Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment

TL;DR

This paper proves that for large environments, the distribution of a quantum system's wave function conditioned on entanglement is typically close to a universal Gaussian-adjusted measure, supporting its role in describing thermal equilibrium.

Contribution

It establishes universality results for the distribution of conditional wave functions in entangled quantum systems, linking them to GAP measures and thermal states.

Findings

Most entangled states yield a conditional wave function distribution close to GAP measures.

In microcanonical and canonical regimes, the distribution approximates GAP measures associated with energy states.

Supports the use of GAP measures as models for thermal equilibrium wave function distributions.

Abstract

A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) is usually not attributed a wave function but only a reduced density matrix $\rho_1$. Nevertheless, there is a precise way of attributing to it a random wave function $\psi_1$, called its conditional wave function, whose probability distribution $\mu_1$ depends on the entangled wave function $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about $\mu_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of $\mathscr{H}_2$, most entangled states $\psi$ with given reduced density matrix $\rho_1$ are such that $\mu_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP(\rho_1)$. We also show that, for most entangled states $\psi$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+\delta E]$) and most orthonormal bases of $\mathscr{H}_2$, $\mu_1$ is close to $GAP(\mathrm{tr}_2 \rho_{mc})$ with $\rho_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then $\mu_1$ is close to $GAP(\rho_\beta)$ with $\rho_\beta$ the canonical density matrix on $\mathscr{H}_1$ at inverse temperature $\beta=\beta(E)$. This provides the mathematical justification of our claim in [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that $GAP$ measures describe the thermal equilibrium distribution of the wave function.