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 $\mathscr{O}{A}$ followed by $\mathscr{O}{B}$, then it necessarily follows that $\langle\mathscr{O}{A}\,,\mathscr{O}{B}\rangle=\langle\mathscr{O}{B}\,,\mathscr{O}{A}\rangle$, where $\langle\mathscr{O}{A}\,,\mathscr{O}{B}\rangle$ is the two-time expectation value corresponding to the product of the measurement outcomes of $\mathscr{O}{A}$ followed by $\mathscr{O}{B}$, and $\langle\mathscr{O}{B}\,,\mathscr{O}{A}\rangle$ is the two-time expectation value associated with the time reversal of the unitary dynamics, where a measurement of $\mathscr{O}{B}$ precedes a measurement of $\mathscr{O}{A}$. 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.