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arXiv:1309.6890 · Physical Review A

Comment on “Sudden change in quantum discord accompanying the transition from bound to free entanglement”

Swapan Rana, Preeti Parashar

Published Sep 16, 2013• 1 citation
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Abstract

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.

DOI
10.1103/physreva.88.036301

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We show that the mistakes pointed out by Rana and Parashar [Phys. Rev. A 87, 016301 (2013)] do not invalidate the main conclusion of our work [Phys. Rev. A 86, 024302 (2012)]. We show that the errors affected only a particular application of our general results, and present the correction.

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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.
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.
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.