stochastic map representing the unitary. ABCL then show that, for the fragment
consisting of all pure states, all measurements in orthonormal bases, and all
unitaries on $\mathbb{C}^{d}$, if the ontic state space is either the
projective Hilbert space of $\mathbb{C}^{d}$ or the set of unitary operators
on $\mathbb{C}^{d}$ and the dynamics is represented by the usual action of the
unitary group on these spaces, then no ABCL symmetric and pairwise
$\psi$-epistemic model exists.

Imposing additional symmetry requirements of the type suggested by ABCL is a
promising direction for $\psi$-ontology theorems, but their setup is so
restricted that one cannot really draw any firm conclusions about ontological
models in general from it. The proof itself is quite technical and not
obviously generalizable to other ontic state spaces, so the details are
omitted here.

##  12 Continuity

The idea of continuity is that if two pure quantum states are close to one
another, in the sense of having large inner product, then there should be
probability measures representing them that have large overlap. Imposing
continuity is not a bad idea, since presumably models in which a small change
in experimental conditions leads to a large change at the ontological level
would be quite contrived. The most basic notion of continuity runs as follows.

######  Definition 12.1.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{P}$ consists of pure states and let
$\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model of it. $\Theta$
is _continuous_ if, for all $\epsilon>0$, there exists a $\delta>0$ such that
if $\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)>1-\delta$ for
some pure states $\left[\psi\right],\left[\phi\right]\in\mathcal{P}$, then
there exists $\mu\in\Delta_{\psi}$ and $\nu\in\Delta_{\phi}$ such that
$L(\mu,\nu)>1-\epsilon$.

This notion of continuity is quite permissive because it imposes no
constraints on how $\delta$ should be related to $\epsilon$. It is easy to see
that any pairwise $\psi$-epistemic model is continuous in this sense, since
all it requires is that the overlap of measures corresponding to nonorthogonal
states should be bounded away from zero. Since ABCL have shown that pairwise
$\psi$-epistemic models exist, this notion of continuity cannot be used to
derive a no-go theorem.

Therefore, it is interesting to look at stronger notions of continuity that
might be reasonable for a $\psi$-epistemic model. In the remainder of this
section, two such notions are discussed. §12.1 discusses a notion of
continuity due to Patra, Pironio and Massar [60] and discusses a theorem that
they proved to rule such models out. I argue that their definition of
continuity is unreasonable for a $\psi$-epistemic model. §12.2 discusses a
different notion based on Lipschitz continuity, which has been used in a
number of recent works to derive measures of the extent to which a model is
$\psi$-epistemic. The best known bound shows that the ratio of the overlap to
the inner product must be zero for Hilbert spaces of dimension $\geq 4$.

###  12.1 The Patra–Pironio–Massar Theorem

PPM’s continuity assumption runs as follows.

######  Definition 12.2.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment, where $\mathcal{P}$ consists of pure states, let $\delta>0$ and, for
$\left[\psi\right]\in\mathcal{P}$, let
$B^{\delta}_{\psi}=\left\\{\left[\phi\right]\in\mathcal{P}\middle|\sqrt{\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)}\geq
1-\delta\right\\}$ be the ball of radius $\delta$ centered at
$\left[\psi\right]$. An ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$
of $\mathfrak{F}$ is _PPM- $\delta$-continuous_ if, for every pure state
$\left[\psi\right]\in\mathcal{P}$ and every set
$\\{\left[\phi_{j}\right]\\}\subseteq\mathcal{P}$ where
$\left[\phi_{j}\right]\in B^{\delta}_{\psi}$, there exist probability measures
$\mu_{j}\in\Delta_{\phi_{j}}$ such that $L(\\{\mu_{j}\\})>0$.

For the case of a finite ontic state space, the PPM-$\delta$-continuity
assumption says that, for every set of quantum states that are sufficiently
close some common state $\left[\psi\right]$, there should be at least one
ontic state to which they all assign nonzero probability. This is stronger
than the basic notion of continuity given in Definition 12.1, which only
constrains pairwise overlaps.

Based on their assumption, PPM prove the following theorem.

######  Theorem 12.3.

Let $\mathfrak{F}=\langle\mathbb{C}^{d},\mathcal{P},\mathcal{M}\rangle$ be a
PM fragment with $\left\\{\left[k\right]\right\\}_{k=0}^{d-1}\in\mathcal{M}$
for some orthonormal basis
$\left\\{\left|k\right\rangle\right\\}_{k=0}^{d-1}$. If
$\left[\psi\right]\in\mathcal{P}$ and
$\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}\subseteq\mathcal{P}$,
where

$\displaystyle\left|\psi\right\rangle$
$\displaystyle=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\left|k\right\rangle$ (276)
$\displaystyle\left|\phi_{j}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{d-1}}\sum_{k\neq j}\left|k\right\rangle,$ (277)

then there is no PPM-$\delta$-continuous ontological model of $\mathfrak{F}$
that reproduces the quantum preclusions for $\delta\geq 1-\sqrt{(d-1)/d}$.

As with the Hardy and Colbeck–Renner Theorems, the constraint on $\delta$ can
be removed by assuming that adding ancillas preserves ontological
distinctness.

###### Proof of Theorem 12.3.

Let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model of
$\mathfrak{F}$ and let $\mu_{j}\in\Delta_{\phi_{j}}$. The states
$\left[\phi_{j}\right]$ all lie within $B^{\delta}_{\psi}$ for
$\delta=1-\sqrt{(d-1)/d}$. However, the states $\left[\phi_{j}\right]$ are
antidistinguished by the measurement
$\left\\{\left[k\right]\right\\}_{k=0}^{d-1}$ and hence, by Theorem 6.5,
$L\left(\left\\{\mu_{j}\right\\}\right)=0$. ∎

This proof is remarkably simple, but it does not mean much as
PPM-$\delta$-continuity is unreasonably strong. In order to see the problem,
it is helpful to consider a simple example. Therefore, consider how the
fragment $\mathfrak{F}=\langle\mathbb{C}^{d},\mathcal{P},\mathcal{M}\rangle$
might be modeled, where $\mathcal{P}$ just contains the states
$\left[\psi\right]$, $\left[\phi_{j}\right]$ and $\mathcal{M}$ only contains
the measurement $M=\\{\left[k\right]\\}_{k=1}^{d}$. A natural way of doing
this is to use the ontic state space $\Lambda=\\{0,1,\ldots,d-1\\}$, model
$\left[\psi\right]$ by the uniform distribution, and model
$\left[\phi_{j}\right]$ by the distribution that is zero on $j$ and uniformly
distributed over the rest of the ontic state space. Then, it is
straightforward to see that setting the conditional probabilities
$\text{Pr}\left(\left[k\right]\middle|M,j\right)=\delta_{jk}$ yields a model
that reproduces the quantum predictions. Furthermore, this model is pairwise
$\psi$-epistemic, and hence satisfies the more basic notion of continuity,
because $D(\mu_{j},\mu_{k})=1/(d-1)$ for $j\neq k$.

From this, it is easy to see that the problem with PPM-$\delta$-continuity
comes from ontological states that are assigned a small weight in the
distribution. If a distribution corresponding to some quantum state assigns a
small weight to all of the ontic states in its support then one way of making
a small change to that distribution is to set the weight assigned to one of
the ontic states to zero and redistribute it over the other ontic states. If
those distributions all happen to represent quantum states that are close to
the original one, and there is no reason why they should not, then these
quantum states will have no ontic state that is common to all of them, and
hence PPM-$\delta$-continuity will be violated.

In fact, the same criticism applies to classical models. Suppose we have a
system consisting of a ball that can be in one of $d$ boxes, labeled
$0,1,\ldots,d-1$. The system is prepared in one of the following ways. Either
a box is chosen uniformly at random and the ball is placed in that box, or
first an integer $j$ from $0$ to $d-1$ is specified and box $j$ is removed,
then the ball is placed uniformly at random in one of the remaining boxes,
before finally returning box $j$ to its place. Suppose you are given a table
which specifies the probabilities of finding the ball in each of the boxes for
each of these $d+1$ preparation procedures. You are, however, not told
anything about where these probabilities come from, i.e. you are not told
about the ball and boxes but rather just given a table that tells you that if
preparation $j$ is made then the probability of getting outcome $k$ is $0$ if
$k=j$ and $1/(d-1)$ otherwise. Your task is to try and come up with a model of
what is going on in reality in order to generate these statistics; a task
perfectly analogous to the project of constructing an ontological model for
some fragment of quantum theory. If you imposed upon yourself the analog of
PPM-$\delta$-continuity, that sets of preparations having close operational
predictions must have an ontic state in common, you would never be able to
come up with the ball and box model that is actually generating the
probabilities and, in fact, no model satisfying this condition exists. That,
in a nutshell, is what is wrong with PPM-$\delta$-continuity. A condition that
rules out a classical probabilistic description of a classical probabilistic
model is clearly way to strong to prove anything meaningful about quantum
theory.

###  12.2 Lipschitz continuity

In order for the $\psi$-epistemic explanation of the indistinguishability of
quantum states to be viable, quantum states that have large inner product
should correspond to probability measures with large overlap, but quantum
states that are almost orthogonal need have very little overlap at the ontic
level. In between these extremes, an amount of overlap that scales with the
inner product in some way is needed. This motivates the definition of
Lipschitz continuity.

