Husimi Q-Function

Quantum phase space distribution Q(α) = |⟨α|ψ⟩|²/π — smoothed Wigner function

Quantum State

Observables

StateFock |3⟩
⟨n⟩3.00
⟨Δx²⟩
⟨Δp²⟩

Physics

Q(α) is the positive-definite quasi-probability distribution in phase space. Unlike Wigner W(x,p), Q(α) ≥ 0 always.

Q(α) = ⟨α|ρ̂|α⟩/π where |α⟩ is a coherent state.

It is the convolution of W with a Gaussian — a smoothed phase space portrait that always looks classical.

Fock state: ring structure (photon number = angular momentum in phase space)
Cat state: interference fringes in Wigner; Q shows two blobs
Squeezed: Q ellipse, sub-shot-noise in one quadrature