Abstract

The canonical correlation, which describes the fundamental correlation between two separated subsystems S₁and S₂, is described by the Schmidt decomposition: ψ^s= ∑_ka_kζ_kη_k, where the {ζ_k} and {η_k} constitute orthonormal sets of states for S₁and S₂, respectively. Any pair of operators Ậin S₁and $\widehat{B}$ in S₂, which have as non-degenerate eigenfunctions the set {ζ_k} and {η_k} are perfectly correlated, where the probability for eigenvalues λ_i, of Ậoperating on ζ_k, arid µ_jof $\widehat{B}$ operating on η_kis given by P(λ_i, and µ_j) = P_ij=| a_k|².The Shannon index of correlation is given by : $I_{\text{Shann}}={{\displaystyle {\sum }_{i,j}P}}_{ij}\log \left(\frac{Pij}{PiPj}\right)$. Another description of correlation is given by the quantum index of correlation which is related to the quantum entropy. The overall state of the two-component system is described by a density operator ρand the states of the component systems are described by the reduced density operators _ρaand _ρb.Using the definition of entropy S= −Trρlnρ, the quantum index of correlation for such system is : I_c= S_a+ S_b− S.As we restrict the discussion to pure states, the entropy Sis precisely zero. For a pure two component system the maximal Shannon index of correlation is equal to half the quantum index of con elation According to hidden variables theory the correlation between the two separated systems was produced during the interaction time in the past by common hidden variables λ. This idea can be represented as: P_ij= ∑_λP_iλP_jλ;P_i= ∑_λP_iλ;P_j=∑_λP_jk.Using information theory, we find that hidden variables theories lead to a refined distribution where the original values of P_iand P_jhave been resolved into a number of values p_iλand P_jλ. Following a calculation which includes hidden variables, the following interesting result has been found: $I_{\text{Shann}}^{\text{HV}}=I_c$This result means that the refinement by hidden variables should increase the amount of information included in the Shannon index of correlation, so that it can be equal to the quantum index of correlation. We find that hidden variable theories lead to mutual information between subsystems which is larger than that obtained by QM. A by-product of refuting the hidden variables theories is reestablishing the quantum limit of mutual information. However, following the last equality a possible unconventional meaning to hidden variables is discussed.

© 2000 IEEE