Connections of Coherent Information, Quantum Discord, and Entanglement

Transcription

1 Commun. Theor. Phys. 57 (212) Vol. 57, No., April 15, 212 Connections of Coherent Information, Quantum Discord, and Entanglement FU Hui-Juan ( ), LI Jun-Gang (Ó ), ZOU Jian (Õ ), and SHAO Bin (ÅÉ) School of Physics, Beijing Institute of Technology, Beijing 181, China (Received November 7, 211; revised manuscript received January, 212) Abstract For a pure non-marovian dephasing model we derive analytic expressions of coherent information, quantum discord, and entanglement. We find that for the cases of the initial Werner states, the dynamical behavior of coherent information is similar to that of quantum discord but different from that of entanglement. Coherent information, as well as quantum discord, can reveal the quantum correlations in some mixed-states, in which the entanglement is zero. PACS numbers: 3.67.Mn, 3.65.Ud, 3.65.Yz Key words: coherent information, quantum discord, entanglement 1 Introduction Quantum information technology [1 2] largely relies on a precious resource, quantum correlation, a highly nontrivial manifestation of the coherent superposition of states of composite quantum systems. Recently, lots of interests have been devoted to the definition and understanding of correlations in quantum systems. Two popular measures to describe such quantum correlations are quantum entanglement and quantum discord. Quantum entanglement, as a main physical resource, plays a central role in the application of quantum communication and quantum information. [1 2] Quantum discord, originally introduced by Olliver and Zure [3] is defined as the difference between the quantum mutual information [] and the classical correlations. [5] It has been shown [6 7] that quantum discord is a more general concept to measure quantum correlation than quantum entanglement since there is a nonzero quantum discord in some separable mixed states. [3] So, quantum discord is considered as a more universal resource than quantum entanglement in some sense and becomes more and more popular. [8 9] In realistic quantum information processing, quantum systems are always subjected to a dynamical evolution, which may represent the transmission of the quantum systems via a noisy quantum channel. [1 11] Thus it is of fundamental importance to now the influence of the channel on the quantum correlation. Imitating the concept of mutual information in classical information theory, Schumacher [12] and Barnum et al. [13] introduced a quantity, called coherent information, to estimate the amount of quantum information conveyed in the noisy channel. It is also found that coherent information can be used to measure the degree of quantum correlation of two qubits while one qubit undergoes some evolution that possibly involves interaction with the environment. Coherent information, as another measure of quantum correlation, may have some similar features and different features with quantum discord and entanglement of formation. The relation between coherent information and entanglement of formation has been discussed. [1] In this paper, we focus on the relationship among quantum discord, entanglement, and coherent information to investigate the connection between these quantities. We perform the study on some special density matrices representing two-level bipartite systems of which the analytic expressions for quantum discord can be obtained. Our principal results are presented here: We find that for the cases of Bell states as the initial states, the dynamics of quantum discord and that of coherent information are identical. When the initial states are Werner states, quantum discord is the same as the coherent information plus a constant. The paper is organized as follows. In Sec. 2 we recall three different definitions of quantum correlation. In Sec. 3, we present the model and its analytical solution. In Sec. the discussions are given. Finally we give a conclusion of our results in Sec Different Measurements of Quantum Correlation In this section we recall three different definitions of quantum correlation, namely, coherent information, quantum discord, and entanglement. 2.1 Coherent Information Coherent information [12] is defined as I c = S(ρ Q ) S e, (1) where S e is the entropy exchange, which is an intrinsic property of the system. It is a measure of the information exchanged between system Q and the environment during Supported by National Science Foundation of China under Grant Nos. 1158, 19716, and jungl@bit.edu.cn c 211 Chinese Physical Society and IOP Publishing Ltd

