Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum thermalization must occur in translation-invariant systems at high temperature

Saúl Pilatowsky-Cameo, Soonwon Choi

Abstract

Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.

Quantum thermalization must occur in translation-invariant systems at high temperature

Abstract

Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.
Paper Structure (5 sections, 7 theorems, 11 equations, 3 figures)

This paper contains 5 sections, 7 theorems, 11 equations, 3 figures.

Table of Contents

  1. Quantum thermalization
  2. EDGE Theorem
  3. EDGEs of shallow complexity at finite temperature
  4. Generalized Gibbs state and finite-time thermalization
  5. Discussion

Key Result

Theorem 1

Consider any translation invariant, geometrically local Hamiltonian for $N$ qubits in $D$-dimensional lattice with nondegenerate spectral gaps. Below a certain threshold inverse temperature $\beta_{*}>0$ (independent of $N$), any ensemble satisfying properties (P1--3) thermalizes [i.e., satisfies Eq

Figures (3)

  • Figure 1: Main idea of quantum thermalization. The local indistinguishability between time-evolved quantum many-body states (left) and the thermal Gibbs states (right) establishes the bridge between quantum and statistical physics.
  • Figure 2: Flowchart for our proof of quantum thermalization. We summarize logical implications among our results (blue), previously-known results (green), and assumptions (orange).
  • Figure 3: Schematic illustrations of product states on 1D or 2D lattices.a Periodic (left) and aperiodic (right) computational basis states in 1D. b Approximately periodic product state in 2D: qubit configurations in regions A and B separated by a period are similar, giving rise to long-range correlations.

Theorems & Definitions (10)

  • Theorem 1: Quantum thermalization
  • Corollary 1
  • Definition 1: Energy dispersed Gibbs ensemble
  • Theorem 2: EDGE Theorem
  • proof
  • Proposition 1: A corollary of the weak ETH
  • Lemma 1
  • Theorem 3: Bound on average IPR
  • Proposition 2
  • proof