Quantized geometry associated with uncertainty and correlation

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Abstract
The geometric structure of the law of quantum-state evolution is studied. The Fubini-Study metric induced on the quantum evolution submanifold of the projective Hilbert space is shown to be completely expressed by the uncertainties and correlations of various generators of evolutions. The Riemannian connection is expressed as a quantum-mechanical expectation value of a certain Hermitian operator. It is discussed that the metric carries some of quantum numbers contained in a given reference state, in general, and consequently the geometry is inherently quantized. These results are demonstrated by the simple examples of the squeezed coherent state, displaced number state, squeezed number state, and generalized coherent spin state.

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