# An example of a quantum system for which Wigner function transitions to negative values

I want to check my understanding of the Wigner transform and try to understand why and how exactly the probabilistic interpretation drops down as the function goes to zero and then to negative values

So, suppose we have a free quantum oscillator with integer eigenvalues of energy and (boson) occupation.

Questions:

• Is it possible to start the system in a state that will have a well defined Wigner phase space probability density (with an analog interpretation as a Liouville phase space density), and it will freely evolve into a state for which the Wigner density becomes negative-valued?. In other words, is the positive-definite property of a Wigner density invariant under free evolution?

• is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

• what happens with the probability interpretation in the boundary of the transition between positive-definite and negative at certain regions in general?

The Wigner function is the fourier transform in one variable of the density matrix for the single particle $\rho(x,y)$. If you Fourier transform in y, you get the Wigner function $\rho(x,p)$. This is important to understand, because it explains why the Wigner function is at all interesting, and why it obeys simple dynamics. It also shows that it doesn't have a probability interpretation away from the classical limit, because only the on-diagonal density matrix elements are probabilities.

### Question 1: is is possible for Wigner's density to be negative after starting positive?

The answer is yes for a general potential, but for the special case of a Harmonic oscillator, the answer is no, because time evolution just rotates phase space. For the other special case of a free particle, a Gaussian wavefunction just spreads into a wider Gaussian, so it is not possible there either. For this reason, you have a hard time getting an example.

It is also true that semiclassically, the motion is along the classical trajectory, with shear according to the change in period with increasing J. So if a semiclassical Wigner matrix is positive near a single non-chaotic trajectory, it won't become negative, near the trajectory, at least not for a long time.

But it is very easy to get any kind of Wigner densities away from the semiclassical regime.

### Question 2: What about the "probability density on phase space" interpretation?

This interpretation is faulty. The density matrix is only a probability for on-diagonal elements. So the Fourier transform of the Wigner density in either x or p has a diagonal probability density interpretation at all times, but the x,p position Wigner function is nothing. It's just a complicated encoding of off-diagonal matrix elements that is not particularly special or useful, away from the semi-classical limit.

To see that it is no good, consider a plane-wave state, whose phase space Wigner density is the plane-wave profile times a delta function at a certain value of p. If you scatter any lump into approximate plane waves, the end state of scattering is a bunch of complex stuff that has no interpretation as a phase-space probability.

The proper interpretation of Wigner's phase space density is that it is a Fourier transform of the density matrix, nothing more.

• You're off by a 45 degree rotation in your definition of the Wigner function. The usual notation is that $\rho(x_1,x_2) = \langle x_1 | \rho | x_2 \rangle$. In this case, the Wigner function is the result of Fourier transforming $\rho(x+\Delta x/2,x-\Delta x/2)$ with respect to the variable $\Delta x$, not $\rho(x_1,x_2)$ with respect to $x_2$. Jul 25, 2014 at 18:41
• I disagree with the latter section. The quasiprobabilistic interpretation of Wigner function is very well established and supported by its marginals (including rotated quadratures) and its use as weight in calculating arbitrary expectation values. The example is flawed as a plane wave has no profile. Once there is a "certain value of p", you are in an eigenstate which is unique (assuming 1 dimension as you did). If you have a superposition of many plane waves the Wigner function is more complicated than that (though not complex, mathematically speaking) but its evolution is perfectly regular. Dec 6, 2015 at 12:15
• Btw. for completeness: free space evolution evolves W(x,p) into W'(x,p) = W(x-pt/m,p) preserving its values. So it can't create negativity for the same reason as the harmonic oscillator. It can even be treated as a limit of harmonic oscillator evolution when the potential is infinitely weak. Dec 6, 2015 at 12:18

The times that the Wigner function is positive does not mean it should be interpreted as a probability distribution. (What events would it be the probability distribution of? Certainly not a fuzzy joint measurement of position and momentum, e.g, with coherent states; that's the Husimi Q function.)

Is there a simple system (hopefully a simple harmonic oscillator based state) for which the Wigner density transitions from being positive-definite to being negative at certain regions?

The only non-negative Wigner functions for pure states are mixtures of pure Gaussian wavepackets. Quadratic Hamiltonians (like the simple Harmonic oscillator) preserve the Gaussianity of pure states, so the Wigner function never becomes partially negative after starting out non-negative, or vice versa.

On the other hand, non-quadratic Hamiltonians do not have this property. So just about any non-quadratic Hamiltonian you pull out of a hat will in general destroy and create strict positivity. However, there really isn't anything profound going on, because you shouldn't be taking a probability interpretation of the Wigner function either way; it's called a quasiprobability distribution for a reason.

The probabilistic interpretation of the Wigner function is already flawed at the level of the Kolmogorov axioms, even for unremittingly positive values. That is to say, two points in phase space within a distance less than $\hbar$ are not mutually exclusive sample space contingencies.

In physics parlance, these two points are not distinguishable in any Heisenberg-UncPrncp allowed way, and are already "fuzzed up" together---negative values for the Wigner function being present or not! Lack of positive semi-definiteness of the WF is not the principal impediment to a strict probabilistic interpretation --- an error often repeated in discussions of the Husimi, and leading to errors further down the stream.

All WFs automatically constrain the phase space variance to exceed $\hbar$ (cf. Ref. 1), whether they have negative regions or not, and so all are immune to strict probabilistic interpretations. Nevertheless, "they do the job" of probability distributions: they serve to provide the phase-space measure to integrals producing expectation values for observables.

Negative values of the WF are actually a major asset in the phase space formulation, and not a liability, as they ensure *-orthogonality of different states (Wigner's observation, op cit). Moreover, they are a trusty hallmark of quantum interference. They are, of course, "small" in terms of phase-space areas, roughly of the order of a few $\hbar$s. (This is evident by convolving with a Gaussian of width larger than $\hbar$, where a theorem ensures the resulting Weierstrass transform must be positive semi-definite. Consequently, hypothetical puddles of uniform negative value larger than a few $\hbar$s would yield negative values when convolved with a positive smaller Gaussian; which is excluded by the above theorem: such puddles cannot exist.)

References:

1. Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.
• PS All three of your specific questions are illustrated magnificently in Bracken & Wood, arxiv.org/pdf/quant-ph/0511227.pdf . May 15, 2016 at 16:09