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Undecidability in quantum thermalization.

Naoto Shiraishi1, Keiji Matsumoto2

  • 1Department of Physics, Gakushuin Univerisity, Toshima-ku, Tokyo, Japan. naoto.shiraishi@gakushuin.ac.jp.

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|August 25, 2021
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Summary
This summary is machine-generated.

The problem of determining if quantum systems thermalize is undecidable. This means no general algorithm can predict thermalization, even for simple systems, revealing fundamental limits in quantum mechanics.

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Area of Science:

  • Quantum mechanics
  • Statistical mechanics
  • Theoretical physics

Background:

  • Thermalization in isolated quantum many-body systems is a long-standing problem.
  • Most systems are expected to thermalize, but some do not, posing a fundamental question.
  • Previous attempts to resolve this have been unsuccessful.

Purpose of the Study:

  • To determine if the problem of thermalization in quantum systems is decidable.
  • To investigate the limits of predicting thermalization in various quantum systems.
  • To explore the connection between computational universality and thermalization.

Main Methods:

  • Constructing a family of Hamiltonians that encode the dynamics of a reversible universal Turing machine.
  • Analyzing the halting problem of Turing machines in relation to quantum system relaxation.
  • Applying these constructions to one-dimensional shift-invariant systems with nearest-neighbor interactions and fixed product states.

Main Results:

  • The problem of determining whether a given quantum many-body system thermalizes is undecidable.
  • This undecidability holds even for restricted systems like 1D shift-invariant models with nearest-neighbor interactions.
  • The thermalization behavior depends on the halting of an encoded universal Turing machine.

Conclusions:

  • There is no general theorem, algorithm, or systematic procedure to determine thermalization for any given Hamiltonian.
  • Fundamental limits exist in predicting thermalization in isolated quantum many-body systems.
  • The study reveals a deep connection between computability theory and quantum statistical mechanics.