Geometry of the Hilbert space and the Quantum Zeno Effect

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.