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Journal of Mathematical Physics

When is the Wigner function of multidimensional systems nonnegative?

Francisco Soto, P. Claverie

Published Jan 1, 1983• 154 citations
Publisher

Abstract

It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one-dimensional systems.

DOI
10.1063/1.525607

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It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one-dimensional systems.
We show that a (generalized) function f has a nonnegative Wigner distribution W(f,f)W(f,f)W(f,f) if and only if f is a Gauss function (possibly degenerate). We prove, more generally, that the convolution of W(f,f)W(f,f)W(f,f) with certain Gauss functions is nonnegative if and only if f is of the special type mentioned. As a consequence we have that the only (generalized) functions whose Wigner distributions are concentrated on a curve of a particular type are delta functions or exponentials exp⁡(−παt2+2πβt+γ)\exp ( - \pi \alpha t^2 + 2\pi \beta t + \gamma )exp(−παt2+2πβt+γ) with $\alpha $, $\beta $, $\gamma $ complex, Re⁡α=0\operatorname{Re} \alpha = 0Reα=0. The main tool used is Moyal’s formula for the Wigner distribution together with Bargmann’s integral transform.
The integral of the Wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one. It is shown that for systems with 1 degree of freedom, the problem of determining the best possible upper and lower bounds on such an integral, over all possible states, reduces to the problem of finding the greatest and least eigenvalues of a Hermitian operator corresponding to the subregion. The problem is solved exactly in the case of an arbitrary elliptical region. These bounds provide checks on experimentally measured quasiprobability distributions.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+ln\ensuremath{\pi}$. We prove this conjecture for the qubits formed by Fock states $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture's validity. Lastly, we elaborate on the states which are in accordance with our condition.
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to non-negative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of prime-power dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.