# A limit relation for entropy and channel capacity per unit cost

## TL;DR

This paper proves a conjecture relating entropy limits and relative entropy in quantum systems, utilizing two proofs—one analytic and one based on channel capacity—leading to broader generalizations.

## Contribution

It provides the first proof of the conjecture for density matrices and clarifies the connection to classical-quantum channel capacity per unit cost.

## Key findings

- Proven limit relation for entropy and relative entropy in quantum systems
- Two distinct proofs: analytic using quantum law of large numbers and channel capacity approach
- Generalizations of the original conjecture achieved

## Abstract

In a quantum mechanical model, Diosi, Feldmann and Kosloff arrived at a conjecture stating that the limit of the entropy of certain mixtures is the relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The first proof is analytic and uses the quantum law of large numbers. The second one clarifies the relation to channel capacity per unit cost for classical-quantum channels. Both proofs lead to generalization of the conjecture.