Imperial College London
Department of Physics
Negative Probabilities in Physics:
a Review
Adam C. Levy
September 2015
Submitted in part fulfilment of the requirements for the degree of
Master of Science in Physics of Imperial College London
1
Abstract
We review some of the literature on negative probabilities in physics focussing primarily on the Wigner function and other related quasi-probability
distribution functions. We suggest some unanswered questions and discuss
the utility of using extended probabilities.
2
Contents
1 Introduction
6
1.1
Why Consider Negative Probabilities? . . . . . . . . . . . . .
6
1.2
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.3
An Elementary Review of the Density Operator . . . . . . . .
9
1.4
The Einstein-Podolsky-Rosen Paradox . . . . . . . . . . . . . 12
1.5
A Review of Bell and CHSH Inequalities . . . . . . . . . . . . 13
2 Review of the Literature on Negative Probabilities
17
2.1
Interpretations of Quantum Mechanics . . . . . . . . . . . . . 17
2.2
Smoothing and Positive Density Functions . . . . . . . . . . . 19
2.3
Non-locality versus Extended Probabilities . . . . . . . . . . . 21
2.4
Further Applications of Extended Probabilities . . . . . . . . 23
3 The Wigner-Weyl Approach to Quantum Mechanics
25
3.1
The Phase-Space Approach . . . . . . . . . . . . . . . . . . . 25
3.2
The Wigner Function
3.3
The Weyl Transform . . . . . . . . . . . . . . . . . . . . . . . 31
3.4
Negative Values of the Wigner Function . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . . . 26
4 General Distribution Functions
4.1
36
Alternatives to the Wigner Function . . . . . . . . . . . . . . 36
4.1.1
The Husimi Function (Q) . . . . . . . . . . . . . . . . 36
3
4.1.2
4.2
The Glauber-Sudarshan Function (P) . . . . . . . . . 37
The Generalised Weight Function . . . . . . . . . . . . . . . . 38
5 Negative Values of the Quasi-probability Distributions
42
5.1
Negative Values of the Wigner Function . . . . . . . . . . . . 42
5.2
Negative Values of the General Distribution Functions . . . . 46
6 Fine’s Theorem and the Discrete Wigner Function
50
6.1
Fine’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2
Viable and Non-viable Quasi-Probabilities . . . . . . . . . . . 50
6.3
The Discrete Wigner Function
7 Conclusion
. . . . . . . . . . . . . . . . . 51
53
7.1
Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8 Bibliography
56
4
List of Figures
5.1
The Wigner Function of the Ground-State of the Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2
The Wigner Function of the First Excited State of the Simple
Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 45
5
1 Introduction
1.1 Why Consider Negative Probabilities?
There are many examples in mathematics where an extension to an existing
concept results in a useful new way of tackling problems. For instance, the
extension from natural numbers to the set of integers and the use of the
square root of negative one, have both proved exceedingly useful in solving
problems.
Similarly the use of negative figures in accounting is useful as a bookkeeping device even though the negative quantities may not be realisable in
practice. The money in someone’s bank account may well be negative but
the physical money in their wallet is strictly positive.
In a similar way, it may be that our intuitive sense, that a probability
must necessarily lie between zero and one, and specifically must always be
positive, may prevent us from solving problems involving probabilities in
the most efficient and logical way.
Another example given by Feynman [1], is that of a particle diffusing in
one dimension. The particle has a probability P (x, t) of being found at
position x at time t. If we consider it diffusing in a tube of length π with
absorbers at both ends so that the boundary conditions on the probability
6
are that P (0, t) = P (π, t) = 0, the particle obeys the diffusion equation,
∂P (x, t) ∂ 2 P (x, t)
−
= 0.
∂t
∂x2
(1.1)
The solution to this equation for initial conditions, P (x, 0) = A(x), is,
P (x, t) =
∞
X
2t
Pn sin(nx)e−n
(1.2)
dxA(x) sin(nx)
(1.3)
n=1
with,
Pn =
2
π
Z
π
0
Since, in this case, any distribution can be written as a superposition of
sine waves, we can regard the Pn as the probability that the particle starts
in state sin(nx) given the initial condition A(x). However, sin(nx) cannot
be a true probability since all the states, apart from the n = 1 state, have
regions of negative values.
Provided that these solutions come from an initial (non-negative) probability, A(x) > 0, we are guaranteed that P (x, t) is a true probability. So
we see here an example where it would be foolish not to allow those Fourier
modes with negative values, but considered separately they give rise to negative probabilities. Here we are ‘saved’ by the initial conditions which hide
these negative values from view.
Since the early days of quantum mechanics, physicists have been grappling
with the apparent paradoxes that it generates. Fundamentally there have
been questions surrounding the status of the wavefunction. Is a quantum
state epistemic or ontic? Does it merely represent our knowledge of the
system, which may be incomplete, and are there hidden variables which
make reality once again intelligible to our macroscopic, classical minds? If
7
the state is ‘real’ then is there something else, for instance many-worlds, or
Bohmian mechanics which explains what we see.
Many of these problems of interpretation are associated with the fact that
the wavefunction is a complex quantity and states may interfere with each
other. This means that quantum processes can act as though there are
negative probabilities involved.
When asked, many physicists are reluctant to agree that negative probabilities exist (see Mückenheim’s review paper [2], for a selection of quotes).
However, this may be no more than a problem of semantics. Defined as the
frequency of the occurence of an event when the number of trials tends to
infinity, a negative probability makes no sense. But if we broaden our definition to include quantities which are always utilised in such a way that the
final result they give is positive and between 0 and 1, then we may regard
these as probabilities nevertheless. They may also give us greater insight
into problems if we accept their utility.
1.2 Overview
Whilst this paper will attempt to cover some broader views and applications
of negative probabilities, it has been appropriate to focus to a large extent
on a key source of negative probabilities, namely those related to quantum
mechanics. In this area the main theme will be the Wigner function and
related distributions. We will generally refer to these as quasi-probability
distributions since either they can be negative, or are not bounded by 1.
They can also be referred to as extended probabilities.
In the rest of this section we will review some elementary topics that will
be useful later, including the density operator, the Einstein-Podolsky-Rosen
8
paradox and the Bell and Clauser-Horne-Shimony-Holt (CHSH) inequalities. In chapter 2, we will briefly review a broad spectrum of the literature
written about negative probabilities and the Wigner function.
In chapter 3, we will cover the main formalism which gives rise to negative probabilities in physics in general, and specifically quantum mechanics,
namely the Wigner-Weyl phase-space approach to quantum mechanics. Following this, we will see in chapter 4, that the Wigner function is one of a
family of possible quasi-distribution functions that may or may not have
negative values.
In chapter 5 we will focus on negative values of the Wigner function and
the more general quasi-probability distributions. We will also explicitly plot
the Wigner function for the first two states of the simple harmonic oscillator.
Having focused somewhat on the Wigner, and related functions, of a
continuous phase-space, in chapter 6, we will look at quasi-probabilities in
the context of the CHSH inequalities and cover a related theorem by Fine
[3, 4]. We will also look at how the Wigner formalism can be applied to a
discrete phase-space.
Finally we will attempt to summarise our findings in chapter 7, and cover
areas that may be of further interest.
1.3 An Elementary Review of the Density
Operator
Since we will later assume familiarity with the density operator it is worthwhile recapping its properties. For a pure state the density operator is given
by,
ρ = |Ψi hΨ| .
