Necessary and sufficient conditions for local creation of quantum correlation
aa r X i v : . [ qu a n t - ph ] M a r Necessary and sufficient conditions for local creation of quantum correlation
Xueyuan Hu, ∗ Heng Fan, D. L. Zhou, and Wu-Ming Liu
Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: November 2, 2018)Quantum correlation can be created by local operation from some initially classical states. Weprove that the necessary and sufficient condition for a local trace-preserving channel to createquantum correlation is that it is not a commutativity-preserving channel. This condition is validfor arbitrary finite dimension systems. We also derive the explicit form of commutativity-preservingchannels. For a qubit, a commutativity-preserving channel is either a completely decohering channelor a mixing channel. For a three-dimension system (qutrit), a commutativity-preserving channel iseither a completely decohering channel or an isotropic channel.
PACS numbers: 03.65.Ud, 03.65.Yz, 03.67.Mn
Quantum correlation is the unique phenomenon ofquantum physics and believed to be a resource for quan-tum information processes which can generally surpassthe corresponding classical schemes. Many previousstudies focus on entanglement, a well-known quantumcorrelation, since its apparent role in teleportation, su-perdense coding [1, 2], etc. Recently, measures of thenonclassicalness of correlation, such as quantum discord[3] and quantum deficit [4, 5], began to attract much at-tention since the discovery that some quantum informa-tion schemes can be realized without entanglement butwith a positive quantum discord [6, 7]. Much progresshas been made to quantify the amount of quantum cor-relation in different physical systems [8, 9] and to giveit intuitive and operational interpretations. It is shownthat quantum discord can be operationally interpreted asthe minimum information missing from the environment[10]. One-way quantum deficit [11, 12] has been found asthe reason for entanglement irreversibility [13] and canbe related to quantum entanglement via an interestingscheme [14, 15].Quantum noise usually plays a destructive role inquantum information process. However, there are sit-uations that local quantum noise can enhance nonlocalquantum properties for some mixed quantum states. Forexample, local amplitude damping can increase the av-erage teleportation fidelity for a class of entangled states[16–18]. Quantum discord can also be increased or cre-ated by local noise [19–21]. An interesting result is thatany separable state with positive quantum discord canbe produced by local positive operator-valued measure(POVM) on a classical state in a larger Hilbert space [22].In fact, almost all states in the Hilbert space containsquantum correlation, and an arbitrary small disturbancecan drive a classical state into a quantum state withnonzero quantum correlation [23]. Counterintuitively, ithas recently been discovered that mixedness is as impor-tant as entanglement for quantum correlation. In partic- ∗ Electronic address: [email protected] ular, some mixed states contain more quantum discordthan that of maximally entangled pure state when thedimension of the system is large enough [15]. Thus it isof interest to know how is the effect of mixedness on thequantum correlation of quantum states. The conditionfor local increase of quantum correlation has been derivedfor the qubit case [24], and it has been pointed out thatthis condition is not valid for high-dimension systems.In this article, we derive a simple necessary and suf-ficient condition for a local channel to create quantumcorrelation in some half-classical states, which is valid forarbitrary finite dimension systems. A trace-preserving lo-cal channel can create quantum correlation if and only ifit is not a commutativity-preserving channel. For qubitcase, we show that a commutativity-preserving channelis either a mixing channel or a completely