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Universal Probability Distribution for the Wave Function of an Open Quantum System

An open quantum system (i.e., one that interacts with its environment) is almost always entangled with its environment; it is therefore usually not attributed a wave function but only a reduced density matrix rho. Nevertheless, there is a precise way of attributing to it a wave function psi_1, called its conditional wave function, which is a random wave function of the system whose probability distribution mu_1 depends on the entangled wave function psi in the Hilbert space of system and environment together. We prove several universality (or typicality) results about mu_1; they show that if the environment is sufficiently large then mu_1 does not depend much on the details of psi and is approximately given by one of the so-called GAP measures. Specifically, for most entangled states psi with given reduced density matrix rho_1, mu_1 is close to GAP(rho_1). We also show that, if the coupling between the system and the environment is weak, then for most entangled states psi from a microcanonical subspace corresponding to energies in a narrow interval [E,E+delta E] (and most bases of Hilbert space 2), mu_1 is close to GAP(rho_beta) with rho_beta the canonical density matrix on Hilbert space 1 at inverse temperature beta=beta(E). This provides the mathematical justification of the claim that GAP(rho_beta) is the thermal equilibrium distribution of psi_1.