Rigorous QSDE representations for stationary quantum Gaussian processes (including quantum quasi-Markov class)

Develop rigorous Hudson–Parthasarathy quantum stochastic differential equation representations for stationary quantum Gaussian processes that satisfy the Kubo–Martin–Schwinger (KMS) condition, including the subclass of quantum quasi-Markov stationary Gaussian processes, by constructing appropriate representation Hilbert spaces and identifying operators and noise processes that realize these laws as solutions of quantum stochastic differential equations.

Background

In the discussion of modeling the quantum noise arising in quantum Brownian motion and related open quantum systems, the notes highlight that while the existence of stationary quantum stochastic processes satisfying the KMS condition has been rigorously established (e.g., Lewis and Thomas, 1975), there is a notable gap regarding their representation via Hudson–Parthasarathy quantum stochastic differential equations (QSDEs).

The author suggests mimicking the classical theory of quasi-Markov Gaussian processes—where finite-dimensional SDE realizations are available—by developing analogous QSDE representations for stationary quantum Gaussian processes, particularly those termed quantum quasi-Markov. This requires dealing with technical issues such as constructing appropriate representation Hilbert spaces for the processes (as developed in the operator-algebraic literature of Araki and Woods; Bratteli and Robinson).

A rigorous QSDE realization would bridge the existence theory of such stationary quantum processes and the widely used dynamical framework of Hudson–Parthasarathy calculus, enabling systematic modeling of non-Markovian quantum noises within the QSDE formalism used for open quantum systems.