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Weak-measurement induced quantum discord and monogamy of X states

Asutosh Kumar

Published Jan 1, 2013• 0 citations
Publisher

Abstract

Weak-measurement induced quantum discord or super quantum discord (SQD) is a generalization of the normal quantum discord and is defined as the difference between quantum mutual information and classical correlation obtained by weak measurements in a given quantum system. This correlation is an information-theoretic measure and is, in general, different from entanglement-separability measures such as entanglement. Super quantum discord may be nonzero even for certain separable states. So far, SQD has been calculated explicitly only for a limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for SQD for X states, a seven real-parameter family of two-qubit states and investigate its monogamy properties. The monogamy behaviour of SQD depends on the measurement strength. The formalism can be easily extended to N-qubit X states.

DOI
10.13140/rg.2.2.25703.16804

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Weak-measurement induced quantum discord or super quantum discord (SQD) is a generalization of the normal quantum discord and is defined as the difference between quantum mutual information and classical correlation obtained by weak measurements in a given quantum system. This correlation is an information-theoretic measure and is, in general, different from entanglement-separability measures such as entanglement. Super quantum discord may be nonzero even for certain separable states. So far, SQD has been calculated explicitly only for a limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for SQD for X states, a seven real-parameter family of two-qubit states and investigate its monogamy properties. The monogamy behaviour of SQD depends on the measurement strength. The formalism can be easily extended to N-qubit X states.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. In general, this correlation is different from entanglement, and quantum discord may be nonzero even for certain separable states. Even in the simple case of bipartite quantum systems, this different kind of quantum correlation has interesting and significant applications in quantum information processing. So far, quantum discord has been calculated explicitly only for a rather limited set of two-qubit quantum states and expressions for more general quantum states are not known. In this article, we derive explicit expressions for quantum discord for a larger class of two-qubit states, namely, a seven-parameter family of so called XXX states that have been of interest in a variety of contexts in the field. We also study the relation between quantum discord, classical correlation, and entanglement for a number of two-qubit states to demonstrate that they are independent measures of correlation with no simple relative ordering between them.
Super-quantum discord (SQD) with weak measurement is regarded as a kind of quantum correlation in quantum information processing. We compare and analyze the dynamical evolutions of SQD, quantum discord (QD), and quantum entanglement (QE) between two qubits in the correlated dephasing environmental model. The results indicate that (i) owing to the much smaller influence of weak measurement on the coherence of the system than that of von Neumann projection measurement, SQD with weak measurement is larger than QD, and (ii) dynamical evolution of QD or QE monotonically goes to zero with time, while SQD monotonically tends to a stable value and a freezing phenomenon occurs. The stable value after freezing mainly depends on the measurement strength and the purity of the initial quantum state.
So far, super quantum discord has been calculated explicitly only for Bell-diagonal states and expressions for more general quantum states are not known. In this paper, we derive explicit expressions for super quantum discord for a larger class of two-qubit states, namely, a 4-parameter family of two-qubit states. We observe that, weak measurements obtain more quantumness of correlations than strong measurements. As an application, the dynamic behavior of the super quantum discord under decoherence channel is investigated. We find that, the super quantum discord decreases monotonically as a function of the measurement strength parameter.
Quantum discord, as introduced by Olliver and Zurek [Phys. Rev. Lett. 88, 017901 (2001)], is a measure of the discrepancy between two natural yet different quantum analogs of the classical mutual information. This notion characterizes and quantifies quantumness of correlations in bipartite states from a measurement perspective, and is fundamentally different from the various entanglement measures in the entanglement vs separability paradigm. The phenomenon of nonzero quantum discord is a manifestation of quantum correlations due to noncommutativity rather than due to entanglement, and has interesting and significant applications in revealing the advantage of certain quantum tasks. We will evaluate analytically the quantum discord for a large family of two-qubit states, and make a comparative study of the relationships between classical and quantum correlations in terms of the quantum discord. We furthermore compare the quantum discord with the entanglement of formation, and illustrate that the latter may be larger than the former, although for separable states, the entanglement of formation always vanishes and thus is less than the quantum discord.
We extend the definition of quantum discord as a quantifier of nonClassical correlations in a quantum state to the case where weak measurements are performed on subsystem AAA of a bipartite system ABABAB. The properties of weak discord are explored for several families of quantum states. We find that in many cases weak quantum discord is identical to normal discord and in general the values of the two are very close to each other. Weak quantum discord reduces to discord in the appropriate limits as well. We also discuss the implications of these observations on the interpretations of quantum discord.
In contrast with entanglement, as measured by concurrence, in general, quantum discord does not possess the property of monogamy; that is, there is no tradeoff between the quantum discord shared by a pair of subsystems and the quantum discord that both of them can share with a third party. Here, we show that, as far as monogamy is considered, quantum discord of pure states is equivalent to the entanglement of formation. This result allows one to analytically prove that none of the pure three-qubit states belonging to the subclass of WWW states is monogamous. A suitable physical interpretation of the meaning of the correlation information as a quantifier of monogamy for the total information is also given. Finally, we prove that, for rank 2 two-qubit states, discord and classical correlations are bounded from above by single-qubit von Neumann entropies.