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International Journal of Quantum Information

Quantum mutual information matrices

Feng Liu

Published Dec 16, 2016• 2 citations
Publisher

Abstract

For any [Formula: see text]-partite state [Formula: see text], we define its quantum mutual information matrix as an [Formula: see text][Formula: see text][Formula: see text][Formula: see text][Formula: see text] matrix whose [Formula: see text]-entry is given by quantum mutual information [Formula: see text]. Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we define the genuine n-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.

DOI
10.1142/s0219749917500058

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For any [Formula: see text]-partite state [Formula: see text], we define its quantum mutual information matrix as an [Formula: see text][Formula: see text][Formula: see text][Formula: see text][Formula: see text] matrix whose [Formula: see text]-entry is given by quantum mutual information [Formula: see text]. Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we define the genuine n-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.
For any nnn-partite state ρA1A2⋅⋅⋅Anρ_{A_{1}A_{2}\cdot\cdot\cdot A_{n}}ρA1​A2​⋅⋅⋅An​​, we define its quantum mutual information matrix as an nnn by nnn matrix whose (i,j)(i,j)(i,j)-entry is given by quantum mutual information I(ρAiAj)I(ρ_{A_{i}A_{j}})I(ρAi​Aj​​). Although each entry of quantum mutual information matrix, like its classical counterpart, is also used to measure bipartite correlations, the similarity ends here: quantum mutual information matrices are not always positive semidefinite even for collections of up to 3-partite states. In this work, we obtain necessary and sufficient conditions for the positive semidefinite quantum mutual information matrix. We further define the \emph{genuine} nnn-partite mutual information which can be easily calculated. This definition is symmetric, nonnegative, bounded and more accurate for measuring multipartite states.
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.
Quantum correlation in bipartite systems has been extensively studied, and in recent years, people have been interested in the study of multipartite systems. Studying quantum correlations in local multipartite systems has also become a hot topic. In this paper, we define the local bipartite conditional mutual information, and further give the definitions of local multipartite interaction (common) information and local genuinely multipartite quantum mutual information. Then, we mainly study some properties and special relations of local bipartite conditional mutual information and local genuinely multipartite quantum mutual information satisfaction. These properties lay the foundation for our understanding of the relationships between microscopic particles.
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.
"Using the relative entropy of total correlation, we derive an expression relating the mutual information of n-partite pure states to the sum of the mutual informa- tions and entropies of its marginals and analyze some of its implications. Besides, by utilizing the extended strong sub- additivity of von Neumann entropy, we obtain generalized monogamy relations for the total correlation in three-partite mixed states. These inequalities lead to a tight lower bound for this correlation in terms of the sum of the bipartite mutual informations. We use this bound to propose a mea- sure for residual three-partite total correlation and discuss the non-applicability of this kind of quantifier to measure genuine multiparty correlations."