decoheringchannel. This confirms the result in Ref. [24]. For thequtrit case, quantum correlation can be created by a lo-cal channel in some half-classical input states if and onlyif the channel is neither a completely decohering channelnor an isotropic channel. We also analyze the reason fora local mixing channel to create quantum correlation inqutrit situation and then give a conjecture to extend theresult of qutrits to arbitrary finite dimension systems.The total correlation between two quantum systems iscomposed of classical and quantum correlations. Fromthis point of view, quantum correlation is defined asthe difference between total and classical correlations.Therefore, various measures of quantum correlation de-fined on one party of a composite system vanish for ex-act the same class of states, which is called half-classicalstates. Because classical correlation is defined by the cor-relation that can be revealed by local measurements, astate ρ AB is half-classical on B if and only if there exista measurement on B that does not affect the total state.As proved in Ref. [25] a half-classical state on B can bewritten as ρ AB = X i p i ρ α i A ⊗ | α i i B h α i | . (1)where {| α i i B } consist an orthogonal basis for the Hilbertspace of subsystem B , and ρ α i A are corresponding densitymatrices of A . The subsystem A can be a single quantumparticle or an ensemble of quantum particles. In the fol-lowing, by quantum correlation, we mean quantum cor-relation defined on subsystem B . The main purpose ofthis paper is to characterize the channel Λ B satisfyingI A ⊗ Λ B ( ρ AB ) ∈ D , ∀ ρ AB ∈ D , (2)where D is the set of half-classical states. Before provid-ing the condition, we first introduce a class of quantumchannels, which we call commutativity-preserving chan-nels.Definition 1 (commutativity-preserving channel): acommutativity-preserving channel Λ CP is the channelthat can preserve the commutativity of any input den-sity operators, i.e.,[Λ CP ( ξ ) , Λ CP ( ξ ′ )] = 0 (3)holds for any density operators satisfying [ ξ, ξ ′ ] = 0.It is worth mentioning an equivalent definition of acommutativity-preserving channel. A channel Λ is acommutativity-preserving channel if and only if[Λ( φ ) , Λ( ψ )] = 0 (4)holds for any pure states satisfying h φ | ψ i = 0. The “onlyif” part is obtained directly by choosing ξ = | φ ih φ | and ξ ′ = | ψ ih ψ | . Conversely, if Eq. (4) holds, by writing ξ and ξ ′ on their common eigenbasis, we arrive at Eq. (3).Now we are ready to prove the first main result of thispaper. It holds for arbitrary finite-dimension systems.Theorem 1: A channel Λ acting on subsystem B cancreate quantum correlation between subsystems A and B for some input half-classical state ρ AB if and only if it isnot a commutativity-preserving channel.Proof: Any separable state can be written as ξ AB = X i p i ξ Ai ⊗ ξ Bi , (5)where ξ Ai are linearly independent. We will first provethat ξ AB is a half-classical state if and only if[ ξ Bi , ξ Bj ] = 0 , ∀ i, j. (6)For proving the “only if” part, we notice that for anyhalf-classical state, there exist a measurement basis Π α i B that does not affect the state. Therefore, X i p i ξ Ai ⊗ ( ξ Bi − Π α j B ξ Bi Π α j B ) = 0 . (7)Because ξ Ai are linearly independent, ξ Bi is diagonal on { Π α j B } and thus satisfies Eq. (6). Conversely, if Eq. (6)holds, ξ Bi and ξ Bj share common eigenvectors for any i and j . By choosing these eigenvectors as the basis forvon Neumann measurement, the state does not changeafter the measurement, which means that ξ AB is a half-classical state. Now consider an arbitrary half-classical state in form of Eqs. (5) and (6) as the input state,the channel Λ acting on subsystem B leads the state to ξ ′ AB ≡ I A ⊗ Λ B ( ξ AB ) = P i p i ξ Ai ⊗ Λ( ξ Bi ), which is still ahalf-classical state if and only if[Λ( ξ Bi ) , Λ( ξ Bj )] = 0 , (8)for arbitrary choice of ξ Bi and ξ Bj satisfying Eq. (6). Thisis just the definition of a