9
(1.4)
Whilst for a mixed state we have a similar expression but this time, assuming
each state |Ψi i occurs with probability pi , we must sum over these states,
thus,
ρ=
X
pi |Ψi i hΨi |
(1.5)
i
That the density operator is Hermitian can be seen by considering the matrix elements of ρ,
ρij = hi|Ψi hΨ|ji .
(1.6)
Taking their Hermitian conjugate we find,
ρ†ij = ρ∗ji
(1.7)
= (hj|Ψi hΨ|ii)∗
(1.8)
= hi|Ψi hΨ|ji
(1.9)
= ρij
(1.10)
If we take the trace of the density operator we find,
trρ =
X
=
X
hi|Ψi hΨ|ii
(1.11)
| hi|Ψi |2
(1.12)
i
i
= 1,
(1.13)
10
by completeness of the states |ii. This trace, for a mixed state is defined as,
trρ =
X
pj hi|Ψi hΨ|ii
(1.14)
pj | hi|Ψi |2
(1.15)
ij
=
X
ij
=
X
pj
X
j
| hi|Ψi |2
(1.16)
i
= 1.
(1.17)
Since ρ acts as a projection operator onto the state |Ψi, for a pure state we
must have,
ρ2 = ρ.
(1.18)
To find the expectation value of an operator O we may conveniently take
the trace,
tr(Oρ) = Oij ρji
X
=
hi| O |ji hj|Ψi hΨ|ii
(1.19)
(1.20)
ij
=
X
hΨ|ii hi| O |ji hj|Ψi
(1.21)
ij
= hΨ| O |Ψi
(1.22)
= hOi .
(1.23)
11
This has the simple extension for a mixed state,
tr(Oρ) = Oij ρji
X
X
=
hi| O |ji
pk hj|Ψk i hΨk |ii
ij
=
X
=
(1.25)
k
pk
X
hΨk |ii hi| O |ji hj|Ψk i
(1.26)
ij
k
X
(1.24)
pk hΨk | O |Ψk i
(1.27)
k
= hOi .
(1.28)
1.4 The Einstein-Podolsky-Rosen Paradox
The Einstein-Podolsky-Rosen (EPR) paradox was intended to demonstrate
the flaws with quantum mechanics. The usually considered set-up is that
first introduced by Bohm where we have two Stern-Gerlach devices operated
by observers A and B. A pair of spin- 21 particles are produced in the singlet
state of zero total spin. If we consider a measurement made by each observer
in either the z or x direction then by conservation of angular momentum
if A, say, measures spin up (+) in the x direction, B will measure spin
down (−) in the same direction. From elementary quantum mechanics we
know that if, on the other hand, B measures in the orthogonal direction
to A, then they will get either spin up or spin down each with probability
1
2.
Einstein proposed that we simply have a mixture of particles with spins
(sx , sz ) ∈ {(+, +), (+, −), (−, +), (−, −)} distributed equally between the
four different types. In this case quantum mechanics agrees entirely with
a hidden variable theory determining the outcome of the experiment. This
is another way of saying, there exists a joint probability distribution for sx
and sz whose possible outcomes all have probability 41 .
12
1.5 A Review of Bell and
Clauser-Horne-Shimony-Holt Inequalities
In order to see the discrepancy between quantum mechanics and a hidden
variable theory as proposed by Einstein, we must move to a slightly more
complex experiment (based on Mermin’s readable account [5]) where A and
B have the choice of 3 different coplanar directions to measure in and again
the pair of particles are produced in the singlet state. Assuming an a priori
set of hidden variables which determines the outcome in each of the three
spin directions, s1 , s2 , and s3 , each pair of particles must be of a particular
type, Ti , whose predetermined spins in each of these directions are,
T1 : A = (+ + +)
B = (− − −)
T2 : A = (+ + −) B = (− − +)
T3 : A = (+ − +) B = (− + −)
T4 : A = (+ − −) B = (− + +)
T5 : A = (− + +)
B = (+ − −)
T6 : A = (− + −) B = (+ − +)
T7 : A = (− − +) B = (+ + −)
T8 : A = (− − −) B = (+ + +)
When measured by A and B in the same directions we get the same results
as in the first experiment so we consider only the case when A and B measure along different directions. In the case that we have a T1 or T2 particle
we find the results are always anticorrelated. In any other case, there are
six different combinations of dissimilar measuring directions of which 2 cor-
13
respond to opposite spins. This gives a predicted probability of
1
3
in the
case that all particles are of types 2 to 7. If there are any particles of type 1
or 8 the probability of anticorrelated results must therefore be greater than
1
3.
In comparison, by quantum mechanics we can express the state of the particles before measurement as being in an arbitrary spin basis. For example,
choosing the s1 axis as our basis,
1
1
|ψA , ψB i = √ |+s1 , −s1 i − √ |−s1 , +s1 i .
2
2
(1.29)
If we measure this state in another basis, for instance, s2 , we project onto
the (single-particle) state,
θ12
θ12
|+s1 i + eiφ sin
|−s1 i .
|+s2 i = cos
2
2
(1.30)
So the projection onto s1 = −1, s2 = +1 gives,
θ12
h−s1 |A h+s2 |B |ψi = h−s1 |A
h+s1 |B + e
sin
h−s1 |B
2
1
1
√
√
|+s1 , −s1 i −
|−s1 , +s1 i
(1.31)
×
2
2
−1
θ12
= √ h−s1 |−s1 iA cos
h+s1 |+s1 iB .
(1.32)
2
2
θ12
cos
2
−iφ
So the probability,
1
2 θ12
P (−s1 , +s2 ) = cos
,
2
2
(1.33)
and similarly for the other relevant probabilities under discussion,
P (∓si , ±sj ) =
θij
1
cos2
.
2
2
14
(1.34)
Hence, the probability of getting anticorrelated spins when the two detectors
are set to two differing directions is,
2
P (−si , +sj ) + P (+si , −sj ) = cos
θij
.
2
(1.35)
This prediction disagrees with that of a hidden variable theory for certain
angles between measurement axes, for instance taking all the, θij =
2π
3 ,
we
get that,
1
P (−si , +sj ) + P (+si , −sj ) = ,
4
(1.36)
which is clearly not greater than or equal to 31 . This indicates that there is
some fundamental difference between the predictions of quantum mechanics
and that of a hidden variable scheme. This is an example of a Bell inequality.
Later in this paper, we will show in more detail what this difference may
imply.
In a variation of the above, we can arrive at a set of 8 inequalities which in
practice are easier to verify. Although they may be found by a more general
scheme, for the sake of clarity here we will consider an experiment similar
to that just given. However, we will measure only two possible directions
for each particle. We can then find correlations between measurements by
finding the expectation values of the spin of A multiplied by the spin of B,
each measured in either the s1 or the s2 direction. The correlations we get
are,
C11 = hs1A s1B i
(1.37)
C12 = hs1A s2B i
(1.38)
..
.
..
.
15
where we take the possible values of s1A , s2A , s1B , s2B to be +1 or −1.
Hence, when the values for each particle are the same, the value of the
measurement of the product of spins will be +1, and when they are opposite
it will be −1. It can be shown [6, 7], that by assuming an a priori set of
probabilities for the outcomes, the correlations obey the following CHSH
inequalities,
−2 6 C11 + C12 + C21 − C22 6 2
(1.39)
−2 6 C11 + C12 − C21 + C22 6 2
(1.40)
−2 6 C11 − C12 + C21 + C22 6 2
(1.41)
−2 6 −C11 + C12 + C21 + C22 6 2.
