Coherence extraction from measurement-induced disturbance Xueyuan Hu1∗ and Heng Fan2 1
arXiv:1508.01978v1 [quant-ph] 9 Aug 2015
School of Information Science and Engineering, and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Jinan, 250100, P. R. China 2 Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: August 11, 2015) We establish an operational connection between coherence and quantum correlation by showing that, Alice can create quantum coherence on Bob’s side using local selective measurement as long as the previously shared bipartite state has non-vanishing quantum correlation on B, and the maximum average coherence she can extract can not surpass the B-side quantum correlation in the initial bipartite state. Steering-induced coherence is introduced to characterize Alice’s ability to extract quantum coherence on Bob’s side. For pure states and the maximally correlated states, the steering-induced relative entropy of coherence is proved to reach the initially shared quantum correlation which is measured by the relative entropy of B-side measurement-induced disturbance. The condition to reach the upper bound of quantum correlation varies for different measures of coherence. While the steering-induced l1 -norm of coherence is shown reach the initially shared quantum correlation (measured by trace-norm of B-side measurement-induced disturbance) for any two-qubit states, an example is found whose steering-induced relative entropy of coherence is strictly less than the relative entropy of B-side measurement-induced disturbance. PACS numbers: 03.67.Hk,03.65.Ta, 03.65.Yz, 03.67.Mn
Introduction.—Quantum coherence, being at the heart of quantum mechanics, enables quantum information tasks such as quantum algorithms [1] and quantum key distribution [2]. Analogs to quantum entanglement [3], coherence is recently identified as a resource of quantum information, which can not be created or increased under certain operations [4]. When generalized to multipartite systems, coherence is closely related to entanglement and discord-type quantum correlations. The coherence of an open system is frozen under the identical dynamical condition where discord-type quantum correlation is shown to freeze [5]. Further, discord-type quantum correlation can be interpreted as the minimum coherence of a multipartite system on tensor-product basis [6]. An operational connection between coherence and entanglement is presented in Ref. [7], where they try to build entanglement between a coherent system and an incoherent ancilla using incoherent operations, and the generated entanglement is bounded from above by the initial coherence. The converse procedure is of equal importance: to extract local coherence from a spatially separated but quantum correlated bipartite system. Quantum steering, which means that Alice can remotely change Bob’s state by her local selective measurement if they are correlated, is a natural candidate method to accomplish the task of coherence extraction. The Einstein-Podolsky-Rosen (EPR) steering, has long been noted as a distinct nonlocal quantum effect [8] and has attracted recent research interest both theoretically and experimentally [9–12]. Meanwhile, the quantum steering ellipsoid [13–15] has been used as an effective tool to explore the nonlocal properties of two-qubit states [16–18]. In a recent work [19], we have studied quantitatively the
power of Alice’s local probabilistic measure to create coherence on Bob’s side, and found that the power reaches unity as long as Alice and Bob share an entangled pure state and vanishes only for B-side classical states. In this paper, we introduce the steering-induced coherence for bipartite quantum states. The eigenbasis of Bob’s reduced state is used as the reference basis for quantifying coherence, and obviously, Bob is initially in an incoherent state. Alice’s local projective measurement can steer Bob to a new state which might be coherent. The steering-induced coherence C¯ is then defined as the maximal average coherent of Bob’s steered states that can be created by Alice’s selective projective measurement. For quantifying coherence, both relative entropy of coherence C r and l1 -norm of coherence C l1 are used. For both measures, we prove that the steering induced coherence can not surpass the initially shared B-side quantum correlation, which is quantified by measurement-induced disturbance (MID) QB [20]. States whose steering-induced relative entropy of coherence C¯r can reach the upper bound QrB are distinguished as pure states and maximally correlated states. For two-qubit states, while C¯l1 can always reach QtB , we find an example of state whose C¯r is strictly less than QrB . This indicates that relative entropy and l1 -norm of coherence are truly different measures of coherence. Coherence and measurement-induced disturbance.—A state is said to be incoherent, if it can be written as [4] X pi |ξi ihξi |, (1) σΞ = i
