way in an ontological model. Preparation noncontextuality is just this idea
applied to states rather than measurements. This section shows that, when
extended to mixed states, non maximally $\psi$-epistemic models must be
preparation contextual, a result that was first pointed out in [56].

######  Definition 5.3.

In an ontological model of a PM fragment, a state $\rho\in\mathcal{P}$ is
_preparation noncontextual_ if $\Delta_{\rho}$ only contains a single measure.
Otherwise $\rho$ is _preparation contextual_. Similarly, the ontological model
itself is preparation noncontextual if every $\rho\in\mathcal{P}$ is
preparation noncontextual, and otherwise it is preparation contextual.

Whenever two preparation procedures result in the same quantum state $\rho$,
there is no measurement that can distinguish between them because, according
to quantum theory, all of the outcome probabilities for every measurement are
exactly the same. Thus, the general principle of noncontextuality implies that
all methods of preparing $\rho$ should result in the same probability measure.

Note that much of the literature on $\psi$-ontology implicitly assumes
preparation noncontextuality for pure states by associating a unique measure
$\mu_{\psi}$ with each pure state $\left[\psi\right]$ under consideration.
This assumption is fairly harmless for a couple of reasons. Firstly, the
necessity of preparation contextuality has so far only been established for
mixed states, so assuming a unique measure for pure states is not ruled out by
any existing no-go theorem. Secondly, adapting results to preparation
contextual models is usually just a matter of modifying the definitions in a
fairly obvious way that does not require proofs to be modified substantively.
Nevertheless, I prefer to take the possibility of preparation contextuality
into account explicitly because it is oddly asymmetric to allow for contextual
measurements but not contextual states, and because it allows results to be
proved under weaker assumptions.

In order to prove preparation contextuality for mixed states, a further
assumption is required about how convex combinations of quantum states should
be represented in ontological models. To understand this, let
$\rho=\sum_{j}p_{j}\sigma_{j}$ be a convex decomposition of a mixed state
$\rho$ into other (pure or mixed) states $\sigma_{j}$, i.e. $0\leq p_{j}\leq
1$ and $\sum_{j}p_{j}=1$. One method of preparing $\rho$ is to generate a
classical random variable that takes value $j$ with probability $p_{j}$ (e.g.
by flipping coins, throwing dice, or any other suitable method), prepare the
state $\sigma_{j}$ if the value of the classical variable is $j$, and then
discard and forget the value of the classical variable.

In an ontological model, it is reasonable to assume that, in the above mixing
procedure, if $\mu_{j}$ is the probability measure corresponding to the method
used to prepare $\sigma_{j}$, then $\sum_{j}p_{j}\mu_{j}$ is the measure
corresponding to preparing $\rho$ by this mixing procedure. This is because
the randomness used to generate the classical variable could come from a
source that is completely independent of the system under investigation, in
which case we would expect that the ontic state of the system only depends on
it via its effect on which of the $\sigma_{j}$ is prepared. More formally,

######  Definition 5.4.

An ontological model _respects convex decompositions_ of a state
$\rho\in\mathcal{P}$ if, for every set of states
$\\{\sigma_{j}\\}\subseteq\mathcal{P}$ such that
$\rho=\sum_{j}p_{j}\sigma_{j}\in\mathcal{P}$ for some coefficients $p_{j}$
satisfying $0\leq p_{j}\leq 1$ and $\sum_{j}p_{j}=1$, for all possible choices
of $\mu_{j}\in\Delta_{\sigma_{j}}$, the measure $\sum_{j}p_{j}\mu_{j}$ is in
$\Delta_{\rho}$. The ontological model _respects convexity_ if it respects
convex decompositions of every $\rho\in\mathcal{P}$.

For the purpose of connecting non maximally $\psi$-epistemic models with
preparation contextuality, recall that an ontological model is non maximally
$\psi$-epistemic iff there are two nonorthogonal pure states
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$, such that, for some
$\mu\in\Delta_{\psi}$, $\nu\in\Delta_{\phi}$, $M\in\mathcal{M}$ containing
$\left[\phi\right]$, and $\text{Pr}\in\Xi_{M}$ there exists an
$\Omega\in\Sigma$ such that $\nu(\Omega)=1$ but

$\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)<\int_{\Lambda}\text{Pr}(\phi|M,\lambda)d\mu(\lambda).$
(60)

Preparation contextuality can now be derived from non maximal
$\psi$-epistemicity via the following theorem.

######  Theorem 5.5.

Consider an ontological model of a PM fragment that reproduces the quantum
predictions, respects convexity and is non maximally $\psi$-epistemic.

Assume that, for some pair of pure states
$\left[\psi\right],\left[\phi\right]\in\mathcal{P}$ that satisfy Eq. (60), the
states $\sigma^{\psi^{\perp}}=\frac{1}{d-1}\left(I-\left[\psi\right]\right)$,
$\sigma^{\phi^{\perp}}=\frac{1}{d-1}\left(I-\left[\phi\right]\right)$ and the
maximally mixed state $I/d$ are also in $\mathcal{P}$. Then, $I/d$ is
preparation contextual in the ontological model.

###### Proof.

The maximally mixed state has convex decompositions

$\displaystyle\frac{I}{d}$
$\displaystyle=\frac{1}{d}\left[\psi\right]+\frac{d-1}{d}\sigma^{\psi^{\perp}}$
(61)
$\displaystyle=\frac{1}{d}\left[\phi\right]+\frac{d-1}{d}\sigma^{\phi^{\perp}}.$
(62)

Since the ontological and quantum models respect convexity, for every
$\mu\in\Delta_{\psi}$ and $\mu^{\perp}\in\Delta_{\sigma^{\psi^{\perp}}}$, the
measure $\frac{1}{d}\mu+\frac{d-1}{d}\mu^{\perp}$ is in $\Delta_{I/d}$.
Similarly, for every $\nu\in\Delta_{\phi}$ and
$\nu^{\perp}\in\Delta_{\sigma^{\phi^{\perp}}}$, the measure
$\frac{1}{d}\nu+\frac{d-1}{d}\nu^{\perp}$ is in $\Delta_{I/d}$. Assume that
$I/d$ is preparation noncontextual, so that $\Delta_{I/d}$ only contains one
probability measure $\zeta$. Then, we have

$\displaystyle\zeta$ $\displaystyle=\frac{1}{d}\mu+\frac{d-1}{d}\mu^{\perp}$
(63) $\displaystyle=\frac{1}{d}\nu+\frac{d-1}{d}\nu^{\perp}.$ (64)

Now, since $\left[\psi\right]$ and $\left[\phi\right]$ satisfy Eq. (60), and
the model reproduces the quantum predictions, there must exist a
$\mu\in\Delta_{\psi}$, a $\nu\in\Delta_{\phi}$, an $M\in\mathcal{M}$
containing $\left[\phi\right]$, a $\text{Pr}\in\Xi_{M}$, and a set
$\Omega\in\Sigma$ such that $\nu(\Omega)=1$, but

$\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\mu(\lambda)<\text{Tr}\left(\left[\phi\right]\left[\psi\right]\right).$
(65)

This means that, in order to reproduce the quantum predictions, there must be
a set $\tilde{\Omega}\subseteq\Lambda\backslash\Omega$ such that
$\text{Pr}(\phi|M,\lambda)$ is nonzero everywhere on $\tilde{\Omega}$ and
$\mu(\tilde{\Omega})>0$. It is also the case that $\nu(\tilde{\Omega})=0$
because $\tilde{\Omega}$ is a subset of $\Lambda\backslash\Omega$ and $\Omega$
is measure one for $\nu$.

Again, since the ontological model reproduces the quantum predictions, we must
have

$\int_{\Omega}\text{Pr}(\phi|M,\lambda)d\nu^{\perp}(\lambda)=\text{Tr}\left(\left[\phi\right]\sigma^{\phi^{\perp}}\right)=0.$
(66)

This implies that $\nu^{\perp}(\tilde{\Omega})=0$ because
$\text{Pr}(\phi|M,\lambda)$ is nonzero everywhere on $\tilde{\Omega}$.
Therefore we have

$\displaystyle\zeta(\tilde{\Omega})$
$\displaystyle=\frac{1}{d}\nu(\tilde{\Omega})+\frac{d-1}{d}\nu^{\perp}(\tilde{\Omega})$
(67) $\displaystyle=0,$ (68)

but also

$\displaystyle\zeta(\tilde{\Omega})$
$\displaystyle=\frac{1}{d}\mu(\tilde{\Omega})+\frac{d-1}{d}\mu^{\perp}(\tilde{\Omega})$
(69) $\displaystyle\geq\frac{1}{d}\mu(\tilde{\Omega})$ (70) $\displaystyle>0,$
(71)

which is a contradiction. Therefore, $I/d$ must be preparation contextual. ∎

Similar arguments can be used to establish the preparation contextuality of
other mixed states. If $\sigma$ and $\tau$ are states such that
$\text{Tr}\left(\left[\phi\right]\sigma\right)=0$ and

$p\left[\psi\right]+(1-p)\tau=q\left[\phi\right]+(1-q)\sigma,$ (72)

for some $0\leq p,q\leq 1$, then the state $\rho=p\left[\psi\right]+(1-p)\tau$
must be preparation contextual by the same argument given above. Banik et. al.
have recently used this method to show that all mixed states must be
preparation contextual in Hilbert spaces of dimension $d\geq 3$ [103]. They
also proved the same result for all Hilbert space dimensions, but via a method
that does not make use of the connection with maximal $\psi$-epistemicity.

###  5.4 Bell’s Theorem

The purpose of this section is to argue that being committed to
$\psi$-ontology implies the conclusion of Bell’s theorem, i.e. any
$\psi$-ontic model must fail to satisfy Bell’s locality condition. It is not
my intention to argue that the implication from $\psi$-ontology theorems to
Bell’s theorem is the best or most elegant way of proving Bell’s theorem. The
argument is quite technical compared to other proofs of Bell’s theorem and
$\psi$-ontology theorems are based on auxiliary assumptions that are not
required to prove Bell’s theorem. Instead, the aim is to compare the relative
strength of the conclusions of $\psi$-ontology theorems with those of Bell’s
theorem.

What we will actually do in this section is to argue that the kind of
preparation contextuality that follows from the failure of maximal
$\psi$-epistemicity implies the conclusion of Bell’s theorem. This means that
$\psi$-ontology implies the same, because it can be used to establish the
failure of maximal $\psi$-epistemicity. In principle, one can also view the
result of this section as a full-blown proof of Bell’s theorem that does not
rely on $\psi$-ontology, since, as established in Appendix C, the
Kochen–Specker theorem can alternatively be used to establish the failure of
maximal $\psi$-epistemicity without assuming the reality of the quantum state.
However, this would be quite a convoluted way of deriving Bell’s theorem
compared to the standard proofs.

From a historical perspective, the Einstein-Podolsky-Rosen argument [104]
could be read as an argument that $\psi$-complete models are nonlocal, but a
$\psi$-ontic model may involve other variables in addition to the
wavefunction. That nonlocality follows from the weaker assumption of
$\psi$-ontology is implicit in an argument that Einstein made in
correspondence with Schrödinger. This was first pointed out by Harrigan and
Spekkens in [74], where they also provide a more formal version of the
argument. However, both the Einstein and Harrigan-Spekkens arguments employ an
additional assumption known as _Einstein separability_ , which is discussed in
more detail in §7.3. Einstein separability is not actually required to prove
Bell’s theorem [105], so I do not make this assumption here. This, in addition
to taking care of measure-theoretic issues, makes the argument presented here
a bit more involved than Harrigan and Spekkens’ treatment. However, once this
is done, weakening their assumption of $\psi$-ontology to the failure of
maximal $\psi$-epistemicity does not significantly change the structure of the
argument. As presented here, the argument is based on an idea due to Barrett
[106] that has appeared in the literature in an informal version [56].

The basic idea runs as follows. Suppose Alice and Bob share a composite system
with Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$,
$\mathcal{H}_{A}=\mathcal{H}_{B}=\mathbb{C}^{d}$, prepared in the entangled
state $\left[\Phi^{+}\right]_{AB}$, where

$\left|\Phi^{+}\right\rangle_{AB}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\left|j\right\rangle_{A}\otimes\left|j\right\rangle_{B},$
(73)

and $\left\\{\left|j\right\rangle\right\\}_{j=0}^{d-1}$ is an orthonormal
basis for $\mathbb{C}^{d}$. If Alice performs a measurement described by the
projectors
$\left\\{\left[\psi\right]_{A}^{T},I_{A}-\left[\psi\right]_{A}^{T}\right\\}$,
where T denotes transpose in the $\left|j\right\rangle$ basis and $I_{A}$ is
the identity operator on $\mathcal{H}_{A}$, then, with probability $1/d$ she
obtains the $\left[\psi\right]_{A}^{T}$ outcome and the state of Bob’s system
gets updated to $\left[\psi\right]_{B}$, and with probability $(d-1)/d$ she
gets the $I_{A}-\left[\psi\right]_{A}^{T}$ outcome and the state of Bob’s
system gets updated to $\sigma^{\psi{\perp}}_{B}$. If we allow Alice to
postselect on her measurement outcome, then we can regard this as a method of
preparing Bob’s system in the states $\left[\psi\right]_{B}$ and
$\sigma^{\psi^{\perp}}_{B}$. If she subsequently forgets and discards her
measurement outcome then we can also regard this as a method of preparing
Bob’s system in the maximally mixed state by mixing $\left[\psi\right]_{B}$
with $\sigma^{\psi^{\perp}}_{B}$. Similar remarks apply if Alice measures
$\\{\left[\phi\right]^{T}_{A},I_{A}-\left[\phi\right]^{T}_{A}\\}$, which, by
postselection, prepares Bob’s system in the state $\left[\phi\right]_{B}$ or
$\sigma^{\phi^{\perp}}_{B}$ and, upon forgetting the measurement outcome,
prepares it in the maximally mixed state by mixing $\left[\phi\right]_{B}$
with $\sigma^{\phi^{\perp}}_{B}$.