(1.42)
16
2 Review of the Literature on
Negative Probabilities
2.1 Interpretations of Quantum Mechanics
Negative probabilities have strong roots in the interpretations of quantum
mechanics that were developed in the early 20th century. The Wigner Function is a good example of this. It was introduced by Wigner in his study
of quantum corrections to thermodynamic systems [8]. It is essentially the
closest analogue of a probability density that we can find in the quantum
realm. The key point is that this ‘probability’ density can be negative and
this is why we refer to it as a quasi-probability. In an early treatment
of phase-space formulations of quantum mechanics, Groenewold discusses
operators on phase space in his 1946 paper [9]. This is one of the first treatments of phase-space formulation of quantum mechanics following Wigner’s
elucidation of his quasi-probability distribution.
Quantum mechanics follows clear mathematical principles, and one of
the first to try to put it on a firm footing was von Neumann. However,
finding a satisfactory interpretation of what the mathematics is actually
trying to say about nature and reality has puzzled many brilliant minds.
The conventional wisdom is the widely accepted Copenhagen interpretation.
It is useful here to recall an alternative interpretation of quantum mechan-
17
ics proposed by Bohm [10]. He argued that this alternative interpretation
of quantum mechanics can be made by assuming that our knowledge of a
system is necessarily incomplete. Thus a particle, say, will initially be in
one of many possible states and its apparent wave-like behaviour is due
to a combination of our incomplete knowledge of its initial conditions and
a quantum potential that acts on the particle in addition to the classical
potential.
In his second paper of 1952 [11], Bohm explains how his “hidden” variable
scheme can result in measurements by classical apparatus that are described
in the same way as those in the standard quantum theory. A measurement
of observable Q results in the system being in a non-overlapping packet
of states and since the initial state cannot be determined for certain, a
particular state of the measuring apparatus corresponds to the qth initial
state having a probability |cq |2 as in standard quantum theory. Bohm also
goes on to effectively say that this hidden variable theory necessitates superluminal transmission in an EPR-type experiment but considers that ‘no
contradictions with relativity arise’ since no signal could be carried by this
means. He concludes by saying that the fact that we cannot observe the
wavefunction and the hidden variables directly in today’s experiments does
not mean that they do not have an objective reality.
It is von Neumann and Bohm’s interpretation of quantum mechanics that
Wan and Sumner [12] use to obtain generalised phase-space distributions
which they interpret as distributions of expectation values. Thus, they
obviate the need for negative probabilities by considering negative values
as refering to negative values of an observable, not a probability. This is
an argument that has been put forward by several authors when faced with
negative probabilities. Wan and Sumner, in their 1991 paper [13], argue
18
that the Wigner function approach, mentioned earlier, is incorrect and that
quasi-probability functions are not necessary. These can be supplanted by a
hidden variable approach which only needs positive semi-definite probability
distributions.
Although work on Wigner functions and interpretations of quantum mechanics has waned somewhat, Bondar et al. [14] have recently considered
the Wigner function in a new light. They view it as a kind of unified
wavefunction that is related to the Koopman-von Neuman wavefunction (a
wavefunction that describes classical mechanics) in the classical limit, and
the usual quantum wavefunction in the limit ~ → 1.
2.2 Smoothing and Positive Density Functions
One way to supplant the need for negative probabilities is to attempt to
smooth a quasi-probability function in such a way that it becomes positive
semi-definite everywhere. Researchers have also grappled with the problem
of when negative values arise and when they can be avoided. This line of
research has been going on for some time.
In 1974, Hudson showed that a necessary and sufficient condition for
the Wigner quasi-probability distribution to be non-negative is that the
corresponding Schrödinger state function be the exponential of a quadratic
polynomial [15]. As will be seen later, these are the coherent or squeezed
states. A link was found between types of coherent states and negative
probabilities. N-th order coherent states are defined in a paper by Gordon
[16]. He looked at when negative values arise and concludes that for the
second order coherent states studied, negative probabilities arise. He goes
on to say that in fact any field of a certain form behaves non-classically.
19
An ongoing assumption has been that it is always possible to smooth the
Wigner function to obtain a non-negative probability. While it had been
assumed for some time that smoothing the Wigner function over phasespace cells that are of some finite size greater than h would always result in
a non-negative distribution, Wlodarz [17] shows that this is not generally
the case for all Wigner distribution functions. Later, Davidovic and Lalovic
studied the same problem [18]. Davidovic and Lalovic also found this is not
the case. In fact they considered using a general smoothing function. They
found this smoothing function itself must be a Wigner function. Due to
this restriction they found that the smoothing function must extend over
all space in order to produce a non-negative distribution function in all
cases.
Mizrahi [19] has worked on the Husimi Q function which he terms the
Wave-Packet Phase Space Representation that may be obtained from the
Wigner-Weyl operators by an integral transform. The advantage of this formulation is that the distribution function obtained, being non-negative, may
be interpreted as a probability distribution function. Mizrahi’s 1988 paper
[20] continues this earlier work on the Wave-Packet Phase Space Representation. In this paper Mizrahi looks at taking a distribution function and
using it to derive a wavefunction. He finds certain restrictions on the distribution function that must be satisfied for a possible quantum wavefunction
to exist which ensures that the uncertainty relationship is not violated.
Holland et al. [21] discuss relativistic distribution functions and suggest
that negative probability densities for the phase space representation of the
Klein-Gordon equation correspond to antiparticles. This is another way of
circumventing the problems of negative probabilities. By assuming a wave
function dependent on proper time, a probability distribution can be found
20
that is non-negative and stays non-negative in the non-relativistic limit.
Another related area of research is interpreting what negative values of a
distribution function tell us. Lütkenhaus and Barnett [22] try to quantify
non-classicality of phase space distributions based on the existence of negative regions of the distributions. They consider a generalisation of the usual
Wigner function parameterized by a variable s (these topics are also dealt
with in more detail in 4.1 and 5.2). On this s-space the distributions may
be well-behaved (ie. positive semi-definite), or not well behaved, over phase
space. They find that their classification of states shows that the Gaussian
states are the ones closest to classical states. This is a finding which had
been generally thought to be the case although had not previously been
proved.
2.3 Non-locality versus Extended Probabilities
The question of non-locality has plagued quantum mechanics for a long time.
Negative probabilities have been viewed as a possible answer to this problem.
For instance, Wodkiewicz notes [23] that quantum mechanics is equivalent
to a hidden variable theory with either negative probabilities or non-locality.
Marshall has argued [24] for a local-realist theory of electrodynamics which
considers the effects of the zero-point field on measurements. In his 1991
paper he contends that the zero-point field can explain the phenomenon
of enhancement in optics. This ‘stochastic optics’ is essentially a hidden
variable theory which can break the Bell inequalities but does not violate
causality and hence provides an explanation for the EPR paradox. He
considers that the prediction of negative probabilities made by quantum
mechanics shows that the theory is necessarily non-realist and thus flawed.
21
Sudarshan and Rothman [7] propose that the violation of the Bell inequalities is not necessarily an indication of non-locality but rather an indication
that the probabilities involved are not necessarily positive. In their second
paper [25], Rothman and Sudarshan argue further that the crucial step in
deriving the Bell inequalities is the assumption that probabilities are positive definite. Quantum mechanics breaks the Bell and CHSH inequalities
because we require simultaneous measurement of non-commuting observables in the derivation of the inequalities. This means that the distribution
over these observables, equivalent to the Wigner distribution function in
the space of these variables, can be negative. This is clear from examples
of the Wigner function for all but the ground state of the simple harmonic
oscillator (see figures 5.1 and 5.2).
