Local hidden variable theoretic measure of quantumness of mutual information

摘要

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 ofmutual 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 ρ~(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(∈_a~A;∈_b~B) (∈_a~A, ∈_b~B= ±1) for the spin A to have the component ∈_a~A /2 in the direction a and the spin B to have the component ∈_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(ρ~(AB0) ? I~(LHV) where I_Q(ρ~(AB)) is the quantum theoretic information content in the state ρ~(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~A〉 and ≠ 0, then I_(LHV) is defined to be the maximum of I(a, b) over a with b = S~B/|S~B|. The Q_(LHV) is then the same as the quantum discord for measurement on A if the eigenstates of S~B.b are also the eigenstates of the operator 〈±, a_m|ρ~(AB)|±, a_m〉 on B where a_m is the direction of optimization of spin A for evaluation of the quantum discord and |±, a_m> are the eigenstates of S~A.a_m.