= Classical mixture of coherent states
{title2=$\rho=\int P(\alpha)|\alpha\rangle\langle\alpha|\,d^2\alpha,\quad P\ge0$}
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 $\operatorname{Var}n=\langle n\rangle+\operatorname{Var}(|\alpha|^2)\ge\langle n\rangle$. Each <optical quadrature> has variance at least its vacuum value because mixing adds nonnegative variance of conditional means. These inequalities provide simple <nonclassicality witnesses>.
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