Geometric classical and total correlations via trace distance

TL;DR

This paper introduces geometric measures of classical and total correlations based on the trace norm, providing analytical results for two-qubit states and applying them to quantum phase transitions in spin chains.

Contribution

It defines trace norm-based geometric correlations, derives analytical formulas for Bell-diagonal states, and demonstrates their effectiveness in detecting quantum phase transitions.

Findings

Trace norm geometric correlations are well-defined and analytically computable for two-qubit states.

Classical geometric correlation uniquely signals infinite-order quantum phase transitions.

Geometric correlations differ from entropic measures in hierarchy and monotonicity properties.

Abstract

We introduce the concepts of geometric classical and total correlations through Schatten 1-norm (trace norm), which is the only Schatten p-norm able to ensure a well-defined geometric measure of correlations. In particular, we derive the analytical expressions for the case of two-qubit Bell-diagonal states, discussing the superadditivity of geometric correlations. As an illustration, we compare our results with the entropic correlations, discussing both their hierarchy and monotonicity properties. Moreover, we apply the geometric correlations to investigate the ground state of spin chains in the thermodynamic limit. In contrast to the entropic quantifiers, we show that the classical correlation is the only source of 1-norm geometric correlation that is able to signaling an infinite-order quantum phase transition.