Undecidability in quantum thermalization

• Published in: arXiv.org
• Main Authors: Shiraishi, Naoto, Matsumoto, Keiji
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 24.08.2021
• Subjects: Algorithms; Hamiltonian functions; Mathematics - Mathematical Physics; Physics - Mathematical Physics; Physics - Quantum Physics; Physics - Statistical Mechanics; Statistical mechanics; Thermalization (energy absorption); Turing machines
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2012.13889

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.