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انجمن اپتیک و فوتونیک ایران

Quantum Discord of two-qubit states

Seyed Javad Akhtarshenas, Hamidreza Mohammadi, Vahid Nassajpour, Fahimeh S. Mousavi

Published Feb 15, 2014• 0 citations
OpenAlex

Abstract

In this paper, we provide a simple relation for the conditional entropy as the difference of two Shannon entropies. This relation leads to an analytical formula for discord of some special two-qubit states and also it presents a tight upper bound for quantum discord of a general two-qubit state. We also present an analytical procedure of optimization and obtain conditions under which the quantum conditional entropy of a general two-qubit state is stationary.

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In this paper, we provide a simple relation for the conditional entropy as the difference of two Shannon entropies. This relation leads to an analytical formula for discord of some special two-qubit states and also it presents a tight upper bound for quantum discord of a general two-qubit state. We also present an analytical procedure of optimization and obtain conditions under which the quantum conditional entropy of a general two-qubit state is stationary.
We present a reliable algorithm to evaluate quantum discord for general two-qubit states, amending and extending an approach recently put forward for the subclass of XXX states. A closed expression for the discord of arbitrary states of two qubits cannot be obtained, as the optimization problem for the conditional entropy requires the solution to a pair of transcendental equations in the state parameters. We apply our algorithm to run a numerical comparison between quantum discord and an alternative, computable measure of nonclassical correlations, namely, the geometric discord. We identify the extremally nonclassically correlated two-qubit states according to the (normalized) geometric discord, at a fixed value of the conventional quantum discord. The latter cannot exceed the square root of the former for systems of two qubits.
Quantum discord quantifies quantum correlations beyond entanglement and assumes nonzero values, which are notoriously hard to compute, for almost all quantum states. Here we provide computable tight bounds for the quantum discord for qubit-qudit states. In the case of two qubits our lower and upper bounds coincide for a 7-parameter family of filtered XXX-states, whose quantum discords can therefore be evaluated analytically. An application to the accessible information of the binary qubit channel is also presented. For the qubit-qudit state output by the circuit of deterministic computation with one qubit, nontrivial lower and upper bounds that respect the zero-discord conditions are obtained for its quantum discord.
We present an analytical solution for classical correlation, defined in terms of linear entropy, in an arbitrary system when the second subsystem is measured. We show that the optimal measurements used in the maximization of the classical correlation in terms of linear entropy, when used to calculate the quantum discord in terms of von Neumann entropy, result in a tight upper bound for arbitrary d⊗2 systems. This bound agrees with all known analytical results about quantum discord in terms of von Neumann entropy and, when comparing it with the numerical results for 10(6) two-qubit random density matrices, we obtain an average deviation of order 10(-4). Furthermore, our results give a way to calculate the quantum discord for arbitrary n-qubit GHZ and W states evolving under the action of the amplitude damping noisy channel.
We derive a general approximate solution to the problem of minimizing the conditional entropy of a qudit-qubit system resulting from a local measurement on the qubit, which is valid for general entropic forms and becomes exact in the limit of weak correlations. This entropy measures the average conditional mixedness of the postmeasurement state of the qudit, and its minimum among all local measurements represents a generalized entanglement of formation. In the case of the von Neumann entropy, it is directly related to the quantum discord. It is shown that at the lowest nontrivial order, the problem reduces to the minimization of a quadratic form determined by the correlation tensor of the system, the Bloch vector of the qubit and the local concavity of the entropy, requiring just the diagonalization of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. A simple geometrical picture in terms of an associated correlation ellipsoid is also derived, which illustrates the link between entropy optimization and correlation access and which is exact for a quadratic entropy. The approach enables a simple estimation of the quantum discord. Illustrative results for two-qubit states are discussed.
Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary d⊗2d\otimes 2d⊗2 quantum states and von Neumann entropy-based classical correlation for arbitrary 2⊗22\otimes 22⊗2 rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.
Two-qubit X-state is a large class of quantum states which plays an important role in the quantification and dynamical study of quantum correlations. However, the corresponding quantification of quantum discord is still missing for bona fide discord measures, like original quantum discord, Bures distance of discord, and relative entropy of discord. In this paper, we consider the calculation of Bures distance of discord, which is a kind of correlation satisfying all criteria of a discord measure, for two-qubit X-states. Firstly, we derive an explicit expression for Bures distance of discord for a kind of five-parameters family of states. Moreover, for general two-qubit X-states, we not only calculate the Bures distance of discord for a subset of two-qubit X-states by classifying and analyzing the optimal local measurements and the optimal projection operators but also provide an analytic upper bound for the entirety.