Matthew F. Pusey, Terry Rudolph
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. We 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 - tr(\rho\sigma)}, 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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http://arxiv.org/abs/1208.2550
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