Quantum smoothing for classical mixtures

We employ a superconducting qubit embedded in a 3D cavity to study quantum smoothing using projective measurements. Whereas the density matrix $\rho(t)$, which depends on the evolution dynamics and measurements performed prior to time $t$ makes predictions about the outcomes of measurements performed at time $t$, further probing of the qubit allows us to refine our prediction in hindsight. We introduce an auxiliary matrix $E(t)$, which is conditioned on the measurement record from $t$ to a final time $T$. The pair of matrices ($\rho(t)$, $E(t)$) exhaust our ability to make a smoothed prediction for the measurement outcome at an earlier time $t$. If the combined dynamics and measurements on a system lead to $\rho(t)$ with only diagonal elements in a given basis $\{|n\rangle\}$, it may be treated as a classical mixture. If continued probing and dynamics of the system lead to $E(t)$ that is also diagonal in the basis $\{|n\rangle\}$, we examine whether the classical mixture description is still valid in determining the smoothed probabilities for the measurement outcome at time $t$. We show experimentally 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 time $t$.