# Is the Quantum State Real?
An Extended Review of $\psi$-ontology Theorems

Matthew Saul Leifer

Towards the end of 2011, Pusey, Barrett and Rudolph derived a theorem that
aimed to show that the quantum state must be ontic (a state of reality) in a
broad class of realist approaches to quantum theory. This result attracted a
lot of attention and controversy. The aim of this review article is to review
the background to the Pusey–Barrett–Rudolph Theorem, to provide a clear
presentation of the theorem itself, and to review related work that has
appeared since the publication of the Pusey–Barrett–Rudolph paper. In
particular, this review: Explains what it means for the quantum state to be
ontic or epistemic (a state of knowledge); Reviews arguments for and against
an ontic interpretation of the quantum state as they existed prior to the
Pusey–Barrett–Rudolph Theorem; Explains why proving the reality of the quantum
state is a very strong constraint on realist theories in that it would imply
many of the known no-go theorems, such as Bell’s Theorem and the need for an
exponentially large ontic state space; Provides a comprehensive presentation
of the Pusey–Barrett–Rudolph Theorem itself, along with subsequent
improvements and criticisms of its assumptions; Reviews two other arguments
for the reality of the quantum state: the first due to Hardy and the second
due to Colbeck and Renner, and explains why their assumptions are less
compelling than those of the Pusey–Barrett–Rudolph Theorem; Reviews subsequent
work aimed at ruling out stronger notions of what it means for the quantum
state to be epistemic and points out open questions in this area. The overall
aim is not only to provide the background needed for the novice in this area
to understand the current status, but also to discuss often overlooked
subtleties that should be of interest to the experts.
Quanta 2014; 3: 67–155.

This is an open access article distributed under the terms of the Creative
Commons Attribution License CC-BY-3.0, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited.

##  1 Introduction

In 1964, John Bell fundamentally changed the way that we think about quantum
theory [1]. Abner Shimony famously referred to tests of Bell’s Theorem as
“experimental metaphysics” [2], but I disagree with this characterization.
What Bell’s Theorem really shows us is that the foundations of quantum theory
is a bona fide field of physics, in which questions are to be resolved by
rigorous argument and experiment, rather than remaining the subject of open-
ended debate. In other words, it is a mistake to crudely divide quantum theory
into its practical part and its interpretation, and to think of the latter as
metaphysics, experimental or otherwise.

In the wake of Bell’s Theorem, the study of entanglement and nonlocality has
become a mainstream field of physics, particularly in light of its practical
applications in quantum information science, but Bell’s broader lesson—that
the interpretation of quantum theory should be approached as a rigorous
science—has rather been missed. This is nowhere more evident than in debates
about the status of the quantum state. The question of just what type of thing
the quantum state, or wavefunction, represents, has been with us since the
beginnings of quantum theory. The likes of de Broglie and Schrödinger
initially wanted to view the wavefunction as a real physical wave, just like
the waves of classical field theory, with perhaps some additional structure to
account for particle-like or “quantum” properties [3]. In contrast, following
Born’s introduction of the probability rule [4], the Copenhagen interpretation
advocated by Bohr, Heisenberg, Pauli et. al. came to view the wavefunction as
a “probability wave” and denied the need for a more fundamental reality to
underlie it [5]. In modern terms, most realist interpretations of quantum
theory; such as many-worlds [6, 7, 8], de Broglie–Bohm theory [9, 10, 11, 12],
spontaneous collapse theories [13, 14], and modal interpretations [15]; view
the wavefunction as part of reality, whereas those that follow more
Copenhagenish lines [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26] tend to view
it as a representation of knowledge, information, or belief. The big advantage
of the latter view is that the notorious collapse of the wavefunction can be
explained as the effect of acquiring new information, no more serious than the
updating of classical probabilities in the light of new data, rather than as
an anomalous physical process that needs to be eliminated or explained away.

The question then is whether this is a necessary dichotomy. Is the only way to
avoid having this weird multidimensional object as part of reality to give up
on reality altogether, or can we reach a compromise in which there is a well-
founded reality, but one in which the wavefunction only represents knowledge?
This seems like a question that is ripe for attacking with the kind of
conceptual rigor that Bell brought to nonlocality, and indeed Pusey, Barrett
and Rudolph have recently proven a theorem to the effect that the wavefunction
must be ontic (i.e. a state of reality), as opposed to epistemic (i.e. a state
of knowledge) in a broad class of realist approaches to quantum theory [27].

Since then, there has been much discussion and criticism of the
Pusey–Barrett–Rudolph Theorem in both formal [28, 29, 30, 31, 32, 33, 34, 35,
36, 37, 38, 39] and informal venues [40, 41, 42, 43, 44, 45, 46, 47, 48, 49],
as well as a couple of attempts to derive the same conclusion as
Pusey–Barrett–Rudolph from different assumptions [50, 51]. The
Pusey–Barrett–Rudolph Theorem and its successors all employ auxiliary
assumptions of varying degrees of reasonableness. Without these assumptions,
it has been shown that the wavefunction may be epistemic [52]. Therefore,
there has also been subsequent work aimed at ruling out stronger notions of
what it means for the wavefunction to be epistemic, without using such
auxiliary assumptions [53, 54, 55, 56, 57, 58, 59, 60]. The aim of this review
article is to provide the background necessary for understanding these
results, to provide a comprehensive presentation and criticism of them, and to
explain their implications.

One of the most intriguing things about proving that the wavefunction must be
ontic is that it would imply a large number of existing no-go results,
including Bell’s Theorem [1] and excess baggage theorems [61, 62, 63](i.e.
showing that the size of the ontic state space must be infinite and must scale
exponentially with the number of systems). Therefore, even apart from its
foundational significance, proving the reality of the wavefunction could
potentially provide a powerful unification of the known no-go theorems, and
may have applications in quantum information theory.

My aim is that this review should be accessible to as wide an audience as
possible, but I have made three decisions about how to present the material
that make my treatment somewhat more involved than those found elsewhere in
the literature. Firstly, I adopt rigorous measure theoretic probability
theory. It is common in the literature to specialize to finite sample spaces
or to adopt a less rigorous approach to continuous spaces, which basically
involves proving all results as if you were dealing with smooth and continuous
probability densities and then hoping everything still works when you throw in
a bunch of Dirac delta functions. Although a measure theoretic approach may
reduce accessibility, there are important reasons for adopting it. It would be
odd to attempt to prove the reality of the wavefunction within a framework
that does not admit a model in which the wavefunction is real in the first
place. Since the wavefunction involves continuous parameters, this means that
there is no option of specializing to finite sample spaces. Furthermore, there
are several special cases of interest for which the optimistic non-rigorous
approach simply does not work, including the case where the wavefunction, and
only the wavefunction, is the state of reality. Therefore, in order to cover
all the cases of interest, there is really no option other than taking a
measure theoretic approach. As an aid to accessibility, I outline how the main
definitions and arguments specialize to the case of a finite sample space,
which should be sufficient for those who do not wish to get embroiled in the
technical details.

Secondly, it is common in the literature to assume that we are interested in
modeling all pure states and all projective measurements on some finite
dimensional Hilbert space, and to specialize results to that context. However,
some results apply equally to subsets of states and measurements, which I call
_fragments_ of quantum theory. In addition, it is known that some fragments of
quantum theory, have natural models in which the wavefunction is epistemic
[64, 65, 66]. Therefore, I think it is important to emphasize the minimal
structures in which the various results can be proved, rather than just
assuming that we are trying to model all states and measurements on some
Hilbert space.

The third presentation decision concerns my treatment of _preparation
contextuality_ (see §5.3 for the formal definition). The main issue we are
interested in is whether pure quantum states must be ontic, since it is
uncontroversial that mixed states can at least sometimes represent lack of
knowledge about which of a set of pure states was prepared. It is common in
the literature to assume that each pure quantum state is represented by a
unique probability measure over the possible states of reality, but I do not
make this assumption. In a preparation contextual model, different methods of
preparing a quantum state may lead to different probability measures. In fact,
this must occur for mixed states [67], so it seems sensible to allow for the
possibility that it might occur for pure states as well. In addition, some of
the intermediate results to be discussed hold equally well for mixed states,
but this can only be established by adopting definitions that are broad enough
to encompass mixed states, which are necessarily preparation contextual.

These three presentation decisions mean that the definitions, statements of
results, and proofs that appear in this review often differ from those in the
existing literature. Generally, this is just a matter of making a few obvious
generalizations, without substantively changing the ideas. For this reason, I
do not explicitly point out where such generalizations have been made.

The review is divided into three parts. Part I is a general review of the
distinction between ontic and epistemic interpretations of the quantum state.
It discusses the arguments that had been given for ontic and epistemic
interpretations of the wavefunction prior to the discovery of the
Pusey–Barrett–Rudolph Theorem. My aim in this part is to convince you that
there is some merit to the epistemic interpretation and that previous
arguments for the reality of the quantum state are unconvincing. In this part,
I also give a formal definition of the class of models to which the
Pusey–Barrett–Rudolph Theorem and related results apply, and define what it
means for the quantum state to be ontic or epistemic within this class of
models. Following this, I give a detailed discussion of the other no-go
theorems that would follow as corollaries of proving the reality of the
wavefunction.

Part II reviews the three theorems that attempt to prove the reality of the
wavefunction: the Pusey–Barrett–Rudolph Theorem, Hardy’s Theorem, and the
Colbeck–Renner Theorem. The treatment of the Pusey–Barrett–Rudolph Theorem is
the most detailed of the three, since it has attracted the largest literature
and has been subject to the largest amount of confusion and criticism. In my
view, it makes the strongest case of the three theorems for the reality of the
wavefunction, although it is still not bulletproof, so I go to some lengths to
sort the silly criticisms from the substantive ones. The assumptions behind
the Hardy and Colbeck–Renner Theorems receive a more critical treatment, but
the theorems are still presented in detail because they are interestingly
related to other arguments about realist interpretations of quantum theory.

Part III deals with attempts to go beyond the rigid distinction between
epistemic and ontic interpretations of the wavefunction by positing stronger
constraints on epistemic interpretations. One of the aims of doing this is to
remove the problematic auxiliary assumptions needed to prove the three main
theorems, whilst still arriving at a conclusion that is morally similar. This
part is shorter than the other two and mostly just summarizes the known
results without proof. The reason for this is that many of the results are
only preliminary and will likely be superseded by the time this review is
published. My main aim in this part is to point out the most promising
directions for future research.

## Part I. The $\psi$-ontic/epistemic distinction

The results reviewed in this paper aim to show that the quantum state must be
_ontic_ (a state of reality) rather than _epistemic_ (a state of knowledge).
What does this mean and why is it important? The word “ontology” derives from
the Greek word for “being” and refers to the branch of metaphysics that
concerns the character of things that exist. In the present context, an _ontic
state_ refers to something that objectively exists in the world, independently
of any observer or agent. In other words, ontic states are the things that
would still exist if all intelligent beings were suddenly wiped out from the
universe. On the other hand, “epistemology” is the branch of philosophy that
studies of the nature and scope of knowledge. An _epistemic state_ is
therefore a description of what an observer currently knows about a physical
system. It is something that exists in the mind of the observer rather than in
the external physical world.

(a)  An ontic state is a point in phase space.

(b)  An epistemic state is a probability density on phase space. Contours
indicate lines of equal probability density.

(c)  An ontic state (cross) is deemed possible in more than one epistemic
state ($f_{1}$ and $f_{2}$). Phase space has been schematically collapsed down
to one dimension for illustrative purposes.

Figure 1: The distinction between ontic and epistemic states in single
particle classical mechanics.

In classical mechanics, the distinction between ontic and epistemic states is
fairly clear. A single Newtonian particle in one dimension has a position $x$
and a momentum $p$ and these are objective properties of the particle that
exist independently of us. All other objective properties of the particle are
functions of $x$ and $p$. The ontic state of the particle is therefore the
phase space point $(x,p)$. This evolves according to Hamilton’s equations

$\displaystyle\frac{dp}{dt}$ $\displaystyle=-\frac{\partial H}{\partial x}$
$\displaystyle\frac{dx}{dt}$ $\displaystyle=\frac{\partial H}{\partial p},$
(1)

where $H$ is the Hamiltonian. On the other hand, if we do not know the exact
position and momentum of the particle then our knowledge about its ontic state
is represented by a probability density $f(x,p)$ over phase space. By applying
Hamilton’s equations to the individual phase space points on which $f(x,p)$ is
supported, it can be shown that $f(x,p)$ evolves according to Liouville’s
equation

$\frac{\partial f}{\partial t}=\frac{\partial H}{\partial x}\frac{\partial
f}{\partial p}-\frac{\partial H}{\partial p}\frac{\partial f}{\partial x}.$
(2)

The probability density $f(x,p)$ is our epistemic state. See Fig. 1 for an
illustration of the distinction between classical ontic and epistemic states.
For other types of classical system the situation is analogous, the only
difference being the dimension of the phase space, e.g. $6N$ dimensions for
$N$ particles in $3$ dimensional space or a continuum for field systems. The
phase space point is still the ontic state and a probability density over
phase space is the epistemic state.

Note that calling a probability density “epistemic” is controversial in some
circles. It presupposes a broadly Bayesian interpretation of probability
theory in which probabilities represent an agent’s knowledge, information, or
beliefs. Fortunately, the issue at stake does not really depend on this as it
also appears in other interpretations of probability under a different name.
What is important is that the states dubbed “epistemic” only have
probabilistic import so they cannot be regarded as intrinsic properties of
individual physical systems. The key property that this implies is that a
given ontic state is deemed possible in more than one epistemic state.

On the Bayesian reading, this is due to the fact that different agents may
have different knowledge about one and the same physical system. For example,
perhaps Alice knows the position of a classical particle exactly but nothing
about its momentum, whilst Bob knows the momentum precisely but nothing about
its position. Alice and Bob would then assign different probability
distributions to the system, with the crucial property that they would overlap
on the ontic state actually occupied by the system.

