

Wigner function’s negativity demystified

Denys I. Bondar Department of Chemistry, Princeton University, Princeton, NJ 08544, USA    Renan Cabrera Department of Chemistry, Princeton University, Princeton, NJ 08544, USA    Dmitry V. Zhdanov Present address: Department of Chemistry, Northwestern University, Evanston, IL 60208, USA Department of Chemistry, Princeton University, Princeton, NJ 08544, USA    Herschel A. Rabitz Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
March 22, 2020

As early as 1932 Wigner Wigner1932 defined the joint distribution for the coordinate and momentum of a quantum particle. Despite a drawback of being sometimes negative, the Wigner distribution has stood the test of time and found many applications. Having demonstrated that the Wigner function of a pure quantum state is a wave function in the specially tuned Dirac bra-ket formalism, we argue that the Wigner function is in fact a probability amplitude for the quantum particle to be at a certain point of the classical phase space. Since probability amplitude need not be positive, our findings elucidate the long-standing mystery of the Wigner function’s negativity. Additionally, we establish that in the classical limit, the Wigner function transforms into a classical Koopman-von Neumann wave function Koopman1931 ; Neumann1932 ; Neumann1932a ; DaniloMauro2002 rather than into a classical probability distribution. As a result, contrary to widespread beliefs, the volume of negative regions in the Wigner distribution cannot quantify the degree of quantum character; while, variations in the negativity (or positivity) during evolution is a genuine signature of quantum dynamics. Our results identify what quantum processes are implementable by classical means, which is an open question in quantum information theory Mari2012 : A negativity (or positivity) preserving process can be realized by classical simulations, whereas non-preserving evolution can only be quantum.


In his seminal work Wigner1932 , Wigner defined the combined distribution of the quantum particle’s coordinate and momentum in terms of the wave function. Since then, this function, bearing his name, plays a paramount role in the phase space formulation of quantum mechanics Zachos2005 , is a standard tool for establishing the quantum-to-classical interface Zurek2001 ; Bolivar2004 ; Haroche2006 , and has a broad range of applications in optics and signal processing Dragoman2005 . Techniques for the experimental measurement of the Wigner function are also developed Kurtsiefer1997 ; Haroche2006 ; Ourjoumtsev2007 ; Deleglise2008 . Despite its ubiquity, the Wigner function is haunted by the obscure feature of possibly being negative. Wigner O'Connell1981 upgraded this peculiarity to a mystery by demonstrating that his function is the only one satisfying a reasonable set of axioms for a joint probability distribution. Subsequently, the Wigner function’s negativity motivated the development of a mathematical framework for handling negative probabilities Muckenheim1986 ; Landsberg1987 . Moreover, it is widely believed that the presence of negative regions in the Wigner distribution is an authentic sign of quantum character. Even though doubts were cast on the validity of such an interpretation Bracken2004 ; Man'ko2004 ; Ferrie2011 , the implications of the negativity has remained unclear.

In this Letter we clarify the origin of the negativity by advocating the following interpretation: The Wigner function is a probability amplitude for a quantum particle to be at a certain point of the classical phase space, i.e., the Wigner function is a wave function analogous to the Koopman von-Neumann (KvN) wave function of a classical particle.

Around the time the Wigner distribution was conceived, Koopman and von Neumann Koopman1931 ; Neumann1932 ; Neumann1932a ; DaniloMauro2002 recast classical mechanics in a form similar to quantum mechanics by introducing classical complex valued wave functions and representing associated physical observables by means of commuting self-adjoint operators. In particular, it was postulated that the wave function of a classical particle obeys the following equation of motion:


Without loss of generality one-dimensional systems are considered throughout. Since the self-adjoint operators representing the classical observables of coordinate and momentum commute, they share a common set of orthogonal eigenvectors such that . In the KvN classical mechanics, all observables are functions of and . The expectation value of an observable at time equals to . The probability amplitude for a classical particle to be at point with momentum at time is found to satisfy


This is the evolution equation for the classical wave function in the - representation, where , , , and in order to satisfy the commutation relations (2). Utilizing the chain rule and equation (3), we obtain the well known classical Liouville equation for the phase-space probability distribution ,


Note that the classical wave function and the classical probability distribution satisfy the same dynamical equation, which reflects the physical irrelevance of the phase of a classical wave function.

