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Reviews in Mathematical Physics

Local conditional quantum mutual information and local multipartite quantum mutual information

Qi Han, Lijie Gou, Shuai Wang, Rong Zhang

Published May 8, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0129055x25500175

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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 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.
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.
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.
Abstract It is shown that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence topology). Some applications of this property are considered. New continuity conditions for the quantum mutual information and for the squashed entanglement in both bipartite and multipartite infinite-dimensional systems are obtained. It is proved, in particular, that the multipartite squashed entanglement of any countably indecomposable separable state with finite marginal entropies is equal to zero. Special continuity properties of the information gain of a quantum measurement with and without quantum side information are established that can be treated as robustness (stability) of these quantities with respect to perturbation of the measurement and the measured state.
Quantum information theory has considerably helped in the understanding of quantum many-body systems. The role of quantum correlations and in particular, bipartite entanglement, has become crucial to characterise, classify and simulate quantum many body systems. Furthermore, the scaling of entanglement has inspired modifications to numerical techniques for the simulation of many-body systems leading to the, now established, area of tensor networks. However, the notions and methods brought by quantum information do not end with bipartite entanglement. There are other forms of correlations embedded in the ground, excited and thermal states of quantum many-body systems that also need to be explored and might be utilised as potential resources for quantum technologies. The aim of this work is to review the most recent developments regarding correlations in quantum many-body systems focussing on multipartite entanglement, quantum nonlocality, quantum discord, mutual information but also other non classical measures of correlations based on quantum coherence. Moreover, we also discuss applications of quantum metrology in quantum many-body systems.