Now, given an ontological model for the composite system that satisfies Bell’s
locality condition, we can use it to derive an ontological model for these
preparation procedures on Bob’s system alone. This model must be preparation
noncontextual for the state $I/d$ because locality implies that the
probability of getting a particular ontic state for the derived model on Bob’s
system cannot depend on whether Alice chooses to measure
$\left\\{\left[\psi\right]^{T}_{A},I_{A}-\left[\psi\right]^{T}_{A}\right\\}$
or
$\left\\{\left[\phi\right]^{T}_{A},I_{A}-\left[\phi\right]^{T}_{A}\right\\}$.
However, in order to construct a model for all possible preparations and
measurements of this form that reproduces the quantum predictions, preparation
noncontextuality must fail for some such pair because a maximally
$\psi$-epistemic model is impossible. Therefore, Bell’s locality condition
must fail. The remainder of this section formalizes this argument.

The type of PM fragment relevant to Bell’s Theorem involves a bipartite system
and local measurements. This is formalized in the concept of a product
measurement fragment.

######  Definition 5.6.

A _product measurement fragment_
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{A}\times\mathcal{M}_{B}\rangle$
is a PM fragment on a composite system with Hilbert space
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, where $\mathcal{M}_{A}$ is a set of
POVMs on $\mathcal{H}_{A}$, $\mathcal{M}_{B}$ is a set of POVMs on
$\mathcal{H}_{B}$ and $\mathcal{M}_{A}\times\mathcal{M}_{B}$ denotes the set
of all POVMs of the form $(M_{A},M_{B})=\\{E_{j}\otimes F_{k}\\}$, where
$M_{A}=\\{E_{j}\\}\in\mathcal{M}_{A}$ and
$M_{B}=\\{F_{k}\\}\in\mathcal{M}_{B}$.

To prove Bell’s Theorem, we only need consider a single bipartite state
$\rho_{AB}$, so for the remainder of this section we assume that
$\mathcal{P}_{AB}$ just contains this single state. Most of the concepts can
be generalized to fragments with more than one state in an obvious way.

The first step is to reinterpret experiments on the composite system as
prepare-and-measure experiments on Bob’s system alone. At the operational
level, when the system is prepared in the state $\rho_{AB}$, Alice makes a
measurement $M_{A}$ on system $A$ and Bob makes a measurement $M_{B}$ on
system $B$, quantum theory predicts the outcome probabilities

$\text{Prob}(E,F|M_{A},M_{B})=\text{Tr}_{AB}\left(E\otimes F\rho_{AB}\right),$
(74)

where $E\in M_{A}$ and $F\in M_{B}$.

These probabilities can be rewritten in terms of states and measurements on
$\mathcal{H}_{B}$ alone by imagining that Alice performs her measurement
before Bob performs his, and considering how the state of Bob’s system gets
updated in light of Alice’s measurement result. Upon learning that the outcome
of $M_{A}$ was $E$, the state of Bob’s system gets updated to

$\rho_{B|E}=\frac{\text{Tr}_{A}\left(E\otimes
I_{B}\rho_{AB}\right)}{\text{Prob}(E|M_{A})},$ (75)

where $\text{Prob}(E|M_{A})=\text{Tr}_{A}\left(E\rho_{A}\right)$ is the
probability of Alice’s measurement outcome and
$\rho_{A}=\text{Tr}_{B}\left(\rho_{AB}\right)$. Eq. (74) can then be rewritten
as

$\text{Prob}(E,F|M_{A},M_{B})=\text{Prob}(E|M_{A})\text{Tr}_{B}\left(F\rho_{B|E}\right).$
(76)

By postselecting on obtaining the outcome $E$, Alice’s measurement can be
viewed as a method of preparing Bob’s system in the state $\rho_{B|E}$. In
doing so, $E$ gets reinterpreted as specifying a preparation procedure for
system $B$, instead of a measurement on $A$, so we now want to condition the
probabilities on $E$. This gives

$\text{Prob}(F|M_{A},E,M_{B})=\frac{\text{Prob}(E,F|M_{A},M_{B})}{\text{Prob}(E|M_{A})}=\text{Tr}_{B}\left(F\rho_{B|E}\right).$
(77)

We also want to consider the case where Alice forgets her measurement outcome
outcome. This prepares Bob’s system in the state
$\rho_{B}=\text{Tr}_{A}\left(\rho_{AB}\right)$ and we have, for every
$M_{A}\in\mathcal{M}_{A}$,

$\rho_{B}=\sum_{E\in M_{A}}\text{Prob}(E|M_{A})\rho_{B|E}.$ (78)

We can now define the conditional fragment on Bob’s system consisting of all
the states that Alice can prepare on his system in this way.

######  Definition 5.7.

Let
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{A}\times\mathcal{M}_{B}\rangle$
be a product measurement fragment, where $\mathcal{P}_{AB}$ contains a single
state $\rho_{AB}$. The _conditional_ fragment on $B$ given $A$ is
$\mathfrak{F}_{B|A}=\langle\mathcal{H}_{B},\mathcal{P}_{B|A},\mathcal{M}_{B}\rangle$,
where $\mathcal{P}_{B|A}$ consists of all states of the form $\rho_{B|E}$ as
given by Eq. (75) where $E\in M_{A}$ for some $M_{A}\in\mathcal{M}_{A}$, as
well as the state $\rho_{B}=\text{Tr}_{A}\left(\rho_{AB}\right)$.

Typically, there are several different methods of preparing the same quantum
state and of implementing the same measurement in a PM fragment. These
different methods need not correspond to the same probability measures nor the
same conditional probability distributions in an ontological model. However,
the derivation of Bell’s theorem is greatly simplified if we focus attention
on one specific method of preparing $\rho_{AB}$ and one one method of
implementing each of the measurements $M_{A}\in\mathcal{M}_{A}$ and
$M_{B}\in\mathcal{M}_{B}$. Then, we can assume that $\rho_{AB}$ is represented
by a unique measure $\mu$ and that each $(M_{A},M_{B})$ is represented by a
unique conditional probability distribution
$\text{Pr}(E,F|M_{A},M_{B},\lambda)$, where $E\in M_{A}$ and $F\in M_{B}$.
This does not weaken the conclusion because the result will hold regardless of
which method of implementation is chosen. Therefore, for the remainder of this
section, we assume a unique measure and conditional probabilities for
ontological models of the product measurement fragment.

Given such an ontological model of $\mathfrak{F}_{AB}$, it is natural to think
that an ontological model for the conditional PM fragment $\mathfrak{F}_{B|A}$
might be derivable from it. However, in general, an ontological model for
$\mathfrak{F}_{AB}$ cannot be cleanly separated into a part that depends on
system $A$ and a part that depends on system $B$ unless it satisfies Bell’s
condition of local causality. To see this, suppose we have an ontological
model of $\mathfrak{F}_{AB}$. If Alice chooses to make the measurement $M_{A}$
and Bob chooses to make the measurement $M_{B}$ then in general, not assuming
locality, this is represented by a conditional probability distribution
$\text{Pr}(E,F|M_{A},M_{B},\lambda)$, where $E\in M_{A}$ and $F\in M_{B}$. The
model then predicts that the observed probability of obtaining the outcomes
$E$ and $F$ is given by

$\text{Pr}(E,F|M_{A},M_{B})=\int_{\Lambda}\text{Pr}(E,F|M_{A},M_{B},\lambda)d\mu(\lambda),$
(79)

where $\mu$ is the measure representing $\rho_{AB}$.

At the quantum level, the conditional probability
$\text{Prob}(F|M_{A},E,M_{B})$ could be rewritten in terms of a state on
$\mathcal{H}_{B}$ that depends on $E$ alone and the measurement operator $F$,
which is also defined on $\mathcal{H}_{B}$. The analogue at the ontological
level would be to absorb the dependence on $M_{A}$ and $E$ into the measure
$\mu$ and be left with a conditional probability distribution that only
depends on $M_{B}$ and $F$. Unfortunately, the conditional probability
distribution $\text{Pr}(E,F|M_{A},M_{B},\lambda)$ need not factor cleanly into
a term that only depends on $E$ and $M_{A}$ and a term that only depends on
$F$ and $M_{B}$. In order to obtain such a decomposition, a locality
assumption is required.

######  Definition 5.8.

An ontological model of a product measurement fragment
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{A}\times\mathcal{M}_{B}\rangle$
is _Bell local_ if, for all $M_{A}\in\mathcal{M}_{A}$,
$M_{B}\in\mathcal{M}_{B}$, $E\in M_{A}$, $F\in M_{B}$, $\lambda\in\Lambda$,

$\text{Pr}(E,F|M_{A},M_{B},\lambda)=\text{Pr}(E|M_{A},\lambda)\text{Pr}(F|M_{B},\lambda)$
(80)

where $\text{Pr}(E|M_{A},\lambda)$ is a valid conditional probability
distribution over $M_{A}$ and $\text{Pr}(F|M_{B},\lambda)$ is a valid
conditional probability distribution over $M_{B}$.

In a Bell local model of $\mathfrak{F}_{AB}$, measures conditional on $M_{A}$
and $E\in M_{A}$ can be defined as follows,

$\mu_{M_{A},E}(\Omega)=\frac{1}{\text{Pr}(E|M_{A})}\int_{\Omega}\text{Pr}(E|M_{A},\lambda)d\mu(\lambda),$
(81)

where
$\text{Pr}(E|M_{A})=\int_{\Lambda}\text{Pr}(E|M_{A},\lambda)d\mu(\lambda)$ is
the observed probability of obtaining the outcome $E$ for the measurement
$M_{A}$ according to the ontological model. Eq. (80) can then be rewritten as

$\text{Pr}(E,F|M_{A},M_{B},)=\text{Pr}(E|M_{A})\int_{\Lambda}\text{Pr}(F|M_{B},\lambda)d\mu_{M_{A},E}(\lambda),$
(82)

and conditioning on $E$ gives

$\text{Pr}(F|M_{A},E,M_{B})=\int_{\Lambda}\text{Pr}(F|M_{B},\lambda)d\mu_{M_{A},E}(\lambda),$
(83)

which is the analogue of Eq. (77) at the ontological level.

Further, in the case where Alice does not report her measurement result we
have,

$\text{Pr}(F|M_{A},M_{B})=\int_{\Lambda}\text{Pr}(F|M_{B},\lambda)d\mu(\lambda),$
(84)

Thus, an ontological model for the conditional theory can be defined as
follows.

######  Definition 5.9.

Let
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{A}\times\mathcal{M}_{B}\rangle$
be a product measurement fragment, where $\mathcal{P}_{AB}$ contains a single
state $\rho_{AB}$, and consider a Bell local ontological model of it in which
$\rho_{AB}$ is represented by a unique measure $\mu$, each
$M_{A}\in\mathcal{M}_{A}$ is represented by a unique conditional probability
distribution $\text{Pr}(E|M_{A},\lambda)$, and each $M_{B}\in\mathcal{M}_{B}$
is represented by a unique conditional probability distribution
$\text{Pr}(F|M_{B},\lambda)$. Then, the _conditional ontological model_ of the
conditional fragment $\mathfrak{F}_{B|A}$ is defined by

  * •

Placing the measure $\mu_{M_{A},E}$ in $\Delta_{\rho_{B|E}}$, where

$\mu_{M_{A},E}(\Omega)=\frac{1}{\text{Pr}_{A}(E|M_{A})}\int_{\Omega}\text{Pr}(E|M_{A},\lambda)d\mu(\lambda).$
(85)

Note that there may be more than one measure in $\Delta_{\rho_{B|E}}$ if $E$
occurs in more than one of the POVMs in $\mathcal{M}_{A}$.

  * •

Associating the unique measure $\mu$ with the state
$\rho_{B}=\text{Tr}_{A}\left(\rho_{AB}\right)$.

  * •

Associating the unique conditional probability distribution
$\text{Pr}(F|M_{B},\lambda)$ as defined in the Bell local model with each
$M_{B}\in\mathcal{M}_{B}$.

It is straightforward to see that if the original ontological model reproduces
the quantum predictions for $\mathfrak{F}_{AB}$ then the conditional
ontological model reproduces the quantum predictions of $\mathfrak{F}_{B|A}$.

