The probability that goes below zero

The return a classical cloud never makes ended with a quantum oscillator whose state, a quarter of the way to its revival, was two copies of a swinging packet on opposite sides of the well — and with a picture that could not say what that meant. A probability density showed two bumps. It would have shown the same two bumps for a single packet whose side had been decided by a coin, and the difference between those two situations is the whole difference between a superposition and ignorance.

The comparison that essay made was with a classical cloud: a distribution of points in the plane of position and momentum, each point a definite state. A cloud is the natural way to describe a classical system known imperfectly, and if quantum states could be written the same way, the correspondence between quantum and classical pictures would be a matter of comparing two clouds. In 1932 Eugene Wigner found the function that does the writing. It is exact, it has every property a phase-space distribution should have but one, and the property it lacks is the answer.

A distribution built to cast the right shadows #

Wigner’s function is built from the wavefunction by an integral that pairs the value of the state a little to one side of a point with the complex conjugate of its value a little to the other side, weighted by a wave in momentum:

$$W(x,p) = \frac{1}{\pi\hbar}\int \psi^*(x+y)\,\psi(x-y)\,e^{2ipy/\hbar}\,dy.$$

It is real. It integrates to one over the whole plane. And its two shadows are exact: integrate it over momentum and the result is the probability density in position, $|\psi(x)|^2$; integrate over position and it is the probability density in momentum. The same holds for a shadow cast along any direction in the plane, which is to say for any combination of position and momentum — and those combinations are what a homodyne detector measures, so a Wigner function can be reconstructed from measurements by the same mathematics a hospital scanner uses to rebuild a slice of a body from its projections.

The figures compute it that way rather than quoting it. The state is carried as a wavefunction on a lattice of points, the integral is a sum over pairs of lattice points, and every closed form the figures cite is compared with the sum.

For a harmonic oscillator, and for any potential no steeper than quadratic, the function also evolves exactly as a classical cloud would: each point of it slides along the classical trajectory. The average that obeys Newton gave the general equation, the classical one plus corrections carrying $\hbar^2$ and the third derivative of the potential. So in the case where the evolution is classical the Wigner function is a cloud, flowing like one. Everything that distinguishes it has to be in its shape.

The stripes are an interference pattern #

The shape in the first figure has two parts. The blobs are the two copies of the packet, each a Gaussian of the ground state’s width — the disc the state that swings like a pendulum described as the picture of a coherent state. Between them, centred on the origin where neither copy is, is a set of stripes running across the momentum axis, alternately positive and negative.

The stripes are the interference between the copies, and their spacing says so. Two sources a distance $d$ apart in position produce fringes in momentum with period $2\pi\hbar/d$, exactly as two slits a distance apart produce fringes in angle; here the copies are 5.66 apart in the figure’s units and the fringes repeat every 1.11 in momentum. The momentum shadow shows the fringes plainly, and it falls to zero at their minima. That is where the negative stripes do their work: at those momenta, adding up the Wigner function along the position axis, the negative band between the blobs cancels exactly the positive contribution of everything else.

A classical cloud cannot do that. Its density is never negative, so its shadows can show only sums of non-negative contributions, and a zero in a shadow requires the cloud to be empty along a whole line. Two clouds of possibilities, one for each copy, added together — which is what a coin deciding the side would give — produce two blobs and no stripes and no zeros. Why two lamps never interfere is the same distinction between adding intensities and adding amplitudes, and the Wigner function makes it visible as a sign.

The stripes are also taller than the blobs: 0.289 against 0.159. The interference is not a small correction riding on two classical objects. It is the largest feature in the picture, and it is where the state’s coherence is stored.

A number that can be measured at a point #

There is a way to read the value of a Wigner function at a single point that makes both its negativity and its bounds unsurprising. Reflect the state through the point $(x, p)$ — send every displacement from that point to its opposite, in position and in momentum at once. The Wigner function at that point is $1/\pi\hbar$ times the expectation value of that reflection.

A reflection applied twice does nothing, so its possible outcomes are $+1$ and $-1$, and its expectation value lies between them. So a Wigner function can never exceed $1/\pi\hbar$ in magnitude, at any point, for any state; and it is negative at a point exactly when the state is more odd than even under reflection through that point. A single packet centred on a point is perfectly even there and reaches the upper bound. The two-packet state, reflected through the origin, becomes itself with its two copies swapped, and whether that is more even or more odd depends on the relative phase of the copies — which is why the stripes alternate in sign as the point moves along the momentum axis.