######  Definition 12.4.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{P}$ consists of pure states and where, for every
$\left[\psi\right]\in\mathcal{P}$, there exists an $M\in\mathcal{M}$ such that
$\left[\psi\right]\in M$. Let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an
ontological model of $\mathfrak{F}$. $\Theta$ is Lipschitz continuous if there
exists a $K>0$ such that, for all
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$,

$L(\mu,\nu)\geq K\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right),$
(278)

for some $\mu\in\Delta_{\psi}$, $\nu\in\Delta_{\phi}$.

The first bounds on $K$ were obtained by Maroney [58], followed by Barrett et.
al. [54] and myself [57]. However, Branciard [55] has recently proved a
theorem to the effect that, for Hilbert spaces of $d\geq 4$, $K$ must be zero,
or, in other words, Lipschitz continuous models are not possible. This could
be regarded as a $\psi$-ontology theorem in its own right, if we regard
Lipschitz continuity as a stronger definition of what it means for a model to
be $\psi$-epistemic.

However, Lipschitz continuity is quite a strong requirement. It says that
there should be a fixed bound on the ratio of $L(\mu,\nu)$ to
$\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$, that is
independent of the states $\left[\psi\right]$ and $\left[\phi\right]$ under
consideration. Is this really required to maintain the viability of
$\psi$-epistemic explanations of quantum phenomena, such as the
indistinguishability of quantum states?

In any ontological model, regardless of whether it is $\psi$-epistemic or
$\psi$-ontic, there are two mechanisms that can account for the
indistinguishability of a pair of quantum states. The first is overlap of the
probability measures corresponding to states, which I have referred to as the
$\psi$-epistemic explanation of indistinguishability. Even if you are told the
exact ontic state occupied by the system, your probability of correctly
guessing whether $\left[\psi\right]$ or $\left[\phi\right]$ was prepared is
bounded by $\frac{1}{2}\left(2-L(\mu,\nu)\right)$, where $\mu\in\Delta_{\psi}$
and $\nu\in\Delta_{\phi}$. Hence, if $L(\mu,\nu)>0$, the states are
necessarily indistinguishable. However, there is also a second mechanism. We
do not usually have complete knowledge about the ontic state, but are instead
only able to infer information about it via the outcomes of quantum
measurements. The conditional probability distributions
$\text{Pr}(E|M,\lambda)$ corresponding to measurements typically do not reveal
the value of $\lambda$ exactly, but only give coarse-grained information about
it. This will render quantum states less distinguishable than they would be if
we knew the exact ontic state, so this mechanism can also explain part of
their indistinguishability. In fact, in a $\psi$-ontic model, the
indistinguishability of quantum states must be explained entirely by this
second mechanism as there is no overlap of probability measures. In general,
there will be a tradeoff between the two effects, with some portion of the
indistinguishability of quantum states being explained by overlap of
probability measures, and some being explained by the coarse-grained nature of
measurements. Clearly, some portion of the indistinguishability must be
explained by overlap in a $\psi$-epistemic theory, as the theory would be
$\psi$-ontic if there were no overlap, but how much of it ought to be
explained in this way?

It is easy to see that measurements must only reveal coarse-grained
information about $\lambda$ in certain types of ontological model, regardless
of whether they are $\psi$-epistemic or $\psi$-ontic. For example, in an
outcome deterministic model, such as the Kochen–Specker model, in which the
probabilities $\text{Pr}(\phi|M,\lambda)$ for measurements in orthonormal
bases are all either $0$ or $1$, exact knowledge of $\lambda$ would entail
exact knowledge of the outcomes of all measurements, which would violate the
uncertainty principle and hence cannot be compatible with reproducing the
quantum predictions. Further, recall that Spekkens’ toy theory was derived
from the knowledge-balance principle, which states that at most half of the
information needed to specify the ontic state can be known at any given time.
This also entails that measurements can only reveal coarse-grained information
about the ontic state. Therefore, even in an archetypal $\psi$-epistemic
theory, which is derived entirely from epistemic principles, measurements only
reveal coarse-grained information about $\lambda$. For these reasons, it is to
be expected that both overlap of probability measures and the coarse-grained
nature of measurements will be present in a viable $\psi$-epistemic theory.

Because of this, it is not obvious that there is some definite value of the
ratio $L(\mu,\nu)/\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$
below which $\psi$-epistemic models ought to be discarded as implausible.
Further, it is also not obvious that this ratio should have a fixed bound,
independent of the choice of $\left[\psi\right]$ and $\left[\phi\right]$. It
is conceivable that the amount of indistinguishability explained by overlap
and the amount explained by the coarse-grained nature of measurements could
vary for different pairs of states, e.g. it might vary with
$\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$. Therefore, it is
interesting to see what can be inferred from the results mentioned above [58,
54, 57, 55] without assuming Lipschitz continuity.

All of these papers consider some family of states
$\mathcal{S}_{n}=\\{\left[\psi_{j}\right]\\}_{j=1}^{n}$, i.e. there are sets
of states with increasing values of $n$. Without Lipschitz continuity, the
results provide an upper bound on the ratio

$R_{n}=\min_{\left[\psi\right],\left[\phi\right]\in\mathcal{S}_{n}}\frac{L(\mu_{\psi},\nu_{\phi})}{\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)},$
(279)

for any choice of $\mu_{\psi}\in\Delta_{\psi}$, $\nu_{\phi}\in\Delta_{\phi}$.
In other words, we can infer that there exists at least one pair of states in
the set for which the ratio is at least this bad, but we can no longer infer
that this holds for all states in the Hilbert space as we could under
Lipschitz continuity. Given that it is difficult to say just how much of the
indistinguishability of quantum states needs to be explained by overlap of
probability measures in a viable $\psi$-epistemic theory, this type of bound
is only really interesting if it shows that
$L(\mu_{\psi},\nu_{\phi})/\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$
must be close to zero for a wide range of different pairs of states. Since the
state families used in these constructions can be unitarily transformed
without changing the result, this means that
$L(\mu_{\psi},\nu_{\phi})/\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$
should be close to zero for a wide range of values of
$\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$.

The best bound so far obtained comes from Branciard’s work [55], in which he
shows that there exists a family of states in any Hilbert space of dimension
$\geq 4$, such that $R_{n}\rightarrow 0$ as $n\rightarrow\infty$. However, all
of the existing results share the feature that
$\min_{\left[\psi\right],\left[\phi\right]\in\mathcal{S}_{n}}\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)\rightarrow
0$ as $n\rightarrow\infty$ as well, so all we can really say without assuming
Lipschitz continuity is that the ratio is close to zero for pairs of states
that are almost distinguishable. It is difficult to tell the difference
between states that are perfectly distinguishable and states that are almost
distinguishable in a practical experiment, so it is arguable whether it
matters that the $\psi$-epistemic explanation of indistinguishability plays
almost no role for such states. It is still an open possibility that the ratio
might be much higher for states with a larger inner product, so what is really
needed is to find families of states for which $R_{n}\rightarrow 0$ as
$n\rightarrow\infty$, but
$\min_{\left[\psi\right],\left[\phi\right]\in\mathcal{S}_{n}}\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)$
takes on a wide range of values.

##  13 Never $\psi$-ontic models

To conclude, I would like to discuss one further strengthening of the notion
of a $\psi$-epistemic model, first introduced informally by Montina [134],
about which very little is currently known. This is the notion of a never
$\psi$-ontic model.

######  Definition 13.1.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment and let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model
of it. $\Theta$ is _sometimes $\psi$-ontic_ if, for all pure states
$\left[\psi\right]\in\mathcal{P}$, there exists a $\Omega\in\Sigma$ such that
$\mu(\Omega)>0$ for some $\mu\in\Delta_{\psi}$, but for all other pure states
$\left[\phi\right]\in\mathcal{P}$, $\left[\phi\right]\neq\left[\psi\right]$,
every $\nu\in\Delta_{\phi}$ has $\nu(\Omega)=0$. Otherwise the model is _never
$\psi$-ontic_.

Roughly speaking, in a sometimes $\psi$-ontic model, each pure state has a
special region of the ontic state space all to itself. If you know that the
ontic state occupies this region then you can identify the quantum state that
was prepared with probability one. However, the ontic state need not always
occupy such a region; hence the terminology “sometimes” $\psi$-ontic. In
contrast, in a never $\psi$-ontic model, every region of the ontic state space
is shared nontrivially by more than one quantum state. Considerations about
the degree of overlap of pairs of states do not really bear on the question of
whether a never $\psi$-ontic model is possible, since arbitrary overlaps may
occur outside the special regions in a sometimes $\psi$-ontic model.

The reason why the notion of a sometimes $\psi$-ontic model is interesting is
that all of the implications of $\psi$-ontology discussed in §5 can be derived
from sometimes $\psi$-ontology instead. Whilst the fact that a maximally
$\psi$-epistemic model is impossible is enough to derive preparation
contextuality and Bell’s Theorem, it does not imply excess baggage. However,
excess baggage follows directly from sometimes $\psi$-ontology, since there
must be at least as many ontic states as there are pure quantum states if each
quantum state is to be assigned its own region of ontic state space. Further,
a sometimes $\psi$-ontic model cannot be maximally $\psi$-epistemic, so we
obtain all the implications of that as well.

######  Theorem 13.2.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment and let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model
of it. If $\Theta$ is sometimes $\psi$-ontic then it is not maximally
$\psi$-epistemic.

###### Proof.

Assume that $\Theta$ is maximally $\psi$-epistemic. Then, for all
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$, all
$\mu\in\Delta_{\psi}$, all $\nu\in\Delta_{\phi}$, and all $\Omega\in\Sigma$
such that $\nu(\Omega)=1$,

$\int_{\Omega}\text{Pr}(E|M,\lambda)d\mu(\lambda)=\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda),$
(280)

for all $M\in\mathcal{M}$, $\text{Pr}\in\Xi_{M}$.

