Correlations in typicality and an affirmative solution to the exact catalytic entropy conjecture

TL;DR

This paper proves that for finite-dimensional quantum states with smaller von Neumann entropy, multiple copies can be transformed into a state with identical marginals, confirming the exact catalytic entropy conjecture and its classical analogue.

Contribution

It provides an affirmative proof of the exact catalytic entropy conjecture for quantum states and extends the result to classical probability vectors.

Findings

Sufficient tensor copies can majorize a state with identical marginals.

Confirmed the exact catalytic entropy conjecture (CEC).

Results apply to both quantum and classical settings.

Abstract

I show that if a finite-dimensional density matrix has strictly smaller von Neumann entropy than a second one of the same dimension (and the rank is not bigger), then sufficiently (but finitely) many tensor-copies of the first density matrix majorize a density matrix whose single-body marginals are all exactly equal to the second density matrix. This implies an affirmative solution of the exact catalytic entropy conjecture (CEC) introduced by Boes et al. [PRL 122, 210402 (2019)]. Both the Lemma and the solution to the CEC transfer to the classical setting of finite-dimensional probability vectors (with permutations of entries instead of unitary transformations for the CEC).