The states of definite energy show the reflection rule at its plainest. A state with $n$ quanta has parity $(-1)^n$ about the centre of the well, so its Wigner function at the origin is $+1/\pi$ for even $n$ and $-1/\pi$ for odd $n$ — the bound, reached exactly, in both directions. The one-quantum state is therefore as negative at its centre as any state can be anywhere. Its negative region is a disc of radius $1/\sqrt2$, which has area $\pi\hbar/2$: small, of the order of the area the motion that cannot be stopped assigns to the ground state, and not small at all compared with the state.

The reflection rule is also how the function is measured at a point, without reconstructing it from shadows. Displace the field so that the chosen point moves to the origin, then measure whether the number of quanta is even or odd: the reflection through the origin is the parity of that number, so the average of the answers is the Wigner function at the point, directly. Proposed in 1997, it was done in 2002 by sending atoms through a microwave cavity holding one photon, which read out the parity and found the value at the centre below zero. Superconducting circuits have since used the same trick to map the Wigner functions of microwave fields in resonators, including superpositions of two coherent states carrying around a hundred photons, stripes and all.

A negative Wigner function does not require a large superposition. A single photon has one, and in 2001 a single photon was prepared in a light pulse and its Wigner function reconstructed from homodyne measurements, with the dip below zero at the centre. The cat state in the first figure is negative because it is a superposition of two distinguishable things; the one-quantum state is negative because it is a state of definite number, which light arrives in lumps makes the defining property of a photon. Neither kind of state is available to a classical cloud.

Only a Gaussian is safe #

Which pure states, then, never go negative? The answer is sharp, and it was proved by Robin Hudson in 1974: exactly the Gaussian ones. A coherent state is a Gaussian, displaced; a squeezed state is a Gaussian, stretched along one direction and compressed along the other; and every Wigner function of that form is positive everywhere. Every other pure state has somewhere a region where it dips below zero.

The figure measures the negativity of three families on one axis. The number states are the most negative for their size and grow steadily more so. The two-packet superpositions start from almost nothing — at a quarter of a quantum the two copies overlap almost completely and there is little between them to interfere — and grow as the copies separate. The Gaussians sit on zero at every size drawn, to within the rounding of the arithmetic.

That theorem turns a picture into a classification. The states called classical-looking in the account of the swinging state — the coherent states, which a laser and a signal generator produce without being asked — are classical-looking in the strongest sense available: there is an honest classical cloud with exactly their statistics for every measurement of position, momentum or any combination. The same holds for anything built from them by the operations that keep Gaussians Gaussian, such as beam splitters, squeezers and energy loss, and there is a corresponding theorem that a computation using only those states and operations can be simulated efficiently on an ordinary computer. Negativity is what such a device lacks, and it has come to be treated as one of the resources a quantum computer consumes.

What losing energy does to the stripes #

The oscillator whose revivals the first figure came from was assumed to be isolated, and that was the last of the ways its model stopped. A real one leaks. A photon leaves the cavity through a mirror; a vibrating membrane passes its energy to its supports. The simplest version is loss to cold surroundings at a rate $\gamma$, which shrinks the energy as $e^{-\gamma t}$, and for a superposition of two packets it can be followed exactly.

The blobs hardly change: each packet loses energy the way a classical oscillator does, and a packet that keeps 95 per cent of its energy has moved in by 2.5 per cent. The stripes change a great deal. The coherence between the packets, which sets their height, falls as $\exp(-2\alpha^2(1 - e^{-\gamma t}))$, and at early times that is a decay at $2\alpha^2$ times the energy decay rate — eight times as fast for these packets.

The reason is that every photon leaving the oscillator carries a little of the oscillator’s field with it, and that field has opposite signs for the two copies. The escaped light therefore holds a partial record of which copy the oscillator is in. A record of which copy is exactly the path knowledge that the measurement that never touched it trades against fringe visibility, and a superposition whose alternatives have been recorded somewhere, even by a stray photon that nobody catches, no longer shows interference between them. The further apart the copies, the more distinguishable the record carried by each photon, and the separation enters squared.

Negativity ends at half the energy #

The rates in that figure are the scaling that makes large superpositions hard to keep: at amplitude 3 the negativity falls eighteen times faster than the energy. Doubling the separation of two packets quarters the time the superposition lasts, and a superposition of two positions of a macroscopic object, with a separation of millions of packet widths, would be gone before any instrument could look.

