Zeno effect and ergodicity in finite-time quantum measurements

Abstract

We demonstrate that an attempt to measure a non-local in time quantity, such as the time average \la A\ra_T of a dynamical variable $A$, by separating Feynman paths into ever narrower exclusive classes traps the system in eigensubspaces of the corresponding operator \a. Conversely, in a long measurement of \la A\ra_T to a finite accuracy, the system explores its Hilbert space and is driven to a universal steady state in which von Neumann ensemble average of \a coincides with \la A\ra_T. Both effects are conveniently analysed in terms of singularities and critical points of the corresponding amplitude distribution and the Zeno-like behaviour is shown to be a consequence of conservation of probability