Quantum dynamics as a pseudo-density matrix

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext class="MJX-tex-mathit" mathvariant="italic">pseudo-density matrix</mml:mtext></mml:mrow></mml:math>. 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.