  In a quantum universe with a strong arrow of time, it is standard to
postulate that the initial wave function started in a particular
macrostate--the special low-entropy macrostate selected by the Past Hypothesis.
Moreover, there is an additional postulate about statistical mechanical
probabilities according to which the initial wave function is a "typical"
choice in the macrostate (the Statistical Postulate). Together, they support a
probabilistic version of the Second Law of Thermodynamics: typical initial wave
functions will increase in entropy. Hence, there are two sources of randomness
in such a universe: the quantum-mechanical probabilities of the Born rule and
the statistical mechanical probabilities of the Statistical Postulate. I
propose a new way to understand time's arrow in a quantum universe. It is based
on what I call the Thermodynamic Theories of Quantum Mechanics. According to
this perspective, there is a natural choice for the initial quantum state of
the universe, which is given by not a wave function but by a density matrix.
The initial density matrix of the universe is exactly the (normalized)
projection operator onto the Past Hypothesis subspace (of the Hilbert space of
the universe). Thus, given an initial subspace, we obtain a unique choice of
the initial density matrix. I call this property "the conditional uniqueness"
of the initial quantum state. The conditional uniqueness provides a new and
general strategy to eliminate statistical mechanical probabilities in the
fundamental physical theories, by which we can reduce the two sources of
randomness to only the quantum mechanical one. I also explore the idea of an
"absolutely unique" initial quantum state, in a way that might realize
Penrose's idea (1989) of a strongly deterministic universe.
