Linear Quantum State Observers

Abstract

This paper considers the use of linear state observers to infer the unknown state (density matrix) of a closed quantum system using generalized quantum measurement—positive operator-valued measures (POVMs). An efficient test for observability is described in terms of two quantum observability matrices. The main contribution of this work is the development of a canonical quantum state observer, which may be utilized to estimate the state of any closed quantum system using POVM measurements. The observer ensures the state estimation error tends to the unobservable space of the system, which is exactly zero when the system is observable. The space of Hermitian matrices is shown to be invariant with respect to the dynamics of the canonical observer, which allows one to project the observer's state onto the space of valid quantum density operators while retaining convergence to the true state.