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.

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

References in corpus (1)

• Optimal Monitoring of Position in Nonlinear Quantum Systems

Cited by in corpus (7)

• Informal Resource Letter - Nonlinear quantum mechanics on arXiv up to August 2004
• Zeno Dynamics in Quantum Statistical Mechanics
• Zeno Dynamics of von Neumann Algebras
• Mathematics of the Quantum Zeno Effect
• Stochastic dynamics and control of a driven nonlinear spin chain: the role of Arnold diffusion
• The geometry of the quantum Zeno effect
• Effect of measurements on quantum speed limit