Continuous Observation of Quantum Systems

Abstract

In a series of papers in the 1980's Alexander Holevo proved a classification theorem for continuous quantum measurement processes, or, as they would today be called, stationary quantum trajectories in continuous time. His main tools were functional analytic in character: starting from a Bochner-type inequality he employed dilation techniques for positive definite kernels. Here we give an alternative, more probabilistic proof: we use weak convergence of measures and employ Levy's Continuity Theorem. We clarify the boundedness conditions in Holevo's theorem, and supply a simple example from quantum optics.

37 pages. This paper grew from a chapter in the book on Quantum Markov Processes that Burkhard Kümmerer and the author are preparing on Quantum Markov Processes. It appeared in "Communicating the Quantum Way, Contributions in Honor of Alexander S Holevo's 80th Birthday" (World Scientific 2026)