Recently, Al-Safi and Short [26] have studied non-signalling systems and
found that by extending the allowed probabilities of the various outcomes
to allow for negative probabilities the non-signalling condition can be met.
EPR bipartite and tripartite systems are considered by Oas et al. [27]. They
make reference to the utility of negative probabilities in the frequentist
interpretation of probabilities where they arise as a result of the violation
of statistical stabilisation in p-adic statistics. In addition, they mention
that they can arise when mapping positive and negative measures into a
joint probability distribution. They also discuss the theorem, previously
proved by others, that the No-Signalling condition implies the existence of a
joint quasi-probabilty distribution. In order to limit the distributions they
consider, Oas et al. propose a minimization condition on the probability
‘mass’.
22
2.4 Further Applications of Extended
Probabilities
Negative, extended, or even complex probabilities have been considered in
other contexts, as well as finding usefulness in further developments related
to the path-integral approach to quantum mechanics and string theory. For
instance, Landsberg argues [28] that in the context of temperature equilibration between bodies, one of which may have a negative heat capacity (eg
a black hole), negative probabilities arise as a natural extension of positive
probabilities.
Atmanspacher [29] considers information transfer, specifically the smallest
possible transfer of information which necessarily occurs in the sub-quantum
regime. He shows that the transfer of information is directly related to
action (δE δt) which in quantum mechanics is quantised in units of h. Hence,
if you wish to consider information transfer that is less than 1 bit, that is,
less than an action of h, you must work with an interval of time less than δt,
usually called a virtual time, tvirt . To expand this idea Atmanspacher goes
on to use a complex time parameter which leads to a complex probability
formed from a bilinear in the wavefunction (ψi ψi ).
Methods that may be applicable to negative probabilities have been studied by many authors. For instance, in 2011, Pulkkinen et al. [30] devised a
computationally efficient way of determining Gaussian modes of a Gaussian
mixture by a convolution method. They noted that their method could
be applied to mixtures with negative weights or negative distributions and
as such may have application to quantum mechanics in addition to signal
processing such as image recognition.
For systems involving spin the phase-space approach can be used and
23
applied in the context of Bell and CHSH inequalities. The Wigner-Weyl
phase space representation, has been developed by Vrilly and Gracia-Bondia
[31] for spin using SU(2) covariance and what they term, ‘traciality’. In
2004, Fu extended the CHSH inequality to higher dimensions [32] which
may lead to interesting applications of negative probabilities and further
situations to examine.
Negative probabilities arise in the context of path integrals. For instance,
Sonego [33] discusses the use of extended probabilities in the context of
determining an explicitly real quasi-probability functional for paths. He
also finds the Wigner and Margenau-Hill functions using a path integral approach. He proposes that negative probabilities for certain paths are not a
problem, provided that the correct answer is obtained at the end, in a similar manner to the pragmatic way many others view negative probabilities.
Sokolovski and Connor [34] found that complex valued probabilities are encountered when applying the Feynman path integral approach to calculating
traversal time distributions. Viewing these complex valued probabilities as
split into real and imaginary distributions they find that each separately
can have negative values.
Recent work has seen quasi-probability distributions appear in the context
of string theory. For instance, in 2014, Cordero et al. employed the phasespace approach in the analysis of quantum spherical 2-branes [35].
24
3 The Wigner-Weyl Approach to
Quantum Mechanics
3.1 The Phase-Space Approach
In classical mechanics it is natural to describe a dynamical system in terms
of a probability distribution in phase-space. Its evolution is given by the
Liouville equation,
∂ρ X
dρ
∂ρ
∂ρ
=
+
p˙i
+ q˙i
.
dt
∂t
∂pi
∂qi
(3.1)
i
To find the expectation value of a variable, knowing the phase-space distribution, it is a simple matter;
Z
hOi =
dp dq O(p, q)ρ(p, q).
(3.2)
It would be convenient if this formalism could be extended directly to quantum mechanics but unfortunately we quickly run into problems. The main
problem we have is that an unambiguous relation that maps functions of
operators to functions on phase space,
Ô(q̂, p̂) → O(p, q),
25
(3.3)
does not exist. Therefore, a joint probability density is not possible.
3.2 The Wigner Function
The Wigner Function is the quantum phase-space quasi-probability density
function. It was introduced by Wigner in his study of quantum corrections
to thermodynamic systems [8]. It is essentially the closest analogue of a
probability density that we can find in the quantum realm. It can be found
using the characteristic function, φ(λ, µ), well known from statistics. The
characteristic function is simply the Fourier transform of the probability
density function, if one exists.
The characteristic function is the expectation value,
φ(λ, µ) = hM (λ, µ)i
(3.4)
M (λ, µ) = ei(λp+µq) .
(3.5)
where,
Written as an operator there are several ways M (λ, µ) can be expressed (see
Tatarski [36] for more details) but the one that will be convenient later is
simply the one we get by promoting p and q to operators in equation (3.5)
to get,
M̂ (λ, µ)) = ei(λp̂+µq̂) .
(3.6)
In quantum mechanics, the expectation value of this characteristic function is given by,
hψ| M̂ (λ, µ) |ψi .
(3.7)
Using the Baker-Campbell-Hausdorff (BCH) identity we can rewrite the
26
characteristic function several ways, by ordering p̂ and q̂ differently, but for
the next derivation we will use,
1
ei(λp̂+µq̂) = e− 2 iλµ~ eiλp̂ eiµq̂ .
(3.8)
Inserting the identity operator we obtain,
Z
M̂ (λ, µ)1 =
1
dq 0 e− 2 iλµ~ eiλp eiµq q 0 q 0 .
(3.9)
And, doing the same again with p states, and some further algebra we find,
Z
M̂ (λ, µ) =
Z
=
Z
=
1
0
dq 0 dp0 e− 2 iλµ~ eiµq eiλp p0 p0 q 0 q 0 (3.10)
ip0 q0 1
0
dq 0 dp0 e− 2 iλµ~ eiµq eiλp p0 e− ~ q 0 (3.11)
ip0
1
0 0
dq 0 dp0 e− 2 iλµ~ eiµq p0 e− ~ (q −~λ) q 0 .
(3.12)
Again inserting the identity operator over q states we get,
Z
M̂ (λ, µ) =
Z
=
ip0 0
1
0 dq 0 dp0 dq 00 e− 2 iλµ~ eiµq q 00 q 00 p0 e− ~ (q −~λ) q 0 (3.13)
ip0 0 ~λ
1
0 00 ~λ
dq 0 dp0 dq 00 e− 2 iλµ~ eiµq q 00 e ~ ((q − 2 )−(q + 2 )) q 0 ,
(3.14)
and after changing variables to q 0 −
Z
M̂ (λ, µ) =
0
0
00 iµq 0
dq dp dq e
~λ
2
→ q 0 , q 00 +
~λ
2
→ q 00 , we get,
0
00 ~λ
ip
~λ (q 0 −q 00 )
0
q −
~
e
q +
.
