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arXiv:1504.07176 · Physical Review A

Multiparty quantum mutual information: An alternative definition

Asutosh Kumar

Published Jul 27, 2017• 33 citations
arXivHTMLPDF

Abstract

Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems. This may, however, no longer be true for three or more party quantum systems. In this paper, we propose an alternative definition of multipartite information, taking into account the shared information between two and more parties. It is non-negative, observes monotonicity under partial trace as well as completely positive maps, and equals the multipartite information measure in literature for pure states. We then define multiparty quantum discord, and give some examples. Interestingly, we observe that quantum discord increases when a measurement is performed on a large number of subsystems. Consequently, the symmetric quantum discord, which involves a measurement on all parties, reveals the maximal quantumness. This raises a question on the interpretation of measured mutual information as a classical correlation.

DOI
10.1103/physreva.96.012332

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Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems. This may, however, no longer be true for three or more party quantum systems. In this paper, we propose an alternative definition of multipartite information, taking into account the shared information between two and more parties. It is non-negative, observes monotonicity under partial trace as well as completely positive maps, and equals the multipartite information measure in literature for pure states. We then define multiparty quantum discord, and give some examples. Interestingly, we observe that quantum discord increases when a measurement is performed on a large number of subsystems. Consequently, the symmetric quantum discord, which involves a measurement on all parties, reveals the maximal quantumness. This raises a question on the interpretation of measured mutual information as a classical correlation.
Mutual information is the reciprocal information that is common to or shared by two or more parties. Quantum mutual information (QMI) for bipartite quantum systems is non-negative, and bears the interpretation of total correlation between the two subsystems.This may, however, be no longer true for three or more party quantum systems. It is possible that three-party QMI can be negative. That QMI can assume negative value is confronting, and we give operational definition of QMI which is non-negative. We then define multiparty operational quantum discord, and give some examples. Interestingly, we observe that quantum discord increases when measurement is performed on larger number of subsystems. Consequently, the symmetric version of quantum discord, which involves measurement on all parties, reveals the maximal quantumness. This raises question on the interpretation of measured mutual information as classical correlation.
A generalization of quantum discord to multipartite systems is proposed. A key feature of our formulation is its consistency with the conventional definition of discord in bipartite systems. It is by construction zero only for systems with classically correlated subsystems and is a non-negative quantity, giving a measure of the total nonclassical correlations in the multipartite system with respect to a fixed measurement ordering. For the tripartite case, we show that the discord can be decomposed into contributions resulting from changes induced by nonclassical correlation breaking measurements in the conditional mutual information and tripartite mutual information. The former gives a measure of the bipartite nonclassical correlations and is a non-negative quantity, while the latter is related to the monogamy of the nonclassical correlations.
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.
Quantum discord is the quantitative difference between two alternative expressions for bipartite mutual information, given respectively in terms of two distinct definitions for the conditional entropy. By constructing a stochastic model of shared states, classical discord can be similarly defined, quantifying the presence of some stochasticity in the measurement process. Therefore, discord can generally be understood as a quantification of the system's state disturbance due to local measurements, be it quantum or classical. We establish an operational meaning of classical discord in the context of state merging with noisy measurement and thereby show the quantum-classical separation in terms of a negative conditional entropy.
Quantum mutual information not only displays the mutual information in the system but also demonstrates some quantum correlation beyond entanglement. We explore here two alternatives of the multipartite quantum mutual information (MQMI) based on the von Neumann entropy according to the framework of the complete measure of the multiparticle quantum system. We show that these two MQMIs are complete and are monogamous on pure states, and one of them (we call it type-1) is not only completely monogamous but also tightly complete monogamous, while the other one (we call it type-2) is not. Moreover, we present another two MQMIs by replacing the von Neumann entropy with the Tsallis qqq-entropy from the former two ones. It is proved that one of them displays some degree of ``completeness'' as a measure of the multiparticle quantum system, but the other one is not even non-negative and thus it cannot be an alternative of MQMI. We also discuss the triangle relation for these three alternatives of MQMI. It is shown that the triangle inequalities hold for the former two MQMIs as that of entanglement measure but the later one fails. We thus can conclude that the type-1 von Neumann entropy MQMI is a ``fine'' measure of multipartite quantum correlation.
We investigate quantum states that possess both maximum entanglement and maximum discord between the pertinent parties. Since entanglement (discord) is defined only for bipartite (two-qubit) systems, we use an appropriate sum over all bipartitions as the associated measure. The ensuing definition---not new for entanglement---is thus extended here to quantum discord. Also, additional dimensions within the parties are considered (qudits). We also discuss quantum correlations that induce Mermin's Bell-inequality violation for all multiqubit systems. One finds some differences when quantum mechanics is defined over the field of real or of complex numbers.