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arXiv:1309.0934 · Quantum Information and Computation

The witness of sudden change of geometric quantum correlation

Chang‐shui Yu, Bo Li, Heng Fan

Published May 1, 2014• 1 citation
arXivHTMLPDF

Abstract

In this paper, we give a sufficient and necessary condition (witness) for the sudden change of geometric quantum discord by considering mathematical definition of the discontinuity of a function. Based on the witness, we can find out various sudden changes of quantum correlation by considering both the Markovian and the non-Markovian cases. In particular, we can accurately find out critical points of the sudden changes even though they are not quite obvious in the graphical representation. In addition, one can also find that sudden change of quantum correlation, like the frozen quantum correlation, strongly depends on the choice of the quantum correlation measure.

DOI
10.26421/qic14.5-6-5

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In this paper, we give a sufficient and necessary condition (witness) for the sudden change of geometric quantum discord by considering mathematical definition of the discontinuity of a function. Based on the witness, we can find out various sudden changes of quantum correlation by considering both the Markovian and the non-Markovian cases. In particular, we can accurately find out critical points of the sudden changes even though they are not quite obvious in the graphical representation. In addition, one can also find that sudden change of quantum correlation, like the frozen quantum correlation, strongly depends on the choice of the quantum correlation measure.
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.
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.
Abstract Sudden changes of quantum correlations in the Bell-diagonal states are well-known effects. They occur when the set of optimal parameters that determine the quantum correlation consists of isolated points and optimal parameters during the evolution of the system jump from one such point to another (e.g., the optimal measurement angle of the quantum discord changes discontinuously from zero to π /2 or vice versa). However, when considering more general X quantum states, we found that quantum discord and one-way quantum work deficit can experience sudden changes of other kinds. Namely, the optimal measurement angle may suddenly start to shift continuously from its stationary value 0 or π /2 to an intermediate optimal measurement angle ϑ ∈ (0, π /2). This leads to a new behavior of quantum correlations, which is mathematically described by the Landau phenomenological theory of second-order phase transitions. In addition, for the one-way quantum work deficit, we found cases where the optimal measurement angle jumps from zero to a nonzero step less than π /2, and then continuously changes its value. This behavior of quantum correlation is similar to a first-order phase transition in Landau’s theory. Dependencies of quantum discord and one-way quantum work deficit near the boundaries, which separate regions with state-dependent (variable) and state-independent (stationary, constant) optimal measurement angles, are examined in detail on an example of the XXZ spin model in an external field at thermal equilibrium.
We study the peculiar dynamics of two qubits quantum discord with a sudden change in behavior. It is widely agreed in the literature nowadays that environment induce sudden change behavior of quantum discord. We explore the sudden change phenomenon under the action of local-independent environments, considering the discord of two-qubit which is obtained by means of the weak measurements on one of the two-qubit. It is shown that the sudden change phenomenon does not appear under weak measurements. We find that the quantum discord with weak measurements decays in a monotonic fashion, and show that quantum discord might exhibit a sudden change only for the strong (projective) measurement. We believe that environment is not the main role of the sudden change, but the quantum correlation itself is.
We investigate the dynamics of geometric quantum correlations for certain decoherence channels and discuss the necessary conditions for the existence of frozen geometric discord. As illustrative examples, we study the phenomenon of double sudden transitions in geometric discord for a system consisting of two noninteracting atoms inserted in two independent dissipative cavities and how the initial state parameters and decay rate of dissipative cavities affect the frozen time during which the geometric discord remains constant. We also explore the dynamics of geometric discord between two noninteracting atoms trapped in a common dissipative cavity and find that the geometric discord exhibits sudden transition between classical and quantum decoherence. Moreover, a nonzero stationary geometric discord can arise in both the independent cavity case and common cavity case.