2 59 Communications in Theoretical Physics Vol. 57 dynamical evolution and its expression is S e = Trρ RQ log ρ RQ. (2) It is the von Neumann entropy of the final state of the joint system RQ (where the logarithm is taen to be base 2). We note here that the coherent information I c is an intrinsic quantity of Q, and depends only on the initial state ρ Q and the dynamical process. It has been shown that I c is a good measure for the capacity of a quantum channel to transmit the entanglement. [12] Different from its analogous quantity for classical systems, I c may be positive, negative, or zero. Thus it can be thought of as measuring the nonclassicity of the joint state ρ RQ, or, in other words, the degree of quantum entanglement retained by R and Q. That is to say, I c is a natural way to measure the degree of quantum coherence in the dynamical process. 2.2 Quantum Discord Quantum discord [3] is defined as the difference between the total correlation and the classical correlation with the following expression D(ρ) = I(ρ) C(ρ). (3) Here the total correlation in a bipartite quantum state ρ is measured by quantum mutual information given by I(ρ) = S(ρ R ) + S(ρ Q ) S(ρ), () where S(ρ) = Tr(ρ log ρ) is the von Neumann entropy, ρ R = Tr Q (ρ) and ρ Q = Tr R (ρ) are the reduced density operators for subsystems R and Q, respectively. And the classical correlation between the two subsystems R and Q can be defined as [3,5] C(ρ) = max [S(ρ R ) S(ρ {Π (Q) } })], (5) {Π (Q) where the maximum is taen over all projective measurements performed locally on qubit Q, described by a set of orthogonal projectors {Π (Q) } corresponding to the outcomes. In Eq. (5), S(ρ {Π (Q) }) is the conditional entropy defined as S(ρ {Π (Q) }) = p S(ρ ), with ρ = [(I (R) Π (Q) )ρ(i (R) Π (Q) )]/p the conditional density operator of qubit R after qubit Q is measured and the outcome is obtained, with probability p = Tr [(I (R) Π (Q) )ρ(i (R) Π (Q) )]. Equation (3) is the so-called quantum discord, and is interpreted as a measure of quantum correlations by Olliver and Zure. It can be shown that the quantum discord is always non-negative by expressing mutual information in terms of quantum relative entropy and invoing the monotonicity property of the latter. We note that the evaluation of quantum discord is a hard tas from a computational point of view. Fortunately, for the conditions considered in this paper, the optimization problem in the definition of the classical correlations can be solved exactly and a simple analytical expression for this quantity can be derived. We will give these results soon afterwards. 2.3 Entanglement of Formation For a given density matrix ρ, the entanglement of formation is given in terms of concurrence C(ρ) by the formula [15] ( C2 (ρ) ) E(ρ) = h, (6) 2 where h(x) = xlog x (1 x)log(1 x) and C(ρ) = λ 1 λ 2 λ 3 λ, (7) with λ i are the eigenvalues, in decreasing order, of the matrix R = ρ(σ y σ y )ρ T (σ y σ y ). 3 Model In this paper we consider two non-interacting qubits one of which is subjected to local decoherence channels without energy exchange. This type of model can be called one-sided quantum channel and has been investigated by in Konrad et al. Ref. [16]. It has been shown that given any one-sided quantum channel, the concurrence of output state corresponding to any initial pure input state of interest can always be equivalently obtained by the product of the concurrence of input state and that of the output state with the maximally entangled state as an input state. [16] Here, we mainly focus on the relationships of the entanglement, quantum discord and Coherent information. In what follows, we first consider the dynamics of the single qubit, and then derive the dynamics of the composite two-qubit system. Recent studies [17 33] have shown that non-marovian quantum processes play an increasingly important role in many fields of physics. In this paper we consider a twolevel system subjected to non-marovian dephasing channels. A model lie this describes, for example, a spin in presence of a magnetic field, having constant intensity but inverting its sign randomly in time. It is possible to write a time-dependent phenomenological Hamiltonian for this ind of system [2] H(t) = Γ(t)σ z, (8) here σ z is the Pauli operator and Γ(t) is a random variable, which can be written as Γ(t) = αn(t). The random variable n(t) has a Poisson distribution with a mean value equal to γ t, while α is a coin-flip random variable assuming the values ±a. The dynamics of this system has been studied in detail by Daffer et al. in Ref. [2] and us in Ref. [3]. It has been shown that this is a dephasing channel with colored noise. The reduced density matrix ρ S (t) for the qubit in the basis { e, g } can be written as [3] ( ) ρ S ρee () ρ eg ()P(t) (t) = ρ ge ()P, (9) (t) ρ gg ()

