Søren Gammelmark, Brian Julsgaard, Klaus Mølmer
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 measurements at t than the conventional quantum state at that time. Our formalism describes systems subject to any quantum measurement scenario, any coherent evolution, and any Markovian dissipation processes. In particular it describes projective measurements and weak value measurements, and in the special case of incoherent quantum systems our theory is shown to be equivalent to the smoothing procedures applied to hidden Markov models in classical probability theory.
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http://arxiv.org/abs/1305.0681
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