On a measure of distance for quantum strategies

TL;DR

This paper introduces a new operator norm to measure the distinguishability of quantum strategies, generalizing state discrimination results and providing a mathematical framework for strategy comparison.

Contribution

It defines and characterizes a novel norm for quantum strategies, including duality and optimization proofs, and extends state discrimination results to strategies and channels.

Findings

Established properties of the new strategy norm

Proved strong duality for the associated semidefinite program

Generalized state discrimination results to quantum strategies and channels

Abstract

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Ref. [GW05]. The generalized result states that for any two convex sets S,T of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of s in S from any choice of t in T with bias proportional to the minimal distance between the sets S and T as measured by this norm. A similar discrimination result for channels then follows as a special case.