Quantum Filtering at Finite Temperature
John Gough
TL;DR
This work extends quantum filtering to finite-temperature environments by modeling the bath with the Araki-Woods representation, enabling a non-demolition, quadrature-based filter. It employs quantum stochastic calculus with non-Fock thermal noise and leverages Tomita-Takesaki dual representations, yielding a thermal Zakai and Kushner-Stratonovich filter for the homodyne output. The main results include explicit SDEs for the unnormalized and normalized filters, including the innovations process $I_t$, and an interpretation of the two representations in terms of fields in the right and left Rindler wedges. The framework reduces to the zero-temperature case when $n=0$ and connects to the Davies-Fulling-Unruh effect, providing state-estimation tools for thermal and accelerated-frame environments.
Abstract
We pose and solve the problem of quantum filtering based on continuous-in-time quadrature measurements (homodyning) for the case where the quantum process is in a thermal state. The standard construction of quantum filters involves the determination of the conditional expectation onto the von Neumann algebra generated by the measured observables with the non-demolition principle telling us to restrict the domain (the observables to be estimated) to the commutant of the algebra. The finite-temperature case, however, has additional structure: we use the Araki-Woods representation for the measured quadratures, but the Tomita-Takesaki theory tells us that there exists a separate, commuting representation and therefore the commutant will have a richer structure than encountered in the Fock vacuum case. We apply this to the question of quantum trajectories to the Davies-Fulling-Unruh model. Here, the two representations are interpreted as the fields in the right and left Rindler wedges.