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

Minimum error state discrimination between two mixed states ρand σcan be aided by the receipt of "classical side information" specifying which states from some convex decompositions of ρand σapply in each run. We quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of ρand σ. The lower bound is simply the trace distance between ρand σ, trivially seen to be tight. The upper bound is \sqrt{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 non-classicality known as preparation contextuality.

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