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 ρ(t) with only diagonal elements in a given basis {|n)}, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states |n) with probabilities ρnn(t). Equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states |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 ρ(t), permit a classical mixture interpretation of the state of the system at the past time t.