Let $\Omega^{\prime}$ be the region uniquely assigned to $\left[\psi\right]$
so that $\nu(\Omega^{\prime})=0$ and $\mu(\Omega^{\prime})>0$. Then $\Omega$
can be chosen such that $\Omega\cap\Omega^{\prime}=\emptyset$. Summing Eq.
(280) over $E$ then gives $\mu(\Omega)=\mu(\Lambda)=1$. However, $\mu(\Omega)$
cannot be equal to one because $\mu(\Omega^{\prime})>0$ and $\Omega$ and
$\Omega^{\prime}$ are disjoint, so this would make $\mu(\Lambda)>1$. Hence,
there is a contradiction, so $\Theta$ cannot be maximally $\psi$-epistemic. ∎

The Kochen–Specker model is never $\psi$-ontic, but all the existing
$\psi$-epistemic models for Hilbert spaces of dimension $\geq 3$ are sometimes
$\psi$-ontic, so whether never $\psi$-ontic models exist in all dimensions is
an open question. Since it is impossible to prove $\psi$-ontology without
auxiliary assumptions, sometimes $\psi$-ontology is perhaps the strongest
result about the reality of the quantum state that we could hope to prove
without such assumptions. For this reason, I consider this to be the most
important open question in this area.

##  14 Discussion and conclusions

###  14.1 Summary

This review article was divided into three parts. In the first part, I
explained the distinction between ontic and epistemic interpretations of the
quantum state, outlined the pre-existing arguments for epistemic and ontic
interpretations, and explained how many existing no-go theorems would follow
from proving that the quantum state is ontic. The aim of this part was to
convince you that the $\psi$-ontic/epistemic distinction is interesting, that
the question is unresolved by qualitative arguments, and that it is the sort
of issue that should be addressed with the conceptual rigor that Bell brought
to nonlocality.

The second part discussed three existing $\psi$-ontology theorems: the
Pusey–Barrett–Rudolph Theorem, Hardy’s Theorem, and the Colbeck–Renner
Theorem. Each of these results involve auxiliary assumptions, of varying
degrees of reasonableness. Hardy’s ontic indifference assumption is not really
appropriate for a $\psi$-epistemic theory, but it does allow the flaw in the
argument for the reality of the wavefunction from interference to be exposed
more clearly. Colbeck and Renner’s assumption of parameter independence is
doubtful in light of Bell’s Theorem. Whilst violations of Bell inequalities
can arise from the failure of outcome independence alone, many viable realist
interpretations of quantum theory, such as de Broglie–Bohm theory, violate
parameter independence instead, and one really wants the scope of a no-go
theorem to include as many viable realist interpretations as possible. In my
view, the preparation independence postulate of the Pusey–Barrett–Rudolph
Theorem is the best of the bunch. It is not completely unassailable, since
weakening it slightly to allow for genuinely nonlocal degrees of freedom does
potentially allow for a viable $\psi$-epistemic interpretation. Nevertheless,
the PIP is satisfied in theories like de Broglie–Bohm, so, unlike the
Colbeck–Renner Theorem, the Pusey–Barrett–Rudolph Theorem does explain why the
wavefunction must be real in theories of that type.

Without auxiliary assumptions, $\psi$-epistemic models do exist, so the third
part of this review discussed strengthenings of the notion of a
$\psi$-epistemic model that may be needed in order to make the
$\psi$-epistemic explanations of phenomena like the indistinguishability of
quantum states and the no cloning theorem go through. Whilst the case is not
yet watertight, I expect that future work on bounding the ratio of the overlap
of probability measures to the quantum probabilities of the states they
represent will eventually make $\psi$-epistemic explanations look implausible
within the ontological models framework.

###  14.2 Open questions

Part of the aim of this review was to provide the necessary background for
researchers who would like to begin working in this area. In this regard,
several open questions were raised in the main text, and solving some of them
would help to put the non viability of $\psi$-epistemic interpretations on a
more secure footing. For easy reference, the following list collects all the
open problems together in one place, and adds a couple of new ones. The
section numbers indicate where in the review you can find a more detailed
description.

  1. 1.

§7.7.2: Can a deterministic $\psi$-ontic model always be converted into a
$\psi$-epistemic one by finding regions of the ontic state space associated
with different pure states that make the exact same predictions? One of
Schlosshauer and Fine’s criticisms of the Pusey–Barrett–Rudolph Theorem was
based on the idea that this can always be done. It is not true for
nondeterministic theories, but it may be true for deterministic theories.

  2. 2.

§11: Can reasonable symmetry requirements be used to rule out pairwise
$\psi$-epistemic models without restrictive assumptions on the nature of the
ontic state space? The theorem of Aaronson et. al. only applies if the ontic
state space is projective Hilbert space or the set of unitary operators, which
is a very restrictive class of models.

  3. 3.

§12.2: Does the overlap of probability measures representing nonorthogonal
states have to tend to be close to zero for pairs of states with a fixed inner
product? The existing results only imply this for families of states that
become orthogonal in the $n\rightarrow\infty$ limit, so this only implies that
the $\psi$-epistemic explanation of indistinguishability is implausible for
states that are almost distinguishable.

  4. 4.

§13: Does a never $\psi$-ontic model exist for the fragment consisting of all
pure states and projective measurements in $\mathbb{C}^{d}$ for $d\geq 3$? In
my view, this is the most important open question, as proving that all models
must be sometimes $\psi$-ontic has the same implications as $\psi$-ontology.

  5. 5.

Are better $\psi$-ontology results possible with POVMs? The known
$\psi$-epistemic models, including the Kochen–Specker model discussed in §4.3
and the ABCL model discussed in §7.5, only reproduce quantum theory for
measurements in orthonormal bases. Morris has shown that the Kochen–Specker
model cannot be extended to all POVMs [113]. Therefore, it is still possible
that there are no $\psi$-epistemic models that reproduce the quantum
predictions for all POVMs, and it may be possible to prove this without
auxiliary assumptions. Further, most of the proofs of $\psi$-ontology results
presented here are based on measurements in orthonormal bases, so it is
possible that simpler proofs and/or better overlap bounds could be derived
using POVMs. This is the case for the Pusey–Barrett–Rudolph theorem, as the
simplified proof presented here does involve POVMs.

  6. 6.

Do $\psi$-ontology results have applications in quantum information theory,
beyond those of the implications discussed in §5? The ontological models
framework is already regarded as implausible by some physicists, but it is
worth investigating anyway because it provides a way of simulating quantum
experiments using classical resources. Results derived in this framework often
go on to have applications in quantum information theory. In this regard,
Montina proved an upper bound on the classical communication complexity of
simulating the identity channel for a qubit using the Kochen–Specker model,
which is maximally $\psi$-epistemic [135]. It is possible that the
nonexistence of such models in higher dimensions could be used to prove lower
bounds on the communication complexity of the identity channel in higher
dimensions. Conversely, Montina et. al.’s recent results on the communication
complexity of simulating the identity channel [136, 137, 138, 139] might be
used to derive $\psi$-ontology results. Some work has also been done on
deriving quantum advantages from antidistinguishability [108, 109].

###  14.3 Experiments

Probably the most important issue not discussed so far in this review is the
question of whether the reality of the quantum state can be established
experimentally. When it comes to experiments in the foundations of quantum
theory, tests of Bell’s Theorem are somewhat of a gold standard. Violation of
a Bell inequality rules out a wide class of local realistic theories
independently of the details of quantum theory. Consequently, the experimenter
does not have to assume much about how their experiment is represented within
quantum theory. From their observed statistics they simply get a yes/no answer
as to whether local hidden variable theories are viable, modulo the known
loopholes. In modern parlance, tests of Bell’s Theorem are _device
independent_ [102].

In contrast, a test of the reality of the quantum state would not be device
independent simply because the “quantum state” is the thing we are testing the
reality of, and that is a theory dependent notion. Consequently, one has to
assume that our quantum theoretical description of the way that our
preparation devices work is more or less accurate, in the sense that they are
approximately preparing the quantum states the theory says they are, in order
to test the existing $\psi$-ontology results. Therefore, it is desirable to
have a more theory independent notion of whether a given set of observed
statistics imply that the “probabilistic state”, i.e. some theory-independent
generalization of the quantum state, must be real. It is not obvious whether
this can be done, but if it can then experimental tests of $\psi$-ontology
results would become much more interesting.

Of course, one can still perform non device independent experimental tests.
This amounts to trying to prepare the states, perform the transformations, and
make the measurements involved in a $\psi$-ontology result and checking that
the quantum predictions are approximately upheld. Due to experimental error,
the agreement will never be exact, but one can bound the overlap between
probability measures representing quantum states instead of showing that it
must be exactly zero. For the special case of the Pusey–Barrett–Rudolph
Theorem given in Example 7.9, this has been done using two ions in an ion trap
[140]. However, the experimental result only shows that the overlap in
probability measures must be a little smaller than the quantum probability,
and not that it must be close to zero, and of course, this only applies if one
assumes the PIP.