In recent work Bondar2011c , we put forth operational dynamical modeling as a systematic theoretical framework for deducing equations of motion from the evolution of average values. First, starting from the Ehrenfest theorems Ehrenfest1927 , we obtained the Schrödinger equation if the momentum and coordinate operators obeyed the canonical commutation relation, and the KvN equation (1) if the momentum and coordinate operators commuted. Then, applying the same technique to the Ehrenfest theorems,


with a generalization () and demanding a smooth classical limit , we have established existence of the uniquely defined operator such that


where and represent the quantum coordinate and momentum respectively, , , , and are the same classical operators as in equation (2), and denotes the degree of quantumness/commutativity: corresponds to quantum mechanics whereas recovers classical mechanics, . See figure 2 for a pictorial summary of these derivations thoroughly presented in Bondar2011c .



Figure 1: The derivation of quantum mechanics, classical mechanics, and quantum mechanical phase space representation within operational dynamical modeling proposed in Bondar2011c .


Figure 2: Negativity in the KvN classical mechanics and quantum mechanics for the Morse potential, in atomic units (a.u.). Note that negativity and positivity are not conserved during quantum evolution. Plot (b) shows the solution of equation (3) at time a.u. corresponding to the initial condition at displayed in figure (a). Plot (c) – the solution of equation (6) at a.u. with the initial condition (a).

A crucial point for our current discussion is that this unified wave function ( is dropped henceforth) in the -representation is proportional to the Wigner function ,


where denotes the density matrix with replaced by Bondar2011c . Straightforward calculations show that the normalization condition for the unified wave function implies that the density matrix must correspond to a pure state,


Setting , we can demonstrate that all expectation values for the state coincide with those for ,


where and are arbitrary functions of the quantum position and momentum, respectively.

Equations (8) and (9) reveal that the Wigner distribution of a pure state indeed behaves like a wave function. As shown in figure 2, the Wigner function’s dynamical equation (6) transforms into the evolution equation for a classical KvN wave function (1). Hence, in the classical limit, the Wigner function maps a quantum wave function into a corresponding KvN classical wave function rather than a classical phase space distribution. Since the vectors are identical in both KvN and Wigner representations, is proportional to the probability amplitude that a quantum particle is located at a point of the classical phase space. Note it is important to distinguish the classical and quantum phase spaces because the notion of a phase-space point arises naturally only in the commutative classical variables . One may take this distinction further and interpret the Wigner function as the KvN wave function of a classical counterpart of the analogous quantum system. Like any wave function, the Wigner function need not be strictly positive, and its negativity does not necessarily imply the system’s quantum nature.

However, the time variation in negativity can be used to discriminate between classical and quantum evolution. Since the Wigner function’s normalization is constant, negativity variations imply positivity variations. The total amount of negativity/positivity as well as its phase-space volume is conserved if the state is propagated according to the classical KvN equation (3). This statement is a KvN extension of the Liouville theorem asserting that the phase-space distribution function is constant along classical trajectories. However, the negativity/positivity need not be conserved if the state is propagated by the quantum Wigner equation (6). Thus, we are lead to the mandatory quantum computation criterion: A negativity/positivity preserving evolution can be realized by classical computations, while non-preserving evolution is mandatory quantum. This criterion is an extension of results reported in Mari2012 .

Consider the following example of dynamics discrimination: Assume the phase-space functions, depicted in figure 2, correspond to the same physical system. Then, we readily infer that evolution from 2(a) to 2(c) is quantum because the negativity and positivity are not preserved; while, propagation from 2(a) to 2(b) cannot be conclusively characterized since the negativity and positivity are conserved. However, according to the mandatory quantum computation criterion, the transition from 2(a) to 2(b) can be efficiently realized by classical computations.

Acknowledgments. The authors acknowledge financial support from DARPA QuBE, NSF, and ARO.