commutativity-preserving chan-nel. Therefore, the channel Λ can create quantum cor-relation for some input half-classical states if and only ifit is not a commutativity-preserving channel. This com-pletes the proof.The rest of this paper is devoted to expose the exactform of a commutativity-preserving channel.Since [I , ρ ] = 0 , ∀ ρ , we obtain a necessary condition fora commutativity-preserving channel[Λ(I) , Λ( ρ )] = 0 , ∀ ρ. (9)When B is a qubit, Eq. (9) is also the sufficient condition.The reason is as follows. By using the linearity of Λ, theleft hand side of Eq. (4) can be written as12 [Λ( | φ ih φ | + | ψ ih ψ | ) , Λ( | ψ ih ψ | − | φ ih φ | )]= 12 [Λ(I) , Λ( uσ z u † )] , (10)where | ψ i = u | i and | φ i = u | i . Since any qubit state ρ can be decomposed as ρ = (I + n x σ x + n y σ y + n z σ z ) / M as S (Λ M ( ρ S )) ≥ S ( ρ S ) , ∀ ρ S , (11)where S ( ρ ) ≡ − Tr( ρ log ρ ) is the von Neumann entropy.It is worth mentioning that when a channel is a mixingchannel, its extension to larger systems I A ⊗ Λ M S is still amixing channel. As proved in Ref. [26], a mixing channelis equivalent to a unital channel.Case 2: Λ(I) = I. Then the diagonal basis of Λ(I) isspecified. According to Eq. (8), the two matrices Λ( ρ )and Λ(I) share common eigenvectors. In other words, thechannel Λ takes any input state ρ to a diagonal form onthe eigenbasis of Λ(I), and is thus a completely decoher-ing channel.Therefore, when B is a qubit, a commutativity-preserving channel is either a mixing channel or a com-pletely decohering channel. This confirms the result inRef. [24].In the following, we will move on to study the ex-act form of a commutativity-preserving channel for high-dimension cases.Definition 2 (isotropic channel) An isotropic channel isof the form Λ iso ( ρ ) = p Γ( ρ ) + (1 − p ) I d , (12)where Γ is any linear channel that preserves the eigen-values of ρ . According to Ref. [27], Γ is either a unitaryoperation or unitarily equivalent to transpose. Parameter p is chosen to make sure that Λ is a completely positivechannel. In particular, − / ( d − ≤ p ≤ − / ( d − ≤ p ≤ / ( d + 1) whenΓ is unitarily equivalent to transpose.Theorem 2: Consider the half-classical input state inEq. (1) with B a qutrit, a channel Λ can not createquantum correlation in any half-classical input state ifand only if Λ is either a completely dechering channel oran isotropic channel.Proof: Writing the eigen-decomposition of Λ(I) asΛ(I) = N X i =1 λ i I r i . (13)Here P Ni =1 r i = P Ni =1 r i λ i = 3, λ i ≥ r i are positive in-tegers, and I r i are identities of the r i -dimension subspace V r i . From Eq. (9) we haveΛ( ρ ) = N X i =1 q i ξ ρr i , ∀ ρ. (14)where ξ ρr i is a density operator on V r i .Clearly, when the eigenvectors of Λ(I) are nondegener-ate, i.e., N = 3 and Eq. (13) becomes Λ(I) = P i =1 λ i Π i ,the channel Λ is a completely decohering channel, since ittakes any input state ρ to a diagonal form on basis { Π i } .When two or three eigenvectors of Λ(I) are degenerate,we study the eigendecomposition of Λ( φ ) for a pure inputstate | φ i Λ( φ ) = N φ X i =1 λ φi I r i ( φ ) , (15)where N φ ≥ N and V r i ( φ ) ⊆ V r j . When none of Λ( φ )breaks the degeneracy of eigenvectors of Λ(I), i.e., N φ = N and V r i ( φ ) = V r i , the channel is also a completelydecohering channel. Now we focus on the case that someΛ( φ ) can break the degeneracy of eigenvectors of Λ(I),i.e., N φ > N and V r i ( φ ) ⊂ V r j for some i . Let {| φ k i} k =0 be a basis of the three-dimension Hilbert space and | φ i be the pure input state whose corresponding output stateΛ( φ ) has the most different eigenvalues. It means that N φ ≥ N φ , ∀ φ .Case 1: For any state | φ ⊥ i = c | φ i + c | φ i which isorthogonal to | φ i , we have N φ ⊥ = N φ and V r i ( φ ⊥ ) = V r i ( φ ) .Then for arbitrary input state ϕ = P i =0 c i | φ i i ,we have [Λ( ϕ ) , Λ( c ∗ | φ i − c ∗ | φ i )] = 0 . (16)Therefore, Λ( ϕ ) is diagonal on the same basis as Λ( φ ).Case 