3 No. Communications in Theoretical Physics 591 where P(t) is a crucial characteristic function, which can be written as e γt[ cosh(µt) + γ ] µ sinh(µt), a < γ 2, P(t) = e γt (1 + t), a = γ 2, (1) e γt[ cos(µt) + γ ] µ sin(µt), a > γ 2, with µ = a 2 γ 2. From Eq. (9) we can find that the two diagonal elements of ρ(t) remain invariant under dephasing channel, and all off-diagonal elements damp with P(t), which approaching to zero at last. Thus, the steady state of a two-qubit system will be a mixture of the two base vectors, and the probabilities are determined by the initial density matrix. It should be noted that when a γ /2, P(t) monotonously decreases with increasing t; when a > γ /2, P(t) may be negative for some time intervals and then increases. The increase of P(t) can be interpreted as the reverse flow of information from the system to the environment, which implies the occurring of the non-marovian dynamics. [18] Let us now consider two non-interacting qubits one of which is coupled to local pure dephasing noise described by Eq. (9). The reduced density matrix ρ(t) for the two-qubit system in the standard product basis B = { 1 = ee, 2 = eg, 3 = ge, = gg } can be written as ρ 11 () ρ 12 () P(t)ρ 13 () P(t)ρ 1 () ρ 21 () ρ 22 () P(t)ρ 23 () P(t)ρ 2 () ρ(t) = P (t)ρ 31 () P (t)ρ 32 () ρ 33 () ρ 3 (). (11) P (t)ρ 1 () P (t)ρ 2 () ρ 3 () ρ () Having this genal density matrix in mind, we can study the dynamics of the quantum correlation for different initial states. Discussions In this section, we discuss some simple examples and give analytic expressions of the coherent information, the quantum discord, and the entanglement. Firstly, we consider a simple case that the initial state is Bell state Ψ + = ( eg + ge )/ 2. For this initial state, the density matrix at time t is 1/2 P(t)/2 ρ Ψ (t) = P (t)/2 1/2. (12) For this density operator, the optimization problem in the definition of the classical correlations can be solved exactly and a simple analytical expression for this quantity can be derived. Indeed, by calculating the action of the one-qubit projectors Π (Q) = I, with = a, b, (13) with a = cosθ e + e iφ sin θ g, b = sin θ e e iφ cosθ g, From Eq. (5), it is straightforward to prove that the classical correlations do not explicitly depend on φ and are maximized for θ = nπ/2 with n Z. The analytic expression for C(ρ) is 1. Therefore, the quantum discord can be written as with D(t) = 1 + λ 1 log λ 1 + λ 2 log λ 2, (1) λ 1 = (1 + P(t) )/2, λ 2 = (1 P(t) )/2, (15) are two non-zero eigenvalues of the density operator (12). Using Eqs. (1) and (2), we can obtain coherent information I c (t) of the density operator (12) as follows I c (t) = 1 + λ 1 log λ 1 + λ 2 log λ 2. (16) For the density operator (12), we can easily obtain the entanglement of formation taes the form: with E(t) = χ 1 log χ 1 χ 2 log χ 2, (17) χ 1 = (1 + 1 P 2 (t))/2, χ 2 = (1 1 P 2 (t))/2. (18) Comparing Eqs. (1) and (16) we can find that coherent information I c (t) and quantum discord D(t) are identical but they are different from the entanglement of formation E(t). The evolutions of the coherent information, the quantum discord, and the entanglement are depicted in Fig. 1(a) for a = 6γ. From Fig. 1(a) we can find that the dynamical behavior of coherent information is indeed identical to that of the quantum discord of the system, but a little different from that of the entanglement. It is obvious that all of them have the phenomenon of revival. The critical reason is that, the reservoir for our model here is non-marovian, so that the reservoir can in principle feedbac part of the information, which they have taen during the interaction with the systems. We note here that the same result can be obtained for the cases of the initial states being the other three Bell states. Then we can conclude that for the cases in which the initial states are Bell states, the coherent information can be used to evaluate the quantum discord.

4 592 Communications in Theoretical Physics Vol. 57 Fig. 1 (Color online) The evolution of coherent information (red circles), quantum discord (green triangles), and entanglement (dar squares) for different parameters and different initial states. (a) a = 6γ and the initial state is Φ = ( ee gg )/ 2. (b) a = 6γ, d =.9 and the initial state is ρ W(). (c) a = 6γ, d =.5 and the initial state is ρ W(). (d) a = 6γ, d =.3 and the initial state is ρ W(). As we now, quantum mechanics and quantum information processing are not constrained to Bell states. So we give the generalization of the discussions. Let us focus now on the Werner states [35] ρ W () = 1 d I + d Φ Φ, (19) where Φ = ( ee gg )/ 2. Werner states ρ W () consist of a mixture of a pure singlet state Φ with probability d ( d 1) and a fully mixed state with probability 1 d, expressed by the unit operator I defined in the -dimensional Hilbert space. Depending on the weight d, Werner states may be entangled (d > 1/3) or separable (d 1/3). [35 36] Moreover, they possess a highly conceptual and historical value because, in the probability range (1/3 < d < 1/ 2), they do not violate any Bell s inequality in spite of being in this range nonseparable entangled states, precisely, negative partial transpose states. [36 37] These states are also important for quantum information, since they model a decoherence process occurring on a Bell state traveling along a noisy channel. [1] From Eqs. (11) and (19), we can obtain the density matrix at time t, which taes the form d + 1 d 2 P(t) 1 d ρ W (t) = 1 d. (2) d 2 P d + 1 (t) For this state, the analytic result for quantum discord can be obtained from the general expression, [38] which is given as follows D(t, d) = 2 + ξ i log ξ i C, (21) where i=1 ξ 1 = (1 d)/, ξ 2 = (1 d)/, ξ 3 = (d + 1 2dP(t))/, ξ = (d dP(t))/ (22) are the eigenvalues of the density matrix, and C is the classical correlation, which can be written as C = 1 d log(1 d) d log(1 + d). (23) 2 2 Equation (23) shows that the classical correlation of the density matrix (2) only depends on the weight d and is independent from the time evolution of the system.

