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Multipartite Quantum Systems: an approach based on Markov matrices and the Gini index
Publication Date
2021-04
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© 2021 IOP Journals. Reproduced in accordance with the publisher's self-archiving policy
This Accepted Manuscript is available for reuse under a CC BY-NC-ND licence after the 12 month embargo period provided that all the terms of the licence are adhered to.
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2021-03-24
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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 de ne 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.
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Accepted manuscript
Citation
Vourdas A (2021) Multipartite Quantum Systems: an approach based on Markov matrices and the Gini index. Journal of Physics A: Mathematical and Theoretical. 54(18): 185201
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