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 $q$-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.
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 q-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.
Although many different entanglement measures have been proposed so far, much less is known in the multipartite case, which leads to the previous monogamy relations in the literature being not complete. We establish here a strict framework for defining the multipartite entanglement measure (MEM): apart from the postulates of the bipartite measure, i.e., vanishing on separable measures and nonincreasing under local operations and classical communication, a complete MEM should additionally satisfy the unification condition and the hierarchy condition. We then come up with a complete monogamy formula for the unified MEM (an MEM is called a unified MEM if it satisfies the unification condition) and a tightly complete monogamy relation for the complete MEM (an MEM is called a complete MEM if it satisfies both the unification condition and the hierarchy condition). Consequently, we propose MEMs which are multipartite extensions of entanglement of formation (EoF), concurrence, tangle, Tsallis q entropy of entanglement, R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, the convex-roof extension of negativity, and negativity. We show that (i) the extensions of EoF, concurrence, tangle, and Tsallis q entropy of entanglement are complete MEMs; (ii) multipartite extensions of R\'enyi $\ensuremath{\alpha}$ entropy of entanglement, negativity, and the convex-roof extension of negativity are unified MEMs but not complete MEMs; and (iii) all these multipartite extensions are completely monogamous, and the ones which are defined by the convex-roof structure (except for the R\'enyi $\ensuremath{\alpha}$ entropy of entanglement and the convex-roof extension of negativity) are not only completely monogamous but also tightly completely monogamous. In addition, as a byproduct, we find a class of states that satisfy the additivity of EoF. We also find a class of tripartite states one part of which can be maximally entangled with the other two parts simultaneously according to the definition of mixed maximally entangled state (MMES) in Li et al. [Z. Li, M. Zhao, S. Fei, H. Fan, and W. Liu, Quantum Inf. Comput. 12, 0063 (2012)]. Consequently, we improve the definition of maximally entangled state (MES) and prove that the only MES is the pure MES.
Multipartite entanglement includes not only the genuine entanglement but also the k-entanglement $(k\ensuremath{\geqslant}2)$. It is known that the most bipartite entanglement measures have been shown to be monogamous, but the monogamy relation is involved in the bipartite entanglement measures rather than the multipartite ones. So how we can explore the monogamy relation for k-entanglement becomes a basic open problem. In this paper we establish an axiomatic definition of the monogamy relation for the k-entanglement measure based on the coarser relation of the system partition. We also present the axiomatic definition of the complete k-entanglement measure and the associated complete monogamy relation according to the framework of the complete multipartite entanglement measure we established in [Phys. Rev. A 101, 032301 (2020)], which is shown to be an efficient tool for characterizing the multipartite quantum correlation as complementary to the monogamy relation associated with the bipartite measure. Consequently, the relation and the difference between monogamy, complete monogamy, and tightly complete monogamy are clearly depicted in light of the three types of coarser relation of the system partition. We then illustrate our approach with two classes of k-entanglement measures in detail. We find that all these k-entanglement measures are monogamous if the reduced function is strictly concave, they are not completely monogamous, and they are not complete generally.
"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."
The unified entropy as a promotion of the von Neumann entropy exhibits distinct diversity which contains the Tsallis entropy, the Rényi entropy, the von Neumann entropy as special cases. The unified-(r,t) entropy entanglement with 0<r<1 and 0<t≤1 was shown to be an entanglement monotone in literature. In this paper, we explore unified-(q,s) entropy entanglement with q>1 and qs≥1 and show that it is also an entanglement monotone and that both of them are monogamous. Going further, we present two kinds of global multipartite entanglement measures (GlMEMs) based on the unified entropy and each kind has two subclasses which are classified by the parameters (q,s) and (r,t). Consequently, from the view of the complete multipartite entanglement measure theory, we show that one of them is a complete multipartite entanglement monotone and is not only completely monogamous but also tightly completely monogamous, but the other three are even not complete. We also explore the genuine entanglement measures induced by the unified entropy and the relations with the bipartite entanglement and the global entanglement are discussed, respectively.
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.
A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory. As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account. The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue. In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function. Especially, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time. By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation. In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.