Local Hidden Variable Theoretic Measure of Quantumness of Mutual Information

TL;DR

This paper introduces a new measure, $Q_{LHV}$, for quantifying the quantumness of mutual information in two spin-1/2 systems by explicitly using local hidden variable theory, simplifying calculations and unifying existing measures.

Contribution

It proposes a novel measure $Q_{LHV}$ based on local hidden variable theory, unifying and simplifying quantumness quantification for two spin-1/2 particles.

Findings

$Q_{LHV}$ matches measurement induced disturbance when Bloch vectors are non-zero.

$Q_{LHV}$ is identical to symmetric discord when both Bloch vectors vanish.

$Q_{LHV}$ unambiguously determines quantumness even when Bloch vectors are zero.

Abstract

Entanglement, a manifestation of quantumness of correlations between the observables of the subsystems of a composite system, and the quantumness of their mutual information are widely studied characteristics of a system of spin-1/2 particles. The concept of quantumness of correlations between the observables of a system is based on incommensurability of the correlations with the predictions of some local hidden variable (LHV) theory. However, the concept of quantumness of mutual information does not invoke the LHV theory explicitly. In this paper, by invoking explicitly the local hidden variable theory, a measure of quantumness of mutual information, $Q_{LHV}$, for a system of two spin-1/2 particles is proposed. It is based on finding the difference between the quantum and classical mutual informations in which the classical mutual information corresponds to the joint probability of the eigenvalues of the spins each along a specified direction. The proposed measure circumvents the need of optimization when the Bloch vector of each spin is non-zero; the optimization is needed but can be performed analytically exactly when the Bloch vector of each spin vanishes and is simplified when the Bloch vector of only one of the spins is zero. In essence, the proposed measure is identical with the measurement induced disturbance when the Bloch vector of each of the spins is non-zero. However, whereas the measurement induced disturbance is non-unique when the Bloch vector of one or both the spins is zero, the proposed measure even then determines the quantumness of mutual information unambiguously. The $Q_{LHV}$ is identical with the symmetric discord if the Bloch vector of each spin vanishes. It is same as the quantum discord if the Bloch vector of only one spin is zero and if the state in question possesses certain additional properties.