Quantum Correlations in Multipartite States. Study Based on the Wootters-Mermin Theorem

TL;DR

This paper explores quantum correlations in multipartite states using the Wootters-Mermin theorem, demonstrating the existence of a finest uncorrelated decomposition and analyzing distant effects within cluster states.

Contribution

It generalizes the Wootters-Mermin theorem and applies it to tensor factorization, establishing the existence and uniqueness of the finest uncorrelated decomposition.

Findings

A finest uncorrelated decomposition always exists.

Coarsenings of the finest decomposition are the only other uncorrelated decompositions.

Distant effects within homogeneous cluster states are discussed.

Abstract

Decomposition of any N-partite state (density operator) into clusters (that do not overlap) is studied in detail with a view to learn as much as possible about the correlations implied by the state. The Wootters-Mermin theorem, stating that the totality of all strings of cluster events (projectors) determines the state in any finite- or infinite-dimensional state space, is a slightly sharpened and generalized form of the original results of Wootters and Mermin. It is applied to tensor factorization of the state into states of clusters (uncorrelated decomposition) and it is shown that a finest uncorrelated decomposition always exists, and that its coarsenings and only they are other possible uncorrelated cluster decompositions. Distant effects witin homogeneous cluster states, which are, by definition, the tensor factors in the finest uncorrelated decomposition, are discussed. The entire study is viewed by the author as a possible further elaboration of Mermin's Ithaca program.