Quantum

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

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Briefings

A new axiomatic framework for multi-time states in operational probabilistic theories (OPTs) is presented, extending Bayesian probability to quantum theory and non-Markovian dynamics. The framework accommodates both Markovian and non-Markovian dynamics through three assumptions: compositionality, reconstructibility, and spatio-temporal marginal consistency. It fully characterizes the unique multi-time state in boxworld and offers a method for Bayesian retrodiction of physical transformations. The work establishes a correspondence between multi-time states and the process-matrix framework for non-Markovian quantum dynamics, enabling causal model inference via semidefinite programming.

Key points

• The framework accommodates both Markovian and non-Markovian dynamics by defining three axioms: compositionality, reconstructibility, and spatio-temporal marginal consistency.
• In non-Markovian quantum dynamics, the framework establishes a correspondence between multi-time states and the process-matrix framework, allowing causal model inference via semidefinite programming.

Context

Quantum states over time were introduced to extend Bayesian probability to quantum theory, with applications in quantum causality and open-system dynamics. While axiomatic characterizations exist for Markovian dynamics, a general framework for non-Markovian dynamics remains elusive. This work fills that gap by developing an axiomatic framework for multi-time states within a class of operational probabilistic theories (OPTs) that includes quantum mechanics as a special case.

Limitations

The framework is limited to operational probabilistic theories (OPTs) and does not cover all physical theories. The characterization of multi-time states in boxworld is specific to that system. The correspondence between multi-time states and the process-matrix framework applies only to non-Markovian quantum dynamics, not general dynamics. The proposed semidefinite programming method serves as a proof of principle for causal order inference.

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