Other interpretations of probability exhibit the same property in a different
way. For example, on a frequentist account of probability, probabilities
represent the relative frequencies of occurrence of some property in an
ensemble of independently and identically prepared systems. In this context,
we would talk about a state being “statistical” rather than “epistemic”. The
statistical state of a system depends upon the choice of ensemble that the
individual system is regarded as being a part of. For example, suppose a
classical particle occupies the phase space point $(1\leavevmode\nobreak\
m,1\leavevmode\nobreak\ kgms^{-1})$. If we regard it as part of an ensemble of
particles that all have positive position, but some have negative momentum,
then it will be assigned a different probability distribution than if we
regard it as part of an ensemble of particles all of which have positive
momentum, but some have negative position. In the former case, the probability
distribution will have support on negative momentum phase space points and in
the latter case it will have support on negative position phase space points.
The point is that ensembles consist of more than one individual system and the
same ontic state may occur as a part of many different ensembles. A
frequentist will not be lead astray by substituting the word “statistical” for
every occurrence of the word “epistemic” in this article, but the latter
terminology is used here because it has become standard.

Interpretations of probability that involve single-case objective chances
present more of a challenge for the ontic/epistemic distinction, since they
imply that probabilities can at least sometimes be ontic. Nevertheless, I
believe that an appropriate distinction can still be made in most of these
theories. This discussion is deferred to Appendix §A since it is mainly of
interest to those concerned with the philosophy of probability. However, it is
worth mentioning that many of those who have felt the need to introduce
objective chances have been motivated in part by the role that probability
plays in physics, and in quantum theory in particular. Since quantum
probabilities are functions of the wavefunction, they only present a novel
issue for the interpretation of probability if the wavefunction itself is
ontic because only then would quantum probabilities need to have a more
objective status than they do in classical physics. Since the status of the
wavefunction is precisely the question at issue, it is perhaps wise to defer
judgment on the necessity of objective chances until the reality of the
wavefunction is decided.

What is at stake then is the following question: When a quantum state
$\left|\psi\right\rangle$ is assigned to a physical system, does this mean
that there is some independently existing property of the individual system
that is in one-to-one correspondence with $\left|\psi\right\rangle$ (up to a
global phase), or is $\left|\psi\right\rangle$ simply a mathematical tool for
determining probabilities, existing only in the minds and calculations of
quantum theorists? This is perhaps the most hotly debated issue in all of
quantum foundations. I refer to it as _the $\psi$-ontic/epistemic distinction_
and use the terms $\psi$-ontic/$\psi$-epistemic to describe interpretations
that adopt an ontic/epistemic view of the quantum state. Holders of the
$\psi$-ontic view have been dubbed $\psi$-ontologists by Christopher Granade
(a masters student in Rob Spekkens’ quantum foundations course at Perimeter
Institute in 2010) and, continuing in this vein, I refer to the reality of the
quantum state as $\psi$-ontology and to theorems that attempt to establish
this view as $\psi$-ontology theorems.

To avoid misunderstanding, note that the $\psi$-ontic/epistemic distinction is
not about whether quantum states are ontic independently of whether quantum
theory is exactly true. It is not about whether the ultimate final theory of
physics, if indeed such a thing exists, will feature quantum states as part of
its ontology. We have little idea of what such a final theory might look like
and consequently we have little idea of what reality is actually made of at
the most fundamental level. Nevertheless, we can still ask what quantum theory
itself _says_ about reality. In other words, we are imagining a hypothetical
world in which quantum theory is in fact a completely correct theory of
physics, and asking whether quantum states would have to be ontological in
that world. That world is very unlikely to be our actual world, so the
question is really about the internal structure of quantum theory. More
specifically, it is about what kinds of explanation are compatible with
quantum theory. For example, a $\psi$-ontic view implies that we should draw
analogies between quantum states and phase space points when comparing quantum
and classical physics, and between the Schrödinger equation and Hamilton’s
equations, whereas a $\psi$-epistemic view says that the appropriate analogies
are between quantum states and probability distributions, and between the
Schrödinger equation and Liouville’s equation. If nothing else, this strongly
impacts how we are to understand the classical limit of quantum theory (e.g.
see [68, 69, 70, 71]). So, whilst the ontic/epistemic question may at first
sight seem abstract and philosophical, it does in fact have concrete
implications for physics.

The remainder of this part is structured as follows. §2 discusses arguments in
favor of the $\psi$-epistemic view, with the aim of convincing you that
$\psi$-ontology theorems are telling us something deep and surprising. For
those that remain unconvinced, §3 reviews the main arguments for the reality
of quantum states that were given prior to the discovery of $\psi$-ontology
theorems. In my view, none of these are particularly compelling, so even
someone who is already convinced of the reality of quantum states needs
something like a $\psi$-ontology theorem if they aspire to defend their
position with the same sort of conceptual force with which Bell derived
nonlocality. Following this, §4 introduces the framework of ontological
models, in which $\psi$-ontology theorems are proven, and gives the rigorous
definition of the $\psi$-ontic/$\psi$-epistemic distinction. Finally, §5
discusses the implications of proving $\psi$-ontology, by showing that several
existing no-go theorems can be derived from it.

##  2 Arguments for a $\psi$-epistemic interpretation

Before getting into the details of $\psi$-epistemic explanations, it is
important to distinguish two kinds of $\psi$-epistemic interpretation. The
most popular type are those variously described as anti-realist,
instrumentalist, or positivist. Since these labels are often intended as terms
of abuse, I prefer to call these approaches _neo-Copenhagen_ in order to avoid
implications for the philosophy of science that go way beyond how we choose to
understand quantum theory. All such interpretations bear a family resemblance
to the Copenhagen interpretation in that they are both $\psi$-epistemic and
they deny the need for any deeper description of reality beyond quantum
theory. Here, by “Copenhagen” I mean the views of Bohr, Heisenberg, Pauli et.
al. (see e.g. [5]), which are clearly $\psi$-epistemic, rather than the view
often found under this name in textbooks, which is actually due to Dirac [72]
and von Neumann [73] and is more ambiguous about whether the wavefunction is
real. If asked what quantum states represent knowledge about, neo-
Copenhagenists are likely to answer that they represent knowledge about the
outcomes of future measurements, rather than knowledge of some underlying
observer-independent reality. Modern neo-Copenhagen views include the Quantum
Bayesianism of Caves, Fuchs and Schack [16, 17, 18], the views of of Bub and
Pitowsky [19], the quantum pragmatism of Healey [21], the relational quantum
mechanics of Rovelli [22], the empiricist interpretation of W. M. de Muynck
[23], as well as the views of David Mermin [24], Asher Peres [25], and Brukner
and Zeilinger [26]. Some may quibble about whether all these interpretations
resemble Copenhagen enough to be called neo-Copenhagen, but for present
purposes all that matters is that these authors do not view the quantum state
as an intrinsic property of an individual system and they do not believe that
a deeper reality is required to make sense of quantum theory.

The second type of $\psi$-epistemic interpretation are those that are realist,
in the sense that they do posit some underlying ontology. They just deny that
the wavefunction is part of that ontology. Instead, the wavefunction is to be
understood as representing our knowledge of the underlying reality, in the
same way that a probability distribution on phase space represents our
knowledge of the true phase space point occupied by a classical particle.
There is evidence that Einstein’s view was of this type [74]. Ballentine’s
statistical interpretation [75] is also compatible with this view in that he
leaves open the possibility that hidden variables exist and only insists that,
if they do exist, the wavefunction remains statistical (as a frequentist,
Ballentine uses the term “statistical” rather than “epistemic”). More
recently, Spekkens has been a strong advocate of this point of view [64].

Neo-Copenhagen and realist $\psi$-epistemic interpretations share much of the
same explanatory structure, since they both view probability measures as the
correct classical analogy for the wavefunction. Many of the arguments for
adopting a $\psi$-epistemic interpretation apply equally to both of them. On
the other hand, $\psi$-ontology theorems only apply to realist
interpretations. This is to be expected as it would be difficult to prove that
the wavefunction must be ontic in a framework that does not admit the
existence of ontic states in the first place. Because of this,
$\psi$-epistemicists always have the option of becoming neo-Copenhagen in the
face of $\psi$-ontology theorems.

Realist $\psi$-epistemic interpretations are already strongly constrained by
existing no-go theorems, such as Bell’s Theorem [1] and the Kochen–Specker
Theorem [76], which go some way to explaining why not many concrete
$\psi$-epistemic models have been proposed. However, there is no reason to
view these results as decisive against realist $\psi$-epistemic
interpretations any more than they are decisive against realist $\psi$-ontic
interpretations. For example, Bohmian mechanics and spontaneous collapse
theories still attract considerable support despite displaying nonlocality and
contextuality, as the existing no-go theorems imply they must. Thus, we would
be guilty of a double-standard if we ruled out realist $\psi$-epistemic
interpretations on the basis of these results but still admitted the
possibility of $\psi$-ontic ones. What is needed is a theorem that explicitly
addresses the $\psi$-ontic/epistemic distinction, and this is the gap that
$\psi$-ontology theorems are intended to fill.

In the remainder of this section, the main arguments in favor of
$\psi$-epistemic interpretations are reviewed. Because we do not have a fully
worked out realist $\psi$-epistemic model that covers the whole of quantum
theory, it is helpful to introduce toy models that are similar to quantum
theory in some respects, but in which the analogous notion to the quantum
state is clearly epistemic. These are intended to demonstrate the kinds of
explanation that are possible in $\psi$-epistemic theories. Spekkens’ toy
theory [64], which reinvigorated interest in realist $\psi$-epistemic models
in recent years, is reviewed in §2.1. There are also $\psi$-epistemic models
that cover fragments of quantum theory, e.g. just pure state preparations and
projective measurements of a single qubit or just continuous variable systems
when restricted to Gaussian states and operations. These are reviewed in §2.2.
Finally, I review three further arguments for the $\psi$-epistemic view based
on the fact that quantum theory can be viewed as a generalization of classical
probability theory in §2.3, on the collapse of the wavefunction in §2.4, and
on the size of the quantum state space in §2.5.

###  2.1 Spekkens toy bit

Spekkens introduced a toy theory [64] that qualitatively reproduces the
physics of spin-$1/2$ particles (or any other instantiation of qubits) when
they are prepared and measured in the $x$, $y$ and $z$ bases. The full version
of Spekkens theory incorporates dynamics and composite systems, including
reproducing some of the phenomena associated with entangled states but, for
illustrative purposes, we restrict attention to the simplest case of a single
_toy bit_. The toy theory is meant to demonstrate the explanatory power of
$\psi$-epistemic interpretations by providing natural explanations of many
quantum phenomena that are puzzling if the quantum state is ontic.

Figure 2: The ontic state space of Spekkens toy-bit. The ontic states are
labeled $(x,y)$ where $x$ and $y$ take values $\pm 1$, which are abbreviated
to $\pm$ for compactness.

A toy bit consists of a system that can be in one of four states, labeled
$(-,-),(-,+),(+,-)$ and $(+,+)$. In Fig. 2, these are depicted laid out as a
grid in the $x-y$ plane, with the origin lying at the center of the grid. The
ontic states can then be thought of as representing the coordinates of the
centers of the grid cells, given by $(x,y)$, with $\pm$ short for $\pm 1$. For
concreteness, one can imagine that the cells of the grid represent four boxes
and that the system is a ball that can be in one of them. The ontic state
$(x,y)$ then represents the state of affairs in which the ball is in the box
centered on the coordinates $(x,y)$.

The most fine grained description of the toy bit is always its ontic state,
but we might not know exactly which of the ontic states is occupied. In
general, our knowledge of the system is described by a probability
distribution over the four ontic states, and this probability distribution is
our epistemic state. Spekkens imagines that there is a restriction on the set
of epistemic states that may be assigned to the system, called the _knowledge-
balance principle_ , which is in some ways analogous to the uncertainty
principle. Roughly speaking, the knowledge balance principle states that at
most half of the information needed to specify the ontic state can be known at
any given time. This means, for example, that if we know the $x$-coordinate
with certainty then we cannot know anything about the $y$-coordinate. Given
this restriction, there are six possible states of maximal knowledge, termed
_pure states_ , as shown in the left hand side of Fig. 3. The pure states are
denoted $\left|x\pm\right),\left|y\pm\right),\left|z\pm\right)$ in analogy to
the quantum states
$\left|x\pm\right\rangle,\left|y\pm\right\rangle,\left|z\pm\right\rangle$ of a
spin-$1/2$ particle.

Figure 3: The allowed states and measurements of Spekkens’ toy bit. For the
states, a red square indicates that the corresponding ontic state has
probability $1/2$ and a white square indicates probability $0$. For the
measurements, a square labeled $+$ gives the $+1$ outcome with certainty and a
square labeled $-$ gives the $-1$ outcome with certainty.

Note that the epistemic states $\left|z\pm\right)$ are not states with a
definite value of the $z$-coordinate. The system is two dimensional so it does
not have a third coordinate. Instead, $\left|z+\right)$ is the state in which
we know only that the $x$ and $y$ coordinates are equal and $\left|z-\right)$
is the state in which we know only that they are different. Defining $z=xy$,
this is equivalent to saying that $\left|z\pm\right)$ is the state in which we
know only that $z=\pm 1$.

Although the knowledge balance principle has been imposed by hand, it is easy
to imagine that it could arise from a lack of fine-grained control over the
system. For example, imagine a preparation device that pushes the ball to the
left along the $x$-axis, but that same device also causes a random disturbance
to the $y$-coordinate, such that the best we can do after operating the device
is to assign the state $\left|x-\right)$.

Having described the epistemic states of the theory, the next task is to
describe the measurements. Spekkens requires that measurements be _repeatable_
, which means that if a measurement is repeated twice in succession then it
should yield the same outcome both times. Also, the measurement should respect
the knowledge balance principle, so that our epistemic state after the
measurement contains at most half of the information required to specify the
ontic state. In order to satisfy this second requirement, the measurement must
necessarily cause a disturbance to the ontic state, since otherwise we could
end up in a situation in which we know the ontic state exactly. For example,
if a measurement of the $x$-coordinate could be implemented without
disturbance then measuring the $x$-coordinate followed by measuring the
$y$-coordinate would tell us the exact ontic state of the system.

There are three nontrivial measurements that can be implemented in such a way
that they satisfy the two requirements: the $X$ measurement reveals the $x$
coordinate, the $Y$ measurement reveals the $y$ coordinate, and the $Z$
measurement reveals the value of $z=xy$. These are illustrated on the right
hand side of Fig. 3. Each of these measurements causes a random exchange
between the pairs of ontic states that give the same outcome in the
measurement. For example, if we perform an $X$ measurement and get the $+1$
outcome, then with probability $1/2$ nothing happens and with probability
$1/2$ the states $(+,-)$ and $(+,+)$ are exchanged. This ensures that we
always end up in an epistemic state that satisfies the knowledge-balance
principle at the end of the measurement, in this case $\left|x+\right)$. It is
easy to see that this is the only type of disturbance that is compatible with
both repeatability and the knowledge-balance principle. For example, for an
$X$-measurement the random disturbance cannot exchange ontic states that have
different values of the $x$-coordinate, e.g. $(+,+)$ and $(-,+)$, since this
would violate repeatability.