2
2 (3.15)
This gives us, finally, a useful form,
Z
M̂ (λ, µ) =
0 iµq 0
dq e
0 ~λ
~λ
0
.
q −
q +
2
2 27
(3.16)
We can then use the characteristic function operator in the form given in
equation (3.16) to find the Wigner function, W (p, q), for a state, |ψi. This is
given by the inverse Fourier transform, as stated earlier, of the characteristic
function,
Z
1
W (p, q) =
dλ dµ e−i(λp+µq) φ(λ, µ)
(2π)2
Z
1
dλ dµ e−i(λp+µq) hψ|M (λ, µ)|ψi
=
(2π)2
(3.17)
(3.18)
Inserting the definition (3.16) into this expression we obtain,
1
W (p, q) =
(2π)2
Z
1
(2π)2
Z
=
=
1
2π
Z
−i(λp+µq) iµq 0
0
dq dλ dµ e
e
0 ~λ
~λ 0
ψ q −
q +
ψ
2
2 (3.19)
~λ
~λ
0
−i(λp+µq) iµq 0 ∗
0
0
dq dλ dµ e
e ψ q −
ψ q +
2
2
(3.20)
~λ
~λ
dq 0 dλ e−iλp δ(q 0 − q)ψ ∗ q 0 −
ψ q0 +
. (3.21)
2
2
Integrating over q 0 we see that the Wigner function can be written in terms
of the wavefunction in q space as,
1
W (p, q) =
2π
Z
−iλp
dλ e
~λ
~λ
ψ q−
ψ q+
.
2
2
∗
(3.22)
This is an important result which we may regard as the definition of the
Wigner function in terms of the wavefunction in q.
The Wigner function has some nice properties that make it, perhaps, the
most useful phase-space weight function. For instance, it can be seen from
28
equation (3.22) that the Wigner function is real since,
Z
~λ
~λ
1
iλp ∗
ψ q−
W (p, q) =
dλ e ψ q +
2π
2
2
Z
~λ
1
~λ
−iλp ∗
=
dλ e
ψ q−
ψ q+
2π
2
2
∗
= W (p, q),
(3.23)
(3.24)
(3.25)
by changing the sign of λ and reversing the integration limits.
The marginals of the Wigner function, found by integrating over one of
the arguments, give the correct probability density for one variable. To find
the probability density for, q, for instance, we integrate over p as follows,
Z
~λ
~λ
1
dλ dp e−iλp ψ ∗ q −
ψ q+
2π
2
2
Z
~λ
~λ
= dλ δ(λ)ψ ∗ q −
ψ q+
2
2
Z
dp W (p, q) =
= ψ ∗ (q)ψ(q).
(3.26)
(3.27)
(3.28)
We recognize this as the usual expression for the quantum mechanical probability density as a function of q. To find the expression as a function of p,
it is easiest to rewrite the Wigner function in terms of the conjugate wavefunctions of q and then carry out the same steps. It is easily shown that
the Wigner function written in these terms has the same form as equation
(3.22), namely,
1
W (p, q) =
2π
Z
iµq
dµ e
~µ
~µ
φ p−
φ p+
.
2
2
∗
29
(3.29)
We will now prove this result using the following relation,
~λ ~λ
= q+
ψ
ψ q+
2
2 Z
~λ 0 0 0
= dp q +
p
p ψ
2 Z
ip0
~λ
1
=√
dp0 e ~ (q+ 2 ) φ(p0 )
2π~
(3.30)
(3.31)
(3.32)
Using (3.32) we can rewrite the Wigner function as,
1
W (p, q) =
2π
=
Z
dλe
Z
1
(2π)2 ~
−iλp
ψ
∗
~λ
q−
2
dλdp0 dp00 e−iλp e
~λ
ψ q+
2
ip0
(q+ ~λ
)
~
2
e−
(3.33)
ip00
(q− ~λ
)
~
2
φ∗ (p00 )φ(p0 )
(3.34)
=
1
(2π)2 ~
Z
p0
dλdp0 dp00 e−iλ(p− 2 −
Making a change of variables, µ =
p0 −p00
~ ,
p00
)
2
ν=
q
0
00
ei ~ (p −p ) φ∗ (p00 )φ(p0 ).
p0 +p00
2
(3.35)
which implies dp0 dp00 =
~dµdν, we get,
1
W (p, q) =
(2π)2 ~
Z
~dµdν e
iµq
~µ
~µ
2πδ (p − ν) φ ν −
φ ν+
2
2
∗
(3.36)
=
1
2π
Z
~µ
~µ
iµq ∗
dµ e φ p −
φ p+
.
2
2
(3.37)
So that, in exactly the same way,
Z
dq W (p, q) = φ∗ (p)φ(p).
(3.38)
Note that it is wrong to think of p and q in the Wigner function as simply
variables that are simultaneously realisable in physical space. In fact, as
pointed out, for instance, by Kira and Koch [37], the q that appears can be
30
shown to be the Fourier transformed variable corresponding to the difference
between wave numbers (eg. q = k − k 0 ), whilst the p in the Wigner function
corresponds to the centre of mass coordinate in p space (eg. p =
k+k0
2 ).
3.3 The Weyl Transform
Once we have the Wigner function for a quantum state, it appears that
we can simply find expectation values of an operator by integrating over
phase-space, as in the classical equation (3.2). As stated earlier this runs
into problems related with the ambiguity of ordering. On the one hand, since
we are integrating over c-numbers in phase space, ordering cannot matter,
but on the other hand we know that different orderings of non-commuting
operators can give different results.
For a function of either p or q no such problem of ordering exists and if
we use expressions such as,
Z
hAi =
dp dqW (p, q)A(q),
(3.39)
we find that,
Z
~λ
~λ
1
ψ q+
A(q)
dp dq e−iλp ψ ∗ q −
2π
2
2
Z
= dq ψ ∗ (q)A(q)ψ(q)
hAi =
(3.40)
(3.41)
the usual form for an expectation value in quantum mechanics. For a func-
31
tion of p only we can write,
Z
~λ
~λ
1
−iλp ∗
ψ q+
A(p)
hAi =
dp dq e
ψ q−
2π
2
2
Z
~µ
1
~µ
iµq ∗
=
dµe φ p −
φ p+
A(p)
2π
2
2
Z
= dp φ∗ (p)A(p)φ(p),
(3.42)
(3.43)
(3.44)
which again is as expected. We can see the problem of finding the correct
function of p and q by considering the expectation value of the commutator
of two non-commuting, conjugate variables, [q̂, p̂], say. If we simply replaced
this with, qp − pq we get,
hqp − pqi =
1
2π
Z
~λ
~λ
dp dq e−iλp ψ ∗ q −
ψ q+
(qp − pq)
2
2
=0
(3.45)
which is clearly not the same as the quantum result h[q̂, p̂]i = i~. This
indicates that as we stated earlier, we should not really regard the p and q
variables as those of ordinary classical phase space since the functions we
use to determine the expectation values of quantum operators are clearly
not the same as the functions of those operators themselves. What we need
is a mapping from the functions of operators to functions on phase space
and back again. This is given by the Wigner, and the Weyl transforms
respectively. The Wigner transform, for the operator ÔA of the observable
A is defined,
Z
A(p, q) = ~
−iλp
dλ e
32
~λ
~λ ÔA q −
.
q+
2 2
(3.46)
It is easy to see that this then gives the correct expectation value of A since,
Z
~λ
~λ
~
−iλp ∗
ψ q+
A(p, q)
hAi =
dp dq dλe
ψ q−
2π
2
2
Z
~
~λ
~λ 0
=
q+
ψ
dp dq dλdλ0 eiλ p e−iλp ψ q −
2π
2
2 ~λ0
~λ0 ÔA q +
× q−
2 2
Z
~
~λ
~λ
ψ
=
q+
dq dλdλ0 2πδ(λ − λ0 ) ψ q −
2π
2
2 ~λ0
~λ0 × q−
Ô
q
+
A
2 2
Z
~λ ~λ ~λ
~λ
= ~ dq dλ ψ q −
q−
ÔA q +
q+
ψ .
