  What is the quantum state of the universe? Although there have been several
interesting suggestions, the question remains open. In this paper, I consider a
natural choice for the universal quantum state arising from the Past
Hypothesis, a boundary condition that accounts for the time-asymmetry of the
universe. The natural choice is given not by a wave function (representing a
pure state) but by a density matrix (representing a mixed state).
  I begin by classifying quantum theories into two types: theories with a
fundamental wave function and theories with a fundamental density matrix. The
Past Hypothesis is compatible with infinitely many initial wave functions, none
of which seems to be particularly natural. However, once we turn to density
matrices, the Past Hypothesis provides a natural choice---the normalized
projection onto the Past Hypothesis subspace in the Hilbert space.
Nevertheless, the two types of theories can be empirically equivalent. To
provide a concrete understanding of the empirical equivalence, I provide a
novel subsystem analysis in the context of Bohmian theories. Given the
empirical equivalence, it seems empirically underdetermined whether the
universe is in a pure state or a mixed state. Finally, I discuss some
theoretical payoffs of the density-matrix theories and present some open
problems for future research.
