Past quantum states of a monitored system.

A density matrix {\rho}(t) yields probabilistic information about the outcome of measurements on a quantum system. We introduce here the past quantum state, which, at time T, accounts for the state of a quantum system at earlier times t < T. The past quantum state {\Xi}(t) is composed of two objects, {\rho}(t) and E(t), conditioned on the dynamics and the probing of the system until t and in the time interval [t, T], respectively. The past quantum state is characterized by its ability to make better predictions for the unknown outcome of projective and weak value measurements at t than the conventional quantum state at that time. On the one hand, our formalism shows how smoothing procedures for estimation of past classical signals by a quantum probe [M. Tsang, Phys. Rev. Lett. 102, 250403, (2009)] apply also to describe the past state of the quantum system itself. On the other hand, it generalizes theories of pre- and post-selected quantum states [Y. Aharonov and L. Vaidman, J. Phys. A: Math. Gen. 24, 2315 (1991)] to systems subject to any quantum measurement scenario, any coherent evolution, and any Markovian dissipation processes.