  While in relativity theory space evolves over time into a single entity known
as spacetime, quantum theory lacks a standard notion of how to encapsulate the
dynamical evolution of a quantum state into a single "state over time".
Recently it was emphasized in the work of Fitzsimons, Jones and Vedral that if
such a state over time is to encode not only spatial but also temporal
correlations which exist within a quantum dynamical process, then it should be
represented not by a density matrix, but rather, by a pseudo-density matrix. A
pseudo-density matrix is a hermitian matrix of unit trace whose marginals are
density matrices, and in this work, we make use a factorization system for
quantum channels to associate a pseudo-density matrix with a quantum system
which is to evolve according to a finite sequence of quantum channels. We then
view such a pseudo-density matrix as the quantum analog of a local patch of
spacetime, and we make an in-depth mathematical analysis of such quantum
dynamical pseudo-density matrices and the properties they satisfy. We also show
how to explicitly extract quantum dynamics from a given pseudo-density matrix,
thus solving an open problem posed in the literature.
