# Quantifying the quantumness of ensembles via unitary similarity invariant norms

## TL;DR

This paper introduces a new way to measure the quantumness of quantum ensembles using unitary similarity invariant norms of commutators, with proven desirable properties and practical applications.

## Contribution

It proposes a novel class of quantumness measures based on invariant norms, extending the theoretical framework for analyzing quantum ensembles.

## Key findings

- Measures satisfy positivity, invariance, concavity, convexity, and monotonicity properties.
- Examples demonstrate the measures' effectiveness in quantum information analysis.
- The approach provides a rigorous mathematical foundation for quantumness quantification.

## Abstract

The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.