2
2 2
2 And then by a change of variables u = q −
~λ
2
and v = q +
~λ
2
(3.47)
(3.48)
(3.49)
(3.50)
we get,
Z
=
du dv hψ|ui hu| ÔA |vi hv|ψi
= hOA i ,
(3.51)
(3.52)
as desired. The reverse mapping from a function on phase-space to an
operator is given by the Weyl transform,
1
ÔA (p̂, q̂) = 2
4π
Z
dp dq dλdµA(p, q)e−i(λp+µq) M̂ (λ, µ).
(3.53)
where M̂ (λ, µ) = ei(λp̂+µq̂) used earlier in the definition of the characteristic
function. As stated by Tatarski [36] using this definition of M̂ gives us what
is known as the Weyl symmetric ordering. This originates from the ideas of
deriving quantum mechanics from classical mechanics by a procedure now
known as Weyl quantisation. These ideas were first set out in Weyl’s 1927
paper [38] although as we have already remarked, this formulation is not
unique and has not generally lead to a bridge between classical systems and
33
quantum ones. Equation (3.53) can be also written [36],
ÔA (p̂, q̂) = A
1 ∂ 1 ∂
,
i ∂λ i ∂µ
M̂ (λ, µ)|λ=µ=0 ,
(3.54)
since the partial differential operators commute, so there is no ambiguity in
this expression.
3.4 Negative Values of the Wigner Function
As has been mentioned previously, the possible values of the Wigner function
are not necessarily positive. We can see that this is the case by considering
the Wigner function for two orthogonal states satisfying,
hψi |ψj i = δij .
(3.55)
The Wigner functions for each of these states is,
1
Wi (p, q) =
2π
Z
−iλp
dλ e
ψi∗
~λ
~λ
q−
ψi q +
.
2
2
34
(3.56)
If we form the product of the two Wigner function and integrate over all
phase-space we find,
Z
Z
1
~λ
0 iλp iλ0 p ∗
dp dq Wi (p, q)Wj (p, q) =
dp dq dλdλ e e ψ q −
(2π)2
2
0
0
~λ
~λ
~λ
∗
×ψ q +
ψ q−
ψ q+
(3.57)
2
2
2
Z
1
~λ
0
=
dp dq dλdλ0 eip(λ−λ ) ψ ∗ q −
2
(2π)
2
0
~λ
~λ0
~λ
∗
ψ q+
ψ q+
(3.58)
×ψ q −
2
2
2
Z
1
~λ
~λ
=
ψ q−
dq dλ ψ ∗ q −
2π
2
2
~λ
~λ
×ψ ∗ q +
ψ q+
(3.59)
2
2
Z
1
=
dq dq 0 ψi∗ (q)ψj (q)ψj∗ (q 0 )ψi (q 0 )
(3.60)
2π~
(δ12 )2
= 0.
(3.61)
=
2π~
This implies that Wi and Wj must have opposite signs over some part of
phase space or else their sum must be identically zero everywhere. Hence
the Wigner function is not non-negative in general. We will discuss the
interpretation of this in more detail in 5.1.
35
4 General Distribution Functions
4.1 Alternatives to the Wigner Function
Not long after Wigner first put forward his ideas on a quantum distribution
function defined over phase-space, several others had proposed alternatively
defined distribution functions with different properties.
4.1.1 The Husimi Function (Q)
In his 1940 paper concerning the quantum density function, Husimi [39]
introduced the function which now commonly bears his name. The Husimi
function, a quasi-probability function on phase-space is defined by,
Q(α) =
1
hα|ρ|αi ,
π
(4.1)
where the states, |αi, are the coherent states that can be defined as eigenstates of the annihilation operator, a. That is, they satisfy the relation,
a |αi = α |αi .
(4.2)
The, α, here is the complex variable that can be formed when we have a
state described by a pair of conjugate variables, p and q say, that satisfy
36
the usual commutation relation, [q̂, p̂] = i~. Then,
α=
1
ip
(λq + ).
2π~
λ
(4.3)
The expectation value of an operator in the Husimi, Q, representation is
given by,
1
Tr[ρ OA ] =
π
Z
d2 α A(α) hα|ρ|αi ,
(4.4)
where the operator OA has the representation on phase-space A(α) and the
two are related by a certain transformation. The nice properties of, Q, is
that it is non-negative definite and bounded since it can be shown to lie
in the range, 0 6 Q(α) 6 π. However, there are still issues arising from
regarding it as a true probability density. This is because two different
points in the phase space do not represent two mutually orthogonal states
since the coherent states are not mutually orthogonal.
4.1.2 The Glauber-Sudarshan Function (P)
The Glauber-Sudarshan function is another alternative to the Wigner function. It is believed to have been first studied by Bopp [40] in relation to
his 1956 paper, written in French, concerning a statistical description of
quantum mechanics. It was studied in detail in the 1960’s by both Cahill
and Glauber [41], and Sudarshan [42]. To find the Glauber-Sudarshan function we require a function such that the density matrix is diagonal in the
coherent states. This means that the density operator can be expanded as,
Z
ρ=
dα P (α) |αi hα| ,
37
(4.5)
where, P (α), is the Glauber-Sudarshan function. Then the expectation
value of an operator, OA , is given by,
Z
Tr(ρ OA ) =
Z
=
d2 αP (α) Tr(|αi hα| OA )
(4.6)
d2 αP (α) hα|OA |αi .
(4.7)
It can be seen that,
Z
Tr(ρ) =
d2 αP (α)
=1
(4.8)
(4.9)
so that P (α) is real and normalised (if indeed it exists). Although the
Glauber-Sudarshan function has some nice properties for certain applications, one problem with it is that in general it is not a regular function and
may be more singular than a delta function.
4.2 The Generalised Weight Function
All of these quasi-probability distribution, or weight functions, can be related to a generalised weight function as described by Cahill and Glauber
in 1969 [41].
In an earlier paper [43], Cahill and Glauber studied generalised operator
orderings. This extends the concepts of normal ordering, (Weyl) symmetric
ordering (which was introduced in 3.3) and anti-normal ordering. Cahill and
Glauber use a parameter, s ∈ C, which correspond to normal ordering when
s = 1, symmetric ordering when, s = 0, and anti-normal ordering when,
s = −1. Having defined, s, Cahill and Glauber use it to define a generalised
38
weight function, W (α, s) which is related to the weight functions,
W (α, −1)
= Q(α)
π
(4.10)
W (α, 0)
= W (α)
π
(4.11)
W (α, +1)
= P (α)
π
(4.12)
where, Q(α), is the Husimi function, W (α), the Wigner function and, P (α)
the Glauber-Sudarshan function. Cahill and Glauber use the function,
D(α, s) ≡ D(α)e
s|α|2
2
,
(4.13)
with,
D(α) = exp αa† − α∗ a ,
(4.14)
the unitary displacement operator which satisfies,
D(α) |0i = |αi .
(4.15)
Hence, D(α), displaces the ground-state to α or, in other words, is the
generator of state, α. Again, α, is defined as in equation (4.3). This implies
that the area element in the complex plane is,
d2 α =
1
dqdp
2~
(4.16)
The complex Fourier transform of, D(α, s), then plays a key role in the
correspondence relation between operators and functions. We define this
by,
T (α, s) ≡
1
π
Z
d2 ξD(α, s) exp(αξ ∗ − α∗ ξ).
