Quantifying total correlations in quantum systems through the Pearson correlation coefficient

TL;DR

This paper introduces a novel approach to quantify total correlations in quantum systems using the Pearson correlation coefficient, distinguishing between classical and quantum correlations as mutually exclusive, and linking these to the entropic uncertainty principle.

Contribution

It proposes an alternative correlation measure for quantum systems, revealing the mutually exclusive nature of classical and quantum correlations and connecting them to the entropic uncertainty principle.

Findings

Pearson correlation coefficient can quantify total correlations in quantum states.

Classical and quantum correlations are mutually exclusive in this framework.

Correlation distributions relate to the entropic uncertainty principle.

Abstract

Conventionally the total correlations within a quantum system are quantified through distance-based expressions such as the relative entropy or the square-norm. Those expressions imply that a quantum state can contain both classical and quantum correlations. In this work, we provide an alternative method to quantify the total correlations through the Pearson correlation coefficient. Using this method, we argue that a quantum state can be correlated in either a classical or a quantum way, i.e., the two cases are mutually exclusive. We also illustrate that, at least for the case of two-qubit systems, the distribution of the correlations among certain locally incompatible pairs of observables provides insight in regards to whether a system contains classical or quantum correlations. Finally, we show how correlations in quantum systems are connected to the general entropic uncertainty principle.