on a fixed reference basis Ξ = {|ξi i}. The incoherent completely positive trace-preserving channel is defined
2 as ΛICPTP(·) =
X
Kn (·)Kn† ,
(2)
n
where the Kraus operators Kn satisfy Kn IΞ Kn† ⊂ IΞ . Here IΞ is the set of incoherent state on basis Ξ. According to Ref. [4], a proper coherence measure C(ρ, Ξ) of a quantum state ρ on a fixed reference basis Ξ should satisfy the following three conditions. (C1) C(ρ, Ξ) = 0 iff ρ ∈ IΞ . (C2) Monotonicity P under selective measurements on average: C(ρ, Ξ) ≥P n pn C(ρn , Ξ), ∀{Kn } satisfying Kn IΞ Kn† ⊂ IΞ and n Kn† Kn = I, where ρn = Kn ρKn† /pn , occurring with probability pn = tr[Kn ρKn† ], is P to outcome n. (C3) Convexity: Pthe state corresponding p C(ρ , Ξ) ≥ C( n n n pn ρn , Ξ). n A candidate of coherence measure is the minimum distance between ρ and an incoherent state C(ρ, Ξ) = min D(ρ, σ), σ∈IΞ
(3)
where D(·, ·) is a distance measure on quantum states. As proved in Ref. [4], when D(·, ·) is the relative entropy Dr (ρ, σ) = S(ρ||σ) ≡ Tr(ρ log2 ρ − ρPlog2 σ) or the l1 matrix norm Dl1 (ρ, σ) = kρ − σkl1 ≡ ij |ρij − σij |, the corresponding coherence C r or C l1 satisfies all of the three conditions (C1-C3). Further, the optimal incoherΞ ent state to reach P the minimum in Eq. (3) is just Λ (ρ), Ξ where Λ (·) ≡ i |ξi ihξi |(·)|ξi ihξi | is the projective measurement on basis Ξ. Hence the coherence of ρ on the reference basis Ξ can be written as C(ρ, Ξ) = D(ρ, ΛΞ (ρ)),
(4)
where the distance D can be chosen as Dr or Dl1 . Introduced in Ref. [20], measurement-induced disturbance (MID) characterizes the quantumness of correlations. MID of a bipartite system ρ is defined as the minimum disturbance caused by local projective measurements that do not change the reduced states ρA ≡ TrB (ρ) and ρB ≡ TrA (ρ) Q(ρ) = inf D(ρ, ΛEAA ⊗ ΛEBB (ρ)), EA ,EB
(5)
where the infimum is taken over projective measurements which satisfy ΛEAA (ρA ) = ρA and ΛEBB (ρB ) = ρB , and D(·, ·) is a distance on quantum states. Comparing Eq. (5) with Eq. (4), we find MID is just the coherence of the bipartite state ρ on the local eigenbasis EA ⊗ EB . For later convenience, we introduce B-side MID as QB (ρ) =
inf
E EB :ΛBB (ρB )=ρB
D(ρ, IA ⊗ ΛEBB (ρ)).
(6)
QB goes to zero for B-side classical states, which can be P B ⊗ |eB written as ρB−cla = i ρA i i ihei |, while Q is strictly positive for ρB−cla if ∃i, [ρA , ρi ] 6= 0. Notice that for QB one do not have a coherence interpretation.
FIG. 1: (color online). Scheme for creating Bob’s coherence by Alice’s local measurement. When Alice implements local projective measurement on basis ΞA = {|ξiA i}, she gets result i with probability pξi and meanwhile steer Bob’s state to ρξBi which can be coherent on Bob’s initial eigenstate EB . Steering-induced coherence is defined as the maximal average coherence of states ρξBi that can be created by Alice’s local selective measurement.
Definition of steering-induced coherence.—As shown in Fig. 1, Alice and Bob initially share a quantum correlated state ρ, and Bob’s reduced state ρB is incoherent on his own basis. Now Alice implement a local projective measurement on basis ΞA and obtain result i with probability pξi , and Bob is “steered” to a coherent state ρξBi . We introduce the concept of steering-induced coherence for characterizing Alice’s ability to create Bob’s coherence on average using her local selective measurement. Definition (Steering-induced coherence). For a bipartite quantum state ρ, Alice’s local projective measurement ξiA = |ξiA ihξiA | can steer Bob’s state to ρξBi = hξiA |ρ|ξiA i/pξi with probability pξi = tr[ρ(ξiA ⊗ I)]. Let EB = {|eB j i} (j = 0, · · · , dB − 1) be the eigenbasis of reduced states ρB . The steering-induced coherence is defined as the maximum average coherence of Bob’s steered states on the reference basis EB # " X ξ ξ i ¯ = inf max (7) p i C(ρ , EB ) . C(ρ) EB
ΞA
B
i
where the maximization is taken over all of Alice’s projective measurement basis ΞA = {ξiA } (i = 0, · · · , dA −1), and the infimum over EB is taken when ρB is degenerate and hence EB is not unique. Since Bob’s initial state ρB is incoherent on its own ¯ basis EB , the steering-induced coherence C(ρ) describes the maximum ability of Alice’s local selective measurement to create Bob’s coherence on average. We prove the ¯ following properties for C(ρ). ¯ ¯ (E1) C(ρ) ≥ 0, and C(ρ) = 0 iff ρ is a B-side classical state. (E2) Non-increasing under Alice’s local completely¯ A ⊗ I(ρ)) ≤ C(ρ). ¯ positive trace-preserving channel: C(Λ (E3) Monotonicity under Bob’s selective measureP local ¯ n ), ∀{KnB } satis¯ C(ρ p ments on average: C(ρ) ≥ n n fying KnB IEB KnB† ⊂ IEB , where ρn = IA ⊗ KnB ρ(IA ⊗ KnB )† /pn and pn = tr[IA ⊗ KnB ρ(IA ⊗ KnB )† ].