The theory described so far makes exactly the same predictions as quantum
theory for sequences of measurements in the $x$, $y$ and $z$ directions of
spin-$1/2$ particles prepared in one of the states $\left|x\pm\right\rangle$,
$\left|y\pm\right\rangle$ and $\left|z\pm\right\rangle$ if we identify these
six states with $\left|x\pm\right)$, $\left|y\pm\right)$ and
$\left|z\pm\right)$ and the Pauli observables $\sigma_{x}$, $\sigma_{y}$ and
$\sigma_{z}$ with $X$, $Y$ and $Z$. It can thus be regarded as a hidden
variable theory for this kind of experiment. Further, the quantum states are
epistemic in this representation, as they are each represented by probability
distributions that have support on two ontic states and nonorthogonal states
overlap, e.g. $\left|x+\right)$ and $\left|y+\right)$ both assign probability
$1/2$ to the ontic state $(+,+)$.

Several features of quantum theory that are puzzling on the $\psi$-ontic view
are present in this theory and have very natural explanations. Firstly,
consider the fact that nonorthogonal pure states cannot be perfectly
distinguished by a measurement, e.g. if either the state
$\left|x+\right\rangle$ or the state $\left|y+\right\rangle$ is prepared, and
you do not know which, then there is no measurement that will enable you to
deduce this information with certainty. If quantum states are ontic then the
two preparations correspond to distinct states of reality and it is puzzling
that we cannot detect this difference. On the other hand, the toy theory
states $\left|x+\right)$ and $\left|y+\right)$ overlap on the ontic state
$(+,+)$ and this will be occupied by the system $50\%$ of the time whenever
$\left|x+\right)$ or $\left|y+\right)$ is prepared. When this does happen,
there is nothing about the ontic state of the system that could possibly tell
you whether $\left|x+\right)$ or $\left|y+\right)$ was prepared. Therefore, we
must fail to distinguish the two preparations at least $50\%$ of the time. The
overlap of the two epistemic states accounts for their indistinguishability.

Another feature of quantum theory that is easily accounted for in Spekkens’
model is the no-cloning theorem. In quantum theory, there is no transformation
that copies both of two nonorthogonal states. For example, there is no device
that operates with certainty and outputs both
$\left|x+\right\rangle\otimes\left|x+\right\rangle$ when
$\left|x+\right\rangle$ is input and
$\left|y+\right\rangle\otimes\left|y+\right\rangle$ when
$\left|y+\right\rangle$ is input. On the $\psi$-ontic view this is puzzling
because the two states represent distinct states of reality, so one might
expect that this distinctness could be detected and then copied over to
another system. Again, this is easily explained in Spekkens’ model in terms of
the overlap between the epistemic states $\left|x+\right)$ and
$\left|y+\right)$. Fig. 4 shows the inputs and outputs of the hypothetical
toy-theory cloning machine. The two input states overlap on the ontic state
$(+,+)$ and this occurs $50\%$ of the time regardless of which input state is
prepared. Since the cloning machine only has access to the ontic state, it
must do the same thing to the state $(+,+)$, regardless of whether it occurs
because $\left|x+\right)$ was prepared or because $\left|y+\right)$ was
prepared. Therefore, $50\%$ of the time, the input must get mapped to the same
set of ontic states, with the same probabilities, regardless of which state
was prepared, so there must be at least a $50\%$ overlap of the output states
of any physically possible device for these two input states. In contrast, the
output states of the hypothetical cloning machine only overlap on the ontic
state $((+,+),(+,+))$ and this must only occur $25\%$ of the time at the
output for either input state. Therefore, the cloning machine is impossible.

Figure 4: Inputs and outputs of a hypothetical toy bit cloning machine. In
order to represent a two toy bit state, the ontic state space of a single toy
bit is represented along one dimension. For the outputs, the horizontal axis
represents the first toy bit and the vertical axis represents the second toy
bit. Dark red represents ontic states occupied with $50\%$ probability and
light red represents those occupied with $25\%$ probability. The inputs
overlap on an ontic state that they both assign $50\%$ probability, but the
outputs only overlap on an ontic state that they both assign $25\%$
probability.

In Spekkens’ toy theory, both indistinguishability and no-cloning follow from
the more general fact that a stochastic map cannot decrease the overlap of two
probability distributions. In quantum theory, there is a similar result that
no transformation that can be implemented with certainty can decrease the
inner product between two pure states [77]. This suggests that the inner
product of two quantum states is analogous to the overlap between two
probability distributions, and this analogy would be most easily explained if
quantum states with nonzero inner product were literally represented by
overlapping probability distributions on some ontic state space, i.e. by a
realist $\psi$-epistemic interpretation.

Finally, consider the fact that mixed states in quantum theory have more than
one decomposition into a convex sum of pure states. For example, the maximally
mixed state of a spin-$1/2$ particle is $I/2$, where $I$ is the identity
operator, and this can be written alternatively as

$\displaystyle\frac{I}{2}$
$\displaystyle=\frac{1}{2}\left|x+\right\rangle\left\langle
x+\right|+\frac{1}{2}\left|x-\right\rangle\left\langle x-\right|$ (3)
$\displaystyle=\frac{1}{2}\left|y+\right\rangle\left\langle
y+\right|+\frac{1}{2}\left|y-\right\rangle\left\langle y-\right|$ (4)
$\displaystyle=\frac{1}{2}\left|z+\right\rangle\left\langle
z+\right|+\frac{1}{2}\left|z-\right\rangle\left\langle z-\right|.$ (5)

Physically speaking, this means that if we prepare a spin-$1/2$ particle in
the $\left|x+\right\rangle$ state with probability $1/2$ and in the
$\left|x-\right\rangle$ state with probability $1/2$ then no experiment can
tell the difference between this ensemble and that formed by preparing it in
the $\left|y+\right\rangle$ state with probability $1/2$ and in the
$\left|y-\right\rangle$ state with probability $1/2$ (and similarly for
$\left|z\pm\right\rangle$). On a $\psi$-ontic view this is puzzling because
the $\left|x\pm\right\rangle$ states are ontologically distinct from the
$\left|y\pm\right\rangle$ (and $\left|z\pm\right\rangle$) states so this
difference should be detectable. However, in Spekkens’ theory this non-
uniqueness of decomposition is easily explained because preparing
$\left|x+\right)$ with probability $1/2$ and $\left|x-\right)$ with
probability $1/2$ leads to exactly the same distribution over ontic states as
preparing $\left|y+\right)$ with probability $1/2$ and $\left|y-\right)$ with
probability $1/2$ (and similarly for $\left|z\pm\right)$). This is illustrated
in Fig. 5. Note that this is only possible because the distributions
corresponding to nonorthogonal quantum states overlap.

Figure 5: Multiple pure-state decompositions of a mixed state in Spekkens toy
theory. The maximally mixed state $\left|I/2\right)$ can be written as a
$50/50$ mixture in three different ways. Light red indicates a probability of
$1/4$.

What has been presented in this section is just a small fraction of the
quantum phenomena that are accounted for in Spekkens’ toy model. Many more can
be found in [64], but I hope the present discussion has conveyed a flavor of
the type of explanation that is possible in a realist $\psi$-epistemic theory.

###  2.2 Models for fragments of quantum theory

Spekkens’ toy model is qualitatively similar to the stabilizer fragment of
quantum theory, which consists of the set of states that are joint eigenstates
of maximal commutative subgroups of the Pauli group (i.e. the group generated
by tensor products of the identity and the three Pauli operators) and has
dynamics given by unitaries that map the Pauli group to itself (see [78] for
details of the stabilizer formalism and [79] for a presentation of Spekkens’
toy theory that closely resembles it). The stabilizer fragment is important in
quantum information theory as it contains all the states and operations needed
for quantum error correction, as well as a number of other quantum protocols.
Spekkens’ model does not exactly reproduce the stabilizer fragment when
dynamics and entanglement are taken into account, but other models have been
proposed that do reproduce fragments of quantum theory exactly in a
$\psi$-epistemic manner.

First of all, Spekkens’ toy theory has been generalized to larger dimensions
[65] and to continuous variable systems [66]. It turns out that for odd
dimensional Hilbert spaces, Spekkens’ model reproduces the stabilizer fragment
of quantum theory exactly. For continuous variable systems, Spekkens’ model
reproduces the Gaussian fragment of quantum theory, in which all states are
Gaussian and the transformations and measurements preserve the Gaussian nature
of the states. A Gaussian state is one that has a Gaussian Wigner function.
For a single particle, the Wigner function is defined in terms of the density
operator $\rho$ as $W(x,p)=\int_{-\infty}^{+\infty}dse^{ips/\hbar}\left\langle
x-\frac{s}{2}\right|\rho\left|x+\frac{s}{2}\right\rangle$ and is a pseudo-
probability distribution, i.e. it is normalized to $1$ but it does not have to
be positive. Gaussian functions are in fact positive so in this case $W(x,p)$
can be regarded as a probability distribution and unsurprisingly these are the
epistemic states in Spekkens’ continuous variable theory, with the ontic
states being the phase space points $(x,p)$.

Kochen and Specker gave a model for a single qubit that is $\psi$-epistemic
[76]. They were not actually trying to generate a $\psi$-epistemic model, but
rather to provide a counterexample to their eponymous theorem in
$2$-dimensions in order to show that the theorem requires a Hilbert space of
$\geq 3$ dimensions for its proof. Nevertheless, their model is a paradigmatic
example of a $\psi$-epistemic theory. The details of this model are presented
in §4.3 after we have introduced the formalism for realist $\psi$-epistemic
models more rigorously. Along similar lines, Lewis et. al. [52] and Aaronson
et. al. [53] have constructed $\psi$-epistemic models that work for all finite
dimensional systems. These models were developed as technical counterexamples
to certain conjectures about $\psi$-ontology theorems and as such they are not
very elegant or plausible. They are discussed in context in §7.5.

###  2.3 Generalized probability theory

Apart from specific models, there are also qualitative arguments in favor of
the $\psi$-epistemic view. The first of these is that quantum theory can be
viewed as a noncommutative generalization of classical probability theory.
Classically, consider the algebra $\mathcal{A}$ of random variables on a
sample space under pointwise addition and multiplication. A probability
distribution can then be regarded as a positive functional
$\mu:\mathcal{A}\rightarrow\mathbb{R}$ that assigns to each random variable
its expectation value. The quantum generalization of this is to replace the
commutative algebra $\mathcal{A}$ by the noncommutative algebra
$\mathfrak{B}\left(\mathcal{H}\right)$ of bounded operators on a Hilbert space
$\mathcal{H}$. A quantum state $\rho$ is isomorphic to a positive functional
$f_{\rho}$ on $\mathfrak{B}(\mathcal{H})$ given by
$f_{\rho}(M)=\text{Tr}\left(M\rho\right)$. In fact, by a theorem of von
Neumann [73], all positive linear functionals on
$\mathfrak{B}\left(\mathcal{H}\right)$ that are normalized such that $f(I)=1$,
where $I$ is the identity operator, are of this form.

Both $\mathcal{A}$ and $\mathfrak{B}\left(\mathcal{H}\right)$ are examples of
von Neumann algebras, and a generalization of classical measure theoretic
probability can be developed by defining generalized probability distributions
to be positive normalized functionals on such algebras [80, 81]. This
generalized theory has both classical probability theory and quantum theory as
special cases. In this theory, quantum states are playing the same role in the
quantum case that probability measures play in the classical case, and so it
is natural to interpret quantum states and classical probabilities as the same
kind of entity. Since classical probabilities are usually interpreted
epistemically, it is natural to interpret quantum states in the same way.

This line of argument would not be too convincing if noncommutative
probability theory were just a formal mathematical generalization with no
practical applications. However, the theory has a rich array of applications
in quantum statistical mechanics, and especially in quantum information
theory. The full machinery of von Neumann algebras is not often needed in
quantum information, as we are usually dealing with finite dimensional
systems. Nevertheless, whenever the analogy is made between classical
probability distributions and density operators, and between stochastic maps
and quantum operations, generalized probability is at play in the background.
For example, in quantum compression theory [82], a density operator on a
finite Hilbert space is viewed as the correct generalization of a classical
information source with finite alphabet, which would be described by a
classical probability distribution. Similarly, a quantum channel is described
by a quantum operation, and this is viewed as generalizing a classical
channel, which would be modeled as a stochastic map.

In fact, it is difficult to find any area of quantum information and computing
in which probabilities are not viewed as the correct classical analogs of
quantum states, and this includes areas that concern themselves exclusively
with pure states and unitary transformations. For example, the standard
circuit model of quantum computing [83] only employs pure states and
unitaries, but quantum computational complexity classes are most often defined
as generalizations of classical probabilistic complexity classes (see [84] for
definitions of the complexity classes mentioned in this section.). The class
BQP, usually thought of as the set of problems that can be solved efficiently
on a quantum computer, is sometimes loosely described as the quantum version
of P, the class of problems that can be solved in polynomial time on a
deterministic classical computer, but in fact it is the generalization of BPP,
the set of problems that can be solved in polynomial time on a probabilistic
classical computer with probability $>2/3$. All over quantum computing theory
we find the analogy made to classical probabilistic computing, and not to
classical deterministic computing.

It seems then, that if we take quantum information and computing seriously, we
must take generalized probability theory seriously as well. On these and other
grounds, I have argued elsewhere [85, 86] that quantum theory is indeed best
viewed as a generalization of probability theory. The details of this would
take us too far afield, but suffice to say there are good reasons for viewing
quantum states as analogous to probability distributions and, if we do that,
we should try to interpret them both in the same sort of way.

###  2.4 The collapse of the wavefunction

A straightforward resolution of the collapse of the wavefunction, the
measurement problem, Schrödinger’s cat and friends is one of the main
advantages of $\psi$-epistemic interpretations. Recall that the measurement
problem stems from the fact that there are two different ways of propagating a
quantum state forward in time. When the system is isolated and not being
observed, the quantum state is evolved smoothly and continuously according to
the Schrödinger equation. On the other hand, when a measurement is made on the
system, the quantum state must be updated according to the projection
postulate, leading to the instantaneous and discontinuous collapse of the
wavefunction. Since a measurement is presumably just some type of physical
interaction between system and apparatus, this poses the problem of why it is
not also modeled by Schrödinger evolution. However, doing so leads to
seemingly absurd situations, such as Schrödinger’s eponymous cat ending up in
a superposition of being alive and dead at the same time.

