g2/2δisthe thediscordmaynevervanish,ormayvanishatafinitenumber effectivecouplingconstant(Rabifrequency).Thefirsttermin ofdiscretetimes,orevencanvanishperiodicallyintime. theHamiltonian(10)representstheintensitydependentStark shift, while the second term is of the form analogous to the familiardipole-dipoleinteractionbetweentheatoms[17]. 0.15 0.20 If we include the dissipation of the cavity mode, the dy- 0.15 0.12 namicsof the system are then determinedby the the density 0.10 0.05 matrixρ(t),whichsatisfiesthefollowingmasterequation 0.09 0.00 0 3 6 9 ρ˙(t)=−i[H ,ρ(t)]+κL ρ(t), (11) 0.06 eff c whereL ρ(t)=2aρ(t)a†−a†aρ(t)−ρ(t)a†arepresentsthe 0.03 c dampingofthecavityfieldbycavitydecaywiththerateκ. 0.00 Using the master equation (11), we find equationsof mo- 0 10 t 20 30 40 tionforthedensitymatrixelements.Fortheinitialconditions FIG.1:Quantumdiscord(solidline)plottedasafunctionofdimen- we choose the field to be in a coherentstate with the ampli- sionlesstimeλtfor|α|2 =0.5922andκ=0.05λ. Thesystemwas tudeα, andtheatomstobeinitiallyina statedeterminedby initiallyinastatedeterminedbyρ11(0)=ρ44(0)=1/4,ρ22(0)= density matrix (1). By tracing over the cavity field, we ob- 3/16,ρ33(0) = 5/16,|ρ23(0)| = 0.05, and|ρ14(0)| = 0.25. The tainthereduceddensitymatrixoftheatoms,andfindthatthe insertshowsthetimeevolutionofthequantumcoherences|ρ14(t)| solutionsforthedensitymatrixelementsare (solidline)and|ρ23(t)|(dashedline). ρ11(t) = ρ11(0), ρ44(t)=ρ44(0), Figure1 shows the time evolutionofthe quantumdiscord ρ (t) = c (0)+c (0)cos(λt)−c (0)sin(λt), for two qubits initially prepared in a non-zero discord state. 22 + − 2 ρ (t) = 1−ρ (t)−ρ (t)−ρ (t), (12) Onecanseethatthediscordvanishesonlyatthefirstperiodof 33 11 22 44 theoscillationswhenthecoherencesρ (t)andρ (t)merge. ρ (t) = c (0)+ic (0)cos(λt)+ic (0)sin(λt), 14 23 23 1 2 − The coherences do not merge during any of the subsequent 2iλ|α|2 ρ (t) = ρ (0)exp −iλt− 1−e−(κ+2iλ)t , periodof the oscillations. This simple exampleshowsthata 41 41 (cid:26) κ+2iλh i(cid:27) purelyclassical state or a pureclassicalnatureof thesystem isaratherashort-livedaffair. wherec (0) = (ρ (0)±ρ (0))/2,andρ (0) = c (0)+ ± 22 33 23 1 Figure2showsthatazero-discordstatecanbecreatednot ic (0). We see that the populations ρ (t) and ρ (t) are 2 11 44 only by a suitable choice of the initial state, but also by a constantsofmotionthattheyretaintheirinitialvaluesforall change of the cavity damping rate. We see that a large cav- times. Bothρ (t)andρ (t)exhibitundampedRabioscilla- 22 33 ity damping rate results in a positive discord at short times tions. Similarly, the coherence ρ (t) oscillates sinusoidally 23 with zeros occurring periodically at longer times. One can in time. Only the coherenceρ (t) exhibits dampedoscilla- 14 noticethatat shorttimes the coherencesand the populations tions. Thus, the loss of the coherenceρ (t) is reversible in 23 crosseachotheratsomediscretetimes,sowhythediscordis time, it does not arise from the cavity damping mechanism. positiveatthattimes. Thereasonisthatthecoherencescross However,the loss ofthe coherenceρ (t) is irreversibledue 41 each other at times differentthan that the populationscross, to thedampingofthe cavityfield. Thecoherenceundergoes sothecondition(9)isnotsatisfied. dampedandphase sifted