# Problem of quantifying quantum correlations with non-commutative discord

## TL;DR

This paper critiques Guo's non-commutativity measure of quantum correlations, identifies its basis-dependence issue, and proposes an optimized, basis-independent version that better quantifies quantum entanglement and correlations.

## Contribution

It introduces a minimization approach to eliminate basis dependence in Guo's measure, making it a valid entanglement measure for pure states and improving correlation quantification.

## Key findings

- Guo's measure can overestimate quantum correlations in most cases.
- The optimized measure reduces to a proper entanglement measure for pure states.
- Guo's measure remains useful as a quick qualitative estimator of quantum correlations.

## Abstract

In this work we analyze a non-commutativity measure of quantum correlations recently proposed by Y. Guo [Sci. Rep. 6, 25241 (2016)]. By recourse to a systematic survey of a two-qubit system, we detected an undesirable behavior of such a measure related to its representation-dependence. In the case of pure states, this dependence manifests as a non-satisfactory entanglement measure whenever a representation other than the Schmidt's is used. In order to avoid this dependence on the basis, we argue that a minimization procedure over the set of all possible representations of the quantum state is required. In the case of pure states, this minimization can be analytically performed and the optimal basis turns out to be that of Schmidt's. In addition, the resulting measure inherits the main properties of Guo's measure and, unlike the latter, it reduces to a legitimate entanglement measure in the case of pure states. Some examples involving general mixed states are also analyzed considering such a optimization. The results show that, in most cases of interest, the use of Guo's measure can result in a overestimation of quantum correlations. However, since Guo's measure has the advantage of being easily computable, it might be used as a qualitative estimator of the presence of quantum correlations.