Geometry of the Hilbert space and the Quantum Zeno Effect

Details Geometry of the Hilbert space and the Quantum Zeno Effect Geometry of the Hilbert space and the Quantum Zeno Effect

1998-03-29

arxiv.org/abs/quant-ph/9803082v1

Physics

Quantum Physics

10 pages, Latex, no figures, submitted to Phys. Rev. A

Scientific paper

10.1103/PhysRevA.58.831

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 Schr\"odinger 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 Schr\"odinger 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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