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

Minimum error state discrimination between two mixed states $\rho$ and $\sigma$ can be aided by the receipt of ``classical side information'' specifying which states from some convex decompositions of $\rho$ and $\sigma$ apply in each run. I will quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of $\rho$ and $\sigma$. The lower bound is simply the trace distance between $\rho$ and $\sigma$, trivially seen to be tight. The upper bound is $\sqrt{1 - {\rm tr}(\rho\sigma)}$, and we conjecture that this is also tight. I will show how to reformulate this conjecture in terms of the existence of a pair of ``unbiased decompositions'', which may be of independent interest. Time permitting, I will outline the evidence for this conjecture. Based on http://arxiv.org/abs/1208.2550