Geometric classical and total correlations via trace distance

^(a) fagner@if.uff.br
^(b) jdmontealegrer@if.uff.br
^(c) amen@if.uff.br
^(d) tro@if.uff.br
^(e) msarandy@if.uff.br

Received: 24 July 2013
Accepted: 5 September 2013

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 signal an infinite-order quantum phase transition.