Gacs Algorithmic Complexity on Hilbert Spaces and Some of its Applications

TL;DR

This paper extends Gacs quantum algorithmic entropy to infinite-dimensional quantum spin chains, explores its relation to quantum dynamical entropies, and proves a quantum version of Brudno's theorem.

Contribution

It introduces a new extension of Gacs entropy to infinite quantum systems and establishes a quantum Brudno's theorem for spin chains.

Findings

Extended Gacs entropy to infinite quantum spin chains

Linked Gacs entropy with quantum dynamical entropies

Proved quantum Brudno's theorem for shift dynamics

Abstract

We extend the notion of Gacs quantum algorithmic entropy, originally formulated for finitely many qubits, to infinite dimensional quantum spin chains and investigate the relation of this extension with two quantum dynamical entropies that have been proposed in recent years. Further, we prove an extension of Brudno's theorem in quantum spin chains with shift dynamics.