System of stochastic interacting wave functions that model quantum measurements

TL;DR

The paper develops a rigorous, non-linear system of stochastic evolution equations (SIWF) to model continuous quantum measurements in infinite-dimensional Hilbert spaces with unbounded operators. It proves well-posedness (existence and uniqueness in law) of the SIWF on $L^2(\mathbb{N};\mathfrak{h})$, and shows that the mixed state density operator $\rho_t$ is independent of the initial ensemble decomposition and satisfies the diffusive Belavkin equation. By constructing a stochastic ensemble of pure states with weights, it links the SIWF to a physically meaningful continuous-measurement framework and proves that the associated quantum master equation has a regular solution for regular initial data, under broadened hypotheses including unbounded generators. The work provides two concrete examples (coordinate representation and circuit QED) to illustrate the applicability and shows the model as a robust alternative to traditional Belavkin SME formulations, with potential numerical advantages for simulating quantum measurement processes. Overall, the results establish a solid mathematical foundation for stochastic, infinite-dimensional quantum measurement models and their unravellings.

Abstract

We develop a system of non-linear stochastic evolution equations that describes the continuous measurements of quantum systems with mixed initial state. We address quantum systems with unbounded Hamiltonians and unbounded interaction operators. Using arguments of the theory of quantum measurements we derive a system of stochastic interacting wave functions (SIWF for short) that models the continuous monitoring of quantum systems. We prove the existence and uniqueness of the solution to this system under conditions general enough for the applications. We obtain that the mixed state generated by the SIWF at any time does not depend on the initial state, and satisfies the diffusive stochastic quantum master equation, which is also known as Belavkin equation. We present two physical examples. In one, the SIWF becomes a system of non-linear stochastic partial differential equations. In the other, we deal with a model of a circuit quantum electrodynamics.