Pure States and thePRepresentation

Abstract

The coherent-state $P$ representation for the density operator of the electromagnetic field is studied for the case in which the density operator represents a pure state, $\rho =\left|\psi 〉〈\psi \right|$. An exact and complete characterization is given of the states for which the $P$ representation exists with a weight function $P\left(\alpha \right)$ that is a tempered distribution. These states $|\varphi 〉$ form an exceedingly narrow class: each may be generated from a particular coherent state $|\alpha 〉$ by the application of a finite number of creation operators, i.e., $|\varphi 〉=[c_0+c_{1}a†+\cdots +c_n{\left(a†\right)}^n\left]\right|\alpha 〉$, where $\alpha$ and the $c_n$ are arbitrary complex numbers. For them the weight function $P\left(\alpha \right)$ is a linear combination of the two-dimensional delta function and a finite number of its derivatives. For other pure states, the function $P\left(\alpha \right)$ has singularities that are not compatible with the form of the $P$ representation.