General covariance for quantum states over time

Quantum states over time are a spatiotemporal generalization of density operators which were first introduced to give a more even-handed treatment of space and time in quantum theory. In particular, quantum states over time encode not only spatial, but also causal correlations associated with the dynamical evolution of a quantum system, and the association of quantum states over time with the dynamical flow of quantum information is in direct analogy with spacetime and its relation to classical dynamics. In this work, we further such an analogy by formulating a notion of general covariance for the theory of quantum states over time. We then associate a canonical state over time with a density operator which is to evolve under a sequence of quantum processes modeled by completely positive trace-preserving (CPTP) maps, and we show that such a canonical state over time satisfies such a notion of covariance. We also show that the dynamical quantum Bayes’ rule transforms covariantly with respect to quantum states over time, and we conclude with a discussion of what it means for a physical law to be generally covariant when formulated in terms of quantum states over time.