Quantum smoothing for classical mixtures

Wave functions and density matrices represent our knowledge about a quantum system and give probabilities for the outcomes of measurements. If the combined dynamics and measurements on a system lead to a density matrix rho(t) with only diagonal elements in a given basis {vertical bar n >}, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states vertical bar n > with probabilities rho(nn) (t). Equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states vertical bar n > may improve our ability to retrodict what was the outcome of a projective state measurement at time t. Here, we show with experiments on a superconducting qubit that the smoothed probabilities do not, in the same way as the diagonal elements of rho(t), permit a classical mixture interpretation of the state of the system at the past time t.