A better approach to deriving overlap bounds is to test the results described
in §12.2. For finite $n$, these can provide overlap bounds for specific sets
of states without auxiliary assumptions. Based on Branciard’s results [55], a
recent experiment with single photons has given a bound on the ratio of
overlap to the quantum probability of $0.690\pm 0.001$ for a set of $10$
states in a Hilbert space of dimension $4$ [141]. This is still quite far from
establishing the reality of the quantum state, for which one would want to
test many pairs of quantum states with a variety of different inner products,
and to show that the overlap ratios are consistently close to zero. Recall
that, due to the fact that part of the indistinguishability of quantum states
may be explained by the coarse-grained nature of measurements in an
ontological model, it is difficult to attach significance to overlap ratios
that are significantly larger than zero. Further theoretical work is required
to derive feasible experiments that can achieve much lower ratios.

Finally, the original version of the Colbeck–Renner Theorem, i.e. the one that
aimed to rule out ontological models with greater predictive power than
quantum theory, has been tested using a quantum optical system [142] and there
has also been an optical test of the Patra–Pironio–Massar Theorem on PPM-
continuous models [143].

###  14.4 Future directions

Assuming that future results drive the final nails into the coffin of
$\psi$-epistemic explanations within the ontological models framework, the
final question I want to address is where to go next. One option is to embrace
the reality of the wavefunction by adopting one of the existing realist
interpretations that fit into the ontological models framework, e.g. de
Broglie–Bohm theory, spontaneous collapse theories or modal interpretations.
Another is to adopt a neo-Copenhagen interpretation.

The first option is unappealing because adopting one of these realist
interpretations opens up an explanatory gap. Namely, given that a
$\psi$-epistemic interpretation would provide compelling explanations of a
whole variety of quantum phenomena, it is puzzling that the quantum state
should nevertheless be ontic. Of course, the existing no-go results, such as
Bell’s Theorem, also imply explanatory gaps, e.g. if a theory is explicitly
nonlocal then why can this not be used to send a superluminal signal? For this
reason, those inclined to the $\psi$-epistemic view are likely to have
rejected these interpretations already on the basis of these other gaps.

Being neo-Copenhagen is always an option, but the merits of such an move
depend on the degree to which one believes that realism is desirable. This is
not the place to get into the debate between realism and antirealism and
whether neo-Copenhagen views are compatible with some weakened notion of
realism. Suffice to say that the viability of these interpretations turns on
issues that are far deeper than the reality of the wavefunction. For my part,
I think that if one denies the existence of an observer-independent reality
then it becomes very difficult to maintain a clear notion of explanation at
all. Closing explanatory gaps by denying the need for any explanation at all
does not seem that appealing to me.

The only remaining option then is to adopt a realist interpretation that does
not fit in to the ontological models framework. There are several
possibilities, most of them highly speculative.

  * •

Many-worlds: The many-worlds interpretation [6, 7, 8] is not contained within
the ontological models framework because the latter assumes that measurements
yield a single unique outcome. Many-worlds is a commonplace retreat for
realists who want to avoid introducing nonlocality in the face of Bell’s
Theorem. However, the conventional many-worlds interpretation is not a retreat
for $\psi$-epistemicists because it is based on the idea that the quantum
state is a literal description of reality. Nevertheless, since $\psi$-ontology
theorems do not apply to theories that involve many-worlds, it is possible
that there is a viable $\psi$-epistemic interpretation that involves them.
This is doubtless an unappealing option to both existing many-worlds advocates
and $\psi$-epistemicists. For many-worlds advocates, the reason to take
multiple worlds seriously is because they are encoded in the branching
structure of the wavefunction. If you take away the reality of the
wavefunction then you take away their reason for believing in them. Similarly,
this is the only real reason for taking many worlds seriously, so someone who
does not believe in the reality of the wavefunction is unlikely to have found
the existence of many-worlds plausible in the first place. Nevertheless, it is
logically possible that the universe could be described by some structure that
does not imply a unique wavefunction, but does support the existence of many-
worlds.

  * •

Histories approaches: Histories interpretations, such as consistent/decoherent
histories [144] and Sorkin’s co-event formalism [145, 146] are based on taking
the space-time picture provided by the path integral seriously. Consistent
histories does not fit into the ontological models framework as it decrees
that not all sets of histories can be assigned a probability and there may not
be a unique description of the universe in terms of a single history.
Griffiths argues that the consistent histories interpretation is appropriately
$\psi$-epistemic [147]. In contrast, the co-event formalism is based on a
modified logic and probability theory. I explained my doubts about realist
interpretations of exotic probability theories in §7.7.4 and, in any case,
Wallden [39] provides evidence that a result analogous to the
Pusey–Barrett–Rudolph Theorem may hold in the coevent formalism.

  * •

Retrocausality: In the ontological models framework, it is assumed that the
probability measure representing a quantum state is independent of the choice
of future measurement setting. If this were not the case then the
$\psi$-epistemic interpretation of quantum phenomena could be maintained by
having the measures corresponding to antidistinguishable states have no
overlap when the antidistinguishing measurement is made, but nonzero overlap
when other measurements are made. One way that dependence on the measurement
setting may occur is if there is a direct causal influence, traveling
backwards in time, from the measurement to the preparation. Several authors
have argued that there are independent reasons for adopting a retrocausal
approach to quantum theory [148, 149, 150], not least because it might allow
an appropriately local resolution to the dilemma posed by Bell’s Theorem. The
transactional interpretation [151, 152] is explicitly retrocausal and the two-
state vector formalism [153] can be read in a retrocausal way. However, both
of these theories posit an ontic wavefunction. If we are to maintain
$\psi$-epistemic explanations then we instead need to look for retrocausal
ontological models that posit a deeper reality underlying quantum theory that
does not include the quantum state.

  * •

Relationalism: The ontological models framework assumes that systems have
their own intrinsic properties, encompassed by the ontological state
$\lambda$. Relationalism posits that systems do not have intrinsic properties,
but only properties relative to other systems. The usual analogy is with the
concept of position. One cannot talk about the position of a particle without
setting down some coordinate system, and this implicitly means that we are
measuring position with respect to some other physical system that provides a
reference frame. There is an obvious commonality with Everett’s relative state
approach [6], except that we want a theory of this type that is
$\psi$-epistemic. Rovelli’s relational quantum mechanics [22] is a theory of
this type in which the wavefunction is supposed to be epistemic, but he
defines relational properties in terms of the global wavefunction so it is not
clear how they are supposed to be determined if the wavefunction is not real.

In conclusion, I think we should try to find a way of understanding quantum
theory that closes as many of the explanatory gaps opened by no-go theorems as
possible. This is because an interpretation that merely accommodates the known
facts about quantum theory, rather than explaining them, is unlikely to yield
principles that can reliably guide us towards future physical theories. Since
$\psi$-ontology implies many of the existing no-go theorems, the gap opened by
$\psi$-ontology results should be taken at least as seriously as the others.
This means that we should investigate the speculative roads less travelled
described above, in addition to others that we have not thought of yet. The
chances that any one of them will bear fruit may be slim, but the rewards if
they do will be worth the effort.

## Acknowledgements

I would like to thank Jonathan Barrett, Sarah Croke, Joseph Emerson, Lucien
Hardy, Teiko Heinosaari, Ravi Kunjwal, Klaas Landsman, Peter Lewis, Owen
Maroney, Matt Pusey, Terry Rudolph, Max Schlosshauer and Rob Spekkens for
useful discussions, and Paul Merriam for a careful proof reading. This
research was supported in part by the FQXi Large Grant “Time and the Structure
of Quantum Theory” and by Perimeter Institute for Theoretical Physics.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of
Research and Innovation.

## Appendix

##  Appendix A The $\psi$-ontic/epistemic distinction and objective chance

In this appendix, the question of whether a distinction equivalent to the
Bayesian distinction between ontic and epistemic states can be made in
theories that involve objective chance is addressed.

Firstly, note that whether or not this issue is relevant to the reality of the
wavefunction depends on exactly which probabilities are deemed to be objective
chances. Most advocates of objective chance support hybrid theories of
probability in which there are epistemic probabilities in addition to
objective chances. Objective chances are invoked to explain lawlike
probabilistic regularities, as they occur in our theories of physics for
example, but epistemic probabilities, usually cashed out in Bayesian or
frequentist terms, also exist in order to make contact with statistics. In the
ontological models framework for a PM fragment, probabilities enter into the
description in two places. Firstly, quantum states correspond to probability
measures over the ontic state space and, secondly, measurements correspond to
conditional probability distributions over their outcomes. A fairly natural
position would be to assert that the probability measures corresponding to
states are epistemic, whereas the conditional probabilities associated to
measurements are objective. The former represent our uncertainty about the
true ontic state, whereas the latter represent our uncertainty about the
response of the measurement device. On a conventional view, it is the response
of the measurement device that represents a “genuinely stochastic” event in
quantum theory, so it would make sense to only place the objective chances
here where they seem to be most needed.

Regardless of whether you agree with this account, the main point is that the
introduction of objective chances into quantum theory only presents a problem
for the ontic/epistemic distinction if the probability measures representing
states are deemed to be objective chances. This is because the definition of
$\psi$-ontic and $\psi$-epistemic models is given entirely in terms of the
overlaps of these probability measures and does not involve the measurement
device probabilities at all. Therefore, if you are happy to put objective
chances only in the measuring device, then the introduction of such chances
does not present any new problem for the ontic/epistemic distinction. It is
only if you think that the probability measures representing states are
objective chances that the distinction needs to be reexamined. Therefore, I
assume that this is what is meant by introducing objective chances into
quantum theory in the remainder of this appendix.