3 (E4) Convexity:
P
n
¯ n ) ≥ C( ¯ P pn ρn ). pn C(ρ n
Proof. Condition (E1) can be proved using the method in Ref. [19], where it is proved that Ci (ρ) ≡ maxξiA C(ρξBi , EB ) vanishes iff ρ is a B-side classical state. (E2) is verified by noticing that the local channel ΛA can not increase the set of Bob’s steered states, and hence the optimal steered states {ρξBi } may not be steered to after the action of channel ΛA . The conditions (E3) and (E4) are directly derived from conditions (C2) and (C3) for coherence. Relation between steering-induced coherence and MID.—Intuitively, Alice’s ability to extract coherence on Bob’s side should depend on the quantum correlation between them. The following theorem gives a quantitative ¯ and relation between the steering-induced coherence C(ρ) quantum correlation measured by B-side MID QB (ρ). Theorem 1. Choosing relative entropy as the distance measure in the definition of MID and coherence, the steering-induced coherence is bounded from above by the B-side measurement-induced disturbance, i.e., C¯r (ρ) ≤ QrB (ρ).
(8)
Proof. We start with the situation that ρB is nondegenerate and hence one do not need to take the infimum in Eqs. (5) and (7). By definition, we have QrB (ρ) = S(ρkρEB ),
(9)
where ρEB = I ⊗ ΛEB (ρ). After Alice implement a selective measurement on basis ΞA , the average coherence of Bob’s state becomes X pξi S(ρξBi kΛEB (ρξBi )) C¯Ξr A (ρ) = i
=
X i
pξi S
A E A† ! B ξiA ρξiA†
ξi ρ ξi . (10) pξi pξi
The second equality holds because S(ρξBi kΛEB (ρξBi )) = ξ A ρξ A†
S(ξiA ⊗ ρξBi kIA ⊗ ΛEB (ξiA ⊗ ρξBi )) and ξiA ⊗ ρξBi = i pξii . Since selective measurement does not increase the relar tive entropy [21], we have C¯Π (ρ) ≤ QrB (ρ), ∀ΠA , and A hence Eq. (8) holds. The generalization to degenerate state is straight forward. We choose EoB to reach the infimum of r QB , which may not be the optimal eigen-basis for Eo r r B ¯
C . Hence we have Q (ρ) = S ρ IA ⊗ ΛB (ρ) ≥ P maxΞA i pξi C r (ρξBi , EoB ) ≥ C¯r (ρ). According to theorem 1, Bob’s maximal coherence that can be extracted by Alice’s local selective measurement is bounded from above by the initial quantum correlation between them. In the following, we prove that the bound can be reached for certain states.
Theorem 2. The steering-induced coherence can reach B-side MID C¯r (ρ) = QrB (ρ).
