Information geometry, dynamics and discrete quantum mechanics
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We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry , the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the Pk , we introduce coordinates Sk canonically conjugate to the Pk and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the Pk and Sk . Consistency between the metric tensor and the symplectic form forces us to introduce a K䨬er geometry . The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the ? k ?=?P k - - - v e iS k K䨬er space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the K䨬er space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters' expression for the statistical distance for quantum systems.
AIP Conference Proceedings, volume 1553, 2013, pages 246-253
© 2013 AIP Publishing LLC. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing.This is an electronic version of an article appeared in the abstract books of Joint MMM/Intermag Conference and may be found at http://www.magnetism.org/
Quantum Information, Computation and Communication
Quantum Physics not elsewhere classified