Undecidability in quantum thermalization

TL;DR

This paper proves that determining whether a quantum many-body system thermalizes is fundamentally undecidable, even under simplified conditions, due to encoding Turing machine computations into Hamiltonian dynamics.

Contribution

It establishes the undecidability of quantum thermalization, showing no general algorithm can predict thermalization outcomes for all Hamiltonians.

Findings

Thermalization problem is undecidable in quantum systems.

Undecidability persists even in 1D shift-invariant systems.

Hamiltonians can encode universal Turing machine dynamics.

Abstract

The investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.