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

A quantum system (with Hilbert space H1) entangled with its environment (with Hilbert space H2) is usually not attributed a wave function but only a reduced density matrix ρ1. Nevertheless, there is a precise way of attributing to it a random wave function ψ1, called its conditional wave function, whose probability distribution μ1 depends on the entangled wave function ψ ∈ H1 ⊗H2 in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of H2 but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about μ1, e.g., that if the environment is sufficiently large then for every orthonormal basis of H2, most entangled states ψ with given reduced density matrix ρ1 are such that μ1 is close to one of the so-called GAP (Gaussian adjusted projected) measures, GAP (ρ1). We also show that, for most entangled states ψ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval [E,E+ δE]) and most orthonormal bases of H2, μ1 is close to GAP (tr2 ρmc) with ρmc the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then μ1 is close to GAP (ρβ) with ρβ the canonical density matrix on H1 at inverse temperature β = β(E). This provides the mathematical justification of our claim in [8] that GAP measures describe the thermal equilibrium distribution of the wave function. Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected] Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected] Dipartimento di Fisica, Università di Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail: [email protected] Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA. E-mail: [email protected] Dipartimento di Fisica, Università di Genova and INFN sezione di Genova, Via Dodecaneso 33, 16146 Genova, Italy. E-mail: [email protected] 1