Local hidden variable theoretic measure of quantumness of mutual information

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, the concept of quantumness of mutual information for a system of two spin-1/2 particles, named A and B, in the state described by the density matrix is formulated by invoking explicitly the LHV theory. To that end, the classical mutual information I(a, b) of the spins is assumed to correspond to the joint probability () for the spin A to have the component in the direction a and the spin B to have the component in the direction b, constructed by invoking the LHV theory. The quantumness of mutual information is then defined as where is the quantum theoretic information content in the state and the LHV theoretic classical information ILHV is defined in terms of I(a, b) by choosing the directions a, b as follows. The choice of the directions a, b is made by finding the Bloch vectors and of the spins A and B where () is the spin vector of spin A (spin B) and . The directions a and b are taken to be along the Bloch vector of A and B respectively if those Bloch vectors are non-zero. In that case ILHV = I(a, b) and QLHV turns out to be identical with the measurement induced disturbance. If , then ILHV is defined to be the maximum of I(a, b) over a and b. The said optimization in this case can be performed analytically exactly and QLHV is then found to be the same as the symmetric discord. If , , then ILHV is defined to be the maximum of I(a, b) over a with . The QLHV is then the same as the quantum discord for measurement on A if the eigenstates of are also the eigenstates of the operator on B where am is the direction of optimization of spin A for evaluation of the quantum discord and | ±, am〉 are the eigenstates of .