Confinement to deterministic manifolds and low-dimensional solution formulas for continuously measured quantum systems

Abstract

Quantum systems subjected to a continuous weak measurement process evolve according to stochastic differential equations (SDE). Depending on the outcomes of these stochastic measurements, the quantum state may diffuse in various directions across the state space. This note points out that in many scenarios relevant to quantum engineering, this diffusion is effectively constrained to a low-dimensional space. Specifically, the quantum state remains confined to a low-dimensional, nonlinear manifold -- often time-dependent, yet independent of the specific measurement outcomes. This note derives the corresponding low-dimensional formulations for expressing the stochastically evolving state in several prototypical cases: quantum non-demolition measurements in arbitrary dimensions; quadrature measurements of a harmonic oscillator (linear quantum system); and measurements of subsystems within multipartite quantum systems. Additionally, it introduces an algebraic criterion to determine whether such low-dimensional manifolds exist or persist when additional dynamics are present.