We now specialize to the product measurement fragment of interest for Bell’s
Theorem. This is
$\mathfrak{F}_{AB}=\left\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{A}\times\mathcal{M}_{B}\right\rangle$,
where $\mathcal{H}_{A}=\mathcal{H}_{B}=\mathbb{C}^{d}$,
$\mathcal{P}_{AB}=\left\\{\left[\Phi^{+}\right]_{AB}\right\\}$,
$\mathcal{M}_{A}$ consists of all projective measurements of the form
$M_{A}=\\{E_{0},E_{1}\\}$ where $E_{0}=\left[\psi\right]$,
$E_{1}=I_{A}-\left[\psi\right]$ for some
$\left|\psi\right\rangle\in\mathbb{C}^{d}$ and $\mathcal{M}_{B}$ consists of
all measurements on $\mathcal{H}_{B}$.

Given the available measurements in $\mathcal{M}_{A}$, $\rho_{B|E_{0}}$ is
always a pure state and in fact Alice can prepare Bob’s system in any pure
state by choosing an appropriate $E_{0}$, i.e. $\left[\psi\right]^{T}_{A}$ if
she wants to prepare $\left[\psi\right]_{B}$.

######  Theorem 5.10.

Consider a Bell local model for the product measurement fragment
$\mathfrak{F}_{AB}$ defined above that reproduces the quantum predictions for
Alice’s measurements, i.e. for all $M_{A}\in\mathcal{M}_{A}$, $E\in M_{A}$,

$\int_{\Lambda}\mathrm{Pr}(E|M_{A},\lambda)d\mu(\lambda)=\mathrm{Tr}_{AB}\left(E\otimes
I_{B}\left[\Phi^{+}\right]_{AB}\right).$ (86)

Then, the conditional ontological model of the conditional fragment
$\mathfrak{F}_{B|A}$ is preparation noncontextual and respects convex
decompositions.

###### Proof.

Given the choice of $\left[\Phi^{+}\right]_{AB}$, there is only one POVM
$M_{A}=\\{E_{0},E_{1}\\}$ that can lead to a given conditional state
$\rho_{B|E_{j}}$. The possible conditional states on Bob’s side are
$\rho_{B|E_{0}}=\left[\psi\right]_{B}$ and
$\rho_{B|E_{1}}=\sigma^{\psi^{\perp}}_{B}=\frac{1}{d-1}(I_{B}-\left[\psi\right]_{B})$,
which occur when Alice makes the measurement
$\\{\left[\psi\right]^{T}_{A},I_{A}-\left[\psi\right]^{T}_{A}\\}$, and so each
of Alice’s measurements determine a unique pair of conditional states for Bob.
Therefore, each $\Delta_{\rho_{B|E_{j}}}$ contains only a single measure, so
the model is preparation noncontextual for these states. Similarly, by
construction, there is a unique measure associated with $I/d$.

The only convex combinations that appear in $\mathfrak{F}_{B|A}$ are those of
the form

$\frac{I_{B}}{d}=\sum_{j=0}^{1}\text{Prob}(E_{j}|M_{A})\rho_{B|E_{j}},$ (87)

where $\text{Prob}(E_{j}|M_{A})=\text{Tr}_{AB}\left(E_{j}\otimes
I_{B}\left[\Phi^{+}\right]_{AB}\right)$. Therefore, the result follows if

$\mu=\sum_{j=0}^{1}\text{Prob}(E_{j}|M_{A})\mu_{M_{A},E_{j}}.$ (88)

To see this, note that $\sum_{j}\text{Pr}(E_{j}|M_{A},\lambda)=1$ for all
$\lambda$, so, for $\Omega\in\Sigma$,

$\displaystyle\mu(\Omega)$ $\displaystyle=\int_{\Omega}d\mu(\lambda)$ (89)
$\displaystyle=\int_{\Omega}\left[\sum_{j}\text{Pr}(E_{j}|M_{A},\lambda)\right]d\mu(\lambda)$
(90)
$\displaystyle=\sum_{j}\int_{\Omega}\text{Pr}(E_{j}|M_{A},\lambda)d\mu(\lambda)$
(91)
$\displaystyle=\sum_{j}\text{Pr}(E_{j}|M_{A})\frac{1}{\text{Pr}(E_{j}|M_{A})}\int_{\Omega}\text{Pr}(E_{j}|M_{A},\lambda)d\mu(\lambda)$
(92) $\displaystyle=\sum_{j}\text{Pr}(E_{j}|M_{A})\mu_{M_{A},E_{j}}(\Omega),$
(93)

where
$\text{Pr}(E_{j}|M_{A})=\int_{\Lambda}\text{Pr}(E_{j}|M_{A},\lambda)d\mu(\lambda)$
is the probability of Alice obtaining outcome $E_{j}$ when measuring $M_{A}$
according to the ontological model. By assumption, the model reproduces the
quantum predictions for Alice’s measurements, so
$\text{Pr}(E_{j}|M_{A})=\text{Prob}(E_{j}|M_{A})$, from which the result
follows. ∎

Theorem 5.10 is enough to show that a $\psi$-ontic model of the above product
measurement fragment that reproduces the quantum predictions cannot be Bell
local. To see this note that Theorem 5.2 shows that a $\psi$-ontic model
cannot be maximally $\psi$-epistemic and hence, by Theorem 5.5, if convexity
is respected then preparation noncontextuality must fail for the maximally
mixed state. However, a Bell local model would give rise to a preparation
noncontextual ontological model for the conditional theory that respects
convexity. Since the conditional fragment contains the states needed for
Theorem 5.5, this implies that Bell locality must fail.

Note that, instead of relying on $\psi$-ontology one could use the
Kochen–Specker theorem to rule out maximally $\psi$-epistemic theories for
$d\geq 3$, as described in Appendix C. Then, Theorem 5.10 can be viewed as a
full proof of Bell’s theorem that does not assume $\psi$-ontology, although
this is quite complicated compared to the standard proofs.

## Part II. $\psi$-ontology theorems

Having understood the implications of $\psi$-ontology, it is finally time to
look at the extent to which it can be established. The first $\psi$-ontology
theorem that was discovered is due to Pusey, Barrett and Rudolph. Of the
available results, it is the most widely discussed in the literature and I
think it makes the strongest case for $\psi$-ontology, so it receives the
longest treatment in §7, including a discussion of generalizations and
criticisms. Following this, I discuss two further $\psi$-ontology theorems:
Hardy’s Theorem in §9, and the Colbeck–Renner Theorem in §10.

Before getting to these theorems, I discuss the concept of
_antidistinguishability_ in §6 because it plays a key role in both the
Pusey–Barrett–Rudolph and Hardy’s theorems, as well as in some of the results
discussed in Part III. Both Hardy’s Theorem and the Colbeck–Renner Theorem
involve assumptions about how dynamics are represented in ontological models,
so this is discussed in §8 after the discussion of Pusey–Barrett–Rudolph is
completed.

##  6 Antidistinguishability

According to Theorem 4.10, sets of operationally distinguishable states are
pairwise ontologically distinct in an ontological model. However,
$\psi$-ontology requires that all pairs of pure states must be ontologically
distinct, and most pairs of pure states are not orthogonal. Therefore, to
prove $\psi$-ontology, it is useful to consider a weaker concept.

######  Definition 6.1.

A finite set of quantum states $\\{\rho_{j}\\}_{j=1}^{n}\subseteq\mathcal{P}$
is _antidistinguishable_ if there exists a POVM
$M=\left\\{E_{j}\right\\}_{j=1}^{n}\in\mathcal{M}$ such that, for each $j$,

$\text{Tr}\left(E_{j}\rho_{j}\right)=0.$ (94)

Recall that the definition of distinguishability states that there should
exist a POVM $M=\\{E_{j}\\}_{j=1}^{n}\in\mathcal{M}$ such that
$\text{Tr}\left(E_{k}\rho_{j}\right)=\delta_{jk}$ for all $j$ and $k$. This is
equivalent to requiring $\text{Tr}\left(E_{j}\rho_{j}\right)=1$ for all $j$ by
the following argument. Firstly, $\text{Tr}\left(E_{j}\rho_{j}\right)=1$ is
obviously a special case of $\text{Tr}\left(E_{k}\rho_{j}\right)=\delta_{jk}$.
For the converse direction note that if there were a $k\neq j$ such that
$\text{Tr}\left(E_{k}\rho_{j}\right)>0$ then
$\text{Tr}\left(\left(E_{j}+E_{k}\right)\rho_{j}\right)>1$, but this cannot be
the case if $\rho_{j}$ is a density operator and $\\{E_{j}\\}_{j=1}^{n}$ is a
POVM because then we must have
$1=\text{Tr}\left(\rho_{j}\right)=\text{Tr}\left(\sum_{m}E_{m}\rho_{j}\right)\geq\text{Tr}\left(\left(E_{j}+E_{k}\right)\rho_{j}\right)$.
Therefore, the difference between distinguishability and
antidistinguishability is simply the replacement of
$\text{Tr}\left(E_{j}\rho_{j}\right)=1$ with
$\text{Tr}\left(E_{j}\rho_{j}\right)=0$. To understand what this means,
suppose that one of the states in a set $\mathcal{D}$ is prepared and you do
not know which. If $\mathcal{D}$ is distinguishable then there is a
measurement for which each outcome identifies that a specific member of
$\mathcal{D}$ was definitely the state prepared. On the other hand, if
$\mathcal{D}$ is antidistinguishable then there is a measurement for which
each outcome identifies that a specific member of $\mathcal{D}$ was definitely
not the state prepared. In this sense, the two concepts are opposites.

Antidistinguishability is a weaker property than distinguishability because,
if the measurement outcome $j$ tells us that $\rho_{j}$ was definitely
prepared, then it also tells us that $\rho_{k}$ was definitely not prepared
for all $k\neq j$. Therefore, a distinguishing measurement can be converted to
an antidistinguishing one just by permuting the outcome indices in such a way
that none of them are left invariant, e.g. if there are $n$ POVM elements then
this can be done by mapping $j\rightarrow j+1$, where $n+1$ is identified with
$1$. For sets of three or more states, antidistinguishability is strictly
weaker than distinguishability, as there exist sets of nonorthogonal quantum
states that can be antidistinguished (see Fig. 11 for an example). For a set
of two states, antidistinguishability is equivalent to distinguishability
because if you know that one of the states was not the one prepared then you
know that it must have been the other one.

Figure 11: A set of nonorthogonal states that can be antidistinguished, and
the POVM that antidistinguishes them. The diagram represents the equator of
the Bloch sphere and the three antidistinguishable states are shown by the
solid arrows. The dashed arrows show three states that are each orthogonal to
one of the states in the antidistinguishable set. A POVM can be formed by
scaling the projectors onto the three dashed states by a factor of $2/3$. By
virtue of the orthogonality relations, this is an antidistinguishing POVM.

Like distinguishability, antidistinguishability has consequences for the
overlaps of probability measures in ontological models. However, whilst
distinguishability constrains the pairwise overlaps, antidistinguishability
only constrains the $n$-way overlaps, where $n$ is the number of states in the
antidistinguishable set. Since the variational distance only applies to two
measures, a more general notion of overlap is needed to capture this.

######  Definition 6.2.

A _countable partition_ $\\{\Omega_{k}\\}$ of a measurable space
$(\Lambda,\Sigma)$ is an at most countable set of disjoint subsets
$\Omega_{k}\in\Sigma$, $\Omega_{k}\cap\Omega_{r}=\emptyset$ for $k\neq r$,
such that $\cup_{k}\Omega_{k}=\Lambda$. $T(\Lambda)$ denotes the set of
countable partitions of $(\Lambda,\Sigma)$.

######  Definition 6.3.

The _overlap_ of an at most countable set $\\{\mu_{j}\\}$ of probability
measures on a sample space $(\Lambda,\Sigma)$ is given by

$L\left(\left\\{\mu_{j}\right\\}\right)=\inf_{\\{\Omega_{k}\\}\in
T(\Lambda)}\left(\sum_{k}\min_{j}\left(\mu_{j}(\Omega_{k})\right)\right).$
(95)

For a set $\\{\mu,nu\\}$ of two measures, we write $L(\mu,\nu)$ for
$L(\\{\mu,nu\\})$.

It can be shown that if $m$ is a measure that dominates $\\{\mu_{j}\\}$ then
the overlap is given by

$L\left(\left\\{\mu_{j}\right\\}\right)=\int_{\Lambda}\min_{j}\left(p_{j}(\lambda)\right)dm(\lambda),$
(96)

where $p_{j}$ is a density that represents $\mu_{j}$ with respect to $m$, i.e.
$\mu_{j}(\Omega)=\int_{\Omega}p_{j}(\lambda)dm(\lambda)$. For a finite set of
measures, a dominating measure always exists. This form of the overlap is
often most useful for calculations.

Note that, if $\\{\mu_{j}\\}$ contains $n$ measures then the optimization in
Eq. (95) can be restricted to partitions that contain $n$ subsets. This is
because, if a partition contains $m>n$ subsets, then the same $\mu_{j}$ is
bound to be the minimum for more than one $\Omega_{k}$ and these sets can then
be replaced by their union without affecting the result. Therefore, the
overlap is given by

$L(\\{\mu_{j}\\})=\min_{\\{\Omega_{k}\\}\in
T_{n}(\Lambda)}\left(\sum_{k=1}^{n}\min_{j}\left(\mu_{j}(\Omega_{k})\right)\right),$
(97)

where $T_{n}(\Lambda)$ is the set of $n$-fold partitions of $\Lambda$. In
fact, this can be simplified further to

$L(\\{\mu_{j}\\})=\min_{\\{\Omega_{j}\\}\in
T_{n}(\Lambda)}\left(\sum_{j}\mu_{j}(\Omega_{j})\right),$ (98)

because the minimizing partition can always be relabeled such that $k$ gets
mapped to the $j$ that minimizes $\mu_{j}(\Omega_{k})$.

