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The foundations of statistical mechanics from entanglement: Individual states vs. averages

Sandu Popescu, Anthony J. Short, Andreas Winter

TL;DR

This work presents a purely quantum, kinematic foundation for thermalisation by treating the universe as a large pure state and analyzing the reduced state of a small subsystem. A General Canonical Principle is proven: for almost all global pure states under a broad class of constraints $R$, the subsystem S is approximately in its canonical state $\Omega_S$, determined by the equiprobable state $\mathcal{E}_R$. The authors derive quantitative bounds using Levy's Lemma and two complementary methods, showing that the trace-norm distance $\|\rho_S-\Omega_S\|_1$ concentrates around zero when the environment is effectively large, with explicit dependence on dimensions $d_S$, $d_R$, and $d_E^{\mathrm{eff}}$. They illustrate the results with a spin-chain example, deriving conditions under which $\rho_S$ closely matches $\Omega_S$ and recovering the familiar Boltzmann-like form in appropriate limits. The findings suggest that almost any individual pure state of the universe suffices to realize canonical statistical behavior in its small subsystems, offering a robust, observer-free justification for thermalisation.

Abstract

We consider an alternative approach to the foundations of statistical mechanics, in which subjective randomness, ensemble-averaging or time-averaging are not required. Instead, the universe (i.e. the system together with a sufficiently large environment) is in a quantum pure state subject to a global constraint, and thermalisation results from entanglement between system and environment. We formulate and prove a "General Canonical Principle", which states that the system will be thermalised for almost all pure states of the universe, and provide rigorous quantitative bounds using Levy's Lemma.

The foundations of statistical mechanics from entanglement: Individual states vs. averages

TL;DR

This work presents a purely quantum, kinematic foundation for thermalisation by treating the universe as a large pure state and analyzing the reduced state of a small subsystem. A General Canonical Principle is proven: for almost all global pure states under a broad class of constraints , the subsystem S is approximately in its canonical state , determined by the equiprobable state . The authors derive quantitative bounds using Levy's Lemma and two complementary methods, showing that the trace-norm distance concentrates around zero when the environment is effectively large, with explicit dependence on dimensions , , and . They illustrate the results with a spin-chain example, deriving conditions under which closely matches and recovering the familiar Boltzmann-like form in appropriate limits. The findings suggest that almost any individual pure state of the universe suffices to realize canonical statistical behavior in its small subsystems, offering a robust, observer-free justification for thermalisation.

Abstract

We consider an alternative approach to the foundations of statistical mechanics, in which subjective randomness, ensemble-averaging or time-averaging are not required. Instead, the universe (i.e. the system together with a sufficiently large environment) is in a quantum pure state subject to a global constraint, and thermalisation results from entanglement between system and environment. We formulate and prove a "General Canonical Principle", which states that the system will be thermalised for almost all pure states of the universe, and provide rigorous quantitative bounds using Levy's Lemma.

Paper Structure

This paper contains 19 sections, 7 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. General Canonical Principle
  3. Quantitative setup and main theorems
  4. Levy's Lemma
  5. Method I: Applying Levy's Lemma to $\left\| \rho_S-\Omega_S \right\|_1$
  6. Calculating $\left\langle \left\| \rho_S-\Omega_S \right\|_1 \right\rangle$
  7. Calculating $\left\langle \operatorname{Tr}(\rho_S^2) \right\rangle$
  8. Bounding $\left\langle \left\| \rho_S - \Omega_S \right\|_1 \right\rangle$
  9. Improved bounds using restricted subspaces
  10. Method II: Applying Levy's Lemma to expectation values
  11. Similarity of expectation values for random and canonical states
  12. Similarity of expectation values for a complete operator basis
  13. Obtaining a probabilistic bound on $\left\| \rho_S - \Omega_S \right\|_1$
  14. Example: Spin chain with $n p$ excitations
  15. Projection on the typical subspace
  16. ...and 4 more sections

Key Result

Theorem 1

For a randomly chosen state $| \phi \rangle \in \mathcal{H}_R \subseteq \mathcal{H}_S \otimes \mathcal{H}_E$ and arbitrary $\epsilon>0$, the distance between the reduced density matrix of the system $\rho_S=\operatorname{Tr}(| \phi \rangle\!\langle \phi |)$ and the canonical state $\Omega_S=\operato where In these expressions, $C$ is a positive constant (given by $C=(18 \pi^3)^{-1}$), $d_S$ and $

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7