Universal Probability Distribution for the Wave Function of a Quantum System Entangled with Its Environment
A quantum system (with Hilbert space $\mathscr{H}_1$) entangled with its environment (with Hilbert space $\mathscr{H}_2$) 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 $ψ\in\mathscr{H}_1\otimes\mathscr{H}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of $\mathscr{H}_2$ 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 $\mathscr{H}_2$, 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 $\mathscr{H}_2$, $μ_1$ is close to $GAP(\mathrm{tr}_2 ρ_{mc})$ 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 $\mathscr{H}_1$ 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.