5 No. Communications in Theoretical Physics 593 Using Eqs. (1) and (2), we obtain the coherent information I c (t, b) = 1 + ξ i log ξ i. (2) i=1 From Eq. (6), we can obtain its entanglement of formation where E(t, d) = X log X (1 X)log(1 X), (25) X = (1 + 1 C 2 )/2, C = 2max{, dp(t)/2 (1 + d)/}. Comparing Eqs. (21) and (2), we find that D(t, d) = I c (t, d) + 1 C. (26) Equation (26) shows that for the case in which the Werner state as the initial state, the expression of quantum discord is equal to that of coherent information plus a constant 1 C. This means that we can use coherent information to evaluate quantum discord in the dynamical process, which is usually difficult to calculate. The evolutions of the quantum discord D(t, d), coherent information I c (t, d), and entanglement E(t, d) are depicted in Figs. 1(b) 1(d) for different values d. We can find that, when d is close to 1 (Fig. 1(b)), the dynamical behavior of the entanglement are very similar to that of the quantum discord, while the coherent information taes negative value and can coincide with the quantum discord when translating a constant. When d decreases, the entanglement shows sudden death and sudden birth phenomena and its dynamical behavior becomes more and more different from that of quantum discord and coherent information (see Fig. 1(c)). Compared to the entanglement, an amazing difference of the quantum discord and coherent information is that, even in the region of d 1/3 ((Fig. 1(d))) where the entanglement is zero, the quantum discord and coherent information still can capture the quantum correlations between the two qubits. That means, the quantum discord and the coherent information can reveal the quantum correlations in some states those are not entangled. We note here that the same result can be obtained for the case of the initial state is the other Werner state ρ W () = 1 d I + d Ψ Ψ, (27) where Ψ = ( eg ge )/ 2. For this initial state, it can be easily obtained that the eigenvalues of the density matrix at time t is equal to that of ρ W (t), so its quantum discord D (t, d), coherent information Ic (t, d) and entanglement E (t, d) are identical to those of ρ W (t). 5 Conclusion In summary, for the case of single-sided pure dephasing channel, we have derived the analytic expressions of coherent information, quantum discord, and entanglement for different initial states. We find that for the cases of Bell states being the initial states, the dynamics of coherent information is the same as that of quantum discord but is different from that of entanglement. When the initial states are Werner states, the dynamics of coherent information is the same to that of quantum discord minus a constant. The quantum discord and the coherent information can reveal the quantum correlations in some states those are not entangled. This implies that both coherent information and quantum discord can be used to evaluate the quantum correlations. References [1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2). [2] C.H. Bennett and D.P. DiVincenzo, Nature (London) (2) 27. [3] H. Ollivier and W.H. Zure, Phys. Rev. Lett. 88 (21) [] B. Schumacher and M.D. Westmoreland, Phys. Rev. Lett. A 7 (26) 235. [5] L. Henderson and V. Vedral, J. Phys. A 3 (21) 6899; V. Vedral, Phys. Rev. Lett. 9 (23) 51. [6] A. Datta, A. Shaji, and C.M. Caves, Phys. Rev. Lett. 1 (28) 552. [7] B.P. Lanyon, M. Barbieri, M.P. Almeida, and A.G. White, Phys. Rev. Lett. 11 (28) 251. [8] X. Hao, T. Pan, J.Q. Sha, and S.Q. Zhu, Commun. Theor. Phys. 55 (211) 1. [9] Z. He, J. Zou, J. Li, and B. Shao, Commun. Theor. Phys. 53 (21) 837. [1] H.P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford (22). [11] C.W. Gardiner and P. Zoller, Quantum Noise, Springer- Verlag, Berlin (1999). [12] B. Schumacher, Phys. Rev. A 5 (1996) 261. [13] H. Barnum, M.A. Nielsen, and B. Schumacher, Phys. Rev. A 57 (1998) 153. [1] Yang Xiang and Shi-Jie Xiong, Phys. Rev. A 76 (27) 136. [15] W.K. Wooters, Phys. Rev. Lett. 8 (1998) 225. [16] T. Konrad, F. de Melo, M. Tiersch, C. Kasztelan, A. Aragão, and A. Buchleitner, Nat. Phys. (28) 99. [17] J. Piilo, S. Maniscalco, K. Härönen, and K.-A. Suominen, Phys. Rev. Lett (28). [18] H.P. Breuer, E.M Laine, and J. Piilo, Phys. Rev. Lett. 13 (29) 211.

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