Multipartite quantum systems: an approach based on Markov matrices and the Gini index

TL;DR

This paper introduces a novel formalism using Markov matrices and the Gini index to analyze probabilities and correlations in multipartite quantum systems, extending classical matrix decompositions to quantum contexts.

Contribution

It generalizes the Birkhoff-von Neumann expansion for stochastic matrices to quantum systems, introducing new probabilistic quantities and dual transforms for multipartite quantum analysis.

Findings

New probabilistic quantities for quantum systems

Application of Gini index to quantify quantum state sparsity

Examples demonstrating the formalism's effectiveness

Abstract

An expansion of row Markov matrices in terms of matrices related to permutations with repetitions, is introduced.It generalises the Birkhoff-von Neumann expansion of doubly stochastic matrices in terms of permutation matrices (without repetitions).An interpretation of the formalism in terms of sequences of integers that open random safes described by the Markov matrices, is presented. Various quantities that describe probabilities and correlations in this context, are discussed. The Gini index is used to quantify the sparsity (certainty) of various probability vectors. The formalism is used in the context of multipartite quantum systems with finite dimensional Hilbert space, which can be viewed as quantum permutations with repetitions or as quantum safes. The scalar product of row Markov matrices, the various Gini indices, etc, are novel probabilistic quantities that describe the statistics of multipartite quantum systems. Local and global Fourier transforms are used to define locally dual and also globally dual statistical quantities. The latter depend on off-diagonal elements that entangle (in general) the various components of the system. Examples which demonstrate these ideas are also presented.