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

Abstract

A quantum system (with Hilbert space ${\mathcal{H}}_1$) entangled with its environment (with Hilbert space ${\mathcal{H}}_2$) is usually not attributed a wave function but only a reduced density matrix ${\rho }_1$. Nevertheless, there is a precise way of attributing to it a random wave function ${\psi }_1$, called its conditional wave function, whose probability distribution ${\mu }_1$ depends on the entangled wave function $\psi \in {\mathcal{H}}_1\otimes {\mathcal{H}}_2$ in the Hilbert space of system and environment together. It also depends on a choice of orthonormal basis of ${\mathcal{H}}_2$ but in relevant cases, as we show, not very much. We prove several universality (or typicality) results about ${\mu }_1$, e.g., that if the environment is sufficiently large then for every orthonormal basis of ${\mathcal{H}}_2$, most entangled states $\psi$ with given reduced density matrix ${\rho }_1$ are such that ${\mu }_1$ is close to one of the so-called GAP (Gaussian adjusted projected) measures, $GAP({\rho }_1)$. We also show that, for most entangled states $\psi$ from a microcanonical subspace (spanned by the eigenvectors of the Hamiltonian with energies in a narrow interval $[E,E+\delta E]$) and most orthonormal bases of ${\mathcal{H}}_2$, ${\mu }_1$ is close to $GAP({\mathrm{tr}}_2{\rho }_{mc})$ with ${\rho }_{mc}$ the normalized projection to the microcanonical subspace. In particular, if the coupling between the system and the environment is weak, then ${\mu }_1$ is close to $GAP({\rho }_{\beta })$ with ${\rho }_{\beta }$ the canonical density matrix on ${\mathcal{H}}_1$ at inverse temperature $\beta =\beta (E)$. 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.

27 pages LaTeX, no figures; v2 major revision with simpler proofs

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