2: There exist a pure state, say | φ i , whose cor-responding output state Λ( φ ) does not break as much degeneracy as Λ( φ ), i.e., N φ < N φ . We will firstprove that for any pure state | ϕ i = | φ i − β | φ i in2-dimension subspace W φ , the output state is diagonalon the same basis as Λ( φ ), say { Π i } . We introduce | ϕ ( β ) i = | ϕ i + β | φ i and | ϕ ( β ) i = β ∗ | φ i − | φ i .Notice that h ϕ ( β ) | ϕ ( β ) i = 0, we have[Λ( ϕ ( β )) , Λ( ϕ ( β ))] = 0 . (17)Because P k =0 | φ k ih φ k | = I, we have N φ = N φ and V r i ( φ ) = V r i ( φ ) . Therefore, Λ( ϕ ( β )) is diago-nal on { Π i } by noticing that [Λ( ϕ ( β )) , Λ( φ )] = 0.Since the channel cannot increase the distance betweenstates, Λ( ϕ ( β )) breaks the same degeneracy as Λ( φ )for sufficiently large | β | . From Eq. (17), we haveΛ( ϕ ( β )) and Λ( ϕ ( − β )) are diagonal on { Π i } . There-fore, Λ( ϕ ) = Λ( ϕ ( β )) + Λ( ϕ ( − β )) − | β | Λ( φ ) is alsodiagonal on { Π i } . Further, we will show that Λ( ϕ ( β ))is diagonal on { Π i } for arbitrary β . From Eq. (17),this is obvious when Λ( ϕ ( β )) is nondegenerate. Forthe case where Λ( ϕ ( β )) is degenerate, Λ( ϕ ( − β )) = | β | Λ( φ ) + Λ( φ ) − Λ( ϕ ( β )) is nondegenerate and con-sequently, Λ( ϕ ( − β )) is diagonal on { Π i } . Therefore,Λ( ϕ ( β )) = Λ( ϕ ) + | β | Λ( φ ) − Λ( ϕ ( − β )) is diagonalon { Π i } . Λ is a completely decohering channel.Case 3: now we are only left with the case that N φ ⊥ = N φ but I r i ( φ ⊥ ) = I r i ( φ ) , which can happen only whenΛ(I) = I and N φ k = 2. Therefore, we haveΛ( φ k ) = p Π( φ k ) + (1 − p ) I3 , (18)where Π( φ k ) is a basis determined by | φ k i . Notices that p is independent of | φ k i because of the linearity of Λ.Consequently, for any input state ρ = P i p i | α i ih α i | , wehave Λ( ρ ) = p P i p i Π( α i ) + (1 − p )I /
3. It means thatchannel Λ is an isotropic channel.Combining the three cases together, we conclude thatfor a qutrit, a commutativity-preserving channel is eithera completely decohering channel or an isotropic channel.Since depolarizing channel is a subset of mixing chan-nel, there exist mixing channels that are able to locallycreate quantum correlation. Therefore, mixedness cancontribute to creation of quantum correlations. Here wegive an example to look more closely at why a mixingchannel can create quantum correlation in states withhigh dimensions. Consider the following mixing channelΛ( · ) = P i E ( i ) ( · )E ( i ) † , where the Kraus operators areE (0) = | ih | , E ( i ) = e i u ( i )2 ( | ih | + | ih | ) , i = 1 , , · · · . (19)Here u ( i )2 are rank-2 unitary operators on basis {| i , | i} .This channel can create quantum correlation in the state ρ = ˜ ρ φA ⊗ | φ i B h φ | + ˜ ρ ψA ⊗ | ψ i B h ψ | if and only if Eq.(4) is violated. Writing the two orthogonal states as | φ i = P i =0 a i | i i and | ψ i = P i =0 b i | i i ( P i =0 a i b ∗ i = 0),we obtain the left hand side of Eq. (4)[ X i e i u ( i )2 | φ ih φ | u ( i ) † , X i e i u ( i )2 | ψ ih ψ | u ( i ) † ] , (20)where | φ i = a | i + a | i and | ψ i = b | i + b | i arereduced states on Hilbert space of dimension 2. There-fore, Eq. (4) is violated if and only if h φ | ψ i 6 = 0 ,
1. Twohigh-dimension orthogonal states may become unorthog-onal when reduced to Hilbert space of dimension two.This is just the reason for creating quantum correlationusing a local mixing channel. Isotropic channels act onall of the states in Hilbert spaces equivalently, so they arelikely the only subset of mixing channels which belongsto the class of commutativity-preserving channels. Thisobservation leads to the following conjecture.Conjecture: Consider the half-classical input state inEq. (1) where B is a d -dimension quantum system (qu-dit) with d ≥
