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

TL;DR

This paper explores the operational meaning of the Hilbert-Schmidt product in quantum state discrimination, establishing bounds on the advantage gained from classical side information and linking it to non-classicality concepts.

Contribution

It introduces bounds on the trace distance with side information, formulates a conjecture on tightness involving unbiased decompositions, and connects these ideas to preparation contextuality.

Findings

Lower bound on trace distance is tight and equals the trace distance between states.

Upper bound on average trace distance is ^{1} - tr( ho \sigma), conjectured to be tight.

Proved the conjecture for special cases and linked to non-classicality.

Abstract

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.