Ma, ShanWoolley, Matthew JPetersen, Ian2023-08-160018-9286http://hdl.handle.net/1885/295618The purpose of this article is to synthesize a linear quantum system, which is strictly stable and has a steady thermal state. Specifically, we give a parameterization of a class of stable linear quantum systems that have V=τI/2 , τ>1 , as their steady covariance matrsices. This is physically important since the covariance matrix τI/2 , τ>1 , corresponds to a quantum thermal state. Hence, we can say that these systems will asymptotically evolve into a quantum thermal state. An extension to the case where V=Sdiag(Λ,Λ)S⊤/2 with Λ>I being a diagonal matrix and S being a symplectic matrix will also be considered. Physically, a covariance matrix of the form V=Sdiag(Λ,Λ)S⊤/2 , Λ>I , corresponds to a mixed Gaussian quantum state. So, we can alternatively say that the corresponding linear quantum systems will asymptotically evolve into a mixed Gaussian quantum state.This work was supported by the National Natural Science Foundation of China under Grants 61803389 and 61973317, the 111 Project (B17048), the Air Force Office of Scientific Research under agreement FA2386-18- 1-4026 and Agreement FA2386-16-1-4065, the ARC Centre of Excellence for Engineered Quantum Systems (CE170100009), the Australian Research Councils Discovery Projects Funding Scheme under Project DP180101805, and the U. S. Office of Naval Research Global under Grant N62909-19- 1-2129application/pdfen-AU© 2022 IEEE202210.1109/TAC.2021.30792912022-07-24