Detector dependency of diffusive quantum monitorings

Continuous measurements play a pivotal role in the study of dynamical open quantum systems. "Dyne" detections are among the most widespread and efficient measurement schemes, and give rise to quantum diffusion of the conditioned state. In this work we study under what conditions the detector dependency of the conditional state of a quantum system subject to diffusive monitoring can be demonstrated experimentally, in the sense of ruling out any detector-independent pure-state dynamical model for the system. We consider an arbitrary number L of environments to which the system is coupled, and an arbitrary number K of different types of dyne detections. We prove that nontrivial necessary conditions for such a demonstration can be determined efficiently by semidefinite programming. To determine sufficient conditions, different physical environmental couplings and Hamiltonians for a qubit and different sets of diffusive monitorings are scrutinized. We compare the threshold efficiencies that are sufficient in the various cases, as well as cases previously considered in the literature, to suggest the most feasible experimental options.