Quantum distinguishability measures: projectors vs. states maximization

TL;DR

This paper introduces a new quantum state distinguishability measure based on maximization over density matrices, offering an alternative to the traditional projectors-based approach, with properties like convexity and invariance under unitary transformations.

Contribution

It proposes an alternative quantum distinguishability measure based on density matrices, extending the operational framework and analyzing its properties and implications for quantum transformations.

Findings

The new measure is based on an infinite-norm rather than 1-norm.

Properties such as convexity, monotonicity, and invariance are satisfied.

Contractivity under unital maps is established, enabling a quantumness measure for non-unital maps.

Abstract

The distinguishability between two quantum states can be defined in terms of their trace distance. The operational meaning of this definition involves a maximization over measurement projectors. Here we introduce an alternative definition of distinguishability which, instead of projectors, is based on maximization over normalized states (density matrices). It is shown that this procedure leads to a distance (between two states) that, in contrast to the usual approach based on a 1-norm, is based on an infinite-norm. Properties such as convexity, monotonicity, and invariance under unitary transformations are fulfilled. Equivalent operational implementations based on maximization over classical probabilities and hypothesis testing scenarios are also established. When considering the action of completely positive transformations contractivity is only granted for unital maps. This feature allows us to introduce a measure of the quantumness of non-unital maps that can be written in terms of the proposed distinguishability measure and corresponds to the maximal possible deviation from contractivity. Particular examples sustain the main results and conclusions.