On the Observability of Quantum Dynamical Systems

Quantum statistical mechanics offers an increasingly relevant theory for a wide variety of probabilistic systems including thermodynamics, particle dynamics, and robotics. Quantum dynamical systems can be described by linear time invariant systems and so there is a need to build out traditional control theory for quantum statistical mechanics. The probability information in a quantum dynamical system evolves according to the quantum master equation, whose state is a matrix instead of a column vector. Accordingly, the traditional notion of a full rank observability matrix does not apply. In this work, we develop a proof of observability for quantum dynamical systems including a rank test and algorithmic considerations. A qubit example is provided for situations where the dynamical system is both observable and unobservable.