The measurement problem is not so much resolved by $\psi$-epistemic
interpretations as it is dissolved by them. It is revealed as a pseudo-problem
that we were wrong to have placed so much emphasis on in the first place. This
is because the measurement problem is only well-posed if we have already
established that the quantum state is ontic, i.e. that it is a direct
representation of reality. Only then does a superposition of dead and alive
cats necessarily represent a distinct physical state of affairs from a
definitely alive or definitely dead cat. On the other hand, if the quantum
state only represents what we know about reality then the cat may perfectly
well be definitely dead or alive before we look, and the fact that we describe
it by a superposition may simply reflect the fact that we do not know which
possibility has occurred yet.

###  2.5 Excess baggage

According to the $\psi$-ontologist, a single qubit contains an infinite amount
of information because a pure state of a qubit is specified by two continuous
complex parameters (ignoring normalization). For example, Alice could encode
an arbitrarily long bit string as the decimal expansion of the amplitude of
the $\left|0\right\rangle$ state. However, according to the Holevo bound [87],
only a single bit of classical information can be encoded in a qubit in such a
way that it can be reliably retrieved. If the quantum state truly exists in
reality, it is puzzling that we cannot detect all of this extra information.
Hardy has coined the term “ontological excess baggage” to refer to this
phenomenon [61]. It seems that $\psi$-ontologists are attributing a lot more
information to the state of reality than required to explain our observations.

The $\psi$-epistemic response to this is to note that a classical probability
distribution is also specified by continuous parameters. A probability
distribution over a single classical bit requires two real parameters (again
ignoring normalization). If probabilities were intrinsic properties of
individual systems then this would present a similar puzzle as there would be
an infinite amount of information in a single bit. However, classical bits are
in fact always either in the state zero or one and the probabilities simply
represent our knowledge about that value. In reality, there is just as much
information in a classical bit as we can extract from it, namely one bit. If
the quantum state is epistemic, then the same resolution is available to the
problem of excess baggage. The continuous parameters required to specify the
state of a qubit simply represent our knowledge about it, and the actual ontic
state of the qubit, whatever that may be, might only contain a finite amount
of information.

The excess baggage problem is exacerbated by considering how the state space
scales with the number of qubits. A pure state of $n$ qubits is specified by
$2^{n}$ complex parameters, but only $n$ bits can be reliably encoded
according to the Holevo bound. However, the number of parameters required to
specify a probability distribution over $n$ bits also scales exponentially, so
the $\psi$-epistemic resolution of the problem is still available.

In response to this, $\psi$-ontologists might be inclined to point out that
the number of bits that can be reliably encoded in $n$ qubits depends on how
exactly the communication task is defined. If Alice and Bob have pre-shared
entanglement then Alice can send $2n$ bits to Bob in $n$ qubits via superdense
coding [88]. Similarly, qubits perform better than classical bits in random
access coding [89], wherein Bob is not required to reliably retrieve all of
the bits that Alice sends, but only a limited number of them of his choice.
However, the amount of information that Alice can send to Bob does not scale
exponentially with the number of qubits in any of these protocols, so there is
still an excess baggage problem.

##  3 Arguments for a $\psi$-ontic interpretation

Having reviewed the arguments in favor of $\psi$-epistemic interpretations, we
now look at those that had been put forward in favor of the reality of quantum
states prior to the discovery of $\psi$-ontology theorems. Despite receiving a
good deal of support, I hope to convince you that they are far from
compelling. Thus, even those who are already convinced of the reality of the
quantum state should be interested in establishing their claim rigorously via
$\psi$-ontology theorems.

A big difficulty in extracting arguments for $\psi$-ontology from the
literature is that the majority of authors neglect the possibility of
_realist_ $\psi$-epistemic theories. Instead, they seem to think that either
the wavefunction must be real, or else we must adopt some kind of neo-
Copenhagen approach. Thus, many purported arguments for the reality of the
wavefunction are really just arguments for the reality of something,
regardless of whether that thing is the wavefunction. Since realist
$\psi$-epistemic interpretations already accept the need for an objective
reality, such arguments can be dismissed in the present context. From amongst
these arguments, I have attempted to sift out those that say something more
substantive about the wavefunction specifically. I have found four broad
classes of argument, each of which is discussed in turn in this section. §3.1
discusses the argument from interference, §3.2 discusses the argument from the
eigenvalue-eigenstate link, §3.3 discusses the argument from existing realist
interpretations of quantum theory, and finally §3.4 discusses the argument
from quantum computation.

###  3.1 Interference

> We choose to examine a phenomenon which is impossible, _absolutely_
> impossible, to explain in any classical way, and which has in it the heart
> of quantum mechanics. In reality, it contains the _only_ mystery. — R. P.
> Feynman [90] [Emphasis in original]

Following Feynman, single particle interference phenomena, such as the double
slit experiment, are often viewed as containing the essential mystery of
quantum theory. The problem of explaining the double slit experiment is
usually presented as a dichotomy between explaining it in terms of a classical
wave that spreads out and travels through both slits or in terms of a
classical particle that travels along a definite trajectory that goes through
only one slit. Neither of these explanations can account for both the
interference pattern and the fact that it is built out of discrete localized
detection events. A wave would not produce discrete detection events and a
classical particle would not be affected by whether or not the other slit is
open. This is taken as evidence that no classical description can work, and
that something more Copenhagen-like must be at work.

Of course, the dichotomy between either classical waves or particles is a
false one. If we allow the state of reality to be something more general, i.e.
some sort of quantum stuff that we do not necessarily understand yet, then
many additional explanations of the experiment become available. For example,
there is the Bohmian picture in which both a wave and a particle exist, and
the motion of the particle is guided by the wave. The wave then explains the
interference fringes, whilst the particle explains the discrete detection
events. This is by no means the only possibility, but it does highlight the
gap in the usual argument. Nevertheless, in a realist picture, it seems that
something wavelike needs to exist in order to explain the interference
fringes, and the obvious candidate is the wavefunction.

However, in order to arrive at the conclusion that the wavefunction must be
real, greater leeway has been given in determining what the ontic state might
be like compared to the original argument, which intended to rule out both
particles and waves. Given this, we should be careful to rule out other
possibilities rigorously, rather than jumping to the conclusion that the
wavefunction must be real. In this broader context, the only thing that the
double slit experiment definitively establishes is that there must be some
sort influence that travels through both slits in order to generate the
interference pattern. It does not establish that this influence must be a
wavefunction.

In fact, interference phenomena occur in some of the previously discussed
$\psi$-epistemic models, so the inference from interference to the reality of
the wavefunction is incorrect. In Spekkens’ toy theory, a notion of coherent
superposition can be introduced such that, for example, $\left|y+\right)$ is a
coherent superposition of $\left|x+\right)$ and $\left|x-\right)$. Such
superpositions are preserved under dynamical evolution, so there is a
superposition principle in the theory (see [64] for details). Further, since
all two-dimensional Hilbert spaces are created equal, there is nothing special
about the interpretation of the toy bit in terms of a spin-$1/2$ particle. It
could equally well be a model of any other two-dimensional system. For
example, consider the two dimensional subspace of an optical mode spanned by
the vacuum state $\left|0\right\rangle$ and the state $\left|1\right\rangle$
where it contains one photon. The toy bit state $\left|x+\right)$ can be
reinterpreted as $\left|0\right\rangle$ and $\left|x-\right)$ as
$\left|1\right\rangle$, and by doing so a whole host of Mach-Zehnder
interferometry experiments can be qualitatively reproduced by the theory [91].
This includes not only basic interferometry, but also such seemingly
paradoxical effects as the delayed choice experiment [92] and the Elitzur-
Vaidman bomb test [93]. In this theory, there is always a fact of the matter
about which arm of the interferometer the photon travels along, but it does
not fall afoul of the standard waves vs. particles argument because the vacuum
state has structure. For example, the situation in which the photon travels
along the left arm of a Mach-Zehnder interferometer would be represented in
quantum theory by $\left|1\right\rangle_{L}\otimes\left|0\right\rangle_{R}$.
The $\left|0\right\rangle_{R}$ factor would be represented by the epistemic
state $\left|x+\right)$ in the toy theory, which is compatible with two
possible ontic states $(+,+)$ and $(+,-)$, and these ontic states travel along
the right arm of the interferometer. Hence, when a photon is in the left arm
of an interferometer, and no photon is in the right arm, there is still a bit
of information traveling along the right arm of the interferometer,
corresponding to whether the ontic state is $(+,+)$ or $(+,-)$, that can be
used to convey information about whether or not its path was blocked. There is
an influence that travels through both arms, but that influence is not a
wavefunction.

Interference phenomena also occur in all of the models discussed in §2.2
simply because they reproduce fragments of quantum theory exactly and those
fragments contain coherent superpositions. It is arguable whether the
mechanisms explaining interference in all these models are plausible, but the
main point is that the direct inference from interference to the reality of
the wavefunction is blocked by them. If there is an argument from interference
to be made then it will need to employ further assumptions. Hardy’s
$\psi$-ontology theorem, discussed in §9, can be viewed as an attempt at doing
this, but, in light of the way that interference is modeled in Spekkens’ toy
theory, its assumptions do not seem all that plausible.

Ultimately, the intuition behind the argument from interference stems from an
analogy with classical fields. Because wavefunctions can be superposed, they
exhibit interference. Prior to the discovery of quantum theory, the only
entities in physics that obeyed a superposition principle and exhibited
interference were classical fields, and these were definitely intended to be
taken as real. For example, the value of the electromagnetic field at some
point in space-time is an objective property that can be measured by observing
the motion of test charges. The interference of wavefunctions is then taken as
evidence that they should be interpreted as something similar to classical
fields.

However, the analogy between wavefunctions and fields is only exact for a
single spinless particle, for which the wavefunction is essentially just a
field on ordinary three dimensional space. This breaks down for more than one
particle, due to the possibility of entanglement. The size of the quantum
state space scales exponentially with the number of systems, leading to the
previously discussed excess baggage problem. The wavefunction can no longer be
viewed as field on ordinary three-dimensional space, so the analogy with a
classical field should be viewed with skepticism. In combination with the fact
that interference phenomena can be modeled $\psi$-epistemically, the argument
from interference is far from compelling.

###  3.2 The eigenvalue-eigenstate link

The eigenvalue-eigenstate link refers to the tenet of orthodox quantum theory
that when a system is in an eigenstate $\left|m\right\rangle$ of an observable
$M$ with eigenvalue $m$ then $M$ is a property of the system that has value
$m$. Conversely, when the state is not an eigenstate of $M$ then $M$ is not a
property of the system. In other words, the properties of a system consist of
all the observables of which the quantum state is an eigenstate and nothing
else. These properties are taken to be objectively real, independently of the
observer.

This leads to an argument for the reality of the wavefunction because the
quantum state of a system is determined uniquely by the set of observables of
which it is an eigenstate. Indeed, it is determined uniquely by just a single
observable, since $\left|\psi\right\rangle$ is an eigenstate of the projector
$\left|\psi\middle\rangle\middle\langle\psi\right|$ with eigenvalue $1$ and
(up to a global phase) it is the only state in the $+1$ eigenspace of
$\left|\psi\middle\rangle\middle\langle\psi\right|$. The argument is then
that, if a system has a set of definite properties, and those properties
uniquely determine the wavefunction, then the wavefunction itself must be
real.

Roger Penrose is perhaps the most prominent advocate of this argument, so here
it is in his own words.

> One of the most powerful reasons for rejecting such a subjective viewpoint
> concerning the reality of $\left|\psi\right\rangle$ comes from the fact that
> whatever $\left|\psi\right\rangle$ might be, there is always—in principle,
> at least–a primitive measurement whose YES space consists of the Hilbert
> space ray determined by $\left|\psi\right\rangle$. The point is that the
> physical state $\left|\psi\right\rangle$ (determined by the ray of complex
> multiples of $\left|\psi\right\rangle$) is uniquely determined by the fact
> that the outcome YES, for this state, is certain. No other physical state
> has this property. For any other state, there would merely be some
> probability, short of certainty, that the outcome will be YES, and an
> outcome of NO might occur. Thus, although there is no measurement which will
> tell us what $\left|\psi\right\rangle$ actually is, the physical state
> $\left|\psi\right\rangle$ is uniquely determined by what it asserts must be
> the result of a measurement that might be performed on it…
>
> To put the point a little more forcefully, imagine that a quantum system has
> been set up in a known state, say $\left|\phi\right\rangle$, and it is
> computed that after a time $t$ the state will have evolved, under the action
> of $U$, into another state $\left|\psi\right\rangle$. For example,
> $\left|\phi\right\rangle$ might represent the state ‘spin up’
> ($\left|\phi\right\rangle=\left|\uparrow\right\rangle$) of an atom of spin
> $\frac{1}{2}$, and we can suppose that it has been put in that state by the
> action of some previous measurement. Let us assume that our atom has a
> magnetic moment aligned with its spin (i.e. it is a little magnet pointing
> to the spin direction). When the atom is placed in a magnetic field, the
> spin direction will precess in a well-defined way, that can be accurately
> computed as the action of $U$, to give some new state, say
> $\left|\psi\right\rangle=\left|\rightarrow\right\rangle$, after a time $t$.
> Is this computed state to be taken seriously as part of physical reality? It
> is hard to see how this can be denied. For $\left|\psi\right\rangle$ has to
> be prepared for the possibility that we might choose to measure it with the
> primitive measurement referred to above, namely that whose YES space
> consists precisely of the multiples of $\left|\psi\right\rangle$. Here, this
> is the spin measurement in the direction $\rightarrow$. The system has to
> know to give the answer YES, with certainty for that measurement, whereas no
> spin state of the atom other than
> $\left|\psi\right\rangle=\left|\rightarrow\right\rangle$ could guarantee
> this. — Roger Penrose, quoted in [94].

