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Physical Review A

Sudden change in quantum discord accompanying the transition from bound to free entanglement

Xiaokun Yan, Guo‐Hua Liu, J. Chee

Published Feb 26, 2013• 14 citations
Publisher

Abstract

We study quantum discord by using the geometric measure of quantum discord for a class of quantum states with bound entanglement. It is found that there exists a nondynamic sudden change in quantum discord accompanying transition from bound entanglement to free entanglement. The calculations presented here should be of interest to the continuing efforts to understand sudden transition of quantum correlations.

DOI
10.1103/physreva.87.022340

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In a recent article [Phys. Rev. A 87, 022340 (2013)], Yan et al. have studied geometric discord for a well-known class of bound entangled states. Based on their calculation, they claim ``It is found that there exists a nondynamic sudden change in quantum discord'' for these states. In this Comment, we criticize their work by pointing out that what has been calculated is actually a bound and is not the exact value of discord. Since, generally, it is not possible to infer the exact value (or property) of a quantity just from its bound, we cannot conclude about (exact) discord and its change. Thus, the above-mentioned conclusion cannot be drawn from the calculations of the paper.

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We study quantum discord by using the geometric measure of quantum discord for a class of quantum states with bound entanglement. It is found that there exists a nondynamic sudden change in quantum discord accompanying transition from bound entanglement to free entanglement. The calculations presented here should be of interest to the continuing efforts to understand sudden transition of quantum correlations.
In a recent article [Phys. Rev. A 87, 022340 (2013)], Yan et al. have studied geometric discord for a well-known class of bound entangled states. Based on their calculation, they claim ``It is found that there exists a nondynamic sudden change in quantum discord'' for these states. In this Comment, we criticize their work by pointing out that what has been calculated is actually a bound and is not the exact value of discord. Since, generally, it is not possible to infer the exact value (or property) of a quantity just from its bound, we cannot conclude about (exact) discord and its change. Thus, the above-mentioned conclusion cannot be drawn from the calculations of the paper.
The dynamics of quantum discord for a class of qutrit-qutrit states under depolarizing and dephasing noise are investigated using the geometric measure of quantum discord. We find that in the quantum discord for these states, a sudden change near the transition point exists from a bound-entangled to a free-entangled state, and that the sudden change point remains unchanged in later evolution under depolarizing noise. Moreover, the behavior of quantum discord is independent of the parameter of the given initial states in many regions.
It is known that quantum discord might experience a sudden transition in its dynamics when calculated for certain Bell-diagonal states (BDS) that are in interaction with their surroundings. We examine this phenomenon, known as the sudden change of quantum discord, considering the case of two qubits independently interacting with dephasing reservoirs. We first demonstrate that, for a class of initial states which can be chosen arbitrarily close to BDS, the transition is in fact not sudden, although it might numerically appear so if not studied carefully. Then, we provide an extension of this discussion covering the X\mathsf{X}X-shaped density matrices. Our findings suggest that the transition of quantum discord might be sudden only for an highly idealized zero-measure subset of states within the set of all possible initial conditions of two qubits.
We investigate the disappearance of discord in 2- and multi-qubit systems subject to decohering influences. We formulate the computation of quantum discord and quantum geometric discord in terms of the generalized Bloch vector, which gives useful insights on the time evolution of quantum coherence for the open system, particularly the comparison of entanglement and discord. We show that the analytical calculation of the global geometric discord is NP-hard in the number of qubits, but a similar statement for global entropic discord is more difficult to prove. We present an efficient numerical method to calculating the quantum discord for a certain important class of multipartite states. In agreement with previous work for 2-qubit cases (Mazzola et al 2010 Phys. Rev. Lett. 104 200401), we find situations in which there is a sudden transition from classical to quantum decoherence characterized by the discord remaining relatively robust (classical decoherence) until a certain point from where it begins to decay quickly whereas the classical correlation decays more slowly (quantum decoherence). However, we find that as the number of qubits increases, the chance of this kind of transition occurring becomes small.
Recent studies show that two completely distinct fields as quantum information and string theory realization can be related and share some of the same properties. So, the knowledge on quantum information of some special quantum systems provides new insights on the sting theory. We analytically study geometric discord and measurement induced nonlocality for a Horodeckis' 33 bound entangled state to find quantum correlation. We find that there is a simultaneous sudden change point in the geometric discord and the measurement-induced nonlocality in the region of the bound entangled states, which is considerable. Moreover, we investigate the lower bound of geometric discord and the upper bound of measurement-induced nonlocality for this system. This study leads to some new and interesting novel results as well.
The dynamics of a geometric measure of the quantum discord (GMQD) under decoherence is investigated. We show that the GMQD of a two-qubit state can be alternatively obtained through the singular values of a 3×43\times43×4 matrix whose elements are the expectation values of Pauli matrices of the two qubits. By using Heisenberg picture, the analytic results of the GMQD is obtained for three typical kinds of the quantum decoherence channels. We compare the dynamics of the GMQD with that of the quantum discord and of entanglement. We show that a sudden change in the decay rate of the GMQD does not always imply that of the quantum discord, and vice versa. We also give a general analysis on the sudden change in behavior and find that at least for the Bell diagonal states, the sudden changes in decay rates of the GMQD and that of the quantum discord occur simultaneously.