Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics
arXiv:2402.01218 · doi:10.22331/q-2024-08-27-1447
Abstract
The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one pair of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.
Accepted in Quantum
References in corpus (16)
- Theoretical framework for quantum networks
- Quantum Mechanics as Quantum Measure Theory
- Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories
- Coherence and non-classicality of quantum Markov processes
- Classical quantum stochastic processes
- Quantum measure and integration theory
- Hidden non-Markovianity in open quantum systems
- Noise representations of open system dynamics
- Non-Markovian quantum control as coherent stochastic trajectories
- Classicality with(out) decoherence: Concepts, relation to Markovianity, and a random matrix theory approach
- Measuring trajectories of environmental noise
- On Markovianity and classicality in multilevel spin-boson models
- Introduction to the theory of open quantum systems
- On the classicality of quantum dephasing processes
- Objectivity of classical quantum stochastic processes
- Hilbert Space Representations of Decoherence Functionals and Quantum Measures