The search for a potential function S allowing us to reconstruct a given metric tensor g and a given symmetric covariant tensor T on a manifold is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on . The connection between this problem, the geometric structure of the space of pure states of quantum mechanics, and the theory of contrast functions of classical information geometry are outlined.
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These are metric tensors satisfying the so-called monotonicity property, i.e., the scalar product they induce on tangent vectors does not increase under the action of completely positive trace-preserving (CPTP) maps.
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