A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the q-z Family
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q-z-relative entropies provide a huge two parameters family of relative entropies which include almost all known relative entropies for suitable values of the parameters. In this paper we use these relative entropies as generalized potential functions to generate metrics on the space of quantum states. We further investigate possible ranges for the parameters q and z which allow to recover known quantities in Information Geometry. In particular, we show that a proper definition of both the Bures metric and the Wigner-Yanase metric can be derived from this family of divergence functions. To easily visualize the results, we first perform the calculation for the qubit case. For q=z=1/2 and q=1/2,z=1, the q-z-relative entropy respectively reduces to the divergence functions for the Bures and Wigner-Yanase metrics and, as we explicitly show, these metrics are actually recovered from the general expression of the q-z-metric for such values of the parameters. Finally, we extend the derivation of the metric tensor to a generic n-level system. This allows us to give explicit expressions both of the Bures and Wigner-Yanase metric also for the n-level case.
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