The overlap has the following operational interpretation. Suppose that a
system is prepared according to one of $n$ preparation procedures $P_{j}$,
where $P_{j}$ corresponds to the measure $\mu_{j}$, each case having equal a
priori probability $1/n$. You are then told the actual value of $\lambda$ and
your task is to announce a $j$ between $1$ and $n$ such that $P_{j}$ was _not_
the procedure used to prepare the system, i.e. you want to antidistinguish the
probability measures. If you adopt a deterministic strategy then the best you
can do is to partition the space into $n$ subsets $\Omega_{j}$ and announce
that $P_{j}$ was not the preparation procedure used when
$\lambda\in\Omega_{j}$. The probability of failure when using this strategy is
$\frac{1}{n}\sum_{j=1}^{n}\mu_{j}(\Omega_{j})$ so the minimal probability of
failure is $\frac{1}{n}L(\\{\mu_{j}\\})$. By convexity, a probabilistic
strategy cannot do any better than this. Therefore, $L(\\{\mu_{j}\\})=0$
corresponds to the case where the probability measures can be
antidistinguished perfectly.

The overlap generalizes the variational distance in the following sense.

######  Theorem 6.4.

For two probability measures $\mu,\nu$ on $(\Lambda,\Sigma)$,
$L(\mu,\nu)=1-D(\mu,\nu)$, where $D$ is the variational distance.

###### Proof.

From Eq. (98),

$\displaystyle L(\mu,\nu)$
$\displaystyle=\inf_{\Omega\in\Sigma}\left(\mu(\Omega)+\nu(\Lambda\backslash\Omega)\right)$
(99)
$\displaystyle=\inf_{\Omega\in\Sigma}\left(\mu(\Omega)+1-\nu(\Omega)\right)$
(100)
$\displaystyle=1-\sup_{\Omega\in\Sigma}\left(\nu(\Omega)-\mu(\Omega)\right)$
(101)
$\displaystyle=1-\sup_{\Omega\in\Sigma}\left|\mu(\Omega)-\nu(\Omega)\right|$
(102) $\displaystyle=1-D(\mu,\nu).$ (103)

∎

Because of this, the condition for two quantum states $\rho$ and $\sigma$ to
be ontologically distinct in an ontological model can be restated as: for all
$\mu\in\Delta_{\rho}$ and $\nu\in\Delta_{\sigma}$, $L(\mu,\nu)=0$.

The reason for introducing the more general $n$-way overlaps is that, being a
weaker concept than distinguishability, antidistinguishability only places
constraints on the $n$-way overlaps rather than on pairwise overlaps. To
understand the intuition behind these constraints, consider the case of a
finite ontic state space and suppose that there is an ontic state $\lambda$
that is assigned nonzero probability by every measure $\mu_{j}$ in a set of
probability measures $\\{\mu_{j}\\}_{j=1}^{n}$ corresponding to an
antidistinguishable set of preparations $\\{\rho_{j}\\}_{j=1}^{n}$. Then,
$\lambda$ must assign zero probability to each of the outcomes of the
antidistinguishing measurement. However, the probabilities that $\lambda$
assigns to the measurement outcomes must form a probability distribution, so
they cannot all be zero. Hence, no such $\lambda$ can exist.

In order to formulate this argument rigorously, we have to deal with measure
zero sets. For this reason, it is better to formulate the argument in terms of
the overlap.

######  Theorem 6.5.

Let $\\{\rho_{j}\\}_{j=1}^{n}$ be an antidistinguishable set of states in a PM
fragment. Then, in any ontological model that reproduces the quantum
predictions,

$L(\\{\mu_{j}\\})=0.$ (104)

for every possible choice of probability measures
$\mu_{j}\in\Delta_{\rho_{j}}$.

###### Proof.

Let $M=\\{E_{j}\\}$ be the antidistinguishing POVM, let
$\mu_{j}\in\Delta_{\rho_{j}}$, and let $m$ be a measure that dominates
$\\{\mu_{j}\\}$ such that, for $\Omega\in\Sigma$,
$\mu_{j}(\Omega)=\int_{\Omega}p_{j}(\lambda)dm(\lambda)$ for some densities
$p_{j}$. In order to reproduce the predictions of $M$, for every
$\text{Pr}\in\Xi_{M}$, the ontological model must satisfy

$\displaystyle\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)d\mu_{j}(\lambda)$
$\displaystyle=$
$\displaystyle\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)p_{j}(\lambda)dm(\lambda)$
(105) $\displaystyle=$ $\displaystyle\text{Tr}\left(E_{j}\rho_{j}\right)=0.$

However,

$\displaystyle\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)p_{j}(\lambda)dm(\lambda)\geq$
$\displaystyle\leavevmode\nobreak\
\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)\min_{k}\left(p_{k}(\lambda)\right)dm(\lambda),$
(106)

so

$\int_{\Lambda}\text{Pr}(E_{j}|M,\lambda)\min_{k}\left(p_{k}(\lambda)\right)dm(\lambda)=0.$
(107)

Now, sum both sides over $j$. Because, $\sum_{j}\text{Pr}(E_{j}|M,\lambda)=1$
for every $\lambda\in\Lambda$, this gives

$\int_{\Lambda}\min_{k}\left(p_{k}(\lambda)\right)dm(\lambda)=0,$ (108)

or in other words, by Eq. (96), $L\left(\left\\{\mu_{k}\right\\}\right)=0$. ∎

Finally, a few remarks are in order about terminology. Antidistinguishability
was first introduced by Caves, Fuchs and Shack in their study of the
compatibility of quantum state assignments, where it was called “post-Peierls
(PP) incompatibility” [107]. The concept did not attract much attention until
its use in the Pusey–Barrett–Rudolph Theorem, but since then more systematic
studies of antidistinguishability have appeared [108, 109], where it goes
under the name “conclusive exclusion of quantum states”. Therefore, if you see
the terms “PP incompatibility” and “conclusive exclusion” in the literature
then you can rest assured that they mean the same thing as
antidistinguishability. I think “antidistinguishability” is a more suggestive
name, but time will tell which terminology becomes standard.

##  7 The Pusey–Barrett–Rudolph Theorem

The Pusey–Barrett–Rudolph Theorem is the most prominent $\psi$-ontology
theorem. It is based on an assumption known as the Preparation Independence
Postulate (PIP), which essentially says that composite systems prepared in a
product state should be independent of one another. It is this postulate that
allows bounds on $n$-way overlaps coming from antidistinguishability to be
leveraged into the bounds on pairwise overlaps needed to establish
$\psi$-ontology. The PIP is presented in its simplest form in §7.1 in order to
facilitate the simplest possible proof of the main result in §7.2. However,
the PIP is the most controversial assumption of the Pusey–Barrett–Rudolph
Theorem, so its motivation is discussed in §7.3, weakenings of it that still
allow the Pusey–Barrett–Rudolph Theorem to go through are discussed in §7.4,
its necessity for proving the theorem is established in §7.5, and criticism of
it is discussed in §7.6. Finally, other criticisms of the
Pusey–Barrett–Rudolph Theorem, not directed at the PIP, are discussed in §7.7.

###  7.1 The Preparation Independence Postulate

The Preparation Independence Postulate (PIP) is an assumption about how
composite systems should be modeled when the subsystems are prepared
independently of one another. Suppose Alice prepares her system in a quantum
state $\rho_{A}$ and Bob prepares his system in a quantum state $\rho_{B}$,
such that their joint state is a product $\rho_{A}\otimes\rho_{B}$. Suppose
further that Alice has an ontological model with ontic state space
$(\Lambda_{A},\Sigma_{A})$ that reproduces the quantum predictions for her
system in isolation and that $\mu_{A}$ is the probability measure that it
assigns to her preparation procedure. Similarly, Bob has an ontological model
for his system with ontic state space $(\Lambda_{B},\Sigma_{B})$ that assigns
the measure $\mu_{B}$ to his preparation procedure. Then, the PIP says that
there ought to be an ontological model for the joint system with ontic state
space $(\Lambda_{A}\times\Lambda_{B},\Sigma_{A}\otimes\Sigma_{B})$ in which
the product measure

$\mu_{A}\times\mu_{B}(\Omega)=\int_{\Lambda_{B}}\mu_{A}(\Omega_{\lambda_{B}})d\mu_{B}(\lambda_{B}),$
(109)

reproduces the operational predictions for $\rho_{A}\otimes\rho_{B}$, where
$\Omega_{\lambda_{B}}=\\{\lambda_{A}\in\Lambda_{A}|(\lambda_{A},\lambda_{B})\in\Omega\\}$.

This assumption looks superficially plausible, and indeed it is not much
commented upon in the Pusey–Barrett–Rudolph paper. However, since it is the
most controversial assumption of the Pusey–Barrett–Rudolph Theorem, it is
worth pausing to define it a bit more rigorously. Discussion of the motivation
for the PIP and criticism of it is deferred until §7.3–§7.6.

To start with, we need to understand how fragments for subsystems can be
combined to form fragments for composite systems.

######  Definition 7.1.

Let
$\mathfrak{F}_{A}=\langle\mathcal{H}_{A},\mathcal{P}_{A},\mathcal{M}_{A}\rangle$
and
$\mathfrak{F}_{B}=\langle\mathcal{H}_{B},\mathcal{P}_{B},\mathcal{M}_{B}\rangle$
be PM fragments. The _direct product_ of $\mathfrak{F}_{A}$ and
$\mathfrak{F}_{B}$ is

$\mathfrak{F}_{A}\times\mathfrak{F}_{B}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{A}\times\mathcal{P}_{B},\mathcal{M}_{A}\times\mathcal{M}_{B}\rangle,$
(110)

where

  * •

$\mathcal{P}_{A}\times\mathcal{P}_{B}$ consists of all product states
$\rho_{A}\otimes\rho_{B}$ with $\rho_{A}\in\mathcal{P}_{A}$ and
$\rho_{B}\in\mathcal{P}_{B}$.

  * •

$\mathcal{M}_{A}\times\mathcal{M}_{B}$ consists of all POVMs of the form
$\\{E_{j}\otimes F_{k}\\}$ with $\\{E_{j}\\}\in\mathcal{M}_{A}$ and
$\\{F_{k}\\}\in\mathcal{M}_{B}$.

A _general product_ (or just product) of $\mathfrak{F}_{A}$ and
$\mathfrak{F}_{B}$ is any PM fragment of the form

$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{AB}\rangle,$
(111)

where

  * •

$\mathcal{P}_{A}\times\mathcal{P}_{B}\subseteq\mathcal{P}_{AB}$.

  * •

$\mathcal{M}_{A}\times\mathcal{M}_{B}\subseteq\mathcal{M}_{AB}$.

$\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$ are called the _factors_ of
$\mathfrak{F}_{AB}$.

A direct product fragment represents a situation in which there are two
separated systems and we can only do separate prepare-and-measure experiments
on them individually. It only allows product states and local measurements, so
there is no possibility of entanglement or even classical correlation between
the two systems. A general product fragment contains all the states and
measurements that are in the direct product, but it may, in addition, include
some extra correlated or entangled states and measurements on the composite
system.

Any two ontological models
$\Theta_{A}=(\Lambda_{A},\Sigma_{A},\Delta_{A},\Xi_{A})$ and
$\Theta_{B}=(\Lambda_{B},\Sigma_{B},\Delta_{B},\Xi_{B})$ for
$\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$ can be combined to form an
ontological model
$\Theta_{AB}=(\Lambda_{AB},\Sigma_{AB},\Delta_{AB},\Xi_{AB})$ for the direct
product $\mathfrak{F}_{A}\times\mathfrak{F}_{B}$. To do so, set

  * •

$\Lambda_{AB}=\Lambda_{A}\times\Lambda_{B}$ and
$\Sigma_{AB}=\Sigma_{A}\otimes\Sigma_{B}$.

  * •

For all $\rho_{A}\in\mathcal{P}_{A}$, $\rho_{B}\in\mathcal{P}_{B}$, let
$\Delta_{AB}[\rho_{A}\otimes\rho_{B}]=\Delta_{A}[\rho_{A}]\times\Delta_{B}[\rho_{B}]$
be the set of measures of the form $\mu_{AB}=\mu_{A}\times\mu_{B}$, where
$\mu_{A}\in\Delta_{A}[\rho_{A}]$ and $\mu_{B}\in\Delta_{B}[\rho_{B}]$.

  * •

For all $M_{A}\in\mathcal{M}_{A}$, $M_{B}\in\mathcal{M}_{B}$, let
$\Xi_{AB}[M_{A},M_{B}]$ be the set of conditional probability distributions of
the form $\text{Pr}_{AB}(E\otimes
F|M_{A},M_{B}\lambda_{A},\lambda_{B})=\text{Pr}_{A}(E|M_{A},\lambda_{A})\text{Pr}_{B}(F|M_{B},\lambda_{B})$,
where $\text{Pr}_{A}\in\Xi_{A}[M_{A}]$ and $\text{Pr}_{B}\in\Xi_{B}[M_{B}]$.

