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

A quantum system (with Hilbert space (Formula presented.)) entangled with its environment (with Hilbert space (Formula presented.)) is usually not attributed to a wave function but only to a reduced density matrix (Formula presented.). Nevertheless, there is a precise way of attributing to it a random wave function (Formula presented.) , called its conditional wave function, whose probability distribution (Formula presented.) depends on the entangled wave function (Formula presented.) in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of (Formula presented.) but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about (Formula presented.) , e.g., that if the environment is sufficiently large then for every orthonormal basis of (Formula presented.) , most entangled states (Formula presented.) with given reduced density matrix (Formula presented.) are such that (Formula presented.) is close to one of the so-called GAP (Gaussian adjusted projected) measures, (Formula presented.). We also show that, for most entangled states (Formula presented.) from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval (Formula presented.)) and most orthonormal bases of (Formula presented.) , (Formula presented.) is close to (Formula presented.) with (Formula presented.) the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then (Formula presented.) is close to (Formula presented.) with (Formula presented.) the canonical density matrix on (Formula presented.) at inverse temperature (Formula presented.). 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.