Exponentially fast selection of sectors for quantum trajectories beyond non demolition measurements

TL;DR

This paper demonstrates that quantum trajectories rapidly select an invariant subspace of the Hilbert space through repeated indirect measurements, extending previous results to more general measurement types with exponential convergence.

Contribution

It generalizes the exponential selection result from non-demolition to arbitrary repeated indirect measurements using a novel deformation of the measurement instrument.

Findings

Quantum trajectories exponentially converge to an invariant subspace

The convergence is almost sure and in average

The method applies to a broad class of indirect measurements

Abstract

We show that, in long time, quantum trajectories select an invariant subspace of the Hilbert space of the system being indirectly measured. This selection is shown to be exponentially fast in an almost sure sense and in average. This result generalizes a known result for non demolition measurements to arbitrary repeated indirect measurements. Our proofs are based on the introduction of a deformation of the original instrument to an equivalent one with a unique invariant state.