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Chinese Physics B

Characterizing the dynamics of quantum discord under phase damping with POVM measurements

Feng-Jian Jiang, Jianfeng Ye, Xin-Hu Yan, Hai-Jiang Lü

Published Sep 29, 2015• 1 citation
Publisher

Abstract

In the analysis of quantum discord, the minimization of average entropy traditionally involved over orthogonal projective measurements may be attained at more optimal decompositions by using the positive-operator-valued measure (POVM) measurements. Taking advantage of the quantum steering ellipsoid in combination with three-element POVM optimization, we show that, for a family of two-qubit X states locally interacting with Markovian non-dissipative environments, the decay rates of quantum discord show smooth dynamical evolutions without any sudden change. This is in contrast to two-element orthogonal projective measurements, in which case the sudden change of the decay rates of quantum and classical decoherences may be a common phenomenon. Notwithstanding this, we find that a subset of X states (including the Bell diagonal states) involving POVM optimization can still preserve the sudden change character as usual.

DOI
10.1088/1674-1056/24/10/100304

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In the analysis of quantum discord, the minimization of average entropy traditionally involved over orthogonal projective measurements may be attained at more optimal decompositions by using the positive-operator-valued measure (POVM) measurements. Taking advantage of the quantum steering ellipsoid in combination with three-element POVM optimization, we show that, for a family of two-qubit X states locally interacting with Markovian non-dissipative environments, the decay rates of quantum discord show smooth dynamical evolutions without any sudden change. This is in contrast to two-element orthogonal projective measurements, in which case the sudden change of the decay rates of quantum and classical decoherences may be a common phenomenon. Notwithstanding this, we find that a subset of X states (including the Bell diagonal states) involving POVM optimization can still preserve the sudden change character as usual.
We present a complete treatment for the quantum discord of two-qubit XXX states, by developing a geometric picture of a quantum steering ellipsoid. It is shown that either a von Neumann measurement or a three-element positive-operator-valued measurement is optimal. The condition for the latter is obtained and expressed in geometric language. We show, by using analytical as well as numerical results, that there is a systematic structure in the optimal decomposition which exists in a class of states including the XXX states. More significantly, we establish the relation to the quantum channel by identifying the steering ellipsoid with the quantum channel ellipsoid. Thus the quantum discord and classical correlation are closely related to the concept of the entanglement entropy of the quantum channel.
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.
The non-Markovian decoherence of quantum and classical correlations is analytically obtained when two qubits are asymmetrically subjected to the bit flip channel and phase flip channel. For one class of initial mixed states, quantum correlations quantified by quantum discord decay synchronously with classical correlations. The discovery that the decaying rates of quantum and classical correlations suddenly change at the characteristic time is physically interpreted by the distance from quantum state to the closest classical states. In a large time interval, quantum correlations are greater than classical correlations. The quantum and classical correlations can be preserved over a longer period of time via the kernel characterizing the environment memory effects.
Quantum discord is a measure based on local projective measurements which captures quantum correlations that go beyond entanglement. A change in the measurement process, achieved by replacing rank-one projectors with a weak positive operator-valued measure (POVM), allows one to define a quantity termed as weak quantum discord. In this work, we experimentally simulate the effect of a weak POVM on a nuclear magnetic resonance quantum information processor. The two-qubit system under investigation is part of a three-qubit system, where one of the qubits is used as an ancillary to implement the phase damping channel. The strength of the weak POVM is controlled by varying the strength of the phase damping channel. We experimentally observed weak discord in two-qubit states and computed a cost function whose purpose is to optimally extract correlations representing the difference between weak and strong quantum discord. The resultant dynamics of the states is investigated as a function of the measurement strength.
Among various definitions of quantum correlations, quantum discord has attracted considerable attention. To find analytical expression of quantum discord is an intractable task. In this paper, we discuss thoroughly the case of two-qubit rank-two states. An analytical expression for the quantum discord is obtained by means of Koashi-Winter relation. A geometric picture is demonstrated by means of quantum steering ellipsoid. We point out that in this case the optimal measurement is indeed the von Neumann measurement, which is usually used in the study of quantum discord. However, for some two-qubit states with the rank larger than two, we find that three-element POVM measurement is more optimal. It means that more careful attention should be paid in the discussion of quantum discord.
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.