Order-preserving condition for coherence measures of projective measurements with One Example

TL;DR

This paper introduces an order-preserving condition for coherence measures of projective measurements, generalizes the concept to subspaces and POVMs, and verifies its validity through mathematical analysis and examples.

Contribution

It proposes a new order-preserving condition for coherence measures of projective measurements and extends the framework to POVMs, enriching the quantification of quantum superposition.

Findings

The order-preserving condition is satisfied by the generalized coherence measure.

The measure adheres to other reasonable coherence criteria.

The framework applies to both projective measurements and POVMs.

Abstract

Superposition is an essential feature of quantum mechanics. From the Schrodinger's cat to quantum algorithms such as Deutsch-Jorsza algorithm, quantum superposition plays an important role. It is one fundamental and crucial question how to quantify superposition. Until now, the framework of coherence has been well established as one typical instance of quantum resource theories. And the concept of coherence has been generalized into linearly independent basis, projective measurements and POVMs. In this work, we will focus on coherence measures for projective measurements or orthogonal subspaces. One new condition, order-preserving condition, is proposed for such measures. This condition is rooted in the mathematical structure of Hilbert spaces' orthogonal decomposition. And by generalizing the 1/2-affinity of coherence into subspace cases, we verify that this generalized coherence measure satisfies the order-preserving condition. And it also satisfies other reasonable conditions to be a good coherence measure. As the partial order relationship exists for not only projective measurements, but also POVMs, it's natural to study the order-preserving condition in POVM cases, which will be the last part of this work.