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Geometry of the Hilbert space and the Quantum Zeno Effect

A. K. Pati·S. V. Lawande
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Abstract

We show that the quadratic short time behaviour of transition probability is a natural consequence of the inner product of the Hilbert space of the quantum system. We prove that Schrödinger time evolution between two successive measurements is not a necessary but only a sufficient condition for predicting quantum Zeno effect. We provide a relation between the survival probability and the underlying geometric structure such as the Fubini-Study metric defined on the projective Hilbert space of the quantum system. This predicts the quantum Zeno effect even for systems described by non-linear and non-unitary evolution equations, within the collapse mechanism of the wavefunction during measurement process. Two examples are studied, one is non-linear Schrödinger equation and other is Gisin's equation and it is shown that one can observe quantum Zeno effect for systems described by these equations.

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A. K. Pati, S. V. Lawande. 1998-03-29. Geometry of the Hilbert space and the Quantum Zeno Effect. https://doi.org/10.1103/physreva.58.831

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