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The theory of quantum states over time provides an approach to the dynamicsof quantum information which is in direct analogy with spacetime and itsrelation to classical dynamics. In this work, we further such an analogy byformulating a notion of general covariance for the theory of quantum statesover time. We then associate a canonical state over time with a densityoperator which is to evolve under a sequence of quantum processes modeled bycompletely positive trace-preserving (CPTP) maps, and we show that such acanonical state over time satisfies such a notion of covariance. We also showthat the dynamical quantum Bayes' rule transforms covariantly with respect tostates over time, and we conclude with a discussion of what it means for aphysical law to be generally covariant when formulated in terms of quantumstates over time.

General covariance for quantum states over time