Existence of maximally correlated states

TL;DR

The paper investigates the conditions under which maximally correlated states exist across multiple systems, revealing that such states do not always exist unless additional measurement invariance conditions are met, in which case maximally entangled states are identified.

Contribution

It demonstrates that maximally correlated states do not universally exist under certain correlation measures, and identifies conditions for their existence in multi-system scenarios.

Findings

Maximally correlated states do not always exist for multiple systems.

Maximally entangled states are maximally correlated for two systems under certain conditions.

Existence of such states depends on the Hilbert space dimensions for more than two systems.

Abstract

A measure of total correlations cannot increase under deterministic local operations. We show that, for any number of systems, this condition alone does not guarantee the existence of maximally correlated states. Namely, there is no state that simultaneously maximizes all the measures satisfying it. If, in addition, the measures do not increase with probability unity under local measurements, then such states exist for two systems. They are the maximally entangled states. For a larger number of systems, it depends on their Hilbert space dimensions.