Relative position between a pair of spin model subfactors
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We start with a pair of distinct 2× 2 complex Hadamard matrices and compute the Pimsner-Popa probabilistic constant and the exact value of the Connes-Størmer relative entropy between the corresponding pair of spin model subfactors of the hyperfinite type II_1 factor R. We have characterized when the subfactors are equal in terms of certain equivalence relation which is finer than the Hadamard equivalence relation. We also prove that the intersection of the two subfactors is a non-irreducible subfactor of R with Jones index 4. Moreover, the angle between the spin model subfactors is ninety degree. Further, for a pair of 4× 4 Hadamard inequivalent complex Hadamard matrices, we compute the Pimsner-Popa probabilistic constant between the corresponding spin model subfactors, and as an application we show that the Connes-Størmer entropy between them is bounded by log 2. Prior to this, we have also computed the Pimsner-Popa probabilistic constant between a pair of Masas of a matrix algebra in terms of the Hamming numbers of the rows of certain naturally arising unitary matrix. As an application, we provide a legitimate bound for their relative entropy.
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