This argument can be easily countered using any of the existing
$\psi$-epistemic models, to which the same reasoning would apply. For example,
consider the epistemic state $\left|x+\right)$ in Spekkens’ toy theory. This
state assigns the definite value $+1$ to the $X$ measurement and indeed it is
the only allowed epistemic state in the theory that does this. In fact, all of
the pure states of a toy bit $\left|x\pm\right),\left|y\pm\right)$ and
$\left|z\pm\right)$ are uniquely determined by the definite value that they
assign to one of the measurements. Following Penrose’s reasoning, we would
then conclude that these states are objective properties of the system.
However, this is not the case since the objective properties of the system are
those that are determined by the ontic state, and each ontic state is
compatible with more than one epistemic state. For example, the ontic state
$(+,+)$ is compatible with $\left|x+\right)$, $\left|y+\right)$ and
$\left|z+\right)$. Given the complete specification of reality, the epistemic
state is underdetermined.

One way of exposing the error in the eigenvalue-eigenstate argument is to note
that, in the toy theory, the fact that observables uniquely determine
epistemic states is a consequence of the knowledge-balance principle and not a
fundamental fact about reality. For example, without the knowledge-balance
principle, it would be permissible to have an epistemic state that assigns
probability $2/3$ to $(+,+)$ and $1/3$ to $(+,-)$. Just like
$\left|x+\right)$, this state assigns probability $1$ to the $+1$ outcome of
the $X$ measurement and this is the only measurement that is assigned a
definite value. If this state were allowed then it would no longer be possible
to mistake $\left|x+\right)$ for an objective property of the system. Penrose
has mistaken the set of states that it is possible to prepare with current
experiments for the set of all logically possible states.

Another way of exposing the error is to look at the restrictions on
measurements in the toy theory. The measurements only reveal coarse-grained
information about the ontic state. Without the knowledge-balance principle it
would be permissible to conceive of a more fine-grained measurement that
reveals the ontic state exactly. This measurement reveals a definite property
of the system because it is determined uniquely by the ontic state. However,
specifying this observable no longer uniquely determines the epistemic state.
For example, if we learn that the ontic state is $(+,+)$ then this is
compatible with $\left|x+\right)$, $\left|y+\right)$ and $\left|z+\right)$.

In conclusion, the mistake in the eigenvalue-eigenstate argument is to assume
that the observables that we can actually measure in experiments form the sum
total of all the properties of the system and to assume that the set of states
that we can actually prepare are the sum total of all logically conceivable
states. Without these assumptions, the argument is simply false.

###  3.3 Existing realist interpretations

There are a handful of fully worked out realist interpretations of quantum
theory, including many-worlds [6, 7, 8], de Broglie–Bohm theory [9, 10, 11,
12], spontaneous collapse theories [13, 14] and modal interpretations [15]. In
each of these interpretations the wavefunction is part of the ontic state, so
there is an argument from lack of imagination to be made: since all the
interpretations of quantum theory that we have managed to come up with that
are uncontroversially realist have a real wavefunction, then the wavefunction
must be real.

I admit that it behooves the realist $\psi$-epistemicist to try to construct a
fully worked out interpretation. However, absence of evidence is not the same
thing as evidence of absence. Nevertheless, I have frequently heard this
argument made in private conversations. Some people seem to think that since
we have a bunch of well worked out interpretations, we ought to simply pick
one of them and not bother thinking about other possibilities. Ever since the
inception of quantum theory we have been beset by the problem of quantum
jumps, by which I mean that quantum theorists are liable to jump to
conclusions.

Despite the obvious weakness of this argument, there is a more subtle point to
be made. John Bell was motivated to work on his eponymous theorem by noting
that de Broglie–Bohm theory exhibited nonlocality. He wanted to know if this
was just a quirk of de Broglie–Bohm theory or an inescapable property of any
realist interpretation of quantum theory. With this motivation, he ended up
proving the latter. The lesson of this is that if we find that all realist
interpretations of quantum theory share a property that some find
objectionable then we ought to determine whether or not this is a necessary
property. However, this is a motivation for developing $\psi$-ontology
theorems, rather than regarding the matter as settled a priori.

###  3.4 Quantum computation

The final argument I want to consider is due to David Deutsch, who put it
forward as an argument in favor of the many-worlds interpretation. However, I
think the argument can be adapted, more generally, into an argument for the
reality of the wavefunction. Here is the argument in Deutsch’s own words.

> To predict that future quantum computers, made to a given specification,
> will work in the ways I have described, one need only solve a few
> uncontroversial equations. But to explain exactly _how_ they will work, some
> form of multiple-universe language is unavoidable. Thus quantum computers
> provide irresistible evidence that the multiverse is real. One especially
> convincing argument is provided by quantum algorithms […] which calculate
> more intermediate results in the course of a single computation than there
> are atoms in the visible universe. When a quantum computer delivers the
> output of such a computation, we shall know that those intermediate results
> must have been computed somewhere, because they were needed to produce the
> right answer. So I issue this challenge to those who still cling to a
> single-universe world view: _if the universe we see around us is all there
> is, where are quantum computations performed?_ I have yet to receive a
> plausible reply. — David Deutsch [95] [Emphasis in original].

Quantum algorithms that offer exponential improvement over existing classical
algorithms, such as Shor’s factoring algorithm [96], start by putting a
quantum system in a superposition of all possible input strings. Then, some
computation is done on each of the strings before using interference effects
between them to elicit the answer to the computation. If each of the branches
of the wavefunction were not individually real, whether or not they are
interpreted in a many-worlds sense, then where does the computation get done?

This is not exactly an argument for the reality of the wavefunction, but it is
at least an argument that the size of ontic state space should scale
exponentially with the number of qubits, and that the ontic state should
contain pieces that look like the branches of a wavefunction. However, whilst
I agree with Deutsch that an interpretation of quantum theory should offer an
explanation of how quantum computations work, it is not at all obvious that
the explanation must be a direct translation of what happens to the
wavefunction. The argument would perhaps be more compelling if there were
known exponential speedups for problems where we think that the best we can do
classically is to just search through an exponentially large set of solutions,
since we could then argue that a quantum computer must be doing just that.
This would be the case if we had such a speedup for the traveling salesman
problem, or any other NP complete problem. The sort of problems for which we
do have exponential speedup, such as factoring, are more subtle than this.
They lie in NP, but are not NP complete. If we were to find an efficient
classical algorithm for these problems then it would not cause the whole
structure of computational complexity theory to come crashing down. If such an
algorithm exists, then whatever deeper theory underlies quantum theory may be
exploiting this same structure to perform the quantum computation.

Even if such a scenario does not play out, Deutsch’s argument is not decisive
against realist $\psi$-epistemic interpretations. Since we have not yet
constructed a viable interpretation of this sort that covers the whole of
quantum theory, who knows what explanations such a theory might provide?
Therefore, as Deutsch says, explaining quantum computation ought to be viewed
as a challenge for the $\psi$-epistemic program rather than an argument
against it.

##  4 Formalizing the $\psi$-ontic/epistemic distinction

Hopefully, by this point I have convinced you that it is worth trying to
settle the question of the reality of the wavefunction rigorously. The aim of
this section is to provide a formal definition of what it means for the
quantum state to be ontic or epistemic within a realist model of quantum
theory. This is usually done within the framework of _ontological models_.
This is really no different from the framework that Bell used to prove his
eponymous theorem, and an ontological model is sometimes alternatively known
as a hidden variable theory. However, I prefer the term “ontological model”
because there is a lot of confusion about the meaning of the term “hidden
variables”. Following the example of Bohmian mechanics, a hidden variable
theory is often thought to be a theory in which some additional variables are
posited alongside the wavefunction, which is itself conceived of as ontic from
the start. Since the reality of the wavefunction is precisely the point at
issue, we definitely want to include models in which it is not assumed to be
real within our framework. In addition, in order to cover orthodox quantum
theory, we want our framework to include models in which the wavefunction is
the _only_ thing that is real, i.e. there are no additional hidden variables.
A further confusion is the commonly held view that a hidden variable theory
must restore determinism, whereas we want to allow for the possibility that
nature might be genuinely stochastic. For these reasons, I prefer to use the
term “ontological model”. It is either the same thing as a hidden variable
theory or more general, depending on how general you thought hidden variable
theories were in the first place.

Whilst the Hardy and Colbeck–Renner Theorems involve assumptions about how
dynamics are represented in an ontological model, the Pusey–Barrett–Rudolph
Theorem only involves prepare-and-measure experiments, i.e. a system is
prepared in some quantum state and is then immediately measured and discarded.
Therefore, we deal with prepare-and-measure experiments first and defer
discussion of dynamics until it is needed. A $\psi$-ontology theorem aims at
proving that any ontological model that reproduces quantum theory must have
ontic quantum states. This does not apply to arbitrary fragments of quantum
theory, since we have seen in §2.2 that there are fragments that can be
modeled with epistemic quantum states. In order to understand both cases, we
need to define ontological models for fragments of quantum theory rather than
just for quantum theory as a whole. The formal definition of a prepare-and-
measure fragment of quantum theory is given in §4.1 and then §4.2 explains how
these are represented in ontological models, with examples given in §4.3.
Based on this, §4.4 gives the formal definition of what it means for the
quantum state to be ontic or epistemic.

###  4.1 Prepare and measure experiments

In a prepare-and-measure experiment, the experimenter performs a preparation
of some physical system and then immediately measures it, records the outcome,
and discards the system. The experimenter can repeat the whole process of
preparing and measuring as many times as she likes in order to build up
frequency statistics for comparison with the probabilities predicted by some
physical theory. Each run of the experiment is assumed to be statistically
independent of the others and it is assumed that the experimenter can choose
which measurement to perform independently of the choice of preparation.

For completeness, we consider the most general type of quantum state—a density
operator—and the most general type of observable —a Positive Operator Valued
Measure (POVM), although we restrict attention to POVMs with a finite number
of outcomes. Readers unfamiliar with these concepts should consult a standard
textbook, such as [83] or [97].

######  Definition 4.1.

A _prepare-and-measure (PM) fragment_
$\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ of quantum
theory consists of a Hilbert space $\mathcal{H}$, a set $\mathcal{P}$ of
density operators on $\mathcal{H}$, and a set $\mathcal{M}$ of POVMs on
$\mathcal{H}$. The probability of obtaining the outcome corresponding to a
POVM element $E\in M$ when performing a measurement $M\in\mathcal{M}$ on a
system prepared in the state $\rho\in\mathcal{P}$ is given by the Born rule

$\text{Prob}(E|\rho,M)=\text{Tr}\left(E\rho\right).$ (6)

For many of the results reviewed here, the PM fragment under consideration is
the one in which $\mathcal{P}$ is the set of all pure states on $\mathcal{H}$
and $\mathcal{M}$ consists of measurements in a set of complete orthonormal
bases. The formalism of PM fragments allows the sets of states and
measurements required to make $\psi$-ontology theorems go through to be made
explicit. Additionally, many of the intermediate results used in proving
$\psi$-ontology theorems apply to PM fragments in general, including those
that feature mixed states and POVMs, so it is worth introducing fragments at
this level of generality.

When only a single PM fragment is under consideration, it is assumed to be
denoted $\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ so that the
notations $\rho\in\mathcal{P}$ and $M\in\mathcal{M}$ can be used without first
explicitly writing down the triple.

###  4.2 Ontological models

The idea of an ontological model of a PM fragment is that there is some set
$\Lambda$ of _ontic states_ that give a complete specification of the
properties of the physical system as they exist in reality. When a quantum
system is prepared in a state $\rho\in\mathcal{P}$, what really happens is
that the system occupies one of the ontic states $\lambda\in\Lambda$. However,
the preparation procedure may not completely control the ontic state, so our
knowledge of the ontic state is described by a probability measure $\mu$ over
$\Lambda$. This means that $\Lambda$ needs to be a measurable space, with a
$\sigma$-algebra $\Sigma$, and that $\mu:\Sigma\rightarrow[0,1]$ is a
$\sigma$-additive function satisfying $\mu(\Lambda)=1$. For those unfamiliar
with measure-theoretic probability, for a finite space $\Sigma$ would just be
the set of all subsets of $\Lambda$ and $\sigma$-additivity reduces to
$\mu(\Omega_{1}\cup\Omega_{2})=\mu(\Omega_{1})+\mu(\Omega_{2})$ for all
disjoint subsets $\Omega_{1}$ and $\Omega_{2}$ of $\Lambda$.

In general, different methods of preparing the same quantum state may result
in different probability measures over the ontic states. This is especially
true of mixed states, since they do not have unique decompositions into a
convex mixture of pure states. If one prepares a mixed state by choosing
randomly from one of the pure states in such a decomposition, then one can
prove that the probability measure must in general depend on the choice of
decomposition. This is known as _preparation contextuality_ , and is discussed
in more detail in §5.3. For this reason, a quantum state $\rho$ is associated
with a set $\Delta_{\rho}$ of probability measures rather than just a single
unique measure. Note that it is possible to find models in which pure states
correspond to unique measures, and much of the literature implicitly assumes
this type of model. However, it turns out that this assumption is not
necessary, so we allow for the possibility that even a pure state is
represented by a set of probability measures for the sake of generality.

Turning now to measurements, the outcome of a measurement $M\in\mathcal{M}$
might not reveal $\lambda$ exactly but depend on it only probabilistically.
This could be because nature is fundamentally stochastic, but it could also
arise in a deterministic theory if the response of the measuring device
depends not only on $\lambda$ but also on degrees of freedom within the
measuring device that are not under the experimenter’s control (see [98] for a
discussion of this). To account for this, each POVM $M$ is represented by a
conditional probability distribution $\text{Pr}(E|M,\lambda)$ over $M$. In a
bit more detail, when a measurement $M\in\mathcal{M}$ is performed, each
$\lambda$ must give rise to a well defined probability distribution over the
outcomes, so, for any fixed $\lambda$, we must have
$\text{Pr}(E|M,\lambda)\geq 0$ for all $E\in M$ and $\sum_{E\in
M}\text{Pr}(E|M,\lambda)=1$. Further, in order to calculate the probabilities
that the model predicts we will observe, we are going to have to average over
our ignorance about $\lambda$ so, for any fixed $E$, $\text{Pr}(E|M,\lambda)$
must be a measurable function of $\lambda$. See Fig. 6 for an illustration of
the conditional probability distribution corresponding to a measurement.

