Photon subtraction ($|{\psi }_{-}⟩=a|\psi ⟩/\sqrt{⟨N⟩}$)

Ideal conditional application of an annihilation operator, defined only for positive initial mean photon number. Number-dependent weighting means that the conditional mean need not fall. Vacuum input gives the zero vector and therefore no successful conditional state.

A subtraction event weights initial occupation $n$ by $n$ before removing one photon. Thus its conditional output mean is $⟨N(N-1)⟩/\mu$, giving the displayed identity for $\mu >0$. If the initial variance exceeds its mean, this biased selection can offset the removed photon. The unconditional ensemble includes unsuccessful events and has a different energy balance.