As in the frequency theory, we are interested in whether objective chances can
be viewed as intrinsic properties of individual systems, or whether they must
be defined with reference to the wider world beyond the individual system,
e.g. in terms of an ensemble of similar systems or the conditions prevailing
in the environment of the system. An intrinsic property of a system is
something like the charge of an electron or the hidden variable state of a
quantum system, if the latter are presumed to exist. In a realist approach to
physics, these correspond to ontological features of the system and they
cannot be changed just by changing our description of the world in which the
system is embedded. Changing them requires an intervention in the system
itself.

Deciding whether or not objective chances are intrinsic properties is
difficult because there is no universally agreed upon theory of objective
chance. Fortunately, the question only depends on a couple of broad features
of the theory.

Firstly, some authors posit that objective chance is compatible with
determinism so that, for example, the probabilities involved in classical
statistical mechanics can be viewed as objective chances. Others think that
objective chances only make sense if there is a genuine stochasticity in
nature, with quantum theory providing the prime example of a theory that
involves such genuine chance. It should be clear that objective chance cannot
be intrinsic in any theory that is compatible with determinism. This is
because, in a deterministic theory, the intrinsic properties of an isolated
system determine its future uniquely, so they could only ever give rise to
objective chances of $0$ or $1$. Therefore, to make such a theory work, one
has to refer either to an ensemble of systems, as in the frequency theory, or
to the conditions surrounding the system. As an example of the latter, one can
imagine a specification of the way in which a coin should be tossed such as “a
strong flip between thumb and forefinger” that is specific enough to license
the assignment of a fixed objective chance, but vague enough that it does not
determine the outcome of the toss uniquely. Of course, it is questionable
whether such a notion makes sense, but the point is that, if one does take
this view, then probabilities are not intrinsic properties of systems. Either
they refer to ensembles or they refer to the conditions of interaction between
a system and its environment, and specifying these in different levels of
detail would lead to different probability assignments, just as in the
frequency case.

However, many philosophers of objective chance, including Popper [154] and
Lewis [155], take the view that objective chances only make sense in a
genuinely stochastic universe. In this case, one can require that the
objective chances of an experimental outcome are only the same if the
intrinsic properties of the system before the experiment are identical in all
relevant details. The prime example of this would be to say that the objective
chances of obtaining a given outcome in a quantum measurement on two different
systems are the same iff the systems are described by the same quantum state
$\left|\psi\right\rangle$ prior to the experiment. This presents more of a
problem for the distinction we are trying to make, since ostensibly the state
$\left|\psi\right\rangle$ only refers to the system itself.

However, it should be noted that the quantum state example rests on
questionable assumptions about the interpretation of quantum theory. It
assumes that $\left|\psi\right\rangle$ can be regarded as an intrinsic
property of the system and, further, that $\left|\psi\right\rangle$ is a
complete description of the system. In an operationalist approach, the
assertion that $\left|\psi\right\rangle$ is an intrinsic property of a system
is denied. Instead, it is a description of those facts about the device that
prepared the system that are relevant for predicting future measurement
outcomes. In other words, it is a condensed description of a set of knob
settings, meter readings, etc. that refer to a piece of experimental apparatus
external to the system. These are not intrinsic properties of the system, so
the relevant distinction is maintained.

Of course, both Popper and Lewis intend a more realist interpretation of
quantum theory. However, even if quantum states are intrinsic properties of
quantum systems, they need not be complete. For example, in de Broglie–Bohm
theory one also needs to specify the positions of particles. The theory is
deterministic when both the quantum state and particle positions are
specified, so we are back in the position of having to define objective
chances in a deterministic theory, in which case they are not intrinsic. In de
Broglie–Bohm theory, the statement that a system is described by a state
$\left|\psi\right\rangle$ really means that it is part of an ensemble of
systems described by the quantum state
$\left|\psi\right\rangle\otimes\left|\psi\right\rangle\otimes\ldots$ and in
which the particle positions are distributed according to
$\left|\psi(x)\right|^{2}$. Since there is freedom to look at subensembles
where the positions are not distributed according to
$\left|\psi(x)\right|^{2}$, and these would allow prediction with greater
accuracy than quantum theory, systems described by the same quantum state do
not have the same intrinsic properties in all relevant detail.

Nevertheless, although quantum theory is a prime motivation for objective
chances, theories of objective chance are usually independent of the details
of physics. Thus, we can ignore the quantum motivation and just look at the
theories of chance actually proposed to determine whether chances are
intrinsic. In this regard, an important distinction is whether or not a theory
of chance is _Humean_. Roughly speaking, a Humean theory is one in which the
chances are defined in terms of the facts on the ground, i.e. facts about the
universe that could form part of our experience of it (see [156] for a more
precise definition). This means that objective chances cannot be defined in
terms of things like $\left|\psi\right\rangle$, which do not form part of our
experience. Lewis [155] was a proponent of Humean theories of chance and his
favoured _best system_ theory is really just a modification of frequentism.
More specifically, he thought that chances were specified via a tradeoff
between accurately capturing the relative frequencies and simplicity. Thus, if
a large number of coin flips is performed several different times, and on each
occasion the relative frequency of heads obtained was close to, but not
exactly, $1/2$, varying in a seemingly random way, Lewis would say that this
licenses assigning an objective chance of $1/2$ to the coin flips. This
differs from the frequency theory in that it does not demand that
probabilities are exactly the relative frequencies in some real or
hypothetical, finite or infinite, sequence of experiments. Of course, defining
an objective tradeoff between predictability and simplicity is difficult, but
for present purposes all that matters is that again objective chances are not
intrinsic properties of systems, but are instead defined with respect to an
ensemble. Likewise, I think that any Humean theory of chance makes chance a
non-intrinsic property because chances would have to be defined in terms of
observable facts, and I do not see how this could be done without referring to
ensembles or to the surroundings of the system.

Therefore, the only chance theories that really pose a problem for the
distinction we are trying to make are those that are non-Humean, i.e. they
posit that chances do not supervene on facts that could form part of our
experience. Of these, the most prominent is Popper’s propensity theory [154].
Propensities are dispositional properties, i.e. a system has a disposition to
produce a certain outcome in an experiment. Propensity theories are broadly
classified as either long-run propensity theories or single-case propensity
theories. In a long-run theory, a propensity is read as a disposition to
produce a certain relative frequency of outcomes in the long run. Since this
refers to an ensemble, again there is no problem distinguishing this type of
property from intrinsic properties of individual systems. On the other hand,
single case propensities are read as a disposition to produce a certain
outcome in a single experiment. These are much more problematic for the
distinction we are trying to make, as they do not refer to entities other than
the individual system. Thus, in a single case propensity theory, it may not be
possible to make a clean distinction that is analogous to the Bayesian
distinction between ontic and epistemic states.

Before concluding, note that many philosophers take a more laissez faire
approach to objective chances. Whilst they believe that objective chances
exist, they do not commit to a specific theory and are instead content to
specify some rules that they must obey, such as Lewis’ principal principle
[155]. This is a perfectly reasonable attitude to take, but it is unreasonable
to expect that the question of whether chances are intrinsic properties can be
settled at this level of generality. Some further features of objective chance
would need to be specified, such as whether or not they ought to be Humean.

In summary, most theories of objective chance seem to admit a distinction
equivalent to the Bayesian distinction between ontic and epistemic states in
that chances refer to non intrinsic facts about a system. The problematic
theories are single-case, such as the single-case propensity theory. However,
it seems a bit of a stretch to adopt this theory in order to avoid
investigating the question of whether quantum states are ontic or epistemic,
particularly since the interpretation of quantum theory is a prime motivation
for introducing objective chances in the first place.

##  Appendix B The Kochen–Specker model

This appendix proves that the Kochen–Specker model of Example 4.6 reproduces
the quantum predictions and is maximally $\psi$-epistemic. Recall that, in the
Kochen–Specker model, the quantum state $\left[\psi\right]$ is represented by
a unique probability measure

$\mu(\Omega)=\int_{\Omega}p(\vec{\lambda})\sin\vartheta d\vartheta d\varphi,$
(281)

over the Bloch sphere, where the density $p$ is given by

$p(\vec{\lambda})=\frac{1}{\pi}H\left(\vec{\psi}\cdot\vec{\lambda}\right)\vec{\psi}\cdot\vec{\lambda},$
(282)

and $H$ is the Heaviside step function. The conditional probability
distribution for a measurement
$M=\left\\{\left[\phi\right],\left[\phi^{\perp}\right]\right\\}$ is given by

$\displaystyle\text{Pr}(\phi|M,\vec{\lambda})$
$\displaystyle=H(\vec{\phi}\cdot\vec{\lambda})$ (283)
$\displaystyle\text{Pr}(\phi^{\perp}|M,\vec{\lambda})$
$\displaystyle=1-\text{Pr}(\phi|M,\vec{\lambda}).$ (284)

To prove that this reproduces the quantum predictions, we need to show that,
for any pair of states $\left[\psi\right]$ and $\left[\phi\right]$,

$\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})=\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right).$
(285)

The corresponding equation for $\left[\phi^{\perp}\right]$ will then be
automatically satisfied because

$\int_{\Lambda}\text{Pr}(\phi^{\perp}|M,\vec{\lambda})d\mu(\vec{\lambda})=1-\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}).$
(286)

To prove Eq. (285), it is convenient to choose a parameterization of the Bloch
sphere such that both $\vec{\psi}$ and $\vec{\phi}$ lie on the equator. We can
further choose $\vec{\psi}$ to point along the $x$ axis so that
$\vec{\psi}=(1,0,0)$ and
$\vec{\phi}=(\cos\varphi_{\phi},\sin\varphi_{\phi},0)$ for some angle
$-\pi<\varphi_{\phi}\leq\pi$. Using Eq. (28), this means that the right hand
side of Eq. (285) is

