Wigner Function
Quantum states live in Hilbert space. The Wigner function W(x, p) is a "quasi-probability distribution" — a real-valued function on phase space (position × momentum) that contains all the same information. Classical probabilities are non-negative. Wigner functions can be negative. Those negative regions are the signature of non-classicality — they have no shadow in any classical world.
why negative probability matters
Hudson's theorem (1974): the only quantum states with everywhere-positive Wigner functions are Gaussian states — coherent states, squeezed states, and thermal mixtures. Every other quantum state has regions where W < 0, and those negative regions cannot be reproduced by any classical "hidden variable" probability distribution.
Coherent state (laser light): perfect Gaussian. Closest thing to a classical sinusoid. Positive everywhere.
Squeezed state: still Gaussian, but stretched in one quadrature and squeezed in the other. Heisenberg uncertainty preserved (area = ℏ/2) but redistributed. Foundation of gravitational-wave detection at LIGO (squeezed light reduces shot noise).
Schrödinger cat: superposition of two coherent states with opposite phases — |α⟩ + |−α⟩. The Wigner function has two Gaussian blobs AND a striking interference pattern between them with negative regions. Those fringes are the quantum coherence; they decohere away when the cat interacts with its environment.
Fock states |n⟩ (n photons, exactly): rotationally symmetric Wigner functions with concentric rings of alternating sign. The vacuum |0⟩ is the unique Gaussian Fock; |1⟩, |2⟩, … all show negativity. Negativity of W is what quantum tomography measures to certify "real quantum advantage" in continuous-variable quantum computing.