Complex Field Formulation of the Quantum Estimation Theory

Abstract

We present a complex field formulation of the quantum estimation theory that works natively with complex statistics on the dependence of complex parameters. This formulation states new complex versions of the main quantities and results of the estimation theory depending on complex parameters, such as Fisher information matrices and Cram\'er-Rao bounds. This can be useful in contexts where the quantum states are described through complex parameters, such as coherent states or squeezed states. We show an example of an application of our theory in quantum communication with coherent states.