Classical mixture of coherent states ($\rho =\int P(\alpha )|\alpha ⟩⟨\alpha |d^2\alpha ,P\ge 0$)

A normalized nonnegative probability measure over coherent states produces a classical mixture. The measure may contain Dirac delta distributions, so a pure coherent state is included. Conditioning on $\alpha$ gives Poisson distribution photon statistics; hence $\mathrm{Var}n=⟨n⟩+\mathrm{Var}(|\alpha {|}^2)\ge ⟨n⟩$. Each optical quadrature has variance at least its vacuum value because mixing adds nonnegative variance of conditional means. These inequalities provide simple nonclassicality witnesses.

The diagonal coherent state representation writes a single-mode density matrix in the form shown. Its weight may be a generalized distribution rather than an ordinary nonnegative probability density. A nonclassical state of light has no nonnegative measure of this form. This use of $P$ differs from the Wigner distribution; a nonnegative Wigner function does not by itself certify classicality.