Time-symmetric correlations for open quantum systems

Abstract

Two-time expectation values of sequential measurements of dichotomic observables are known to be time symmetric for closed quantum systems. Namely, if a system evolves unitarily between sequential measurements of dichotomic observables ${\mathcal{O}}_A$ followed by ${\mathcal{O}}_B$, then it necessarily follows that $⟨{\mathcal{O}}_A,{\mathcal{O}}_B⟩=⟨{\mathcal{O}}_B,{\mathcal{O}}_A⟩$, where $⟨{\mathcal{O}}_A,{\mathcal{O}}_B⟩$ is the two-time expectation value corresponding to the product of the measurement outcomes of ${\mathcal{O}}_A$ followed by ${\mathcal{O}}_B$, and $⟨{\mathcal{O}}_B,{\mathcal{O}}_A⟩$ is the two-time expectation value associated with the time reversal of the unitary dynamics, where a measurement of ${\mathcal{O}}_B$ precedes a measurement of ${\mathcal{O}}_A$. In this work, we show that a quantum Bayes' rule implies a time symmetry for two-time expectation values associated with open quantum systems, which evolve according to a general quantum channel between measurements. Such results are in contrast with the view that processes associated with open quantum systems -- which may lose information to their environment -- are not reversible in any operational sense. We give an example of such time-symmetric correlations for the amplitude-damping channel, and we propose an experimental protocol for the potential verification of the theoretical predictions associated with our results.

17 pages, 5 figures. Comments welcome!