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

Abstract

A quantum system (with Hilbert space ${\mathcal {H}_{1}}$) entangled with its environment (with Hilbert space ${\mathcal {H}_{2}}$) is usually not attributed to a wave function but only to 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 ${\psi \in \mathcal {H}_{1} \otimes \mathcal {H}_{2}}$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of ${\mathcal {H}_{2}}$ 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 ${\mathcal {H}_{2}}$, 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 ${\mathcal {H}_{2}}$, ${\mu_{1}}$ is close to ${GAP(\rm {tr}_{2} \rho_{mc})}$ 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 ${\mathcal {H}_{1}}$ at inverse temperature ${\beta=\beta(E)}$. This provides the mathematical justification of our claim in Goldstein et al. (J Stat Phys 125: 1193–1221, 2006) that GAP measures describe the thermal equilibrium distribution of the wave function.