Figure 6: A possible conditional probability distribution
$\text{Pr}(E_{j}|M,\lambda)$ for a two outcome measurement
$M=\\{E_{1},E_{2}\\}$ in an ontological model. $\text{Pr}(E_{j}|M,\lambda)$ is
the conditional probability of obtaining the outcome $E_{j}$ when $M$ is
measured and the ontic state is $\lambda$. As a result, it must satisfy
$\text{Pr}(E_{1}|M,\lambda)+\text{Pr}(E_{2}|M,\lambda)=1$ for all values of
$\lambda$. For illustrative purposes, the ontic state space is represented as
a $1$-dimensional line, but it may actually be an arbitrary measurable space.

Again, different methods of implementing the same measurement may result in
different conditional probability distributions. This is known as _measurement
contextuality_ and it must occur in certain types of model due to the
Kochen–Specker theorem [76]. Measurement contextuality is discussed further in
Appendix C. For this reason, a measurement $M$ is associated with a set
$\Xi_{M}$ of conditional probability distributions rather than just a single
one.

In order to compute the probabilities that the ontological model predicts for
the observable outcomes of measurements, the conditional probabilities
$\text{Pr}(E|M,\lambda)$ have to be averaged over our ignorance of the true
ontic state as specified by $\mu$. This gives,

$\text{Pr}(E|\rho,M)=\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda).$ (7)

Note that, the notation Pr is used for the probabilities in an ontological
model to distinguish them from the probabilities Prob predicted by quantum
theory.

Finally, if the ontological model is to reproduce the predictions of quantum
theory, then; for each $\rho\in\mathcal{P}$, $M\in\mathcal{M}$, each
$\mu\in\Delta_{\rho}$, $\text{Pr}\in\Xi_{M}$ must satisfy

$\text{Pr}(E|\rho,M)=\text{Prob}(E|\rho,M),$ (8)

or in other words

$\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda)=\text{Tr}\left(E\rho\right).$
(9)

Summarizing, we have

######  Definition 4.2.

An _ontological model_ of a PM fragment is a quadruple
$(\Lambda,\Sigma,\Delta,\Xi)$ that consists of:

  * •

A measurable space $(\Lambda,\Sigma)$, where $\Lambda$ is called the _ontic
state space_.

  * •

A function $\Delta$ that maps each quantum state $\rho\in\mathcal{P}$ to a set
of probability measures $\Delta[\rho]=\Delta_{\rho}$ on $(\Lambda,\Sigma)$.

  * •

A function $\Xi$ that maps each POVM $M\in\mathcal{M}$ to a set of conditional
probability distributions over $M$, $\Xi[M]=\Xi_{M}$, i.e. each
$\text{Pr}\in\Xi_{M}$ is a function from $M\times\Lambda$ to $\mathbb{R}$ that
is measurable as a function of $\lambda\in\Lambda$ and satisfies, for all
$\lambda\in\Lambda$,

$\forall E\in M,\,\,\,\,\text{Pr}(E|M,\lambda)\geq 0$ (10)

and

$\sum_{E\in M}\text{Pr}(E|M,\lambda)=1.$ (11)

The ontological model _reproduces the quantum predictions_ if, for all
$\rho\in\mathcal{P}$ and $M\in\mathcal{M}$, each $\mu\in\Delta_{\rho}$ and
$\text{Pr}\in\Xi_{M}$ satisfies

$\forall E\in
M,\,\,\,\,\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda)=\text{Tr}\left(E\rho\right).$
(12)

In what follows, it is convenient to use projectors
$\left|\psi\middle\rangle\middle\langle\psi\right|$ to represent pure states
instead of vectors $\left|\psi\right\rangle$ to avoid the global phase
ambiguity. The set of projectors onto the pure states of $\mathcal{H}$ is
known as the _projective Hilbert space_ of $\mathcal{H}$. The notation
$\left[\psi\right]=\left|\psi\middle\rangle\middle\langle\psi\right|$ and
$\Delta_{\psi}=\Delta_{\left[\psi\right]}$ is used to reduce clutter.
Similarly, if a measurement $M$ consists of projectors onto pure states
$\left[\phi\right]$ then we use the notation $\text{Pr}(\phi|M,\lambda)$ as
shorthand for $\text{Pr}(\left[\phi\right]|M,\lambda)$.

###  4.3 Examples of Ontological Models

######  Example 4.3 (Spekkens’ toy bit).

Spekkens’ toy bit can be recast as an ontological model of the PM fragment
$\langle\mathbb{C}^{2},\mathcal{P},\mathcal{M}\rangle$ where

$\mathcal{P}=\left\\{\left[x+\right],\left[x-\right],\left[y+\right],\left[y-\right],\left[z+\right],\left[z-\right],I/2\right\\},$
(13)

and

$\mathcal{M}=\left\\{\left\\{\left[x+\right],\left[x-\right]\right\\},\left\\{\left[y+\right],\left[y-\right]\right\\},\left\\{\left[z+\right],\left[z-\right]\right\\}\right\\}.$
(14)

The ontic state space is $\Lambda=\\{(+,+),(+,-),(-,+),(-,-)\\}$. Since
$\Lambda$ is a finite set, the integral in Eq. (12) is just a sum

$\sum_{\lambda\in\Lambda}\text{Pr}(E|M,\lambda)\mu(\lambda)=\text{Tr}\left(E\rho\right),$
(15)

where, abusing notation slightly, we write $\mu(\lambda)$ for
$\mu(\\{\lambda\\})$.

In this model, each quantum state is represented by a unique probability
measure. The six probability functions
$\left|x\pm\right),\left|y\pm\right),\left|z\pm\right)$ illustrated in Fig. 3
represent the states $\left[x\pm\right],\left[y\pm\right],\left[z\pm\right]$,
and $\left|I/2\right)$ from Fig. 5 represents the maximally mixed state $I/2$.
Each measurement is associated with a unique conditional probability
distribution as described in Fig. 3. For example, for the
$X=\\{\left[x+\right],\left[x-\right]\\}$ measurement we have

$\begin{array}[]{lllll}\text{Pr}(x+|X,(+,+))&=&\text{Pr}(x+|X,(+,-))&=&1\\\
\text{Pr}(x+|X,(-,+))&=&\text{Pr}(x+|X,(-,-))&=&0\\\
\text{Pr}(x-|X,(+,+))&=&\text{Pr}(x-|X,(+,-))&=&0\\\
\text{Pr}(x-|X,(-,+))&=&\text{Pr}(x-|X,(-,-))&=&1.\end{array}$ (16)

It is easy to check that this reproduces the quantum predictions for the
fragment.

######  Example 4.4 (The Beltrametti–Bugajski model [99]).

This model is essentially a translation of the orthodox interpretation of
quantum theory into the language of ontological models. The PM fragment is
$\langle\mathbb{C}^{d},\mathcal{P},\mathcal{M}\rangle$ where $\mathcal{P}$
contains all the pure states on $\mathbb{C}^{d}$ and $\mathcal{M}$ consists of
all POVMs on $\mathbb{C}^{d}$. The idea of the model is that the quantum
state, and only the quantum state, represents reality. Therefore, the ontic
state space $\Lambda$ is just the set of pure states $\left[\lambda\right]$
for $\left|\lambda\right\rangle\in\mathbb{C}^{d}$, i.e. states differing by a
global phase are identified so $\Lambda$ is the projective Hilbert space of
$\mathbb{C}^{d}$. This space carries a natural topology induced by the inner
product, and we take $\Sigma$ to be the Borel $\sigma$-algebra of this
topology (see [100] for details).

A pure quantum state $\left[\psi\right]$ is then represented by the point
measure

$\mu(\Omega)=\delta_{\psi}(\Omega)=\begin{cases}1&\text{if}\,\,\left[\psi\right]\in\Omega\\\
0&\text{if}\,\,\left[\psi\right]\notin\Omega.\end{cases}$ (17)

Each POVM is represented by a unique conditional probability distribution and,
since the quantum state is the ontic state in this model, it should come as no
surprise that the response functions simply specify the quantum probabilities,
i.e. if a POVM $M\in\mathcal{M}$ contains the operator $E$ then

$\text{Pr}(E|M,\left[\lambda\right])=\text{Tr}\left(E\left[\lambda\right]\right).$
(18)

We then trivially have

$\displaystyle\int_{\Lambda}\text{Pr}(E|M,\left[\lambda\right])d\mu(\left[\lambda\right])$
$\displaystyle=$
$\displaystyle\int_{\Lambda}\text{Tr}\left(E\left[\lambda\right]\right)d\delta_{\psi}(\left[\lambda\right])$
(19) $\displaystyle=$ $\displaystyle\text{Tr}\left(E\left[\psi\right]\right),$

so the model reproduces the quantum predictions.

The model can be extended to mixed states by writing them as convex
combinations of pure states. For example, the maximally mixed state of a
spin-$1/2$ particle can be written as

$\frac{I}{2}=\frac{1}{2}\left[x+\right]+\frac{1}{2}\left[x-\right].$ (20)

This means that $I/2$ can be prepared by flipping a coin, preparing
$\left[x+\right]$ if the outcome is heads or $\left[x-\right]$ if the outcome
is tails, and then forgetting or erasing the result of the coin flip. Let’s
call this preparation procedure $P$. It follows that the predictions of $I/2$
can be reproduced by the measure

$\mu=\frac{1}{2}\delta_{x+}+\frac{1}{2}\delta_{x-}.$ (21)

Note however that this is a preparation contextual model because the maximally
mixed state can also be prepared by mixing a different set of pure states,
e.g. the decomposition

$\frac{I}{2}=\frac{1}{2}\left[y+\right]+\frac{1}{2}\left[y-\right],$ (22)

yields the measure

$\mu=\frac{1}{2}\delta_{y+}+\frac{1}{2}\delta_{y-},$ (23)

which reproduces the quantum predictions just as well. In general,
$\Delta_{\rho}$ consists of one convex combination of point measures for each
of the different ways of writing $\rho$ as a mixture of pure states.

######  Example 4.5 (The Bell Model [101]).

In his review of no-go theorems for hidden variable theories [101], Bell
introduced an ontological model for measurements in orthonormal bases on
systems with a two-dimensional Hilbert space. The obvious generalization to
arbitrary finite dimensional systems is presented here, as it is needed in
§7.5. Bell intended his model as a pedagogical device to point out the flaws
in previous no-go theorems, and he was primarily interested in whether a
deterministic theory could reproduce the quantum predictions. In modern terms,
the Bell model can be thought of as a minimal modification of the
Beltrametti–Bugajski model, intended to make it deterministic.

The PM fragment of interest is
$\mathfrak{F}=\langle\mathbb{C}^{d},\mathcal{P},\mathcal{M}\rangle$, where
$\mathcal{P}$ consists of all pure states and $\mathcal{M}$ consists of all
measurements of the form
$\mathcal{M}=\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}$, where
$\left\\{\left|\phi_{j}\right\rangle\right\\}_{j=0}^{d-1}$ is an orthonormal
basis for $\mathbb{C}^{d}$.

The generalized Bell model employs an ontic state space
$\Lambda=\Lambda_{1}\times\Lambda_{2}$ that is the Cartesian product of two
state spaces. As in Beltrametti–Bugajski, $\Lambda_{1}$ is the projective
Hilbert space of $\mathbb{C}^{d}$, with Borel $\sigma$-algebra $\Sigma_{1}$.
$\Lambda_{2}$ is the unit interval $[0,1]$, with Borel $\sigma$-algebra
$\Sigma_{2}$, representing an additional hidden variable. The measurable space
is then $(\Lambda_{1}\times\Lambda_{2},\Sigma_{1}\otimes\Sigma_{2})$, where
the tensor product $\sigma$-algebra $\Sigma_{1}\otimes\Sigma_{2}$ is the
$\sigma$-algebra generated by sets of the form $\Omega_{1}\times\Omega_{2}$
with $\Omega_{1}\in\Sigma_{1}$, $\Omega_{2}\in\Sigma_{2}$.

The quantum state $\left[\psi\right]$ is represented by a product measure

$\mu(\Omega)=\int_{\Lambda_{2}}\mu_{1}(\Omega_{\lambda_{2}})d\mu_{2}(\lambda_{2}),$
(24)

where $\mu_{1}$ and $\mu_{2}$ are probability measures on $\Lambda_{1}$ and
$\Lambda_{2}$ respectively, $\Omega\in\Sigma_{1}\otimes\Sigma_{2}$, and

$\Omega_{\lambda_{2}}=\left\\{\left[\lambda_{1}\right]\in\Lambda_{1}\middle|\left(\left[\lambda_{1}\right],\lambda_{2}\right)\in\Omega\right\\}.$
(25)

As in the Beltrametti–Bugajski model, $\left[\lambda_{1}\right]$ represents
the quantum state so $\mu_{1}=\delta_{\psi}$. The other variable $\lambda_{2}$
is uniformly distributed so $\mu_{2}$ is just the uniform measure on $[0,1]$.

The outcome of a measurement $M=\\{\left[\phi_{j}\right]\\}_{j=0}^{d-1}$ is
determined as follows. For each $\left[\lambda_{1}\right]$, the unit interval
is divided into $d$ subsets of length
$\text{Tr}\left(\left[\phi_{j}\right]\left[\lambda_{1}\right]\right)$. If
$\lambda_{2}$ is in the subset corresponding to $\left[\phi_{j}\right]$ then
the $\left[\phi_{j}\right]$ outcome occurs with certainty. It does not matter
how we choose the subsets so long as they are disjoint and of the right
length. One way of doing it is to pick the $j$th set to be the interval

$\sum_{k=0}^{j-1}\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right)\leq\lambda_{2}<\sum_{k=0}^{j}\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right),$
(26)

as illustrated in Fig. 7.

Figure 7: Example of the conditional probabilities in the Bell model for a
measurement in $\mathbb{C}^{4}$. The unit interval is divided into $4$
subintervals of length
$\text{Tr}\left(\left[\phi_{j}\right]\left[\lambda_{1}\right]\right)$. If
$\lambda_{2}$ is in the $j$th interval then the outcome will be $j$. In the
case depicted, the outcome is $\left[\phi_{2}\right]$.

This corresponds to the conditional probabilities

$\text{Pr}(\phi_{j}|M,\left[\lambda_{1}\right],\lambda_{2})=\begin{cases}1&\text{if}\,\,\begin{array}[]{c}\sum_{k=0}^{j-1}\limits\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right)\leq\lambda_{2};\\\
\lambda_{2}<\sum_{k=0}^{j}\limits\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right)\end{array}\\\
0&\text{otherwise}.\end{cases}$ (27)

Note that, in this model, measurement-outcome pairs represented by the same
projector generally correspond to different response functions. This is
because the way that the unit interval is divided up depends on which other
projectors are present in $\\{\left[\phi_{j}\right]\\}_{j=0}^{d-1}$. This is
an example of measurement contextuality.