Then, we have, for $\mu_{AB}\in\Delta_{AB}[\rho_{A}\otimes\rho_{B}]$,

$\displaystyle\text{Pr}_{AB}(E,F|\rho_{A}\otimes\rho_{B},M_{A},M_{B})$
$\displaystyle=\int_{\Lambda_{AB}}\text{Pr}_{AB}(E\otimes
F|M_{A},M_{B},\lambda_{A},\lambda_{B})d\mu_{AB}(\lambda_{A},\lambda_{B})$
(112)
$\displaystyle=\int_{\Lambda_{A}}\text{Pr}_{A}(E|M_{A},\lambda_{A})d\mu_{A}(\lambda_{A})$
$\displaystyle\quad\quad\quad\times\int_{\Lambda_{B}}\text{Pr}_{B}(F|M_{B},\lambda_{B})d\mu_{B}(\lambda_{B})$
(113)
$\displaystyle=\text{Pr}_{A}(E|\rho_{A},M_{A})\text{Pr}_{B}(F|\rho_{B},M_{B}).$
(114)

From this, it follows that if $\Theta_{A}$ and $\Theta_{B}$ reproduce the
quantum predictions for $\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$
respectively, then $\Theta_{AB}$ reproduces the quantum predictions for
$\mathfrak{F}_{A}\times\mathfrak{F}_{B}$.

The key point is that, in order to model a direct product fragment, we only
need to take the Cartesian product of the subsystem ontic state spaces and we
only need to use product measures. The PIP states that this should remain true
even if the product fragment is expanded to include entangled measurements
(but still no entangled states). A little more terminology is required to
state this formally.

######  Definition 7.2.

Let
$\mathfrak{F}_{A}=\langle\mathcal{H}_{A},\mathcal{P}_{A},\mathcal{M}_{B}\rangle$
and
$\mathfrak{F}_{B}=\langle\mathcal{H}_{B},\mathcal{P}_{B},\mathcal{M}_{B}\rangle$
be PM fragments. A _product state fragment_ is a product fragment
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{AB},\mathcal{M}_{AB}\rangle$
where $\mathcal{P}_{AB}=\mathcal{P}_{A}\times\mathcal{P}_{B}$, i.e. it only
includes product states, but it may have entangled measurements.

######  Definition 7.3.

A pair of ontological models
$\Theta_{A}=(\Lambda_{A},\Sigma_{A},\Delta_{A},\Xi_{A})$,
$\Theta_{B}=(\Lambda_{B},\Sigma_{B},\Delta_{B},\Xi_{B})$ for PM fragments
$\mathfrak{F}_{A}=\left\langle\mathcal{H}_{A},\mathcal{P}_{A},\mathcal{M}_{A}\right\rangle$
and
$\mathfrak{F}_{B}=\left\langle\mathcal{H}_{B},\mathcal{P}_{B},\mathcal{M}_{B}\right\rangle$
are _compatible_ with an ontological model
$\Theta_{AB}=(\Lambda_{AB},\Sigma_{AB},\Delta_{AB},\Xi_{AB})$ for a product
state fragment
$\mathfrak{F}_{AB}=\left\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{A}\times\mathcal{P}_{B},\mathcal{M}_{AB}\right\rangle$
if $\Theta_{AB}$ satisfies

  * •

The Cartesian Product Assumption (CPA):

$\displaystyle\Lambda_{AB}$ $\displaystyle=\Lambda_{A}\times\Lambda_{B},$
$\displaystyle\Sigma_{AB}$ $\displaystyle=\Sigma_{A}\otimes\Sigma_{B}.$ (115)

  * •

The No-Correlation Assumption (NCA): For all $\rho_{A}\in\mathcal{P}_{A}$,
$\rho_{B}\in\mathcal{P}_{B}$,
$\Delta_{AB}[\rho_{A}\otimes\rho_{B}]=\Delta_{A}[\rho_{A}]\times\Delta_{B}[\rho_{B}]$
is the set of measures of the form $\mu_{A}\times\mu_{B}$, where
$\mu_{A}\in\Delta_{A}[\rho_{A}]$ and $\mu_{B}\in\Delta_{B}[\rho_{B}]$. This
means that $\mu_{AB}\in\Delta_{AB}[\rho_{A}\otimes\rho_{B}]$ is of the form

$\mu_{AB}(\Omega)=\int_{\Lambda_{B}}\mu_{A}(\Omega_{\lambda_{B}})d\mu_{B}(\lambda_{B}),$
(116)

where
$\Omega_{\lambda_{B}}=\\{\lambda_{A}\in\Lambda_{A}|(\lambda_{A},\Lambda_{B})\in\Omega\\}$,
$\mu_{A}\in\Delta_{A}[\rho_{A}]$ and $\mu_{B}\in\Delta_{B}[\rho_{B}]$.

The CPA says that, when modeling independent local preparations, there are no
additional properties of the joint system that are not derived from the
properties of the individual systems. In other words, there are no “genuinely
nonlocal properties” that are irrelevant for local measurements but come into
play when making joint measurements of the two systems. The NCA says that,
further, local preparations can be modeled as product measures.

######  Definition 7.4.

A pair of ontological models $\Theta_{A}$ and $\Theta_{B}$ for PM fragments
$\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$ that reproduce the quantum
predictions satisfy the _Preparation Independence Postulate (PIP)_ if, for all
product state fragments $\mathfrak{F}_{AB}$ with $\mathfrak{F}_{A}$ and
$\mathfrak{F}_{B}$ as factors, there exists an ontological model $\Theta_{AB}$
that is compatible with $\Theta_{A}$ and $\Theta_{B}$ and reproduces the
quantum predictions.

Note that, it is very important that the PIP only applies to product state
fragments and not to general product fragments. In other words, it does not
have any implications for how entangled states should be represented in an
ontological model. Suppose we have an ontological model
$(\Lambda_{AB},\Sigma_{AB},\Delta_{AB},\Xi_{AB})$ for a product state fragment
$\mathfrak{F}_{AB}$ that satisfies the PIP and we want to extend it to a more
general product fragment $\mathfrak{F}^{\prime}_{AB}$ that differs from
$\mathfrak{F}_{AB}$ by the addition of some entangled states. Then, we have to
allow for the fact that the extended ontological model
$(\Lambda^{\prime}_{AB},\Sigma^{\prime}_{AB},\Delta^{\prime}_{AB},\Xi^{\prime}_{AB})$
might have extra ontic states that have zero measure for all product states,
but nonzero measure for some entangled state. Therefore, all we can conclude
about the ontic state space of the extended model from the PIP is that
$\Lambda_{A}\times\Lambda_{B}\subseteq\Lambda^{\prime}_{AB}$, that
$\Sigma_{A}\otimes\Sigma_{B}$ is a $\sigma$-subalgebra of
$\Sigma^{\prime}_{AB}$, and that independent local preparations have measure
zero outside the $\Lambda_{A}\times\Lambda_{B}$ subset, but there may be other
ontic states that come into play when we prepare entangled states. Similarly,
probability measures that induce correlations between $\Lambda_{A}$ and
$\Lambda_{B}$ may come into play when we prepare entangled states.

The reason this is important is that $\psi$-ontic models do not satisfy
$\Lambda^{\prime}_{AB}=\Lambda_{A}\times\Lambda_{B}$ and it would be poor form
to prove a $\psi$-ontology theorem only to find that $\psi$-ontic models are
also ruled out by its assumptions. To understand this, consider the
Beltrametti–Bugajski model for two Hilbert spaces $\mathcal{H}_{A}$,
$\mathcal{H}_{B}$ and for the composite system
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. For the two systems individually, we
have the two projective Hilbert spaces consisting of all pure states
$\left[\psi\right]_{A}$ and $\left[\phi\right]_{B}$ as the ontic state spaces.
If we only want to model a product state fragment then we only need states of
the form $\left[\psi\right]_{A}\otimes\left[\phi\right]_{B}$, and the set of
such states is isomorphic to the Cartesian product of the two projective
Hilbert spaces. Similarly, the point measure on
$\left[\psi\right]_{A}\otimes\left[\phi\right]_{B}$ can be written as the
product of a point measure at $\left[\psi\right]_{A}$ with a point measure at
$\left[\psi\right]_{B}$. Therefore, the model satisfies the PIP. On the other
hand, if we want to model entangled states then we need additional ontic
states $\left[\psi\right]_{AB}$ that cannot be written as products, and the
point measure at $\left[\psi\right]_{AB}$ cannot be written as a product of
measures on the ontic state spaces of the factors. Therefore, modeling
entangled states requires additional ontic states and probability measures,
but this is not a violation of the PIP because the PIP only applies to product
state fragments.

For ease of exposition, we have so far confined attention to composites of two
subsystems. The Pusey–Barrett–Rudolph Theorem actually requires the
generalization to $n$ systems, but only in the case of $n$ identical factors.
The required generalization should be obvious, but for completeness here are
the definitions.

######  Definition 7.5.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment. An _$n$ fold product state fragment_ with factors $\mathfrak{F}$ is
any fragment $\mathfrak{F}^{n}$ of the form

$\mathfrak{F}^{n}=\langle\mathcal{H}^{\otimes n},\mathcal{P}^{\times
n},\mathcal{M}^{\prime}\rangle,$ (117)

where $\mathcal{P}^{\times n}$ consists of states of the form
$\otimes_{j=1}^{n}\rho_{j}$ for $\rho_{j}\in\mathcal{P}$, and
$\mathcal{M}^{\times n}\subseteq\mathcal{M}^{\prime}$ where
$\mathcal{M}^{\times n}$ consists of all POVMs of the form
$\\{\otimes_{j=1}^{n}E^{(j)}_{k_{j}}\\}$ for
$\\{E^{(j)}_{k_{j}}\\}\in\mathcal{M}$.

######  Definition 7.6.

An ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of a fragment
$\mathfrak{F}$ is _compatible_ with an ontological model
$\Theta^{n}=(\Lambda^{\prime},\Sigma^{\prime},\Delta^{\prime},\Xi^{\prime})$
for an $n$-fold product state fragment $\mathfrak{F}^{n}$ with factors
$\mathfrak{F}$ if it satisfies:

  * •

The Cartesian Product Assumption (CPA):

$\displaystyle\Lambda^{\prime}$ $\displaystyle=\Lambda^{\times n},$
$\displaystyle\Sigma^{\prime}$ $\displaystyle=\Sigma^{\otimes n}.$ (118)

  * •

The No-Correlation Assumption (NCA):
$\Delta^{\prime}_{\otimes_{j=1}^{n}\rho_{j}}$ consists of all product measures
of the form $\times_{j=1}^{n}\mu_{j}$, where $\mu_{j}\in\Delta_{\rho_{j}}$.

######  Definition 7.7.

An ontological model for a PM fragment $\mathfrak{F}$ that reproduces the
quantum predictions satisfies the PIP if it is compatible with an ontological
model for any $n$-fold product state fragment $\mathfrak{F}^{n}$ with factors
$\mathfrak{F}$ that reproduces the quantum predictions.

###  7.2 The main result

Recalling Definition 4.11 of a $\psi$-ontic model, we are now in a position to
prove the main result of the Pusey–Barrett–Rudolph paper.

######  Theorem 7.8 (The Pusey–Barrett–Rudolph Theorem).

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{P}$ contains all pure states on $\mathcal{H}$. Then,
any ontological model of $\mathfrak{F}$ that reproduces the quantum
predictions and satisfies the PIP is $\psi$-ontic.

The proof strategy adopted here is due to Moseley [110], and is simpler than
that of the original Pusey–Barrett–Rudolph paper. It is based on two lemmas.
The first establishes a connection between antidistinguishability of product
states and ontological distinctness, and the second establishes that the
required antidistinguishability holds. Because they are quite technical, it is
worth pausing to sketch a special case.

######  Example 7.9.

Let $\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$ be a basis
for $\mathbb{C}^{2}$ and consider the two states $\left[0\right]$ and
$\left[+\right]$, where
$\left|+\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)$.
These two states are not orthogonal, so we cannot use distinguishability to
establish their ontological distinctness. Since there are only two of them,
they cannot be antidistinguishable either. However, now consider two copies of
the system and the four states $\left[0\right]\otimes\left[0\right]$,
$\left[0\right]\otimes\left[+\right]$, $\left[+\right]\otimes\left[0\right]$
and $\left[+\right]\otimes\left[+\right]$. It turns out that these four states
are antidistinguishable. The measurement required to antidistinguish them is
entangled and consists of the projectors onto the following four orthonormal
vectors.