$\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right)=\left|\left\langle\phi\middle|\psi\right\rangle\right|^{2}=\frac{1}{2}\left(1+\cos(\varphi_{\phi})\right).$
(287)

Now, expanding the left hand side of Eq. (285) gives

$\displaystyle\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle\quad\quad=\frac{1}{\pi}\int_{\Lambda}H(\vec{\phi}\cdot\vec{\lambda})H\left(\vec{\psi}\cdot\vec{\lambda}\right)\vec{\psi}\cdot\vec{\lambda}\sin\vartheta
d\vartheta d\varphi.$ (288)

Since
$\vec{\lambda}=(\sin\vartheta\cos\varphi,\sin\vartheta\sin\varphi,\cos\vartheta)$,
we have

$\displaystyle\vec{\psi}\cdot\vec{\lambda}$
$\displaystyle=\sin\vartheta\cos\varphi$ (289)
$\displaystyle\vec{\phi}\cdot\vec{\lambda}$
$\displaystyle=\sin\vartheta\cos\varphi\cos\varphi_{\phi}+\sin\vartheta\sin\varphi\sin\varphi_{\phi}$
(290) $\displaystyle=\sin\vartheta\cos\left(\varphi-\varphi_{\phi}\right).$
(291)

Due to he Heaviside step functions, we only need to integrate over the region
where both of these are positive. This is the intersection of
$-\frac{\pi}{2}<\varphi<\frac{\pi}{2}$ and
$-\frac{\pi}{2}+\varphi_{\phi}<\varphi<\frac{\pi}{2}+\varphi_{\phi}$. When
$\varphi_{\phi}$ is positive this is the interval
$-\frac{\pi}{2}+\varphi_{\phi}<\varphi<\frac{\pi}{2}$ and when
$\varphi_{\phi}$ is negative this is the interval
$-\frac{\pi}{2}<\vartheta<\frac{\pi}{2}+\varphi_{\phi}$. Consider first the
case where $\varphi_{\phi}$ is positive. Then Eq. (B) reduces to

$\displaystyle\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle=\int_{0}^{\pi}\sin^{2}\vartheta
d\vartheta\int_{-\frac{\pi}{2}+\varphi_{\phi}}^{\frac{\pi}{2}}\cos\varphi
d\varphi$ (292)
$\displaystyle=\frac{1}{2}\left[\sin\varphi\right]_{-\frac{\pi}{2}+\varphi_{\phi}}^{\frac{\pi}{2}}$
(293)
$\displaystyle=\frac{1}{2}\left(1+\sin\left(\frac{\pi}{2}-\varphi_{\phi}\right)\right)$
(294) $\displaystyle=\frac{1}{2}\left(1+\cos\varphi_{\phi}\right),$ (295)

as required. The case where $\varphi_{\phi}$ is negative gives the same result
because $\sin$ is an odd function.

To prove that the model is maximally $\psi$-epistemic, let $\nu$ be the
probability measure associated with $\left[\phi\right]$, i.e.

$\nu(\Omega)=\int_{\Omega}q(\vec{\lambda})\sin\vartheta d\vartheta d\varphi,$
(296)

where

$q(\vec{\lambda})=\frac{1}{\pi}H\left(\vec{\phi}\cdot\vec{\lambda}\right)\vec{\phi}\cdot\vec{\lambda}.$
(297)

We then need to show that

$\int_{\Omega}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})=\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}),$
(298)

for any $\Omega$ such that $\nu(\Omega)=1$. Assume that $\varphi_{\phi}$ is
positive (the negative case follows the same logic). Let,

$\displaystyle\Omega_{\psi}$
$\displaystyle=\left\\{\vec{\lambda}\in\Lambda\middle|0<\vartheta<\pi,-\frac{\pi}{2}<\varphi<\frac{\pi}{2}\right\\}$
(299) $\displaystyle\Omega_{\phi}$
$\displaystyle=\left\\{\vec{\lambda}\in\Lambda\middle|0<\vartheta<\pi,-\frac{\pi}{2}+\varphi_{\phi}<\varphi<\frac{\pi}{2}+\varphi_{\phi}\right\\}.$
(300)

Note that, for any measurable set $\Omega$

$\int_{\Omega}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})=\int_{\Omega\cap\Omega_{\psi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}),$
(301)

because $p(\vec{\lambda})$ is zero outside $\Omega_{\psi}$. Note also that
$\Omega_{\phi}$ is a measure one set according to $\nu$ because
$q(\vec{\lambda})$ is zero outside this set. However, in proving that the
model reproduces the quantum predictions, we showed that

$\displaystyle\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle=\int_{\Omega_{\psi}\cap\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
(302)
$\displaystyle=\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right),$
(303)

and hence

$\displaystyle\int_{\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle=\int_{\Omega_{\psi}\cap\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
(304)
$\displaystyle=\int_{\Lambda}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}),$
(305)

so we have the required property for the special case of the set
$\Omega_{\phi}$.

Now let $\Omega$ be any other set that is of measure one according to $\nu$.
We can write $\Omega$ as the union of two disjoint sets via

$\Omega=\left(\Omega\cap\Omega_{\phi}\right)\cup\left(\Omega\backslash\Omega_{\phi}\right).$
(306)

The set $\Omega\backslash\Omega_{\phi}$ is of measure zero according to $\nu$
because $q(\vec{\lambda})$ is zero outside $\Omega{\phi}$. This means that
$\Omega\cap\Omega_{\phi}$ is of measure one. Further,
$\text{Pr}(\phi|M,\vec{\lambda})$ is also zero outside $\Omega_{\phi}$ so

$\int_{\Omega\backslash\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})=0.$
(307)

Therefore, we only need to show that

$\int_{\Omega\cap\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})=\int_{\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}).$
(308)

To see this, note that $\mu$ and $\nu$ are absolutely continuous with respect
to one another on $\Omega_{\psi}\cap\Omega_{\phi}$. Since,
$\Omega\cap\Omega_{\phi}$ is of measure one according to $\nu$,
$\Omega_{\phi}\backslash\left(\Omega\cap\Omega_{\phi}\right)$ is of measure
zero according to $\nu$ and hence, by absolute continuity,
$\Omega_{\psi}\cap\left(\Omega_{\phi}\backslash\left(\Omega\cap\Omega_{\phi}\right)\right)$
is of measure zero according to both $\nu$ and $\mu$. Thus,

$\displaystyle\int_{\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle\quad=\int_{\Omega_{\psi}\cap\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
(309)
$\displaystyle\quad=\int_{\Omega_{\psi}\cap\Omega_{\phi}\cap\Omega}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
$\displaystyle\quad\quad\quad+\int_{\Omega_{\psi}\cap\left(\Omega_{\phi}\backslash\left(\Omega\cap\Omega_{\phi}\right)\right)}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})$
(310)
$\displaystyle\quad=\int_{\Omega_{\psi}\cap\Omega_{\phi}\cap\Omega}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda})+0$
(311)
$\displaystyle\quad=\int_{\Omega\cap\Omega_{\phi}}\text{Pr}(\phi|M,\vec{\lambda})d\mu(\vec{\lambda}),$
(312)

as required.

##  Appendix C Kochen–Specker contextuality

Kochen–Specker contextuality is not directly related to $\psi$-ontology, but
many of the consequences of $\psi$-ontology can alternatively be derived from
it.

The notion of contextuality first arose in Kochen and Specker’s attempt to
prove a no-go theorem for hidden variable theories [76]. Kochen and Specker’s
definition of contextuality only deals with projective measurements, but this
has since been generalized and given a more operational spin by Spekkens [67],
and we follow his approach here. The basic idea is that, if two things are
operationally equivalent in quantum theory, i.e. if they always give rise to
the exact same observable probabilities, then they should be represented the
same way in an ontological model. Applied to measurements, this is formally
defined as follows.

######  Definition C.1.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment and let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model
of it. $\Theta$ is _measurement noncontextual_ if, for each $M\in\mathcal{M}$,
$\Xi_{M}$ consists of a single unique conditional probability distribution,
and, whenever there exists $M,M^{\prime}\in\mathcal{M}$ and a POVM element $E$
such that $E\in M$ and $E\in M^{\prime}$, then, for all $\lambda$,

$\text{Pr}(E|M,\lambda)=\text{Pr}(E|M^{\prime},\lambda).$ (313)

Otherwise, $\Theta$ is _measurement contextual_.

If a measurement $M$ contains the POVM element $E$, then, for a system
prepared in the state $\rho$, the outcome $E$ always occurs with probability
$\text{Tr}\left(E\rho\right)$, regardless of how the measurement is
implemented. Additionally, if another measurement $M^{\prime}$ also contains
$E$ then the outcome $E$ still has the same quantum probability
$\text{Tr}\left(E\rho\right)$ in the measurement $M^{\prime}$ as it did in
$M$. Since there is nothing in the quantum predictions that distinguishes $E$
occurring in these different contexts, a noncontextual model should represent
them all in the same way. Note that, since $\Xi_{M}$ is a singleton for every
$M\in\mathcal{M}$ in a noncontextual model, we can unambiguously refer to a
unique conditional probability distribution $\text{Pr}(E|M,\lambda)$
associated to each measurement.

