Local hidden variable theoretic measure of quantumness of mutual information

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 (rho) over cap (AB) 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 p(epsilon(A)(a); epsilon(B)(b)) (epsilon(A)(a), epsilon(B)(b) = +/- 1) for the spin A to have the component epsilon(B)(b)/2 in the direction b, constructed by invoking the LHV theory. The quantumness of mutual information is then defined as Q(LHV) = I-Q (vertical bar(rho) over cap (AB)) - I-LHV where I-Q((rho) over cap (AB)) is the quantum theoretic information content in the state (rho) over cap (AB) and the LHV theoretic classical information I-LHV 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 <(S) over cap (A)> and <(S) over cap (B)> of the spins A and B where (S) over cap (A) <(S) over capB > is the spin vector of spin a (spin B) and <(P) over cap > = Tr ((P) over cap (AB)((rho) over cap)). 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 I-LHV = I(a,b) and Q(LHV) turns out to be identical with the measurement induced disturbance. If <(S) over cap (A)> = <(S) over cap (B)> =0, then I-LHV 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 Q(LHV) is then found to be the same as the symmetric discord. IF <(S) over cap (A)> = 0, <(S) over cap (B)> not equal 0, then I-LHV is defined to be the maximum of I(a, b) over a with b = <(S) over cap (B) >/vertical bar(S) over cap (B)vertical bar. The Q(LHV) is then the same as the quantum discord for measurement on A if the eigenstates of S-(B) over cap.b are also the eigenstates of the operator <+/-, a(m)|vertical bar(rho) over cap (AB)vertical bar +/-, a(m)> on B where a(m) is the direction of optimization of spin A for evaluation of the quantum discord and vertical bar +/-, am > are the eigenstates of S-(Lambda) over cap . a(m).