The endpoint is the surprise. Every curve stops at the same time, $\gamma t = \ln 2$, independent of the amplitude, and the figure checks that the minimum is exactly zero there and still negative just before. The explanation is the reflection rule again, seen through a beam splitter. Losing half the energy is equivalent to sending the field through a half-silvered mirror and discarding one output, and what emerges is the input’s Wigner function shrunk by a factor of √2 and blurred by exactly the ground state’s disc. That blur is the smoothest one quantum mechanics permits, the one that corresponds to measuring position and momentum at once as well as the questions that can be asked together allows — and a function blurred that much is never negative, for any input. The same half-silvered mirror is how position and momentum can be measured together. A heterodyne detector splits the signal in two, measures one combination in one half and the conjugate combination in the other, and the distribution of pairs of answers it records is precisely that blurred function — the Husimi function. So the best simultaneous measurement quantum mechanics permits sees every state through the blur that removes all negativity, and no amount of repeating it can reveal a negative region. The stripes are visible only to measurements that give up one variable to learn the other exactly, or that ask about parity; which is the uncertainty principle’s own account of why a classical picture has always seemed to work.

Past half the energy lost, every state of a single oscillator is indistinguishable in all its position and momentum statistics from some classical cloud, though it may still carry faint stripes.

The Paris group of Serge Haroche watched this happen in 1996, with superpositions of two coherent fields in a superconducting cavity probed by atoms passing through it: the interference between the components decayed faster than the field, and faster for components further apart. In 2008 the same group reconstructed the full Wigner functions of such states, negative stripes included, and followed the stripes as they faded.

Where the model stops #

The surroundings are at zero temperature. Energy leaks out and none leaks in. A warm environment also kicks energy into the oscillator at random, which blurs the Wigner function faster, and the negativity then disappears before half the energy is lost.

The oscillator has one mode. A field in a real cavity, or the vibration of a real object, has many, and the Wigner function of several modes lives in a phase space of twice as many dimensions. Hudson’s theorem survives the extension; the pictures do not.

The Wigner function is not the only phase-space representation. Smoothing it by the ground state’s disc gives the Husimi function, which is never negative and is not a true distribution either, because its shadows are blurred; other choices give functions that cannot be written as ordinary functions at all for squeezed states. Each trades one property for another, and only the Wigner function has exact shadows.

And the lattice is a lattice. The integrals are sums over points a twentieth of an oscillator length apart, which is fine for the smooth states drawn and would need refining for states with structure much finer than that.

What the pictures cannot show #

The Wigner function is not itself something any single measurement records. Its shadows are probabilities for outcomes, and its value at a point is an expectation value, but a heat map of it is a reconstruction from many measurements on many identically prepared copies. The negative stripes are real in the sense that they are forced by the shadows; they are not a density of anything.

Nor can a still picture show the term that the Wigner function’s evolution carries and a cloud’s does not. For a potential steeper than quadratic the function does not merely slide along classical trajectories: the third derivative of the potential makes it grow new stripes out of smooth regions, which is how the collapse and the fractional revivals of an anharmonic oscillator appear in phase space, one fraction of the revival time at a time.

Still open: whether anything besides the surroundings erases superpositions #

The account above is complete within quantum mechanics: stripes fade because the environment records which alternative is realised, and they fade faster the more different the alternatives. That makes the absence of superpositions of large objects a matter of how well they can be isolated, not of any law.

Whether it is only that is still being tested. Proposals going back to the 1980s, from Ghirardi, Rimini and Weber and later from Diósi and Penrose, add a small intrinsic collapse to quantum mechanics that grows with mass, so that sufficiently massive superpositions would destroy themselves even in perfect isolation — with gravity, on Penrose’s version, as the cause. Experiments have been closing in from two sides: molecules of more than two thousand atoms, above twenty-five thousand atomic mass units, sent through interferometers and seen to interfere; and in 2023 a crystal of sixteen micrograms vibrating in a superposition of two opposite phases of motion, with negative regions in its Wigner function. No departure from quantum mechanics has been seen, and the masses at which the collapse models would begin to bite have not yet been reached.

The habit worth keeping is the one the sign of the function teaches. Ask whether a state has a classical cloud with the same statistics before calling its behaviour quantum. A coherent state has one, and the correspondence between it and a classical oscillator is exact in every measurement. A single photon, a superposition of two packets or any other pure state that is not a Gaussian does not, and the negative regions of its Wigner function mark exactly where the cloud would have to be less than empty.