TY - JOUR
TI - Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
DO - https://doi.org/doi:10.7282/T3MS3VQH
AU - Goldstein, Sheldon
AU - Lebowitz, Joel L.
AU - Mastrodonato, Christian
AU - Tumulka, Roderich
AU - Zanghì, Nino
PY - 2016
T2 - Communications in Mathematical Physics
VL - 342
IS - 3
SP - 965
EP - 988
AB - 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(tr2ρmc) 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 [J. Statist. Phys. 125:1193 (2006), http://arxiv.org/abs/quant-ph/0309021] that GAP measures describe the thermal equilibrium distribution of the wave function.
KW - Gaussian measures
KW - Gaussian adjusted projected (GAP) measures
KW - Scrooge measures
KW - Haar measure on the unitary group
KW - Thermodynamic limit
KW - Canonical ensemble in quantum mechanics
KW - Typicality theorems
KW - Conditional wave function
KW - Typical wave function
KW - Wave functions
LA - English
ER -