$\displaystyle\left|\phi_{00}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle\left|1\right\rangle+\left|1\right\rangle\left|0\right\rangle\right)$
(119) $\displaystyle\left|\phi_{01}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle\left|-\right\rangle+\left|1\right\rangle\left|+\right\rangle\right)$
(120) $\displaystyle\left|\phi_{10}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|+\right\rangle\left|1\right\rangle+\left|-\right\rangle\left|0\right\rangle\right)$
(121) $\displaystyle\left|\phi_{11}\right\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\left|+\right\rangle\left|-\right\rangle+\left|-\right\rangle\left|+\right\rangle\right),$
(122)

where
$\left|-\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0\right\rangle-\left|1\right\rangle\right)$.
It is easy to check that $\left|\phi_{00}\right\rangle$ is orthogonal to
$\left|0\right\rangle\otimes\left|0\right\rangle$,
$\left|\phi_{01}\right\rangle$ is orthogonal to
$\left|0\right\rangle\otimes\left|+\right\rangle$,
$\left|\phi_{10}\right\rangle$ is orthogonal to
$\left|+\right\rangle\otimes\left|0\right\rangle$, and
$\left|\phi_{11}\right\rangle$ is orthogonal to
$\left|+\right\rangle\otimes\left|+\right\rangle$, as required.

The second lemma shows that a similar thing happens for any pair of pure
states $\left[\psi_{0}\right],\left[\psi_{1}\right]$ for which
$\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)\leq\frac{1}{2}$.
Specifically, the set of states

$\left\\{\left[\psi_{0}\right]\otimes\left[\psi_{0}\right],\left[\psi_{0}\right]\otimes\left[\psi_{1}\right],\left[\psi_{1}\right]\otimes\left[\psi_{0}\right],\left[\psi_{1}\right]\otimes\left[\psi_{1}\right]\right\\},$
(123)

is always antidistinguishable.

Returning to the example, by Theorem 6.5, antidistinguishability implies that,
in an ontological model, the probability measures corresponding to
$\left[0\right]\otimes\left[0\right]$, $\left[0\right]\otimes\left[+\right]$,
$\left[+\right]\otimes\left[0\right]$ and
$\left[+\right]\otimes\left[+\right]$ can have no four-way overlap. However,
if the PIP is assumed, this implies that $\left[0\right]$ and $\left[+\right]$
must be ontologically distinct. As a rough sketch of this argument, consider
the case of a finite ontic state space, let $\mu_{0}$ and $\mu_{+}$ be a pair
of measures associated with $\left[0\right]$ and $\left[+\right]$ in the
ontological model, and assume that there is some $\lambda$ to which they both
assign nonzero probability. According to the PIP, the four product states must
be represented by measures of the form $\mu_{0}\times\mu_{0}$,
$\mu_{0}\times\mu_{+}$, $\mu_{+}\times\mu_{0}$, and $\mu_{+}\times\mu_{+}$.
Each of these assigns a nonzero probability to $(\lambda,\lambda)$, since both
$\mu_{0}$ and $\mu_{+}$ assign nonzero probability to $\lambda$, so there is a
common ontic state to which all four measures assign nonzero probability.
However, antidistinguishability implies that there can be no four-way overlap
between the measures so there is a contradiction, and hence $\left[0\right]$
and $\left[+\right]$ must be ontologically distinct.

The first lemma is a more formal and slightly more general version of this
argument. The generalization is needed to adapt these ideas to the case where
$\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)>\frac{1}{2}$.

######  Lemma 7.10.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment with ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ that
reproduces the quantum predictions, and let $\rho_{0},\rho_{1}\in\mathcal{P}$.
Let $\mathfrak{F}^{n}=\langle\mathcal{H}^{\otimes n},\mathcal{P}^{\times
n},\mathcal{M}^{\prime}\rangle$ be an $n$-fold product state fragment with
factors $\mathfrak{F}$ and let $\Theta^{n}=(\Lambda^{\times n},\Sigma^{\otimes
n},\Delta^{\prime},\Xi^{\prime})$ be an ontological model of it that is
compatible with $\Theta$.

For $\bm{j}\in\\{0,1\\}^{n}$, let

$\rho_{\bm{j}}=\rho_{j_{1}}\otimes\rho_{j_{2}}\otimes\ldots\otimes\rho_{j_{n}},$
(124)

where $\bm{j}=(j_{1},j_{2},\ldots,j_{n})$.

If $L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$
for all possible choices of $\mu_{\bm{j}}\in\Delta^{\prime}_{\rho_{\bm{j}}}$
then $\rho_{0}$ and $\rho_{1}$ are ontologically distinct in $\Theta$.

###### Proof.

Consider any choice of $\mu_{j}\in\Delta_{\rho_{j}}$. By the NCA,

$\mu_{\bm{j}}=\mu_{j_{1}}\times\mu_{j_{2}}\times\cdots\times\mu_{j_{n}},$
(125)

is in $\Delta^{\prime}_{\rho_{\bm{j}}}$ and hence, by assumption
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$. Let
$m$ be a measure that dominates $\mu_{0}$ and $\mu_{1}$, and let $p_{0}$ and
$p_{1}$ be corresponding densities. Then,

$\displaystyle
L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=\int_{\Lambda_{1}}dm(\lambda_{1})\times\int_{\Lambda_{2}}dm(\lambda_{2})\times\ldots$
$\displaystyle\quad\quad\times\int_{\Lambda_{n}}dm(\lambda_{n})\min_{\bm{j}\in\\{0,1\\}^{n}}\left[\prod_{k=1}^{n}p_{j_{k}}(\lambda_{k})\right].$
(126)

Here, the subscripts are just being used to identify the different copies of
$\Lambda$, so $\Lambda_{1}=\Lambda_{2}=\ldots=\Lambda_{n}=\Lambda$ and
$\lambda_{k}\in\Lambda_{k}$. Since each term $p_{j_{k}}(\lambda)$ in the
product is positive, the minimum is attained by minimizing each term
individually. We therefore have

$\displaystyle
L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}}\right)$
$\displaystyle=\left(\int_{\Lambda}dm(\lambda)\min_{j\in\\{0,1\\}}\left[p_{j}(\lambda)\right]\right)^{n}$
(127) $\displaystyle=\left[L\left(\mu_{0},\mu_{1}\right)\right]^{n},$ (128)

and since
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$, this
implies that $L\left(\mu_{0},\mu_{1}\right)=0$ because it is a positive
function. Since this holds for any choice of $\mu_{j}\in\Delta_{\rho_{j}}$,
this implies that $\rho_{0}$ and $\rho_{1}$ are ontologically distinct in
$\Theta$. ∎

######  Lemma 7.11.

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment and let $\left[\psi_{0}\right],\left[\psi_{1}\right]\in\mathcal{P}$
be pure states that satisfy
$\mathrm{Tr}\left(\left[\phi\right]\left[\psi\right]\right)\leq\frac{1}{2}$.
Then there exists a two fold product state fragment with factors
$\mathfrak{F}$ in which the set of states

$\mathcal{D}=\left\\{\left[\psi_{0}\right]\otimes\left[\psi_{0}\right],\left[\psi_{0}\right]\otimes\left[\psi_{1}\right],\left[\psi_{1}\right]\otimes\left[\psi_{0}\right],\left[\psi_{1}\right]\otimes\left[\psi_{1}\right]\right\\}$
(129)

is antidistinguishable.

###### Proof.

The result follows if there exists a POVM $\\{E_{00},E_{01},E_{10},E_{11}\\}$
on $\mathcal{H}\otimes\mathcal{H}$ that antidistinguishes $\mathcal{D}$, i.e.

$\text{Tr}\left(E_{jk}\left[\psi_{j}\right]\otimes\left[\psi_{k}\right]\right)=0$
(130)

Let $\left|\psi_{0}\right\rangle$ and $\left|\psi_{1}\right\rangle$ be vector
representatives of $\left[\psi_{0}\right]$ and $\left[\psi_{1}\right]$ such
that $\left\langle\psi_{0}\middle|\psi_{1}\right\rangle$ is real and positive.
Such vectors can always be found by appropriate choice of global phases. Then
we have
$\left\langle\psi_{0}\middle|\psi_{1}\right\rangle\leq\frac{1}{\sqrt{2}}$ so
we can write
$\left\langle\psi_{0}\middle|\psi_{1}\right\rangle=\sin\vartheta$, where
$0\leq\vartheta\leq\frac{\pi}{4}$. We can then choose an orthonormal basis
$\left\\{\left|0\right\rangle,\left|1\right\rangle\right\\}$ for the two
dimensional subspace spanned by $\left|\psi_{0}\right\rangle$ and
$\left|\psi_{1}\right\rangle$ such that

$\displaystyle\left|\psi_{0}\right\rangle$
$\displaystyle=\left|0\right\rangle,$
$\displaystyle\left|\psi_{1}\right\rangle=\sin\vartheta\left|0\right\rangle+\cos\vartheta\left|1\right\rangle.$
(131)

Note that the POVM elements $E_{jk}$ only need to be defined on this subspace
because the projector onto its orthogonal complement can be added on to any
one of them without changing the measurement probabilities given in Eq. (130).
Hence, the special case $\vartheta=\frac{\pi}{4}$ is just the case dealt with
in Example 7.9, for which the projectors $E_{jk}=\left[\Phi_{jk}\right]$
provide the required antidistinguishing POVM.

The basic idea to deal with the case $\vartheta<\frac{\pi}{4}$ is to note that
in this case $\left[\psi_{0}\right]$ and $\left[\psi_{1}\right]$ are more
distinguishable than $\left[0\right]$ and $\left[+\right]$, and there is
always a physically allowed transformation to map a pair of states to any pair
of less distinguishable states. The antidistinguishing measurement therefore
consists of performing the transformation that maps $\left[\psi_{0}\right]$ to
$\left[0\right]$, i.e. leaves it invariant, and maps $\left[\psi_{1}\right]$
to $\left[+\right]$, followed by performing the projective measurement
$\left\\{\left[\Phi_{jk}\right]\right\\}_{j,k\in\\{0,1\\}}$. Taken together,
these two steps define the required POVM.

More explicitly, consider the linear map $\mathcal{E}$ on the set of density
operators on $\mathcal{H}$ defined by

$\mathcal{E}(\rho)=M_{0}\rho M_{0}^{\dagger}+M_{1}\rho M_{1}^{\dagger},$ (132)

where the operators $M_{0}$ and $M_{1}$ are given by

$\displaystyle M_{0}=\left|0\middle\rangle\middle\langle
0\right|+\tan\vartheta\left|1\middle\rangle\middle\langle 1\right|,$ (133)
$\displaystyle
M_{1}=\sqrt{\frac{1-\tan^{2}\vartheta}{2}}\left(\left|0\right\rangle+\left|1\right\rangle\right)\left\langle
1\right|.$ (134)

This is an example of a completely-positive, trace-preserving map, which is a
dynamically allowed transformation of the set of quantum states. It does not
matter if you do not know this formalism, since for present purposes all that
matters is that

$\displaystyle\mathcal{E}(\left[\psi\right])$ $\displaystyle=\left[0\right],$
$\displaystyle\mathcal{E}(\left[\phi\right])=\left[+\right],$ (135)

which can easily be verified, and that, when
$0\leq\vartheta\leq\frac{\pi}{4}$,

$M_{0}^{\dagger}M_{0}+M_{1}^{\dagger}M_{1}=I_{\mathcal{H}},$ (136)

where $I_{\mathcal{H}}$ is the identity operator on $\mathcal{H}$, which is
again easily verified.

Next, define the operators

$E_{jk}=\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}\left[\Phi_{jk}\right]M_{r}\otimes M_{s}.$ (137)

These operators form the required antidistinguishing POVM. To see this, we
first have to check that $\\{E_{jk}\\}_{j,k\in\\{0,1\\}}$ is a POVM and then
that Eq. (130) is satisfied. To see that the operators $E_{jk}$ are positive
note that any operator of the form $N^{\dagger}N$ is positive and hence
setting $N=\left[\Phi_{jk}\right]M_{r}\otimes M_{s}$ shows that

$M_{r}^{\dagger}\otimes
M_{s}^{\dagger}\left[\Phi_{jk}\right]\left[\Phi_{jk}\right]M_{r}\otimes
M_{s}=M_{r}^{\dagger}\otimes M_{s}^{\dagger}\left[\Phi_{jk}\right]M_{r}\otimes
M_{s},$ (138)

is positive, where we have used the fact that $\left[\Phi_{jk}\right]$ is a
projector, and hence idempotent. Further, a sum of positive operators is
positive, and hence the $E_{jk}$’s are positive. We next need to check that
they sum to the identity on $\mathcal{H}\otimes\mathcal{H}$. This follows from
the fact that
$\sum_{jk}\left[\Phi_{jk}\right]=I_{\mathcal{H}\otimes\mathcal{H}}$ together
with Eq. (136) as follows.

$\displaystyle\sum_{j,k=0}^{1}E_{jk}$
$\displaystyle=\sum_{j,k=}^{1}\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}\left[\Phi_{jk}\right]M_{r}\otimes M_{s}$ (139)
$\displaystyle=\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}\left(\sum_{j,k=0}^{1}\left[\Phi_{jk}\right]\right)M_{r}\otimes
M_{s}$ (140) $\displaystyle=\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}I_{\mathcal{H}\otimes\mathcal{H}}M_{r}\otimes M_{s}$ (141)
$\displaystyle=\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}M_{r}\otimes M_{s}$ (142)
$\displaystyle=\sum_{r,s=0}^{1}M_{r}^{\dagger}M_{r}\otimes
M_{s}^{\dagger}M_{s}$ (143)
$\displaystyle=\left(\sum_{r=0}^{1}M_{r}^{\dagger}M_{r}\right)\otimes\left(\sum_{s=0}^{1}M_{s}^{\dagger}M_{s}\right)$
(144) $\displaystyle=I_{\mathcal{H}}\otimes I_{\mathcal{H}}$ (145)
$\displaystyle=I_{\mathcal{H}\otimes\mathcal{H}}.$ (146)

Finally, the $E_{jk}$’s satisfy Eq. (130) by virtue of Eq. (135) and the
cyclic property of the trace. For example,
$\text{Tr}\left(E_{00}\left[\psi_{0}\right]\otimes\left[\psi_{0}\right]\right)$
is zero by the following argument.

