  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 $\rho_1$. 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 $\psi\in\mathscr{H}_1\otimes\mathscr{H}_2$ 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 $\mu_1$, e.g., that if the environment is sufficiently large then for
every orthonormal basis of $\mathscr{H}_2$, 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 $\mathscr{H}_2$,
$\mu_1$ 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
$\mathscr{H}_1$ 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.