39
(4.17)
The correspondence relation for operators and functions is then,
h
i
ÔA ↔ A(α, −s) = tr ÔA T (α, −s) .
(4.18)
It should be carefully noted that this is not the correspondence relation for
finding the weight function from the density operator which is,
ρ̂ ↔ W (α, s) = tr[ρ̂ T (α, s)].
(4.19)
The reason that in the Wigner function representation, both the density
operator and arbitrary operators transform to phase-space functions in the
same way is that the Wigner representation corresponds to a value of, s = 0.
By putting s = 0 into equation (4.18) and equation (4.19) we can clearly
see that the two expressions are the same.
As in the previously seen formalisms, the expectation value of an operator
is found by integrating the weight function with the operator’s corresponding function over phase-space,
Z
Tr ρ ÔA = d2 αW (α, s)A(α, −s).
(4.20)
If we suppose that the operator, ÔA , can be expanded in a normal ordered
series,
ÔA =
∞
X
cnm (s = 1)(a† )n am ,
(4.21)
n,m=0
then operating on the coherent states, using the eigenvalue equation (4.2),
gives,
A(α, −1) =
∞
X
cnm (s = 1)(α∗ )n αm .
(4.22)
n,m=0
Cahill and Glauber show that this is the case for a broad set of operators,
40
and is particularly applicable in the field of optics, where the Optical Equivalence theorem [42], can be employed for a normally ordered operator.
41
5 Negative Values of the
Quasi-probability Distributions
5.1 Negative Values of the Wigner Function
In order to understand what these negative values of the Wigner distribution actually mean, it is easiest to use an example. Here we will use
the archetypal, but very useful, example of the simple harmonic oscillator
(SHO). The solutions to the SHO are well known,
ψn =
mω 1/4
π~
r
1
mωq 2
mω
√
exp −
Hn
q
2~
~
2n n!
(5.1)
where Hn are the Hermite polynomials defined by,
dn −x2
e .
dxn
(5.2)
mωq 2
exp −
,
2~
(5.3)
Hn (x) = (−1)n ex
2
Thus, the ground-state wavefunction is,
ψ0 (q) =
mω 1/4
π~
and the wavefunction for the first excited state is,
ψ1 (q) = π
−1/4
mω 3/4 √
~
42
mωq 2
2 q exp −
.
2~
(5.4)
Using our definition of the Wigner function in terms of the wavefunction in
position space, given by equation (3.22), we can find the Wigner function
for the wavefunctions of the SHO. For the ground-state we get,
Z
mω
~λ 2
mω
~λ 2
1 mω 1/2
dλ e−iλp e− 2~ (q− 2 ) e− 2~ (q+ 2 )
W0 (p, q) =
2π π~
Z
2
1 mω 1/2
− mω q 2 +( ~λ
2 )
=
dλ e−iλp e ~
2π π~
4π 1/2 − p2
1 mω 1/2 − mωq2
=
e ~
e mω~
2π π~
mω~
1 − mωq2 − p2
mω~
=
e ~
~π
(5.5)
(5.6)
(5.7)
(5.8)
So we see that the ground-state Wigner function is a Gaussian function in
p and q and is always positive. In the same manner we may calculate the
first excited state Wigner function. We find,
Z
~λ
~λ
1 mω 3/2 −1/2
−iλp
2π
dλ e
q−
q+
W1 (p, q) =
2π
~
2
2
mω
mω
~λ 2
~λ 2
(5.9)
×e− 2~ (q− 2 ) e− 2~ (q+ 2 )
mωq 2
p2
1
2mωq 2
2p2
=
−1 +
+
e− ~ − mω~
(5.10)
π~
~
mω~
The Wigner functions for the first two states of the SHO are plotted below.
As can be seen in figure 5.1 the ground-state is positive everywhere, a consequence of it being a pure Gaussian state in p and q. In comparison, the
first excited state, shown in figure 5.2, is not non-negative everywhere in
phase space and achieves a minimum value at the origin of −2/h. Looking
at these Wigner function plots, as a first guess, we may regard the groundstate as being close to a classical state. Similarly we may regard the first
excited state, due to its significant region of negative values, as being highly
non-classical. This is not entirely the full story since even for the ground-
43
Figure 5.1: The Wigner function W0 (p, q) for the ground-state of the simple
harmonic oscillator plotted for m, ω, ~ = 1 with the z axis scaled
in units of 1/π.
44
Figure 5.2: The Wigner function W1 (p, q) for the first excited state of the
simple harmonic oscillator plotted for m, ω, ~ = 1 with the z axis
scaled in units of 1/π.
45
state, some quantum behaviour will be present. For instance, p and q will
obey the Heisenberg uncertainty principle and this arises since we must use
the Wigner transform of an operator to calculate expectation values.
5.2 Negative Values of the General Distribution
Functions
Lütkenhaus and Barnett [22] have considered the family of general distribution functions parameterised by s, as introduced by Cahill and Glauber [41],
and considered when the distribution function is non-negative and hence
may be regarded as a true probability. This work is quite enlightening and
is worth going into in more detail here.
Lütkenhaus and Barnett show that a distribution function, W (α, s), (where
α is the complex phase-space coordinate defined previously and s is the same
as in 4.2) is related to another distribution function, differing in its value of
parameter s, by a convolution,
Z
W (α, s1 ) =
2
d β W (β, s2 )
2|α−β|2
2
−
e s2 −s1
π(s2 − s1 )
(5.11)
with s2 > s1 . Comparing this with the Green function for the heat equation,
Φ(x, t) = √
x2
1
e− 4kt
4πkt
(5.12)
for initial value,
U (x, 0) = δ(x)
(5.13)
∂ 2 U (x, t)
∂U (x, t)
−k
=0
∂t
∂x2
(5.14)
which satisfies,
46
we see that W (α, s) satisfies a sourceless heat equation with the parameter
s playing the role of negative time. Effectively, as s decreases W (α, s) is
gradually smoothed out. This agrees with our discussion of general distribution functions in 4.2 where we saw that the Husimi Q function with s = −1
is the only function guaranteed to be non-negative for all states.
Equation (5.11) implies the theorem (proved by Lütkenhaus and Barnett)
that, if W (α, s1 ) > 0 for all α, and W (α0 , s1 ) = 0 for at least one α0 , then
there is no s2 > s1 such that W (α, s2 ) > 0 for all α.
The proof of this follows from the convolution in equation (5.11). Since
the resulting function is a convolution of the distribution function with a
Gaussian function, the fact that the Gaussian is strictly positive implies
that in order to obtain the required zero of W (α0 , s1 ) at α0 , some values of
W (α, s2 ) must be negative.
Hence, we can divide the parameter space of s into regions where the
distribution function is positive definite and indefinite. At a critical value
of s = sc the distribution is positive semi-definite.
In effect if sc = +1 the state is very ‘classical’ and if sc = −1 the state
is very ‘non-classical’. This also marries with the less general concept that
the negative values of the Wigner function or of the Glauber-Sudarshan
function indicate non-classicality.
For states with minimum sc = −1, the Husimi Q function has zeros and
therefore all distribution functions with s > −1 have negative regions.