$\displaystyle\text{Tr}\left(E_{00}\left[\psi_{0}\right]\otimes\left[\psi_{0}\right]\right)$
$\displaystyle=\text{Tr}\left(\sum_{r,s=0}^{1}M_{r}^{\dagger}\otimes
M_{s}^{\dagger}\left[\Phi_{00}\right]M_{r}\otimes
M_{s}\left[\psi_{0}\right]\otimes\left[\psi_{0}\right]\right)$ (147)
$\displaystyle=\text{Tr}\left(\left[\Phi_{00}\right]\left[\sum_{r=0}^{1}M_{r}\left[\psi\right]M_{r}^{\dagger}\right]\otimes\left[\sum_{s=0}^{1}M_{s}\left[\psi_{0}\right]M_{s}^{\dagger}\right]\right)$
(148)
$\displaystyle=\text{Tr}\left(\left[\Phi_{00}\right]\mathcal{E}(\left[\psi_{0}\right])\otimes\mathcal{E}(\left[\psi_{0}\right])\right)$
(149)
$\displaystyle=\text{Tr}\left(\left[\Phi_{00}\right]\left[0\right]\otimes\left[0\right]\right)=0.$
(150)

The other antidistinguishability conditions follow from similar arguments. ∎

###### Proof of Theorem 7.8.

Consider any pair of pure states
$\left[\psi_{0}\right],\left[\psi_{1}\right]\in\mathcal{P}$ and an ontological
model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ that reproduces
the quantum predictions and satisfies the PIP. If
$\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)\leq\frac{1}{2}$
then, by Lemma 7.11, there exists a two fold product state fragment
$\mathfrak{F}^{2}$ in which the states

$\left\\{\left[\psi_{0}\right]\otimes\left[\psi_{0}\right],\left[\psi_{0}\right]\otimes\left[\psi_{1}\right],\left[\psi_{1}\right]\otimes\left[\psi_{0}\right],\left[\psi_{1}\right]\otimes\left[\psi_{1}\right]\right\\}$
(151)

are antidistinguishable. Consider an ontological model
$\Theta^{2}=(\Lambda\times\Lambda,\Sigma\otimes\Sigma,\Delta^{\prime},\Xi^{\prime})$
of $\mathfrak{F}^{2}$ that is compatible with $\Theta$. Then, by Theorem 6.5
for any choice of measures
$\mu_{jk}\in\Delta^{\prime}_{\left[\psi_{j}\right]\otimes\left[\psi_{k}\right]}$,

$L\left(\left\\{\mu_{jk}\right\\}_{j,k\in\\{0,1\\}}\right)=0.$ (152)

Then, by Lemma 7.10, $\left[\psi_{0}\right]$ and $\left[\psi_{1}\right]$ must
be ontologically distinct in any ontological model of $\mathfrak{F}$ that
satisfies the PIP and reproduces the quantum predictions.

It remains to deal with the case where
$\frac{1}{2}<\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)<1$.
For this, note that

$\text{Tr}\left(\left[\psi_{0}\right]^{\otimes
n}\left[\psi_{1}\right]^{\otimes
n}\right)=\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)^{n}.$
(153)

Because of this, we can always find an $n$ such that

$\text{Tr}\left(\left[\psi_{0}\right]^{\otimes
n}\left[\psi_{1}\right]^{\otimes n}\right)\leq\frac{1}{2}.$ (154)

Now, consider an $n$-fold product state fragment $\mathfrak{F}^{n}$. This must
contain the states $\left[\psi_{0}\right]^{\otimes n}$ and
$\left[\psi_{1}\right]^{\otimes n}$. For any ontological model
$\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of $\mathfrak{F}$ that satisfies the PIP,
we can construct a compatible ontological model $\Theta^{n}=(\Lambda^{\times
n},\Sigma^{\otimes n},\Delta^{\prime},\Xi^{\prime})$ of $\mathfrak{F}^{n}$.
This model will also satisfy the PIP (with respect to taking products of
$\mathfrak{F}^{n}$ with itself), since a model compatible with $\mathfrak{F}$
is necessarily also compatible with $\mathfrak{F}^{n}$. Therefore, we can
apply the argument used in the
$\text{Tr}\left(\left[\psi_{0}\right]\left[\psi_{1}\right]\right)\leq\frac{1}{2}$
case to the states $\left[\psi_{0}\right]^{\otimes n}$ and
$\left[\psi_{1}\right]^{\otimes n}$ and deduce that
$\left[\psi_{0}\right]^{\otimes n}$ and $\left[\psi_{1}\right]^{\otimes n}$
must be ontologically distinct in $\Theta^{n}$. This means that

$L\left(\left\\{\mu_{\bm{0}},\mu_{\bm{1}}\right\\}\right)=0,$ (155)

for any choice of
$\mu_{\bm{0}}\in\Delta^{\prime}_{\left[\psi_{0}\right]^{\otimes n}}$ and
$\mu_{\bm{1}}\in\Delta^{\prime}_{\left[\psi_{1}\right]^{\otimes n}}$, where
$\bm{0}$ denotes the $n$-bit string $(0,0,\ldots,0)$ and $\bm{1}$ denotes the
$n$-bit string $(1,1,\ldots,1)$. Now note that

$L\left(\left\\{\mu_{\bm{0}},\mu_{\bm{1}}\right\\}\right)\geq
L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right),$ (156)

where $\mu_{\bm{j}}\in\Delta_{\psi_{\bm{j}}}$ and

$\left[\psi_{\bm{j}}\right]=\left[\psi_{j_{1}}\right]\otimes\left[\psi_{j_{2}}\right]\otimes\cdots\otimes\left[\psi_{j_{n}}\right].$
(157)

This is because computing
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)$
involves minimizing over a larger set than computing
$L\left(\left\\{\mu_{\bm{0}},\mu_{\bm{1}}\right\\}\right)$. Since $L$ is a
positive function, we therefore have that
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$ and
hence, by Lemma 7.10, $\left[\psi_{0}\right]$ and $\left[\psi_{1}\right]$ must
be ontologically distinct. ∎

###  7.3 Motivation for the PIP

The PIP says that ontological models for subsystems should be compatible with
models for product states on the composite system. Compatibility breaks down
into two assumptions: the CPA, which says that the ontic state space of the
composite system should be a Cartesian product of the ontic states spaces of
the subsystems, and the NCA, which says that product states should be modeled
by product measures. We consider the motivations for each of these
requirements in turn.

The CPA is a weakening of a kind of locality assumption known as _Einstein
separability_. The terminology is due to Howard [111], who defines it as the
idea that “two spatially separated systems possess their own separate real
states”. Einstein formulated this idea in the context of his arguments for the
incompleteness of quantum theory, and expressed it as follows.

> If one asks what is characteristic of the realm of physical ideas
> independently of the quantum-theory, then above all the following attracts
> our attention: the concepts of physics refer to a real external world, i.e.
> ideas are posited of things that claim a ‘real existence’ independent of the
> perceiving subject (bodies, fields, etc.), and these ideas are, on the other
> hand, brought into as secure a relationship as possible with sense
> impressions. Moreover, it is characteristic of these physical things that
> they are conceived of as being arranged in a space-time continuum. Further,
> it appears to be essential for this arrangement of the things introduced in
> physics that, at a specific time, these things claim an existence
> independent of one another, insofar as these things ‘lie in different parts
> of space’. Without such an assumption of the mutually independent existence
> (the ‘being-thus’) of spatially distant things, an assumption which
> originates in everyday thought, physical thought in the sense familiar to us
> would not be possible. Nor does one see how physical laws could be
> formulated and tested without such a clean separation. Field theory has
> carried out this principle to the extreme, in that it localizes within
> infinitely small (four-dimensional) space-elements the elementary things
> existing independently of one another that it takes as basic, as well as the
> elementary laws it postulates for them. — Albert Einstein [112].

The Pusey–Barrett–Rudolph argument only makes use of product state
preparations, so we can always imagine that the individual systems are
prepared very far away from each other and are only later brought together to
perform the entangled measurement. If implemented this way, the individual
systems would occupy spatially separated regions at their point of preparation
and so, according to separability, they ought to possess their own separate
real states. Implicit in this is the idea that there are no inherently global
joint properties of the composite system that are not determined by the
properties of the individual systems. In the language of ontological models,
this means that ontic state spaces should compose according to the Cartesian
product.

Einstein thought that separability should always hold, regardless of whether
we are preparing product states or entangled states. In light of Bell’s
Theorem, the case for such a general separability assumption is significantly
weakened. Separability is not actually required to prove Bell’s Theorem [105],
but, if we are contemplating a world in which the effects of measurement can
be transmitted instantaneously across space, then it makes sense to also
contemplate a world in which there are inherently global properties as well.
Additionally, since we are in the business of proving $\psi$-ontology theorems
and the quantum state of an entangled system would, if ontic, be such an
inherently global property, we had better not introduce any assumptions that
rule them out. For this reason, the CPA is restricted to product state
preparations. It says that separability should hold, not in general, but only
when systems are prepared in product states.

The motivation for assuming separability for product states is that we
generally think that experiments on separated systems are independent of one
another. It should not be necessary to gather our system here on Earth
together with one on Mars in order to determine all of the parameters relevant
to our Earth-bound experiment. Of course, when performing experiments
involving systems on Earth that are correlated with those on Mars, what
happens on Mars is very relevant, but the CPA only applies to product states.
If we allow that genuinely global properties may be relevant even to an
isolated system then we open up a Pandora’s box. It could well be necessary to
gather together every system in the universe in order to determine all the
parameters that are relevant for our system here on Earth.

At this point, it is worth noting that the assumptions behind no-go theorems
are often designed to mirror operational features of quantum theory at the
ontological level. This is perhaps clearest in the case of contextuality.
Preparation noncontextuality says that if there is no difference between two
preparation procedures in terms of the observable statistics they predict,
i.e. they are represented by the same quantum state, then there should be no
difference between them at the ontological level either, i.e. they should be
represented by the same probability measure over the ontic states. If this
assumption does not hold then an explanatory gap is opened because, if two
preparations are represented by different probability measures, then one would
generally expect to be able to pick up this difference in the observed
statistics. To prevent this from happening, the way that measurements are
represented by conditional probability distributions has to be fine-tuned so
that the difference between the two probability measures is washed out by
averaging. This fine-tuning has no natural explanation within the ontological
model, other than that it is needed in order to reproduce the quantum
predictions. Similarly, on an operational level, local measurements on
entangled states cannot be used to send signals, so it makes sense to demand
locality when modeling them ontologically. Otherwise, we would have to explain
why the purported nonlocal influences cannot be used to signal. Proofs of
preparation contextuality and Bell’s Theorem therefore expose explanatory gaps
in the ontological models framework.

In this vein, separability can be motivated by the operational principle known
as _local tomography_. This says that the state of a joint system, and hence
the probabilities it predicts for any measurement, is completely determined by
the statistics of local measurements made on the subsystems. For example, the
state of a two-qubit system is completely determined by the statistics of
measurements of the Pauli observables $\sigma_{x}$, $\sigma_{y}$, and
$\sigma_{z}$ made on each qubit individually and their correlations. There are
no parameters of the joint state that require bringing the two subsystems
together and making a joint measurement in order to determine them. Therefore,
it makes sense to posit an underlying ontology that does not involve genuinely
global properties either. Without this assumption, we would have to explain
why the genuinely global properties do not lead to observable parameters that
can only be determined by bringing the systems together.

Of course, the whole point of no-go theorems, and the reason they are
surprising, is that they show that such operationally motivated assumptions
are not actually viable at the ontological level, at least within the usual
framework. They each imply fine-tunings for which we currently have no good
explanation. My point here is just that separability bears a family
resemblance to the assumptions behind other no-go theorems, so if you think
that preparation noncontextuality and Bell locality have intuitive appeal then
the same should hold for separability as well. If we find, as we have, that
some of these assumptions are not actually viable then it makes sense to
explore the consequences of weakened versions of them, which maintain some of
the appeal but are not yet ruled out. From this perspective the CPA, as a
weakening of separability, is a reasonable thing to posit.

The NCA can similarly be motivated as mirroring an operational feature of
quantum theory at the ontological level. When two systems are prepared in a
product state they are completely uncorrelated from each other. No quantum
measurement will ever reveal any pre-existing correlation. Therefore, it makes
sense to think that the systems are uncorrelated at the ontological level as
well. If not, then it is puzzling that this correlation does not show up in
measurement statistics.