It is straightforward to see that this model reproduces the quantum
predictions. If the state prepared is $\left[\psi\right]$ then the point
measure $\mu_{1}$ implies that $\left[\lambda_{1}\right]=\left[\psi\right]$.
Therefore, the length of the $j$th subset of the unit interval will be
$\text{Tr}\left(\left[\phi_{j}\right]\left[\psi\right]\right)$. Since
$\mu_{2}$ is the uniform measure on $[0,1]$, the probability of $\lambda_{2}$
being in a subset of $[0,1]$ is just the total length of the subset, so the
observed probability of obtaining outcome $j$ is
$\text{Tr}\left(\left[\phi_{j}\right]\left[\psi\right]\right)$, as required.

######  Example 4.6 (The Kochen–Specker model [76]).

Kochen and Specker introduced an ontological model for measurements in
orthonormal bases on a two-dimensional system. The PM fragment is
$\langle\mathbb{C}^{2},\mathcal{P},\mathcal{M}\rangle$ where $\mathcal{P}$
consists of all pure states on $\mathbb{C}^{2}$ and $\mathcal{M}$ consists of
all measurements of the form
$M=\\{\left[\phi\right],\left[\phi^{\perp}\right]\\}$ with
$\\{\left|\phi\right\rangle,\left|\phi^{\perp}\right\rangle\\}$ an orthonormal
basis.

A pure state $\left[\psi\right]$ in a two dimensional Hilbert space can be
represented as a point
$\vec{\psi}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\allowbreak\cos\vartheta)$
on the surface of a unit $2$-sphere $S_{2}$ by choosing a representative
vector $\left|\psi\right\rangle$ via

$\left|\psi\right\rangle=\cos(\frac{\vartheta}{2})\left|z+\right\rangle+e^{\imath\varphi}\sin(\frac{\vartheta}{2})\left|z-\right\rangle,$
(28)

where $0\leq\vartheta<\pi$ and $-\pi<\varphi\leq\pi$. This is known as the
Bloch sphere representation (see Fig. 8).

Figure 8: The Bloch sphere representation of a qubit. The quantum state is
represented as a point $\vec{\psi}$ on the surface of a unit 2-sphere.

In the Kochen–Specker model, the ontic state space is the unit sphere
$\Lambda=S_{2}$, which can be thought of as the Bloch sphere. The quantum
state $\left[\psi\right]$ corresponds to a measure $\mu$ that can be written
as

$\mu(\Omega)=\int_{\Omega}p(\vec{\lambda})\sin\vartheta d\vartheta d\varphi,$
(29)

where
$\vec{\lambda}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$
and the density $p$ is given by

$p(\vec{\lambda})=\frac{1}{\pi}H(\vec{\psi}\cdot\vec{\lambda})\vec{\psi}\cdot\vec{\lambda},$
(30)

where $H$ is the Heaviside step function

$H(x)=\begin{cases}1&\text{if}\,\,x>0\\\ 0&\text{if}\,\,x\leq 0.\end{cases}$
(31)

This density is nonzero on the hemisphere defined by all vectors that subtend
an angle of less than $\pi/2$ with $\vec{\psi}$ and it takes values
proportional to the cosine of the angle between $\vec{\lambda}$ and
$\vec{\psi}$.

For a measurement
$M=\left\\{\left[\phi\right],\left[\phi^{\perp}\right]\right\\}$, the outcome
$\left[\phi\right]$ is obtained if the angle between $\vec{\lambda}$ and
$\vec{\phi}$ is smaller than the angle between $\vec{\lambda}$ and
$\vec{\phi^{\perp}}$. Otherwise the outcome $\left[\phi^{\perp}\right]$ is
obtained. This corresponds to the conditional probabilities

$\displaystyle\text{Pr}(\phi|M,\vec{\lambda})$
$\displaystyle=H(\vec{\phi}\cdot\vec{\lambda})$ (32)
$\displaystyle\text{Pr}(\phi^{\perp}|M,\lambda)$
$\displaystyle=1-\text{Pr}(\phi|M,\vec{\lambda}).$ (33)

A proof that this reproduces the quantum predictions is given in Appendix B.

###  4.4 Defining $\psi$-ontic/epistemic models

We are now in a position to formally define what it means for the quantum
state to be real within an ontological model. The definition presented here is
the one used by Pusey–Barrett–Rudolph [27], which is a slightly more rigorous
version of a definition originally introduced by Harrigan and Spekkens [74].
It is uncontroversial that mixed states at least sometimes represent knowledge
about which of a set of pure states was prepared, so they are at least
partially epistemic. For this reason, $\psi$-ontology theorems are only
concerned with proving the reality of pure quantum states. A $\psi$-ontic
model is then one in which, if the pure state $\left[\psi\right]$ is prepared,
then $\left[\psi\right]$ is part of the ontic state of the system. In other
words, the ontic state space can be thought of as being composed of the set of
pure quantum states along with possibly some extra hidden variables. This will
be the case if the measures corresponding to distinct pure state preparations
do not overlap with one another (see Fig. 9). Conversely, the $\psi$-epistemic
explanations of quantum phenomena discussed in §2.1 depend crucially on having
overlap between the measures representing different quantum states, so whether
or not there is overlap is the key issue.

(a)  Ontic model with nonoverlapping quantum states.

Old label New label $1$ $(\psi_{1},1)$ $2$ $(\psi_{1},2)$ $3$ $(\psi_{2},1)$
$4$ $(\psi_{3},1)$ $5$ $(\psi_{3},2)$ $6$ $(\psi_{1},3)$ $7$ $(\psi_{3},3)$
$8$ $(\psi_{2},2)$ $9$ $(\psi_{3},4)$ $10$ $(\psi_{3},5)$

(b)  Relabeling the ontic states.

(c)  Equivalent model in which the quantum state is explicitly part of the
ontology.

Figure 9: In an ontological model, if the probability measures corresponding
to distinct quantum states do not overlap, then the ontic states can be
relabeled such that the quantum state is explicitly part of the ontology.
Panel LABEL:sub@fig:POEM:relabel:start depicts the measures corresponding to
three quantum states on a discrete ontic state space consisting of the
integers from $1$ to $10$. Red boxes indicate the ontic states on which the
measures have support. Panel LABEL:sub@fig:POEM:relabel:relabel shows a one-
to-one map to a new ontic state space in which the quantum states are an
explicit part of the ontology. The result is shown in Panel
LABEL:sub@fig:POEM:relabel:end.

The notion of when two probability measures overlap can be formalized using
the variational distance.

######  Definition 4.7.

The _variational distance_ between two probability measures $\mu$ and $\nu$ on
a measurable space $(\Lambda,\Sigma)$ is

$D(\mu,\nu)=\sup_{\Omega\in\Sigma}\left|\mu(\Omega)-\nu(\Omega)\right|.$ (34)

Note that taking the absolute value in Eq. (34) is optional, since if
$\nu(\Omega)>\mu(\Omega)$ for some $\Omega\in\Sigma$ then

$\mu(\Lambda\backslash\Omega)-\nu(\Lambda\backslash\Omega)=1-\mu(\Omega)-1+\nu(\Omega)=\nu(\Omega)-\mu(\Omega),$
(35)

so there is always another measurable set for which the difference of the
measures is the same, but $\mu$ is larger than $\nu$. Thus, the variational
distance can equivalently be defined as

$D(\mu,\nu)=\sup_{\Omega\in\Sigma}\left(\mu(\Omega)-\nu(\Omega)\right),$ (36)

or indeed

$D(\mu,\nu)=\sup_{\Omega\in\Sigma}\left(\nu(\Omega)-\mu(\Omega)\right).$ (37)

The variational distance is a metric on the set of probability measures and it
has the following operational interpretation. Suppose that a system is
prepared according to one of two preparation procedures, $P_{1}$ or $P_{2}$,
where $P_{1}$ corresponds to the measure $\mu$ and the $P_{2}$ to the measure
$\nu$, each case having an equal a priori probability. You are then told the
actual value of $\lambda$ and you wish to make the best possible guess as to
which of the two preparation procedures was used. Your best strategy will be
to choose a set $\Omega\subseteq\Lambda$ and guess $P_{1}$ if
$\lambda\in\Omega$ and $P_{2}$ if $\lambda\notin\Omega$. More generally, you
could use a probabilistic procedure, but convexity implies that this cannot
increase your probability of success. Your probability of success is then

$\displaystyle\text{Prob}(P_{1})\text{Prob}(\Omega|P_{1})+\text{Prob}(P_{2})\text{Prob}(\Lambda\backslash\Omega|P_{2})$
$\displaystyle=\frac{1}{2}\mu(\Omega)+\frac{1}{2}\nu(\Lambda\backslash\Omega)=\frac{1}{2}\left(1+\mu(\Omega)-\nu(\Omega)\right),$
(38)

so the maximum probability of success over all such strategies is
$\frac{1}{2}\left(1+D(\mu,\nu)\right)$. Success can occur with unit
probability only when $D(\mu,\nu)=1$, so in this case $\lambda$ effectively
determines which probability density was prepared uniquely.

More rigorously, $D(\mu,\nu)=1$ is equivalent to the existence of a measurable
set $\Omega\in\Sigma$ such that $\mu(\Omega)=1$ and $\nu(\Omega)=0$. An
optimal guessing strategy then consists of guessing $\mu$ if
$\lambda\in\Omega$ and $\nu$ if $\lambda\notin\Omega$.

If $\mu$ and $\nu$ are dominated by a third measure $m$, i.e.

$\displaystyle\mu(\Omega)$ $\displaystyle=\int_{\Omega}p(\lambda)dm(\lambda)$
(39) $\displaystyle\nu(\Omega)$
$\displaystyle=\int_{\Omega}q(\lambda)dm(\lambda),$ (40)

for some $m$-measurable densities $p$ and $q$, then

$D(\mu,\nu)=\frac{1}{2}\int_{\Lambda}\left|p(\lambda)-q(\lambda)\right|dm(\lambda),$
(41)

which is often the most convenient form for computation.

For those unfamiliar with measure-theoretic probability, a measure $m$
dominates a measure $\mu$ if, whenever $m(\Omega)=0$ then $\mu(\Omega)=0$. If
this holds then $\mu$ can be written as a density with respect to $m$. The
measure $m=\frac{1}{2}\left(\mu+\nu\right)$ dominates both $\mu$ and $\nu$,
and a similarly for any finite set of probability measures. For an uncountable
set of measures there need not exist a measure that dominates all of them. For
example, there is no measure that dominates all of the point measures
$\delta_{\psi}$ occurring in the Beltrametti–Bugajski model, which is one of
the reasons why we work with probability measures instead of probability
densities in the present treatment.

Based on the variational distance, we can define what it means for two quantum
states to have no overlap in an ontic model.

######  Definition 4.8.

A pair of quantum states $\rho,\sigma\in\mathcal{P}$ are _ontologically
distinct_ in an ontological model if, for all
$\mu\in\Delta_{\rho},\nu\in\Delta_{\sigma}$,

$D(\mu,\nu)=1,$ (42)

otherwise they are _ontologically indistinct_.

The idea here is that, if $\rho$ and $\sigma$ are ontologically distinct then,
regardless of how they are prepared, they can be perfectly distinguished given
knowledge of the exact ontic state. In order to take preparation contextuality
into account, all possible pairs of preparations that are represented by
$\rho$ and $\sigma$ must correspond to measures that have zero overlap.

If a set of preparations is operationally distinguishable, i.e. there exists a
measurement that perfectly distinguishes them, then they must also be
ontologically distinct. The intuition behind this is straightforward. Consider
the case of a finite ontic state space. If the measures corresponding to two
operationally distinguishable preparations did have nontrivial overlap then
there would be a finite probability of an ontic state occurring that is
assigned a finite probability according to both of them, and if this happens
then the ontic state cannot be used to deduce which preparation was performed
with certainty. Hence, such a model could not reproduce the statistics of the
distinguishing measurement, which does allow this deduction. More formally,

######  Definition 4.9.

A finite set of quantum states $\left\\{\rho_{j}\right\\}\subseteq\mathcal{P}$
is _operationally distinguishable_ if there exists a POVM
$\\{E_{j}\\}\in\mathcal{M}$ such that

$\text{Tr}\left(E_{j}\rho_{k}\right)=\delta_{jk}.$ (43)

######  Theorem 4.10.

Let $\mathcal{D}\subseteq\mathcal{P}$ be an operationally distinguishable set
of states. Then, every pair $\rho,\sigma\in\mathcal{D}$, $\rho\neq\sigma$, is
ontologically distinct in any ontological model that reproduces the
operational predictions.

###### Proof.

Let $M\in\mathcal{M}$ be a measurement that distinguishes the states in
$\mathcal{D}$. Then, for $\rho,\sigma\in\mathcal{D}$, $\rho\neq\sigma$, there
exists a POVM element $E\in M$ such that

$\text{Tr}\left(E\rho\right)=1\qquad\text{and}\qquad\text{Tr}\left(E\sigma\right)=0.$
(44)

In an ontological model that reproduces the quantum predictions, this implies
that

$\displaystyle\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda)$
$\displaystyle=1$ (45)
$\displaystyle\int_{\Lambda}\text{Pr}(E|M,\lambda)d\nu(\lambda)$
$\displaystyle=0,$ (46)

for all $\mu\in\Delta_{\rho}$, $\nu\in\Delta_{\sigma}$, and for any
$\text{Pr}\in\Xi_{M}$.

Assume that $\rho$ and $\sigma$ are ontologically indistinct. Then, there
exist $\mu\in\Delta_{\rho}$, $\nu\in\Delta_{\sigma}$ such that $D(\mu,\nu)<1$.
This is equivalent to saying that all sets that are of measure $1$ according
to $\mu$ are of nonzero measure according to to $\nu$. In order to satisfy Eq.
(45), $\text{Pr}(E|M,\lambda)$ must be equal to $1$ on a set $\Omega$ that is
of measure one according to $\mu$, and thus by ontological indistinctness
$\nu(\Omega)>0$. However, from Eq. (46) we have

$\displaystyle 0$
$\displaystyle=\int_{\Lambda}\text{Pr}(E|M,\lambda)d\nu(\lambda)$ (47)
$\displaystyle\geq\int_{\Omega}\text{Pr}(E|M,\lambda)d\nu(\lambda)$ (48)
$\displaystyle=\int_{\Omega}d\nu(\lambda)=\nu(\Omega)>0,$ (49)

which is a contradiction. ∎

Although operationally distinguishable quantum states are always ontologically
distinct, this argument does not imply that arbitrary pairs of states are. If
all pairs of pure states are nonetheless ontologically distinct then the
ontological model is $\psi$-ontic.

