Pure States and the P Representation
Kevin E. Cahill · Physical Review · 1969
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, $\ensuremath{\rho}=|\ensuremath{\psi}〉〈\ensuremath{\psi}|$. An exact and complete characterization is given of the states for which the $P$ representation exists with a weight function $P(\ensuremath{\alpha})$ that is a tempered distribution. These states $|\ensuremath{\phi}〉$ form an exceedingly narrow class: each may be generated from a particular coherent state $|\ensuremath{\alpha}〉$ by the application of a finite number of creation operators, i.e., $|\ensuremath{\phi}〉=[{c}_{0}+{c}_{1}a\ifmmode\dagger\else\textdagger\fi{}+\ensuremath{\cdots}+{c}_{n}{(a\ifmmode\dagger\else\textdagger\fi{})}^{n}]|\ensuremath{\alpha}〉$, where $\ensuremath{\alpha}$ and the ${c}_{n}$ are arbitrary complex numbers. For them the weight function $P(\ensuremath{\alpha})$ is a linear combination of the two-dimensional delta function and a finite number of its derivatives. For other pure states, the function $P(\ensuremath{\alpha})$ has singularities that are not compatible with the form of the $P$ representation.