Quantum lost property: A possible operational meaning for the Hilbert-Schmidt product

Minimum-error state discrimination between two mixed states ρ and σ can be aided by the receipt of “classical side information” specifying which states from some convex decompositions of ρ and σ apply in each run. We quantify this phenomena by the average trace distance and give lower and upper bounds on this quantity as functions of ρ and σ. The lower bound is simply the trace distance between ρ and σ, trivially seen to be tight. The upper bound is 1-tr(ρσ), and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of “unbiased decompositions,” which may be of independent interest, and prove it for a few special cases. Finally, we point towards a link with a notion of nonclassicality known as preparation contextuality.