In usual quantum theory, the information available about a quantum system is
defined in terms of the density matrix describing it on a spacelike surface.
This definition must be generalized for extensions of quantum theory which do
not have a notion of state on a spacelike surface. It must be generalized for
the generalized quantum theories appropriate when spacetime geometry fluctuates
quantum mechanically or when geometry is fixed but not foliable by spacelike
surfaces. This paper introduces a four-dimensional notion of the information
available about a quantum system's boundary conditions in the various sets of
decohering histories it may display. The idea of spacetime information is
applied in several contexts: When spacetime geometry is fixed the information
available through alternatives restricted to a spacetime region is defined. The
information available through histories of alternatives of general operators is
compared to that obtained from the more limited coarse- grainings of
sum-over-histories quantum mechanics. The definition of information is
considered in generalized quantum theories. We consider as specific examples
time-neutral quantum mechanics with initial and final conditions, quantum
theories with non-unitary evolution, and the generalized quantum frameworks
appropriate for quantum spacetime. In such theories complete information about
a quantum system is not necessarily available on any spacelike surface but must
be searched for throughout spacetime. The information loss commonly associated
with the ``evolution of pure states into mixed states'' in black hole
evaporation is thus not in conflict with the principles of generalized quantum
mechanics.Comment: 47pages, 2 figures, UCSBTH 94-0
