Phase Space and the Wigner Function

A classical particle is a dot in phase space: one position, one momentum. A quantum state cannot be a dot, because x and p do not commute, but it can be a landscape — a function W(x, p) whose shadow along either axis is exactly the probability distribution you would measure for that variable. Integrate over p and you get |ψ(x)|²; integrate over x and you get |φ(p)|². The whole landscape integrates to 1, which the ∫W readout confirms for every state here.

You start on a coherent state, the packet that swings without spreading on the harmonic-oscillator page. Drag |α| and watch the vacuum bump slide outward without changing shape; drag the phase and it circles the origin — that circle is the oscillation, seen from above. Then switch to Fock |1⟩.

The single-photon state has a crater at the origin: W(0, 0) = −1/π, as deep as the bound allows, and the min W readout goes sky-blue to say so. No probability distribution can do that — a probability is a fraction of trials, and no fraction of trials is negative. That is why W is called a quasi-probability. Its shadows are honest probabilities; the landscape itself is not.

The number at the origin is not an accident. W(0, 0) is the mean parity of the state divided by π: +1/π for anything with only even photon numbers, −1/π for anything with only odd ones, and in between for mixtures. The ⟨parity⟩/π readout tracks the origin exactly for every state — try the odd cat, the thermal state, and a coherent state as |α| grows and its parity washes out.

Now pick the cat and pull the separation out. Two coherent lobes, and between them a washboard of fringes that plunge below zero — interference between "here" and "there", written into phase space. The fringe spacing shrinks as the lobes part, which is precisely why big cats are hard to keep alive: the finer the fringes, the less noise it takes to wash them out.

The Negativity readout is ∫|W| − 1: how much of the landscape lies below zero, doubled. It is exactly zero for the vacuum, for every coherent state, for every squeezed state and for the thermal state — all of them are Gaussians, and a theorem of Hudson says the only pure states whose Wigner function stays non-negative are Gaussian. Anything else must dip below zero somewhere.

That dip is the price of being genuinely quantum. A state whose W is everywhere non-negative can be treated as a classical ensemble of phase-space points, and there is a theorem for that too: circuits built from such states and Gaussian operations can be simulated efficiently on a classical computer. Negativity is where the classical description runs out — it is a resource, in the same sense that entanglement is. Squeeze as hard as you like and you get no negativity at all; a single photon gives you the maximum.

Toggle Husimi Q. Same state, same grid — and every crater fills in. Q is W blurred with a Gaussian one vacuum wide, which is enough to smooth away any negative region, so Q is a genuine probability distribution: never below zero and never above 1/2π. It is the distribution you record if you measure x and p simultaneously, which the uncertainty principle only permits at the cost of exactly that much added noise.

The lesson is that "the" phase-space distribution of a quantum state does not exist. There is a family of them, ordered by how much of the noncommutativity you smear over, and W is the one in the middle that gets the marginals right. The fringes of the cat state are still there in Q, faintly; the negative ones just cannot show. Toggle back and forth on the cat with the separation at maximum and the smoothing is unmistakable.

Every one of these surfaces has been measured. Quantum-state tomography reconstructs W from homodyne data, and the first negative Wigner function of a single photon was published in 2001. Cat states with the fringes you see here live in superconducting microwave cavities, where they encode logical qubits whose errors are mostly one kind (photon loss) and therefore easier to correct — the bosonic-code approach to fault tolerance.

Squeezed vacuum is the state LIGO injects to hear black-hole mergers below the vacuum noise floor. Its Wigner function is the ellipse on this page: one quadrature quieter than the vacuum, the other louder, area unchanged. The function itself is from 1932, decades before anyone imagined measuring one: Eugene Wigner introduced it as a bookkeeping device for quantum corrections to thermodynamics, and the negativity he could not avoid turned out to be the whole point.