Multi-Time States in Quantum and Operational Probabilistic Theories: An Axiomatic Framework

arXiv:2610.22489v1 Announce Type: new Abstract: Quantum states over time were introduced, in part, to extend Bayesian probability to quantum theory and have since found applications across diverse areas, ranging from quantum causality and open-system dynamics to tests of macroscopic realism. While axiomatic characterizations of these states exist for Markovian dynamics, a general framework accommodating non-Markovian dynamics is still lacking. Moreover, extending this framework beyond quantum mechanics to general physical theories also remains elusive. In this work, we develop an axiomatic framework for multi-time states within a class of operational probabilistic theories (OPTs) that includes quantum mechanics as a special case. Based on three assumptions---compositionality, reconstructibility, and spatio-temporal marginal consistency---the framework accommodates both Markovian and non-Markovian dynamics, and provides sufficient conditions to certify signaling and temporal memory within OPT. As a concrete application of our OPT framework, we fully characterize the unique multi-time state in boxworld and develop a method for Bayesian retrodiction of physical transformations there. Our general framework is consistent with quantum theory, reproducing several existing results. In the non-Markovian quantum dynamics, we establish a correspondence between multi-time states and the process-matrix framework, and fully characterize the multi-time states compatible with general dynamics. This correspondence also lets us infer quantum causal models from multi-time states, and as a proof of principle, we propose a semidefinite program to infer causal order from quantum multi-time states. These results provide a unified framework for studying spatio-temporal states, causal relations, and Bayesian reasoning within OPTs, with implications for both fundamental and applied research.