######  Definition 4.11.

An ontological model is $\psi$_-ontic_ if all pairs of pure quantum states
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$,
$\left[\psi\right]\neq\left[\phi\right]$, are ontologically distinct.
Otherwise the model is $\psi$_-epistemic_.

This definition captures the idea that quantum states are real in an
ontological model if the probability measures corresponding to distinct pure
states do not overlap. Defining a $\psi$-epistemic model to be the negation of
this is extremely permissive. For example, a model in which only a single pair
of pure states have any overlap, all other pairs being ontologically distinct,
would be $\psi$-epistemic according to this definition. The $\psi$-epistemic
explanations of quantum phenomena discussed in §2.1 would not apply to such a
model. The $\psi$-ontology theorems discussed in Part II aim to rule out even
this permissive notion of a $\psi$-epistemic model, so they obviously rule out
any less permissive definition as well. Nevertheless, because $\psi$-ontology
theorems require questionable auxiliary assumptions, it is still interesting
to consider stronger notions of what it means for the quantum state to be
epistemic. This is done in in §5.2 and Part III.

Note that a $\psi$-ontic model is not necessarily _$\psi$ -complete_, where
the latter means that the wavefunction, and only the wavefunction, is the
ontic state. $\psi$-complete models are obviously $\psi$-ontic, but in general
$\psi$-ontic models may involve other variables in addition to the
wavefunction. A lot of confusion may be avoided by clearly separating the
notions of $\psi$-ontic and $\psi$-complete models, particularly since much of
the literature on hidden variable theories focuses on the question of whether
quantum theory is complete, i.e. it is assumed that the wavefunction is real a
priori and the only question is whether anything else needs to be added to it.
Obviously, such a framework is not suited to discussing the question of
whether the wavefunction must be real in the first place.

Of the examples considered in §4.3, Spekkens’ toy theory and the
Kochen–Specker model are $\psi$-epistemic, and the Beltrametti–Bugajski and
Bell models are $\psi$-ontic. In Spekkens’ theory, the ontic state space is
finite, so the integral in Eq. (41) can be performed with respect to the
counting measure, which yields the sum

$D(\mu,\nu)=\frac{1}{2}\sum_{\lambda}|\mu(\lambda)-\nu(\lambda)|.$ (50)

For toy bit, $\lambda$ takes the values $(+,+),(+,-),(-,+)$ and $(-,-)$. If
$\mu$ is the $\left|x+\right)$ measure, corresponding to the quantum state
$\left[x+\right]$, and $\nu$ is $\left|y+\right)$, corresponding to
$\left[y+\right]$ then we have

$D(\mu,\nu)=\frac{1}{2}\left(\left|\frac{1}{2}-\frac{1}{2}\right|+\left|\frac{1}{2}-0\right|+\left|0-\frac{1}{2}\right|+\left|0-0\right|\right)=\frac{1}{2}.$
(51)

Thus, the model is $\psi$-epistemic because $D(\mu,\nu)<1$, as we expect for
overlapping measures. Appendix B provides a proof that the Kochen–Specker
model is $\psi$-epistemic by showing that it satisfies the stronger notion of
being maximally $\psi$-epistemic to be discussed in §5.2.

The Beltrametti–Bugajski model is in fact $\psi$-complete, because its ontic
state space consists of the set of pure states and preparing a given pure
state causes the corresponding ontic state to be occupied with certainty. It
follows that the model is $\psi$-ontic because each pure state corresponds to
a distinct point measure. Thus, for $\left[\psi\right]\neq\left[\phi\right]$
the corresponding measures are $\delta_{\psi}$ and $\delta_{\phi}$. The set
$\left\\{\left[\psi\right]\right\\}$ is measure one according to
$\delta_{\psi}$ and measure zero according to $\delta_{\phi}$. Similarly, the
Bell model is $\psi$-ontic, where now the set
$\left\\{\left[\psi\right]\right\\}\times[0,1]$ is measure one for the measure
corresponding to $\left[\psi\right]$ and measure zero for the measure
corresponding to $\left[\phi\right]$. It is, however, not $\psi$-complete
because of the additional component $[0,1]$ of the ontic state space.

##  5 Implications of $\psi$-ontology

Before discussing $\psi$-ontology theorems, it is worth pausing to consider
some of their implications. One of the most interesting things about
establishing the reality of the quantum state within the ontological models
framework is that it would imply a lot of existing no-go results as simple
consequences. Thus, even if you are a neo-Copenhagenist who rejects the
ontological models framework outright, you should still be interested in
$\psi$-ontology theorems as potentially the most powerful class of no-go
results that we currently have. Additionally, the ontological models framework
can be thought of as an attempt to simulate quantum theory using classical
resources. Just as Bell’s Theorem has implications for the difference between
quantum and classical communication complexity, and for device independent
quantum cryptography [102], $\psi$-ontology theorems might become a useful
tool in quantum information theory.

The main implications of $\psi$-ontology are illustrated in Fig. 10, and each
of them is discussed in this section. There are two main strands of
implications, one based on the size of the ontic state space and one based on
results related to contextuality. The results of the contextuality strand can
alternatively be derived as consequences of the Kochen–Specker Theorem. Since
$\psi$-ontology is our main concern here, the connection to the Kochen–Specker
theorem is discussed in Appendix C.

Figure 10: Hierarchy of constraints on ontological models of quantum theory.
Properties that appear higher in the diagram imply those that they are
connected to lower down. Non maximal $\psi$-epistemicity can be derived from
either $\psi$-ontology or Kochen–Specker contextuality.

###  5.1 Excess baggage

The idea of excess baggage has already been discussed in §2.5 as the tension
between the infinite amount of information required to specify the quantum
state of a qubit and the fact that it can only be used to reliably transmit a
single bit. Similarly, the number of parameters required to specify a quantum
state scales exponentially with the number of systems, but the amount of
information that can be transmitted scales only linearly.

Excess baggage theorems show that this tension cannot be avoided in an
ontological model by providing lower bounds on the size of the ontic state
space required to reproduce quantum theory. The first such result, from which
the “excess baggage” terminology originates, was due to Hardy, who proved that
an infinite number of ontic states are required to reproduce the predictions
of any quantum system, even just a qubit [61]. Subsequently, Montina showed
that the number of parameters required to specify an ontic state must scale
exponentially with the number of systems [62, 63].

If it can be proved that an ontological model of quantum theory must be
$\psi$-ontic then it follows immediately that the ontic state space must be
uncountably infinite and that the number of parameters required to specify an
ontic state must scale exponentially with the number of systems. This is
because, in a $\psi$-ontic model, there must be at least as many ontic states
as there are quantum states.

###  5.2 Maximally $\psi$-epistemic models

The concept of a maximally $\psi$-epistemic model was introduced by Maroney as
a stronger notion of what it means for an ontological model to be
$\psi$-epistemic [58]. An equivalent concept was introduced earlier by
Harrigan and Rudolph [98], under the terminology “non deficient model”. As you
might imagine from the name, a $\psi$-ontic model cannot be maximally
$\psi$-epistemic, but, even without $\psi$-ontology, maximally
$\psi$-epistemic models are ruled out by the Kochen–Specker theorem, as
discussed in Appendix C.

The real interest in the concept of a maximally $\psi$-epistemic model is that
it enables one to devise measures of the extent to which a model is
$\psi$-epistemic, and hence to go beyond the sharp dichotomy between
$\psi$-epistemic and $\psi$-ontic implied by the definitions adopted so far
[58, 54, 57, 55]. This aspect is discussed in §12.2, but the concept of a
maximally $\psi$-epistemic model is also useful as a stepping stone between
$\psi$-ontology and the next step in the contextuality strand, which is
preparation contextuality.

The basic idea is that, in order to justify the $\psi$-epistemic explanation
of indistinguishability, we need more than just that the probability measures
corresponding to two different quantum states, $\left[\psi\right]$ and
$\left[\phi\right]$, should have nonzero overlap. Ideally, when measuring a
system prepared in the state $\left[\psi\right]$, all of the probability of
obtaining the outcome corresponding to the projector $\left[\phi\right]$
should be accounted for by the overlap region between the probability measures
corresponding to the two states. Since we are comparing the overlap of
$\left[\psi\right]$ and $\left[\phi\right]$, considered as states, with the
probability of obtaining $\left[\phi\right]$ as a measurement outcome, this
only makes sense if we are dealing with a PM fragment for which, for every
pure state $\left[\phi\right]\in\mathcal{P}$, there exists a measurement
$M\in\mathcal{M}$ such that $\left[\phi\right]\in\mathcal{M}$. Therefore, this
is a standing assumption for the remainder of this section.

######  Definition 5.1.

An ontological model of a PM fragment is _maximally $\psi$-epistemic_ if, for
every pair of pure states $\left[\psi\right],\left[\phi\right]\in\mathcal{P}$,
for all $\mu\in\Delta_{\psi}$, $\nu\in\Delta_{\phi}$, $M\in\mathcal{M}$ with
$\left[\phi\right]\in M$, and $\text{Pr}\in\Xi_{M}$,

$\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)=\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda),$
(52)

for all sets $\Omega\in\Sigma$ such that $\nu(\Omega)=1$.

To unpack this definition a little, let $\mu\in\Delta_{\psi}$, let
$\nu\in\Delta_{\phi}$, let $M$ be a measurement that contains
$\left[\phi\right]$, and let $\text{Pr}\in\Xi_{M}$. Suppose $\Omega$ is a set
of measure one according to $\nu$ and $\Omega^{\prime}$ is a set of measure
one according to $\mu$. Then, Eq. (52) is equivalent to

$\int_{\Omega\cap\Omega^{\prime}}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)=\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda).$
(53)

Whatever one might mean by the overlap region between $\mu$ and $\nu$, it
should be the intersection of a set that is measure one according to $\mu$
with a set that is measure one according to $\nu$, and since Eq. (53) must
hold for all such sets, it guarantees that the probability of obtaining
outcome $\left[\Phi\right]$ when measuring a system prepared in the state
$\left[\psi\right]$ is entirely accounted for by any such region.

Note that, instead of Eq. (52), previous works [98, 58, 59, 56] imposed the
requirement that

$\int_{\Lambda_{q}}\text{Pr}(\phi|M,\lambda)p(\lambda)d\lambda=\int_{\Lambda}\text{Pr}(\phi|M,\lambda)p(\lambda)d\lambda,$
(54)

where $d\lambda$ is a measure that dominates $\mu$ and $\nu$, $p$ and $q$ are
densities that represent them, and
$\Lambda_{q}=\\{\lambda\in\Lambda|q(\lambda)>0\\}$ is the support of the
probability density $q$. However, the probability measure $\nu$ does not
correspond to a unique density $q$, since densities that differ on a set of
measure zero according to $d\lambda$ represent the same probability measure.
This means that $\Lambda_{q}$ is not uniquely specified by $\nu$, since
setting the values of $q$ to zero on a measure zero set would change
$\Lambda_{q}$ but not $\nu$. Therefore, to use this definition, one has to
imagine that the ontological model specifies a particular density
representation for each measure, rather than just the measure itself. On its
own, this older definition is also not strong enough to entail that a
$\psi$-ontic model cannot be maximally $\psi$-epistemic. To remedy this, one
can adopt a standing assumption that all probability measures appearing in the
model are dominated by, i.e. absolutely continuous with respect to, a
canonical measure $d\lambda$, as was done explicitly in [56], and this must be
regarded as an implicit assumption in previous works [98, 58, 59]. However, we
do not really want to make this assumption, because some models of interest do
not satisfy it. For example, an uncountable set of point measures on distinct
points is not dominated by any measure, and these occur in the
Beltrametti–Bugajski and Bell models.

It is straightforward to verify that Spekkens’ toy theory is maximally
$\psi$-epistemic and a proof that the Kochen–Specker model is maximally
$\psi$-epistemic is given in Appendix B. On the other hand, a $\psi$-ontic
model that reproduces the quantum predictions cannot be maximally
$\psi$-epistemic, so the Beltrametti–Bugajski and Bell models are not
maximally $\psi$-epistemic.

######  Theorem 5.2.

Consider a PM fragment that contains at least one pair of nonorthogonal pure
states. If an ontological model of this fragment that reproduces the quantum
predictions is maximally $\psi$-epistemic then it is $\psi$-epistemic.

###### Proof.

Assume the ontological model is $\psi$-ontic. Then, for every pair of
nonorthogonal pure states $\left[\psi\right],\left[\phi\right]\in\mathcal{P}$,
every pair $\mu\in\Delta_{\psi}$, $\nu\in\Delta_{\phi}$ satisfies
$D(\mu,\nu)=1$, which is equivalent to saying that there exists a set $\Omega$
that is measure $0$ according to $\mu$ and measure $1$ according to $\nu$.
However, if the ontological model is maximally $\psi$-epistemic then, for all
$\mu\in\Delta_{\psi}$, $\nu\in\Delta_{\phi}$, $M\in\mathcal{M}$ containing
$\left[\phi\right]$, and $\text{Pr}\in\Xi_{M}$,

$\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)=\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda).$
(55)

This implies that,

$\displaystyle\mu(\Omega)$ $\displaystyle=\int_{\Omega}d\mu(\lambda)$ (56)
$\displaystyle\geq\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)$ (57)
$\displaystyle=\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)$ (58)
$\displaystyle=\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)>0,$
(59)

where the second line follows from the fact that
$\text{Pr}(\phi|M,\lambda)\in[0,1]$ and the fourth from the fact that the
ontological model reproduces the quantum predictions. Thus, $\Omega$ cannot be
of measure zero according to $\mu$, which contradicts the assertion that the
model is $\psi$-ontic. ∎

###  5.3 Preparation contextuality

In [67], Spekkens introduced an operational approach to contextuality. In
addition to generalizing the usual Kochen–Specker approach to contextuality of
measurements, as discussed in Appendix C, Spekkens introduced a notion of
contextuality for preparations. In Spekkens’ approach, the fundamental idea of
noncontextuality is that things that are operationally equivalent, i.e. always
lead to the same observable probabilities, should be represented in the same