The classic example of the same measurement operator occurring in more than
one POVM is to take two orthonormal bases,
$\left\\{\left|\phi_{j}\right\rangle\right\\}_{j=0}^{d-1}$ and
$\left\\{\left|\phi^{\prime}_{j}\right\rangle\right\\}_{j=0}^{d-1}$, such that
$\left|\phi_{0}\right\rangle=\left|\phi^{\prime}_{0}\right\rangle$. Such a
pair can be constructed from a unitary $U$ that leaves
$\left|\phi_{0}\right\rangle$ invariant via
$\left|\phi^{\prime}_{j}\right\rangle=U\left|\phi_{j}\right\rangle$. Then, the
two measurements $M=\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}$ and
$M^{\prime}=\left\\{\left[\phi^{\prime}_{j}\right]\right\\}_{j=0}^{d-1}$ share
the common projector $\left[\phi_{0}\right]$. This can only happen
nontrivially if $d\geq 3$, since if $d=2$ and
$U\left|\phi_{0}\right\rangle=\left|\phi_{0}\right\rangle$ then
$U\left|\phi_{1}\right\rangle$ can only differ from
$\left|\phi_{1}\right\rangle$ by a global phase. For non projective POVMs,
nontrivial examples can be constructed for $d=2$ as well (see [67] for
details), but we are only concerned with the traditional Kochen–Specker notion
of contextuality here, which only applies to projective measurements.

Note that, given a POVM $M=\\{E_{0},E_{1},E_{2}\\}$, we can construct another
POVM $N=\\{E_{0},E_{1}+E_{2}\\}$ by coarse-graining the second and third
outcomes. One way of implementing $N$ is to perform the measurement $M$, but
only record whether or not the zeroth outcome occurred, i.e. lump outcomes $1$
and $2$ together into a single outcome. Because of this, it is natural to
assume that the conditional probability distribution representing $N$
satisfies

$\text{Pr}(E_{0}|N,\lambda)=\text{Pr}(E_{0}|M,\lambda),$ (314)

since the coarse-graining is just a classical post-processing of the outcome
of the measurement that happens after the measurement is made. If this holds
then measurement contextuality implies that the conditional probability
distribution representing a measurement depends on more than just which POVM
is measured. In addition to this, different methods of implementing the exact
same POVM must also sometimes be represented by different conditional
probability distributions.

To see this, consider again the two measurements
$M=\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}$ and
$M^{\prime}=\left\\{\left[\phi_{j}^{\prime}\right]\right\\}_{j=0}^{d-1}$ with
$d=3$ and suppose that

$\text{Pr}\left(\phi_{0}\middle|M,\lambda\right)\neq\text{Pr}\left(\phi_{0}\middle|M^{\prime},\lambda\right).$
(315)

Then, since
$E^{\perp}=\left[\phi_{1}\right]+\left[\phi_{2}\right]=\left[\phi^{\prime}_{1}\right]+\left[\phi^{\prime}_{2}\right]$,
we have two ways of implementing the coarse-grained measurement
$N=\left\\{\left[\phi_{0}\right],E^{\perp}\right\\}$, either by measuring $M$
and then coarse-graining over $\left[\phi_{1}\right]$ and
$\left[\phi_{2}\right]$ or by measuring $M^{\prime}$ and coarse-graining over
$\left[\phi_{1}^{\prime}\right]$ and $\left[\phi_{2}^{\prime}\right]$. Both of
these procedures correspond to the same POVM, but the $\left[\phi_{0}\right]$
outcome is represented by $\text{Pr}\left(\phi_{0}\middle|M,\lambda\right)$ in
the first case and $\text{Pr}\left(\phi_{0}\middle|M^{\prime},\lambda\right)$
in the second. Since these are not equal, the same POVM is represented by two
different conditional probability distributions, depending on how the
measurement is implemented. It is for this reason that, generally, a
measurement $M$ has to be represented by a set $\Xi_{M}$ of conditional
probability distributions rather than just a single one.

######  Definition C.2.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{M}$ consists of projective measurements. An
ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ is
_Kochen–Specker (KS) noncontextual_ if it is both:

  * •

Outcome deterministic: for all $M\in\mathcal{M},E\in M,\lambda\in\Lambda$,
every $\text{Pr}\in\Xi_{M}$ satisfies

$\text{Pr}(E|M,\lambda)\in\\{0,1\\}.$ (316)

  * •

Measurement noncontextual.

Otherwise the model is _KS contextual_.

In other words, KS noncontextuality only applies to projective measurements
and is the combination of Spekkens’ notion of measurement noncontextuality
with outcome determinism, i.e. the idea that the ontic state should determine
the outcome of a projective measurement with certainty. The Kochen–Specker
Theorem [76], and other proofs of KS contextuality [157, 158, 159, 160, 161,
162, 163, 164], show that it is impossible to construct a KS noncontextual
model for all projective measurements in Hilbert spaces of dimension $\geq 3$.

Note that a model may be KS contextual either by being measurement contextual
or by being nondeterministic. It can of course be both, but either one on its
own is sufficient to reproduce the quantum predictions. The
Beltrametti–Bugajski model is an example of a model that is nondeterministic,
but measurement noncontextual because the conditional probabilities
$\text{Pr}(E|M,\left[\lambda\right])=\text{Tr}\left(E\left[\lambda\right]\right)$
just mimic the quantum probabilities, which only depend on $E$, and not on
which $M$ containing $E$ is measured nor on how $M$ is implemented. On the
other hand, the Bell model is deterministic, but measurement contextual
because the way in which the unit interval is divided up depends on all of the
basis projectors in the measurement and their ordering.

Unlike $\psi$-ontology, KS contextuality does not obviously imply excess
baggage, but many of the other implications of $\psi$-ontology are implied by
KS contextuality. Therefore, one might wonder whether $\psi$-ontology implies
KS contextuality. This is not the case, as the Bell model in two dimensions is
$\psi$-ontic but can be converted into a KS noncontextual model by reordering
the way in which the unit interval is divided, e.g. by making the first
interval always correspond to the measurement outcome in the northern
hemisphere of the Bloch sphere. Similarly, there are $\psi$-epistemic models
that are KS contextual, such as the ABCL model described in §7.5, so
$\psi$-ontology and KS contextuality are inequivalent constraints on
ontological models.

The following characterization of KS noncontextual models will be useful in
what follows.

######  Definition C.3.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment and let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model
of it. For every $M\in\mathcal{M}$, $E\in M$, $\text{Pr}\in\Xi_{M}$, let

$\Lambda^{(E,M,\text{Pr})}=\left\\{\lambda\in\Lambda\middle|\text{Pr}(E|M,\lambda)=1\right\\},$
(317)

and let

$\Lambda^{(E,M)}=\cap_{\text{Pr}\in\Xi_{M}}\Lambda^{(E,M,\text{Pr})}.$ (318)

The _cosupport_ $\Lambda^{E}$ of a POVM element $E$ is then

$\Lambda^{E}=\cap_{\\{M\in\mathcal{M}|E\in M\\}}\Lambda^{(E,M)}.$ (319)

In other words, the cosupport of $E$ is the set of ontic states that always
return the outcome $E$ whenever a POVM that contains it is measured, for all
possible methods of implementing the measurement.

######  Theorem C.4.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment, where $\mathcal{M}$ consists of projective measurements. If an
ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ is KS
noncontextual then, for all $M\in\mathcal{M}$, $E\in M$, $\rho\in\mathcal{P}$,
every $\mu\in\Delta_{\rho}$ satisfies

$\int_{\Lambda}\mathrm{Pr}(E|M,\lambda)d\mu(\lambda)=\mu(\Lambda^{E}).$ (320)

###### Proof.

If $\Theta$ is KS noncontextual then there is only one conditional probability
distribution $\text{Pr}\in\Xi_{M}$ and $\text{Pr}(E|M,\lambda)$ is independent
of which measurement containing $E$ is performed, so, for any $M$ containing
$E$,
$\Lambda^{E}=\Lambda^{(E,M,\text{Pr})}=\left\\{\lambda\in\Lambda\middle|\text{Pr}(E|M,\lambda)=1\right\\}$.
Furthermore, $\text{Pr}(E|M,\lambda)$ is either equal to $0$ or $1$
everywhere, so we only need integrate over those $\lambda$ for which it is
equal to $1$. This gives

$\displaystyle\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda)$
$\displaystyle=\int_{\Lambda^{E}}\text{Pr}(E|M,\lambda)d\mu(\lambda)$ (321)
$\displaystyle=\int_{\Lambda^{E}}d\mu(\lambda)$ (322)
$\displaystyle=\mu(\Lambda^{E}),$ (323)

as required. ∎

The converse to this theorem does not hold, i.e. if Eq. (320) always holds
then the model may still be KS contextual. This is because there may be
subsets of the ontic state space that are of measure zero according to all the
probability measures that occur in the model. The values of the conditional
probability distributions on these sets are not constrained by Eq. (320).
However, a partial converse can be obtained by excising some of the measure
zero sets from the ontic state space. Since they are of measure zero, their
removal does not substantively affect the structure of the model.

######  Definition C.5.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment. An ontological model
$\tilde{\Theta}=(\tilde{\Lambda},\tilde{\Sigma},\tilde{\Delta},\tilde{\Xi})$
of $\mathfrak{F}$ is a _measure zero revision_ of another ontological model
$\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ if

  * •

The ontic state space $\tilde{\Lambda}$ differs from $\Lambda$ only by the
removal of a measure zero set. Formally, $\tilde{\Lambda}\in\Sigma$ and for
all $\rho\in\mathcal{P}$ and $\mu\in\Delta_{\rho}$, $\mu(\tilde{\Lambda})=1$.
Further, $\tilde{\Sigma}$ is the sub-$\sigma$-algebra of $\Sigma$ containing
all $\Omega\in\Sigma$ such that $\Omega\subseteq\tilde{\Lambda}$.

