# Quantum entropy and complexity

## TL;DR

This paper explores the relationship between quantum complexity and entropy, demonstrating that Gacs complexity aligns with classical and quantum entropy rates in ergodic systems, extending classical results to quantum spin chains.

## Contribution

It establishes a connection between quantum complexity and entropy, generalizing classical ergodic theorems to quantum systems using Gacs complexity.

## Key findings

- Gacs complexity equals Shannon entropy rate in classical ergodic systems.
- The equality between complexity and entropy extends to ergodic quantum spin chains.
- The results unify classical and quantum notions of complexity and entropy in ergodic contexts.

## Abstract

We study the relations between the recently proposed machine-independent quantum complexity of P. Gacs~\cite{Gacs} and the entropy of classical and quantum systems. On one hand, by restricting Gacs complexity to ergodic classical dynamical systems, we retrieve the equality between the Kolmogorov complexity rate and the Shannon entropy rate derived by A.A. Brudno~\cite{Brudno}. On the other hand, using the quantum Shannon-Mc Millan theorem~\cite{BSM}, we show that such an equality holds densely in the case of ergodic quantum spin chains.