We define the quantum Young function corresponding to the classical one used by Pistone and Sempi, namely, cosh x - 1. Thus, let H be a positive self=adjoint operator on a separable Hilbert space, and suppose that exp -H is of trace class, and also that exp -aH is also of trace class for all a larger than some b less than 1. We take it that the trace of exp -H is equal to 1, and so defines a normal state of the quantum system. Let X be an H-bounded form, of small enough H-bound. Then the quantum analogue of the Young function cosh x -1 is the expression
Y(X)=1/2{Tr exp(-H+X) + Tr exp(-H-X)} - 1.
The Orlicz norm defined by Y is indeed a norm.
We show that the Orlicz norms defined at H and H + X are equivalent. This point was omitted from a previous paper Duality in quantum information geometry . This equivalence shows that the sets used previously for the space of the manifold do indeed form a Banach manifold modelled on the Orlicz space at say exp -H, as promised.
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© by Ray Streater, 5/October/2006