  * •

The probability measures of $\tilde{\Theta}$ are the restrictions of the
probability measures of $\Theta$ to $(\tilde{\Lambda},\tilde{\Sigma})$.
Formally, for all $\rho\in\mathcal{P}$, there is a surjective mapping
$f:\Delta_{\rho}\rightarrow\tilde{\Delta}_{\rho}$ defined by

$f[\mu](\Omega)=\mu(\Omega),$ (324)

for all $\Omega\in\tilde{\Sigma}$.

  * •

The conditional probability distributions of $\tilde{\Theta}$ are the
restrictions of the conditional probability distributions of $\Theta$ to
$\tilde{\Lambda}$. Formally, for all $M\in\mathcal{M}$, there is a surjective
mapping $g:\Xi_{M}\rightarrow\tilde{\Xi}_{M}$ defined by,

$g[\text{Pr}](E|M,\lambda)=\text{Pr}(E|M,\lambda),$ (325)

for all $E\in M$, $\lambda\in\tilde{\Lambda}$.

Performing a measure zero revision does not represent a substantive change in
the following sense. Imagine you are passively observing an experimenter who
is performing a sequence of prepare-and-measure experiments in the fragment
$\mathfrak{F}$. Suppose that, in addition to observing the choices of
preparation and measurement that the experimenter makes and the measurement
outcomes she obtains, you also get to see the exact ontic state $\lambda$ in
every run of the experiment. You write down a long list of data consisting of
the preparation performed, the ontic state, the choice of measurement, and the
measurement outcome in each run of the experiment. This data would allow you
to distinguish between an ontological model $\Theta$ and a measure zero
revision of it $\tilde{\Theta}$ only if an ontic state in
$\Lambda\backslash\tilde{\Lambda}$ happens to be occupied in at least one run
of the experiment, since this can only happen in $\Theta$ and the
probabilistic predictions made by the two models are otherwise exactly the
same. However, the probability of this happening is zero because the set of
ontic states removed has zero probability according to all the probability
measures in the theory. To all intents and purposes then, the two models tell
the same story about reality.

A partial converse to Theorem C.4 can be obtained as follows.

######  Theorem C.6.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{M}$ is at most countable and consists of projective
measurements. Let $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model
of $\mathfrak{F}$. If, for all $M\in\mathcal{M}$, $E\in M$,
$\mathrm{Pr}\in\Xi_{M}$, $\rho\in\mathcal{P}$, every $\mu\in\Delta_{\rho}$
satisfies

$\int_{\Lambda}\mathrm{Pr}(E|M,\lambda)d\mu(\lambda)=\mu(\Lambda^{E}),$ (326)

then there exists a measure zero revision of $\Theta$ that is KS
noncontextual.

###### Proof.

The measure zero revision
$\tilde{\Theta}=(\tilde{\Lambda},\tilde{\Sigma},\tilde{\Delta},\tilde{\Xi})$
is constructed as follows. First, let $M=\\{E_{j}\\}\in\mathcal{M}$ and note
that, for all $\rho\in\mathcal{P}$ and $\text{Pr}\in\Xi_{M}$, every
$\mu\in\Delta_{\rho}$ satisfies

$\displaystyle 1$ $\displaystyle=\mu(\Lambda)=\int_{\Lambda}d\mu(\lambda)$
(327)
$\displaystyle=\int_{\Lambda}\sum_{j}\text{Pr}(E_{j}|M,\lambda)d\mu(\lambda)$
(328)
$\displaystyle=\sum_{j}\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)d\mu(\lambda)$
(329) $\displaystyle=\sum_{j}\mu(\Lambda^{E_{j}}),$ (330)

where the second line follows from $\sum_{j}\text{Pr}(E_{j}|M,\lambda)=1$ and
the fourth from Eq. (326).

The sets $\Lambda^{E_{j}}$ are disjoint because
$\Lambda^{E_{j}}\subseteq\Lambda^{(E_{j},M,\text{Pr})}$ for every $M$ that
contains $E$ and every $\text{Pr}\in\Xi_{M}$, and if
$\text{Pr}(E_{j}|M,\lambda)=1$ then it must be the case that
$\text{Pr}(E_{k}|M,\lambda)=0$ for all $k\neq j$ in order to satisfy
$\sum_{j}\text{Pr}(E_{j}|M,\lambda)=1$. Therefore,
$\Lambda^{M}=\cup_{j}\Lambda^{E_{j}}$ is a set of measure one according to
$\mu$. Now set $\tilde{\Lambda}=\cap_{M\in\mathcal{M}}\Lambda^{M}$. This is
also measure one according to $\mu$, since it is the intersection of an at
most countable number of measure one sets. The probability measures in the
revision are then obtained by restriction to $\tilde{\Lambda}$.

For any measurement $M=\\{E_{j}\\}\in\mathcal{M}$ and $\text{Pr}\in\Xi_{M}$,
conditional probability distributions $\tilde{\text{Pr}}(E_{j}|M,\lambda)$
obtained by restriction of $\text{Pr}(E_{j}|M,\lambda)$ to $\tilde{\Lambda}$
are equal to $1$ on $\Lambda^{E_{j}}$ and are zero elsewhere on
$\tilde{\Lambda}$. The latter follows because
$\tilde{\Lambda}\subseteq\cup_{j}\Lambda^{E_{j}}$ and we already saw that
$\text{Pr}(E_{j}|M,\lambda)$ is equal to zero on $\Lambda^{E_{k}}$ for $k\neq
j$. Thus, the model is outcome deterministic. Further, it is measurement
noncontextual because $\Lambda^{E_{j}}$ does not depend on the measurement
context. Thus, the model is KS noncontextual. ∎

In §5, we showed that $\psi$-ontic models cannot be maximally $\psi$-epistemic
and that this implies preparation contextuality. KS contextuality can be used
in place of $\psi$-ontology to derive these results, in the following sense.
If we restrict attention to models of measurements in complete orthonormal
bases such that each measurement outcome occurs in an at most countable set of
contexts, a maximally $\psi$-epistemic model has a measure zero revision that
is KS noncontextual. This means that the Kochen–Specker theorem, which shows
that KS noncontextual models that reproduce the quantum predictions are
impossible, can be used in place of $\psi$-ontology to derive these results.

######  Theorem C.7.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{M}$ consists of measurements in complete orthonormal
bases and, for each rank-one projector $\left[\phi\right]$, the set of
$M\in\mathcal{M}$ that contains $\left[\phi\right]$ is at most countable. Let
$\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ be an ontological model of $\mathfrak{F}$
that reproduces the quantum predictions and is such that, for each
$M\in\mathcal{M}$, $\Xi_{M}$ is at most countable. If $\Theta$ is maximally
$\psi$-epistemic then, for all pure states
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$, every
$\mu\in\Delta_{\psi}$ satisfies

$\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)=\mu(\Lambda^{\phi}),$
(331)

for every $M\in\mathcal{M}$ that contains $\left[\phi\right]$ and every
$\text{Pr}\in\Xi_{M}$, and where $\Lambda^{\phi}$ is the cosupport of
$\left[\phi\right]$.

###### Proof.

Let $\nu\in\Delta_{\phi}$ and consider an $M\in\mathcal{M}$ that contains
$\left[\phi\right]$. Then, in order to reproduce the quantum predictions, for
every $\text{Pr}\in\Xi_{M}$ we must have

$\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\nu(\lambda)=\text{Tr}\left(\left[\phi\right]\left[\phi\right]\right)=1.$
(332)

In order for this to be true, there must be a set $\Omega\in\Sigma$ such that
$\nu(\Omega)=1$ and $\text{Pr}(\phi|M,\lambda)$ is equal to one on $\Omega$.
Any such set satisfies $\Omega\subseteq\Lambda^{(\phi,M,\text{Pr})}$ and so
$\Lambda^{(\phi,M,\text{Pr})}$ is also a set of measure one according to
$\nu$. Since this applies to any $\text{Pr}\in\Xi_{M}$,
$\Lambda^{(\phi,M)}=\cap_{\text{Pr}\in\Xi_{M}}\Lambda^{(\phi,M,\text{Pr})}$ is
also a set of measure one according to $\nu$ because it is the intersection of
an at most countable set of measure one sets. Finally,
$\Lambda^{\phi}=\cap_{\\{M\in\mathcal{M}|\left[\phi\right]\in
M\\}}\Lambda^{(\phi,M)}$ is a measure one according to $\nu$, since it too is
the intersection of an at most countable collection of measure one sets. Since
$\text{Pr}(\phi|M,\lambda)$ is equal to $1$ on $\Lambda^{\phi}$, we have

$\int_{\Lambda^{\phi}}\text{Pr}(\phi|M\lambda)d\mu(\lambda)=\int_{\Lambda^{\phi}}d\mu(\lambda)=\mu(\Lambda^{\phi}).$
(333)

Thus, Eq. (331) is a special case of Eq. (52), so it must hold for a maximally
$\psi$-epistemic model. ∎

######  Corollary C.8.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment in which $\mathcal{M}$ is at most countable and consists of
measurements in complete orthonormal bases. Suppose that, for all
$M\in\mathcal{M}$, $\left[\phi\right]\in M$, it is also the case that
$\left[\phi\right]\in\mathcal{P}$. Then, any maximally $\psi$-epistemic
ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ in
which $\Xi_{M}$ is at most countable for each $M\in\mathcal{M}$ has a measure
one revision that is KS noncontextual.

The proof just consists of combining Theorem C.7 with Theorem C.6.

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