Minimum guesswork discrimination between quantum states

TL;DR

This paper introduces a quantum state discrimination method based on minimizing guesswork, offering a new perspective beyond traditional error probability, with practical conditions and bounds for optimal measurements.

Contribution

It formulates minimum guesswork discrimination as a semidefinite program and explores its relation to error probability, providing conditions for optimal measurements and strategies.

Findings

Minimum guesswork can be optimized via semidefinite programming.

Minimum guesswork and error probability criteria generally differ, except for two states.

No-measurement strategy can be optimal under certain conditions.

Abstract

Error probability is a popular and well-studied optimization criterion in discriminating non-orthogonal quantum states. It captures the threat from an adversary who can only query the actual state once. However, when the adversary is able to use a brute-force strategy to query the state, discrimination measurement with minimum error probability does not necessarily minimize the number of queries to get the actual state. In light of this, we take Massey's guesswork as the underlying optimization criterion and study the problem of minimum guesswork discrimination. We show that this problem can be reduced to a semidefinite programming problem. Necessary and sufficient conditions when a measurement achieves minimum guesswork are presented. We also reveal the relation between minimum guesswork and minimum error probability. We show that the two criteria generally disagree with each other, except for the special case with two states. Both upper and lower information-theoretic bounds on minimum guesswork are given. For geometrically uniform quantum states, we provide sufficient conditions when a measurement achieves minimum guesswork. Moreover, we give the necessary and sufficient condition under which making no measurement at all would be the optimal strategy.