Lütkenhaus and Barnett also show that since,
Q(α) =
hα|ρ|αi
,
π
(5.15)
the states for which Q(α0 ) = 0 must have hα0 |ψi = 0. That is to say that
47
these states have zero overlap with at least one coherent state. Moreover
all states, barring the coherent and squeezed states (coherent states where
one conjugate variable’s uncertainty has been reduced at the expense of an
increase in the other), have zeros.
Writing,
|ψi =
∞
X
cn |ni
(5.16)
n=0
where the states |ni are definite number states of a single mode, Lütkenhaus
and Barnett then define a complex analytic function,
f (α) = e
|α|2
2
hψ|αi
(5.17)
Using the definition of the coherent states in terms of the number states,
|αi = e−
|α|2
2
∞
X
αn
√ |ni ,
n!
n=0
(5.18)
we can see that f (α) is given by a sum,
f (α) =
∞
X
αn
c∗n √ .
n!
n=0
(5.19)
Hence from the definition of f (α), for it to have zeros it is necessary and
sufficient that Q must also have zeros. Lütkenhaus and Barnett further
show that since,
|f (α)|2 6 e|α|
2
(5.20)
the sum has growth of order 6 2 and if it has no zeros then it fulfils
Hadamard’s theorem which implies that,
f (α) = eAα
48
2 +Bα+C
(5.21)
where A, B, C ∈ C. The only states satisfying this are the coherent and
squeezed states since they can be obtained from the ground-state by the
unitary transformation,
|ζ, βi = e(βa
† −β ∗ a)
1
†2 −ζ ∗ a2 )
e 2 (ζa
.
(5.22)
So we see that the most classical states are the coherent, followed by the
squeezed states. By this measure, all other states are more non-classical in
nature and in general have regions of negative values of the quasi-probability
distribution functions.
49
6 Fine’s Theorem and the Discrete
Wigner Function
6.1 Fine’s Theorem
As we saw in 1.5, to derive the Bell and CHSH inequalities, we assumed
that an underlying set of probabilities exists for the various outcomes of
the experiment. Fine’s theorem [3, 4] states that if the 8 CHSH inequalities
hold then there exists a probability distribution with the correct marginals
to construct the CHSH inequalities. It is the converse of the statement that
an underlying probability distribution implies the CHSH inequalities must
hold.
6.2 Viable and Non-viable Quasi-Probabilities
Halliwell and Yearsley [44] considered what quasi-distributions are in some
ways closest to true probability distributions by making the division into viable and non-viable quasi-distributions. They define viable quasi-probabilities
as ones which share the same marginals with a non-negative probability distribution. Similarly they define non-viable quasi-probabilities as ones which
do not have a non-negative probability sharing the same marginals.
Since we have seen that there is a close link between an underlying non-
50
negative probability and a hidden variables scheme we may draw the conclusion that a viable quasi-distribution can be associated with hidden variables
whilst a non-viable distribution cannot. Furthermore, if a hidden variable
system does not exist, then a non-viable distribution could also imply action
at a distance in an EPR-type experiment.
For the CHSH situation, by Fine’s theorem, we see that if the CHSH
inequalities are not satisfied then there is no viable quasi-probability which
matches the given marginals. If they are satisfied then there are viable
quasi-probabilities which means that a true non-negative probability can
then be found.
6.3 The Discrete Wigner Function
The Wigner function formalism has been extended to a finite phase-space by
Wootters [45, 46]. He shows that by properly defining the discrete Wigner
function certain key properties can be preserved from the continuous case.
He considers a discrete N ×N phase space of points where it may be seen that
there are (N +1) different directions possible and each direction has a family
of N parallel lines that can be drawn on the phase space by connecting the
points (if one is drawing this, one must be careful to pick out all the points
on a line which will wrap-around to points on the opposite side of the phase
space). The key properties that Wootters found necessary to preserve are:
1. Normalisation.
R
dpdqW (p, q) = 1
2. Overlap of states.
R
dpdqWi (p, q)Wj (p, q) = tr(ρi ρj )
3. Projection of marginals. The marginal for the observable ap̂ + bq̂
R
having a value between c1 and c2 is given by, C dpdqW (p, q), where
C is the strip in phase space bounded by ap+bq = c1 and ap+bq = c2 .
51
Analogously to the Weyl transform given in 3.3, we can define an operator
Ôα on a finite phase space with coordinate α from which we can construct
the Wigner function satisfying,
ρ̂ =
X
Wα Ôα .
(6.1)
By this definition, Wootters shows that we preserve the conditions for the
discrete case equivalent to the ones given above which are namely,
1. Normalisation.
P
2. Overlap of states.
α Wα
P
=1
α Wiα Wjα
= tr(ρi ρj )
3. Projection of marginals. The marginal for the observable that is defined by its value on each of a set of parallel lines, τ , in phase space
is given by,
pτ =
X
Wα ,
(6.2)
α∈τ
where pτ is the probability that the observable has the value that it
takes on the line τ .
52
7 Conclusion
7.1 Future Directions
The field of negative probabilities is a wide area and even though we have
limited our study in this paper mainly to the Wigner function and related
functions there are still many topics that are worthy of further research in
this area alone.
The link to classical mechanics provided by the phase-space representation
of quantum mechanics has been, and no doubt will continue to be, a fruitful
area of research. Work can still be done to relate the two, as has been
shown by Bondar et al. [14] and it would be interesting to see further how
the Koopman-von Neumann relation to the Wigner function may provide
new insights.
It would be instructive to look at further examples of Wigner functions
for different states to see how negative regions are related to the type of
state and how they evolve. This could lead to another method of classifying
states according to their non-classicality. The method used by Lütkenhaus
and Barnett [22], may be able to be extended further for states other than
the coherent and squeezed states.
Other questions that have arisen are, for instance, do the positive probability distributions that match the marginals for viable quasi-probabilities,
53
as defined by Halliwell and Yearsley [44], relate to each other in a similar
way to the family of general weight functions discussed in chapter 4? Also
does the critical value of the parameter s correspond in some way to the
sets of viable and non-viable probabilities? Furthermore, can the generalised
distribution function be successfully applied to the discrete case discussed
in 6.3?
7.2 Summary
In this paper we have attempted to review a broad field related to negative
probabilities. Since the primary example of this is the Wigner function, it
is on this topic that we have spent most of our time. We have covered some
elementary quantum mechanics and talked about the EPR paradox and the
Bell and CHSH inequalities in order to provide the necessary background
for some of the later topics. We have covered a diverse range of the published literature in order to, hopefully, provide a flavour of how negative
probabilities may be used and where they may be encountered.
The Wigner function and some of its properties have been covered in
quite some depth and we have discussed how even Wigner functions that
are positive everywhere are not really classical because of the way we must
interpret them. We have then seen how the Wigner function is actually just
one of a broad family of possible quasi-probability functions which include
the Husimi, Q, function and the Glauber-Sudarshan, P , function.
We have covered the example of the simple harmonic oscillator which
helps one visualise what the formalism actually means in practice. The
classification of the classicality of states is an interesting topic, and we have
focused on one possible method concerning the critical value of the variable
54
that parameterizes the quasi-probability distribution.
After this we have briefly looked at viable and non-viable quasi-probabilities
and Fine’s theorem before covering the discrete Wigner function.
In conclusion, the question of whether negative probabilities exist may
be the wrong question to ask. No-one doubts the usefulness of negative
numbers, and there is no call for them not to be categorized as numbers,
even though in the real world they cannot be used to count real objects. In
a similar way, whilst it seems inconceivable that true probabilities can ever
be negative, their use certainly seems to have a place. The only problem is
working out how we may consistently apply them.
55
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