(11)
for the following states. (a) Pure states ρ = |ΨihΨ|. (b) Mixed states whose pure state decompositions share the basis, i.e., ρ = P P same Schmidt decomposition λ |jji. These are the soq |Ψ ihΨ | with |Ψ i = ij i i i i j i called maximally correlated state ρmc [22]. Proof. (a) Any pure state P can be written in a Schmidt decomposition form |Ψi = i λi |iii with λi 6= 0. Hence P ρB = i |λi |2 |iihi| has eigenbasis EB = {|ii}. Let the P projective measurement ΛEBB (·) = j |jiB hj|(·)|jiB hj|, which does not change the reduced state ρB , be implemented, and the bipartite state ρ becomes ρEB = P 2 EB is incoherent on basis {|iji}, i |λi | |iiihii|. Since ρ the B-side MID X |λi |2 log2 |λi |2 . QrB (ρ) = S(ρkρEB ) = C r (ρ, {|iji}) = − i
(12) In order to extract the maximum average coherence on Bob’s side, Alice measures her quantum system on Pd−1 2πkj basis ΞA , where |ξkA i = √1d j=0 e−ı d |ji and d is the number of λi . On the measurement result k, Bob’s state P 2πkj is steered to |φξk i = i eı d λi |ii, which happens with probability pξk = d1 . The coherence of steered state |φξk i on basis EB is then X |λi |2 log2 |λi |2 = QrB (ρ), (13) C r (|φξk i, EB ) = − i
and hence we arrive at Eq. (11). (b) Following similar method of (a), we check that C¯r (ρ) = QrB (ρ) = C r (ρ, {|iji}) X (qi |λij |2 ) log2 (qi |λij |2 ) − S(ρ). (14) = − i
Two-qubit case, relation between steering-induced l1 norm of coherence and trace-norm distance of MID.— For single-qubit states ρBpand σB , both the trace-norm , σB )† (ρB , σB ) and the l1 distance Dt (ρB , σB ) ≡ tr (ρBP B B matrix norm ||ρB − σB ||l1 ≡ i,j |ρi,j − σi,j | equal to ρ σ ρ ρ |r − r |, where r and r are Bloch vectors of ρB and σB respectively. Therefore, the l1 norm of coherence for a single-qubit state ρB can be written as C l1 (ρB , Ξ) = Dt (ρB , ΛΞ (ρB )).
(15)
Now we consider a two-qubit state ρ, and employ C l1 in ¯ as in Eq. (7). We prove that, when the definition of C(ρ) the trace-norm distance Dt is chosen in the definition of QB , the steering-induced coherence C¯l1 (ρ) of a two-qubit state can always reach the trace-norm distance of B-side MID QB .
4 Theorem 3. For a two-qubit state ρ, we have C¯l1 (ρ) = QtB (ρ).
(16)
Proof. P The state of a two-qubit state can be written as 3 ρ = 14 i,j=0 Θij σiA ⊗ σjB , where the coefficient matrix Θij = tr(ρσiA ⊗ σjB ) can be written in the block form 1 bT Θ= . a T For non-degenerate case b 6= 0, we choose the eigenbasis of ρB for the basis of density matrix and hence b = (0, 0, b3 ). Further, we can choose a proper basis of qubit A such that the matrix T is in a triangle form with T11 = T12 = T21 = 0. We calculated the explicit form of QtB (ρ) and C¯l1 (ρ) and obtain 2 2 2 T22 + T31 + T32 t l1 ¯ QB (ρ) = C (ρ) = 2 p 12 2 + T 2 )2 + 2T 2 (T 2 − T 2 ) + T 4 (T32 22 31 32 22 31 (. 17) + 2 For degenerate case with b = 0, we can always chose proper local basis such that T is diagonal. Here we impose T11 ≥ T22 ≥ T33 without loss of generality. Direct calculations lead to QtB (ρ) = C¯l1 (ρ) = T22 .
(18)
We check that, for the state ρ = 12 |Φ+ ihΦ+ |+ 21 |01ih01|, we have C¯r (ρ) < QrB (ρ), but according to theorem 3, C¯t (ρ) = QtB (ρ). It means that relative entropy of coherence and l1 -norm of coherence are truly different measures of coherence. Conclusions.—In this paper, we introduced the notion of steering-induced coherence which characterizes the power of Alice’s selective measurement to remotely create quantum coherence on Bob’s side. To quantify the coherence, we use both relative entropy of coherence C r and l1 -norm of coherence C l1 . When C r is employed, we prove that steering-induced coherence is bounded above by the initial quantum correlation which is measured by B-side MID QrB . The bound can be reached when the initial state shared between Alice and Bob is a pure state or a maximally mixed state. Later, steering-induced l1 norm of coherence C¯l1 is studied for two-qubit states. We prove that C¯l1 can always reach the initially shared quantum correlation QtB . However, we find an example of two-qubit state whose steering-induced relative entropy of coherence is strictly less than its quantum correlation QrB . It indicates that the relative entropy and l1 -norm of coherence are truly different measures of coherence. Note added.—On accomplishing this paper, we became aware of the related Ref. [23], in which the so-called local quantum-incoherent operations and classical commu-
nication (LQICC) is employed in the task of coherence extraction. This work was supported by NSFC under Grant No. 11447161, the Fundamental Research Funds of Shandong University under Grant No. 2014TB018, and the National Key Basic Research Program of China under Grant No. 2015CB921003.
∗ Electronic address:
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