Quantum lost property: A possible operational meaning for the Hilbert-Schmidt product
Matthew F. Pusey, Terry Rudolph · Physical Review A · 2012
Minimum-error state discrimination between two mixed states $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ can be aided by the receipt of ``classical side information'' specifying which states from some convex decompositions of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ 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 $\ensuremath{\rho}$ and $\ensuremath{\sigma}$. The lower bound is simply the trace distance between $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, trivially seen to be tight. The upper bound is $\sqrt{1\ensuremath{-}\mathrm{tr}(\ensuremath{\rho}\ensuremath{\sigma})}$, 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 nonclassicality known as preparation contextuality.