Quantum smoothing for classical mixtures

In quantum mechanics, 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 $\{|n\rangle\}$, it may be treated as a classical mixture, i.e., a system which randomly occupies the basis states $|n\rangle$ with probabilities $ρ_{nn}(t)$. Fully equivalent to so-called smoothing in classical probability theory, subsequent probing of the occupation of the states $|n\rangle$ improves 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 $ρ$, permit a classical mixture interpretation of the state of the system at the past time $t$.