3, a channel Λ can not create quantumcorrelation in any half-classical input state if and only ifΛ is either a completely dechering channel or an isotropicchannel.We further prove that mixing channel can not increasethe teleportation fidelity of any two-qudit state. Theaverage teleportation fidelity f is related to the maximumsinglet fraction (MSF) [28] F = max Φ h Φ | ρ | Φ i as f =( dF + 1) / ( d + 1). After the action of mixing channel on B , the MSF becomes F ′ = Tr( ρ Ξ) (21) where Ξ = P i I ⊗ E ( i ) † | Φ ih Φ | I ⊗ E ( i ) . Notice thatfor a mixing channel Λ( · ) = P i E ( i ) ( · )E ( i ) † , its conjec-ture Λ ∗ ( · ) = P i E ( i ) † ( · )E ( i ) is also a mixing channel.Therefore, Ξ A = Ξ B = I /
2, so Ξ can be decomposedas a mixture of maximally entangled pure states Ξ = P i p i | Φ i ih Φ i | . Then we have F ′ = P i p i h Φ i | ρ | Φ i i ≤ F .Therefore, average teleportation fidelity can never be in-creased by mixing channel. This result suggests thatquantum correlation created by mixing channel may notbe a useful resource for quantum information tasks.In summary, we have proved that the necessary andsufficient condition for a local operation to create quan-tum correlation in some half-classical state is that itis not a commutativity-preserving channel. When thesubsystem B affected by the local channel is a qubit,a commutativity-preserving channel is either a mixingchannel or a completely decohering channel. This resultconfirms the results in Ref. [24]. When B is a qutrit,we have proved that a commutativity-preserving channelis either an isotropic channel or a completely decoheringchannel. This result is likely to be extended to arbitraryfinite dimension situation.Hu thanks Sixia Yu and Chengjie Zhang for help-ful discussions. This work is supported NSFC un-der grants Nos. 10934010, 60978019, the NKBRSFCunder grants Nos. 2009CB930701, 2010CB922904,2011CB921502, 2012CB821300, NSFC-RGC undergrants Nos. 11061160490, 1386-N-HKU748/10, andCNSF under grants Nos. 10975181 and 11175247. [1] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[2] P. Hausladen, R. Jozsa, B. Schumacher, M. Westmore-land, and W. K. Wootters, Phys. Rev. A , 1869 (1996).[3] H. Ollivier and W. H. Zurek, Phys. Rev. Lett. , 017901(2001).[4] J. Oppenheim, M. Horodecki, P. Horodecki, andR. Horodecki, Phys. Rev. Lett. , 180402 (2002).[5] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppen-heim, A. Sen(De), U. Sen, and B. Synak-Radtke, Phys.Rev. A , 062307 (2005).[6] A. Datta, A. Shaji, and C. M. Caves, Phys. Rev. Lett. , 050502 (2008).[7] L. Roa, J. C. Retamal, and M. Alid-Vaccarezza, Phys.Rev. Lett. , 080401 (2011).[8] S. Luo, Phys. Rev. A , 042303 (2008).[9] M. Ali, A. R. P. Rau, and G. Alber, Phys. Rev. A ,042105 (2010).[10] P. J. Coles, arXiv: 1110.1664v1 (2011).[11] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. , 1 (2009).[12] W. H. Zurek, Phys. Rev. A , 012320 (2003).[13] M. F. Cornelio, M. C. de Oliveira, and F. F. Fanchini,Phys. Rev. Lett. , 020502 (2011).[14] A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev.Lett. , 160401 (2011). [15] M. Piani, S. Gharibian, G. Adesso, J. Calsamiglia,P. Horodecki, and A. Winter, Phys. Rev. Lett. ,220403 (2011).[16] P. Badzi¸ag, M. Horodecki, P. Horodecki, andR. Horodecki, Phys. Rev. A , 012311 (2000).[17] Y. Yeo, Phys. Rev. A , 022334 (2008).[18] X. Hu, Y. Gu, Q. Gong, and G. Guo, Phys. Rev. A ,054302 (2010).[19] X. Hu, Y. Gu, Q. Gong, and G. Guo, Phys. Rev. A ,022113 (2011).[20] F. Ciccarello and V. Giovannetti, Phys. Rev. A ,010102 (2012).[21] F. Ciccarello and V. Giovannetti, Phys. Rev. A ,022108 (2012).[22] N. Li and S. Luo, Phys. Rev. A , 024303 (2008).[23] A. Ferraro, L. Aolita, D. Cavalcanti, F. M. Cucchietti,and A. Ac´ın, Phys. Rev. A , 052318 (2010).[24] A. Streltsov, H. Kampermann, and D. Bruß, Phys. Rev.Lett. , 170502 (2011).[25] A. Datta, arXiv:0807.4490v1 (2008).[26] A. Wehrl, Rev. Mod. Phys. , 221 (1978).[27] M. Marcus and B. N. Moyls, Canad. J. Math. , 61(1959).[28] M. Horodecki, P. Horodecki, and R. Horodecki, Phys.Rev. A , 1888 (1999). rXiv:1112.3141v2 [quant-ph] 7 Mar 2012 ( D a t e d : N o v e m b e r , ) A b s tr a c t1