Universal Probability Distribution for the Wave Function of an Open Quantum System (original) (raw)

2011, arXiv (Cornell University)

A quantum system (with Hilbert space H 1) entangled with its environment (with Hilbert space H 2) is usually not attributed a wave function but only a reduced density matrix ρ 1. Nevertheless, there is a precise way of attributing to it a random wave function ψ 1 , called its conditional wave function, whose probability distribution µ 1 depends on the entangled wave function ψ ∈ H 1 ⊗ H 2 in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of H 2 but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about µ 1 , e.g., that if the environment is sufficiently large then for every orthonormal basis of H 2 , most entangled states ψ with given reduced density matrix ρ 1 are such that µ 1 is close to one of the so-called GAP (Gaussian adjusted projected) measures, GAP (ρ 1). 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 H 2 , µ 1 is close to GAP (tr 2 ρ mc) with ρ mc the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then µ 1 is close to GAP (ρ β) with ρ β the canonical density matrix on H 1 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.