Canonical Typicality

TL;DR

This paper proves that for most pure quantum states within an energy shell, the reduced state of a subsystem is canonical, strengthening the understanding of thermalization in quantum systems.

Contribution

It establishes that the canonical state emerges for the overwhelming majority of pure states, extending the typicality concept beyond mixed states.

Findings

Most pure states in an energy shell have a canonical reduced density matrix.

The result generalizes and justifies Schrödinger's 1952 remarks.

Supports the idea of typicality in quantum thermalization.

Abstract

It is well known that a system, S, weakly coupled to a heat bath, B, is described by the canonical ensemble when the composite, S+B, is described by the microcanonical ensemble corresponding to a suitable energy shell. This is true both for classical distributions on the phase space and for quantum density matrices. Here we show that a much stronger statement holds for quantum systems. Even if the state of the composite corresponds to a single wave function rather than a mixture, the reduced density matrix of the system is canonical, for the overwhelming majority of wave functions in the subspace corresponding to the energy interval encompassed by the microcanonical ensemble. This clarifies, expands and justifies remarks made by Schr\"odinger in 1952.