Unified view of correlations using the square-norm distance

Abstract

The distance between a quantum state and its closest state not having a certain property has been used to quantify the amount of correlations corresponding to that property. This approach allows a unified view of the various kinds of correlations present in a quantum system. In particular, using relative entropy as a distance measure, total correlations can be meaningfully separated in a quantum and a classical part thanks to an additive relation involving only distances between states. Here, we investigate a unified view of correlations using as distance measure the square norm, already used to define the so-called geometric quantum discord. We thus consider geometric quantifiers also for total and classical correlations finding, for a quite general class of bipartite states, their explicit expressions. We analyze the relationship among geometric total, quantum and classical correlations and we find that they do not satisfy anymore a closed additivity relation.

Unified scheme for correlations using linear relative entropy

“…Inspired by the definition of the geometric discord based on Hilbert-Schmidt distance [35], Bellomo et al investigated a geometrical unified scheme in which the total correlation T (2), the quantum correlation D (4), the classical correlation C (5) and the quantity L (6) are redefined as [42] T…”

“…The symmetric (9) and antisymmetric (11) [41,42] using the Hilbert-Schmidt distance) can be derived. This issue is discussed, in the next subsection, for an arbitrary two qubit system.…”

“…We show that the linear relative entropy provides us with a simple approach to treat the different kinds of bipartite correlations in a common framework. This approach can be seen, in some sense, interpolating between the relative entropy-based [23,24] and Hilbert-Schmidt-based [41,42] classification schemes. More specifically, it provides us with a very simple way to perform the optimizations required in deriving closest product, classical and classical product states.…”

“…In this respect, a purely geometrical unified framework to classify the correlations in a given quantum state was discussed in [41,42]. Using the Hilbert-Schmidt norm and paralleling the definition of the geometric discord, the geometric measures of total and classical correlations in a two qubit system were derived in [41,42]. In contrast with the relative entropy, the additivity relation of the type…”

“…In the other hand, from an analytical point of view, the derivation of closed expressions of relative entropy based measures involves optimization procedures that are in general very complicated to perform. In this respect, a purely geometrical unified framework to classify the correlations in a given quantum state was discussed in [41,42]. Using the Hilbert-Schmidt norm and paralleling the definition of the geometric discord, the geometric measures of total and classical correlations in a two qubit system were derived in [41,42].…”