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arXiv:1010.3521 · Journal of Physics A Mathematical and Theoretical

Comparison of different measures for quantum discord under non-Markovian noise

Z Y Xu, Wan Li Yang, Xianbo Xiao, Min Feng

Published Sep 8, 2011• 20 citations
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Abstract

Two geometric measures for quantum discord were recently proposed by Modi et al (2010 Phys. Rev. Lett. 104 080501) and Dakić et al (2010 Phys. Rev. Lett. 105 190502). We study the similarities and differences for total quantum correlations of Bell-diagonal states using these two geometry-based quantum discord and the original quantum discord. We show that, under non-Markovian dephasing channels, quantum discord and one of the geometric measures remain constant for a finite amount of time, but not the other geometric measure. However, all the three measures share a common sudden change point. Our study on critical point of sudden transition might be useful for keeping long-time total quantum correlations under decoherence.

DOI
10.1088/1751-8113/44/39/395304

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Two geometric measures for quantum discord were recently proposed by Modi et al (2010 Phys. Rev. Lett. 104 080501) and Dakić et al (2010 Phys. Rev. Lett. 105 190502). We study the similarities and differences for total quantum correlations of Bell-diagonal states using these two geometry-based quantum discord and the original quantum discord. We show that, under non-Markovian dephasing channels, quantum discord and one of the geometric measures remain constant for a finite amount of time, but not the other geometric measure. However, all the three measures share a common sudden change point. Our study on critical point of sudden transition might be useful for keeping long-time total quantum correlations under decoherence.
Quantum discord, as introduced by Ollivier and Zurek (2001 Phys. Rev. Lett. 88 017901), is a measure of the discrepancy between quantum versions of two classically equivalent expressions for mutual information and is found to be useful in quantification and application of quantum correlations in mixed states. It is viewed as a key resource present in certain quantum communication tasks and quantum computational models without containing much entanglement. An early step toward the quantification of quantum discord in a quantum state was by Dakic et al (2010 Phys. Rev. Lett. 105 190502) who introduced a geometric measure of quantum discord and derived an explicit formula for any two-qubit state. Recently, Luo and Fu (2010 Phys. Rev. A 82 034302) introduced a generic form of the geometric measure of quantum discord for a bipartite quantum state. We extend these results and find generic forms of the geometric measure of quantum discord and total quantum correlations in a general N -partite quantum state. Further, we obtain computable exact formulas for the geometric measure of quantum discord and total quantum correlations in an N -qubit quantum state. The exact formulas for the N -qubit quantum state can be used to get experimental estimates of the quantum discord and the total quantum correlation.
Geometric quantum discord is a well-defined measure of quantum correlation if Schatten one-norm (trace norm) is adopted as a distance measure. Here, we analytically investigate the dynamical behavior of the one-norm geometric quantum discord under the effect of decoherence. By starting from arbitrary Bell-diagonal mixed states under Markovian local noise, we provide the decays of the quantum correlation as a function of the decoherence parameters. In particular, we show that the one-norm geometric discord exhibits the possibility of double sudden changes and freezing behavior during its evolution. For nontrivial Bell-diagonal states under simple Markovian channels, these are new features that are in contrast with the Schatten two-norm (Hilbert-Schmidt) geometric discord. The necessary and sufficient conditions for double sudden changes as well as their exact locations in terms of decoherence probabilities are provided. Moreover, we illustrate our results by investigating decoherence in quantum spin chains in the thermodynamic limit.
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.
We investigate the dynamics of quantum and classical correlations in a system of two qubits under local colored-noise dephasing channels. The time evolution of a single qubit interacting with its own environment is described by a memory kernel non-Markovian master equation. The memory effects of the non-Markovian reservoirs introduce new features in the dynamics of quantum and classical correlations compared to the white noise Markovian case. Depending on the geometry of the initial state, the system can exhibit frozen discord and multiple sudden transitions between classical and quantum decoherence [L. Mazzola, J. Piilo and S. Maniscalco, Phys. Rev. Lett. 104 (2010) 200401]. We provide a geometric interpretation of those phenomena in terms of the distance of the state under investigation to its closest classical state in the Hilbert space of the system.
We discuss some properties of the quantum discord based on the geometric distance advanced by Dakic, Vedral, and Brukner [Phys. Rev. Lett. {\bf 105}, 190502 (2010)], with emphasis on Werner- and MEM-states. We ascertain just how good the measure is in representing quantum discord. We explore the dependence of quantum discord on the degree of mixedness of the bipartite states, and also its connection with non-locality as measured by the maximum violation of a Bell inequality within the CHSH scenario.
Some properties of the quantum discord based on the geometric measure advanced by Dakic et al (2010 Phys. Rev. Lett. 105 190502 ) are discussed here by recourse to a systematic survey of the two-qubit state-space, with emphasis on Werner and MEM states. We explore the dependence of quantum discord on the degree of mixedness of the bipartite states, and also its connection with non-locality as measured by the maximum violation of a Bell inequality within the CHSH scenario. Some considerations regarding the XY model are also made.