  Discussions of quantum mechanics often loosely claim that time evolution
logically must be unitary, in order for the probabilistic interpretation of the
amplitudes of the state vector to make sense at all times. We discuss from
first principles whether this claim is true: if we assume only that the
time-evolution operator is *linear*, then does the stronger requirement that it
be *unitary* follow from the other axioms of quantum mechanics? The answer is
subtle. We discuss two mathematically distinct but physically equivalent
formulations of the axioms of quantum mechanics, and consider generalizing each
to postulate only that time evolution is linear. Within one formulation, the
unitarity of time evolution follows logically from the other axioms -- but
within the other formulation, it does not. Allowing the time-evolution operator
be (a priori) arbitrarily linear does not change the physical observables in
one formulation of quantum mechanics, but changes the other formulation to a
*distinct* (internally consistent) physical theory that allows new
phenomonology like (e.g.) faster-than-light communication. Therefore, the
unitarity of time evolution is arguably better thought of as a logically
independent and experimentally falsifiable axiom of quantum mechanics, not as a
tautological consequence of the other axioms.
