  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 $\rho(t)$ 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
$\rho_{nn}(t)$. 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 $\rho$, permit a classical mixture interpretation
of the state of the system at the past time $t$.