oscillationssuch that itdecaysto a Itisinterestingthatthepossibilityforthediscordtovanish non-zerostationaryvalue dependsonwhethertheinitialstatewasentangledornot.This 4λ2|α|2 is illustrated in Fig. 3, where the plot the time evolution of lim |ρ (t)|→|ρ (0)|exp − . (13) t→∞ 41 41 (cid:20) κ2+(4λ)2(cid:21) the discordfor two differentinitialBell-diagonalstates [18], 4 the phase damping of the coherence ρ (t) breaks the con- 0.025 14 dition of |ρ (t)| = |ρ (t)|. Thus the phase damping, it 0.28 14 23 0.020 turnsout, is sufficientto wipe outany zero-discordstate. In 0.21 other words, the dissipation can build up quantum correla- 0.015 0.14 tions, leavingpurelyclassical statestooccuronlyduringthe 0.07 initialtime. 0.010 0.00 0 10 20 30 Finally, we discuss a possible experimental arrangement 0.005 wherethezero-discordstatescouldbemeasured.Agoodcan- didate is the scheme used by Osnaghi et al. [19] to observe 0.000 0 10 t 20 30 40 entanglement between two Rydberg atoms crossing a non- FIG.2:Quantumdiscord(solidline)plottedasafunctionofdimen- resonant microwave cavity. The scheme involves the Tavis- sionlesstimeλtfor|α|2 =1.1434andκ=0.25λ. Thesystemwas CummingsmodeldeterminedbytheHamiltonianofthesame initiallyinastatedeterminedbyρ11(0)=ρ44(0)=1/4,ρ22(0)= form as Eq. (10). Therefore, we anticipate no difficulty to 3/16,ρ33(0) = 5/16,|ρ23(0)| = 0.05, and|ρ14(0)| = 0.25. The modifytheschemeforthemeasurementofthediscord. insert shows the time evolution of the populations ρ22(t) (dashed Insummary,wehavederivedconditionsfornullityofquan- line) and ρ33(t) (dashed-dotted line), and also the quantum coher- tumdiscordofanarbitrarytwo-qubitsystemwhosedynami- ences|ρ14(t)|(solidline)and|ρ23(t)|(dottedline). calbehaviourisgivenbyanX-stateformdensitymatrix.The resultsderivedshowthatthequantumdiscordcanvanisheven mixedstateswhoseeigenstatesarefourmaximallyentangled in the presence of quantum coherences between the qubits. states,theBellstates.Fortheinitialunentangledstate,thedis- The condition is related physically to the preparation of the cordvanishesperiodicallyintimeduetoamodulationofthe spin correlated and anti-correlated states in equal superposi- Rabi oscillations. The modulation results from the crossing tionstates. Wehaveshownthattheconditioncanberealized of the coherences periodically at discrete times. The coher- in a practical system of the coherent-state Tavis-Cummings ence|ρ |isconstantintimeand,asthetimeprogresses,the model in which two distant qubits interact dispersively with 23 coherence|ρ |evolvestowardsitsnon-zerostationaryvalue thecavityfield. 14 of |ρ14| = |ρ23|. In contrast, the discord decreases in time Thiswork was supportedby the NationalNaturalScience andvanishesatthesteadystate. Notethatthecollapseofthe Foundationof China (GrantNos. 60878004and 11074087), Rabioscillationsofthe coherence|ρ14| is farfromcomplete the Ministry of Education under project SRFDP (Grant No. whenthediscorddisappears. Thebehaviourofthediscordis 200805110002),andtheNaturalScienceFoundationofHubei differentfor the initial entangled state. In this case, the dis- Province(GrantNo. 2010CDA075). corddiffersfromzeroforalltimesthatmakesustoconclude thattheinitialentangledstatewipesoutzerosofthediscord. 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