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Geodesics and the best measurement for distinguishing quantum states
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From statistical distinguishability of probability distributions, one can define distinguishability of quantum states. A corresponding measurement to perform, optimal in a definite sense, for distinguishing between two given states rho_A and rho_B, has been derived by Fuchs and Caves. We show that the Bures-Uhlmann geodesic through the two states singles out this measurement. The geodesic `bounces' at the boundary of the set of quantum states. Whenever the geodesic hits the boundary, the state orthogonal to that boundary state is one of the basis states for the measurement.
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A thermofield-double model of Uhlmann anholonomy
For a thermofield double model, Uhlmann parallel transport acts as optimal filtering measurements on one side and as holonomic quantum gates on the other, with an iSWAP gate constructible from geodesic triangles.
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