Perhaps the CPA and NCA are not so self-evidently true that they must never be
questioned, but they are at least as solid as the assumptions that go into
other no-go theorems. Therefore, the Pusey–Barrett–Rudolph Theorem presents us
with a dilemma. Either we must give up the CPA, the NPA, or
$\psi$-epistemicism, and each choice opens up an explanatory gap. I think this
is at least as interesting as the dilemmas posed by Bell’s Theorem and
contextuality.

###  7.4 Weakening the assumptions

The assumptions of the Pusey–Barrett–Rudolph Theorem have so far been
presented in their most intuitively accessible form. However, it is possible
to weaken them somewhat without affecting the conclusion. This has been
pointed out by Hall [33] and by Schlosshauer and Fine [38]. The cost of doing
this is that the weakened assumptions are less clearly motivated by
operational properties of quantum theory. Nevertheless, it is interesting to
identify the weakest set of assumptions under which the theorem can be proved.

First of all, note that the only quantum predictions used in the
Pusey–Barrett–Rudolph argument involve measurement outcomes that are assigned
probability zero by some quantum state via antidistinguishability. It would
make no difference to the argument if ontological models were only required to
reproduce the probability zero predictions of quantum theory, instead of
requiring that they reproduce all of the quantum predictions exactly. Thus,
the requirement of reproducing the quantum predictions may be replaced by the
following.

######  Definition 7.12.

An ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ of a PM fragment
$\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ _reproduces
the quantum preclusions_ if, for all $\rho\in\mathcal{P}$, $M\in\mathcal{M}$
and $E\in M$ such that $\text{Tr}\left(E\rho\right)=0$, each
$\mu\in\Delta_{\rho}$ and $\text{Pr}\in\Xi_{M}$ satisfies

$\int_{\Lambda}\text{Pr}(E|M,\lambda)d\mu(\lambda)=0.$ (158)

Secondly, in the Pusey–Barrett–Rudolph argument, the PIP is only used in Lemma
7.10, where it licenses the following inference. Given a pair of quantum
states $\rho_{0}$ and $\rho_{1}$, if the product states
$\\{\rho_{j_{1}}\otimes\rho_{j_{2}}\otimes\cdots\otimes\rho_{j_{n}}\\}_{\bm{j}\in\\{0,1\\}^{n}}$
correspond to nonoverlapping measures $\mu_{\bm{j}}$, i.e.
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$, then
the measures $\mu_{0}$ and $\mu_{1}$ corresponding to $\rho_{0}$ and
$\rho_{1}$ should also not overlap, i.e. $L(\mu_{0},\mu_{1})=0$. Therefore, if
we want to avoid using the PIP, then we can simply make this assumption
directly.

######  Definition 7.13.

An ontological model $\Theta=(\Lambda,\Sigma,\Delta,\Xi)$ for a PM fragment
$\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ is _compact_
with respect to an ontological model
$\Theta^{n}=(\Lambda^{\prime},\Sigma^{\prime},\Delta^{\prime},\Xi^{\prime})$
for the $n$-fold product state fragment
$\mathfrak{F}^{n}=\langle\mathcal{H}^{\otimes n},\mathcal{P}^{\times
n},\mathcal{M}^{\prime}\rangle$ with factors $\mathfrak{F}$ if, whenever
$L\left(\left\\{\mu_{\bm{j}}\right\\}_{\bm{j}\in\\{0,1\\}^{n}}\right)=0$ for
all choices of $\mu_{\bm{j}}\in\Delta^{\prime}_{\rho_{\bm{j}}}$ (where
$\rho_{\bm{j}}=\otimes_{k=1}^{n}\rho_{j_{k}}$), then $L(\mu_{0},\mu_{1})=0$
for all choices of $\mu_{j}\in\Delta_{\rho_{j}}$.

######  Definition 7.14.

An ontological model for a fragment $\mathfrak{F}$ that reproduces the quantum
preclusions satisfies _compactness_ if it is compact with respect to an
ontological model that reproduces the quantum preclusions for any $n$-fold
product state fragment $\mathfrak{F}^{n}$ with factors $\mathfrak{F}$.

The condition of compactness is originally due to Schlosshauer and Fine [38],
and is a slight generalization of a condition that Hall called “compatibility”
[33]. Here, I have presented it in a slightly more rigorous form in order to
take care of measure zero issues and to allow for preparation contextuality.
In this form, the meaning of compactness may be somewhat obscure, so it is
helpful to consider the special case of a finite ontic state space. In this
case, when $n=2$, compactness says is that, if a measure $\mu_{0}$
corresponding to $\rho_{0}$ assigns a nonzero probability to the ontic state
$\lambda$ and a measure $\mu_{1}$ corresponding to $\rho_{1}$ likewise assigns
a nonzero probability to $\lambda$, then there ought to be probability
measures $\mu_{00}$, $\mu_{01}$, $\mu_{10}$ and $\mu_{11}$ representing the
states $\rho_{0}\otimes\rho_{0}$, $\rho_{0}\otimes\rho_{1}$,
$\rho_{1}\otimes\rho_{0}$ and $\rho_{1}\otimes\rho_{1}$ of the composite
system that all assign nonzero probability to some ontic state
$\lambda^{\prime}$. The advantage of this formulation is that it does not
assume that the ontic state space of the composite system has a Cartesian
product structure because it does not specify how $\lambda^{\prime}$, which is
an ontic state of the composite, is related to $\lambda$, which is an ontic
state of the subsystem. It also does not rule out the possibility of
correlations between the two systems at the ontological level.

However, the motivation for assuming compactness is somewhat obscure. Unlike
the PIP, it is not a weakened form of separability, and has no obvious
operational motivation. It is doubtful that one would come up with compactness
without having the idea of Cartesian products and product measures in mind in
the first place. It amounts to simply assuming that Lemma 7.10 is true by
fiat. Without any examples of natural models that satisfy compactness but not
the PIP, it is not clear why one would make such an assumption. Life would be
very simple if we always just raised the lemmas needed to prove theorems into
assumptions instead of proving them. Nevertheless, the Pusey–Barrett–Rudolph
Theorem can be stated in the more general form.

######  Theorem 7.15 (The “Generalized” Pusey–Barrett–Rudolph Theorem).

Let $\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ be a PM
fragment where $\mathcal{P}$ contains all pure states on $\mathcal{H}$. Then,
any ontological model of $\mathfrak{F}$ that reproduces the quantum
preclusions and satisfies compactness is $\psi$-ontic.

The proof is just the same as the original Pusey–Barrett–Rudolph Theorem,
except for the removal of Lemma 7.10.

###  7.5 Necessity of the PIP

The PIP is the most substantive assumption that goes into the
Pusey–Barrett–Rudolph Theorem. All of the rest of its assumptions are standard
for the ontological models framework and are common to the vast majority of
no-go theorems for realist models of quantum theory, such as Bell’s Theorem
and the Kochen–Specker Theorem. Therefore, any criticism of
Pusey–Barrett–Rudolph that is not directed against the PIP could equally well
be directed against these other results. Whilst I do not wish to belittle such
criticisms, and will deal with them in §7.7, it is more important to deal with
objections that are specific to the Pusey–Barrett–Rudolph Theorem, and hence
to the PIP. Before doing so, we should check that the PIP is a necessary
assumption, since if $\psi$-ontology can be established without the PIP then
the question of whether to accept it is moot.

The Kochen–Specker model shows that the PIP is necessary for the case of a
qubit. However, the case of a qubit is rather special and the existence of
this model does not rule out the possibility that ontological models for
systems of larger dimension might have to be $\psi$-ontic. Lewis, Jennings,
Barrett and Rudolph (LJBR) showed that it is in fact possible to construct a
$\psi$-epistemic model for measurements in orthonormal bases in all dimensions
[52]. This was based on an idea due to Rudolph, as discussed in Morris’
master’s thesis [113]. Subsequently, Aaronson, Bouland, Chua and Lowther
(ABCL) provided an alternative construction based on the same basic idea [53].
Of course, none of these models satisfy the PIP. Whilst the LJBR model is
$\psi$-epistemic, the ABCL construction additionally shows that, for any pair
$\left[\psi\right]$, $\left[\phi\right]$ of nonorthogonal states, a
$\psi$-epistemic model can be constructed in which measures corresponding to
$\left[\psi\right]$ and $\left[\phi\right]$ have nonzero overlap. Since the
main idea is the same, we focus on the ABCL result here.

Both LJBR and ABCL start from the Bell model described in Example 4.5. This is
a $\psi$-ontic model, but by a simple modification it can be made
$\psi$-epistemic. Recall that, in the Bell model, the ontic state space is
$\Lambda=\Lambda_{1}\times\Lambda_{2}$, where $\Lambda_{1}$ is the set of pure
states on $\mathbb{C}^{d}$ and $\Lambda_{2}$ is the unit interval. A state
$\left[\psi\right]$ is represented by the product measure
$\mu=\mu_{1}\times\mu_{2}$, where $\mu_{1}=\delta_{\psi}$ is the point measure
at $\left[\psi\right]$ on $\Lambda_{1}$ and $\mu_{2}$ is the uniform measure
on $\Lambda_{2}=[0,1]$. For each $[\lambda_{1}]\in\Lambda_{1}$, the response
functions are set by dividing up the interval $[0,1]$ into subsets of length
$\text{Tr}\left(\left[\phi_{j}\right]\left[\lambda_{1}\right]\right)$ that
give the $\left[\phi_{j}\right]$ outcome with certainty. In Example 4.5 this
was done simply by dividing $[0,1]$ up into consecutive intervals of the form

$\sum_{k=0}^{j-1}\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right)\leq\lambda_{2}<\sum_{k=0}^{j}\text{Tr}\left(\left[\phi_{k}\right]\left[\lambda_{1}\right]\right),$
(159)

as illustrated in Fig. 7. However, this division could be done in a different
way, providing the total length of the subset that gives
$\left[\phi_{j}\right]$ with certainty remains equal to
$\text{Tr}\left(\left[\phi_{j}\right]\left[\lambda_{1}\right]\right)$. The
first step in converting the Bell model into a $\psi$-epistemic model is to
change this division.

Following ABCL, let $\left[a\right]$ and $\left[b\right]$ be two nonorthogonal
states that we would like to make ontologically indistinct. Given a
measurement $M=\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}$, let
$\sigma$ be a permutation of $(0,1,\ldots,d-1)$ such that

$\displaystyle\min\left(\text{Tr}\left(\left[\phi_{\sigma(0)}\right]\left[a\right]\right),\text{Tr}\left(\left[\phi_{\sigma(0)}\right]\left[b\right]\right)\right)$
$\displaystyle\geq\min\left(\text{Tr}\left(\left[\phi_{\sigma(1)}\right]\left[a\right]\right),\text{Tr}\left(\left[\phi_{\sigma(1)}\right]\left[b\right]\right)\right)\geq\ldots$
(160)
$\displaystyle\geq\min\left(\text{Tr}\left(\left[\phi_{\sigma(d-1)}\right]\left[a\right]\right),\text{Tr}\left(\left[\phi_{\sigma(d-1)}\right]\left[b\right]\right)\right).$
(161)

Now, the unit interval is divided up in exactly the same way as before, except
with respect to the permuted indices instead of the original ordering of
outcomes i.e. the outcome $\left[\phi_{\sigma(j)}\right]$ is obtained with
certainty if

$\sum_{k=0}^{j-1}\text{Tr}\left(\left[\phi_{\sigma(k)}\right]\left[\lambda_{1}\right]\right)\leq\lambda_{2}<\sum_{k=0}^{j}\text{Tr}\left(\left[\phi_{\sigma(k)}\right]\left[\lambda_{1}\right]\right).$
(162)

So far, the model is still $\psi$-ontic because we have only modified the
conditional probability distributions representing measurements, so the
probability measures representing states still have their point measure terms.
The next step is to show that there is an $\epsilon>0$ such that all ontic
states in the set

$\Omega_{a,b}=\\{\left[a\right],\left[b\right]\\}\times[0,\epsilon],$ (163)

always return the $\left[\phi_{\sigma(0)}\right]$ outcome in any measurement.
Thus, any weight that a probability measure assigns to this region of the
ontic state space can be redistributed in an arbitrary way without affecting
the observed probabilities, and by doing so the probability measures
associated with $\left[a\right]$ and $\left[b\right]$ can be made to overlap.

To prove this, let $\left|a\right\rangle$, $\left|b\right\rangle$ and
$\left|\phi_{j}\right\rangle$ be vector representatives of $\left[a\right]$,
$\left[b\right]$ and $\left[\phi_{j}\right]$. It is sufficient to show that
there exists an $\epsilon>0$ such that, for all measurements
$M=\left\\{\left[\phi_{j}\right]\right\\}_{j=0}^{d-1}$, there exists a $j$
such that $\left|\left\langle\phi_{j}\middle|a\right\rangle\right|>\epsilon$
and $\left|\left\langle\phi_{j}\middle|b\right\rangle\right|>\epsilon$. Then,
whatever $\left[\phi_{\sigma(0)}\right]$ is, it must satisfy
$\left|\left\langle\phi_{\sigma(0)}\middle|a\right\rangle\right|>\epsilon$ and
$\left|\left\langle\phi_{\sigma(0)}\middle|b\right\rangle\right|>\epsilon$
because $\sigma$ orders the outcomes in decreasing order of
$\min\left(\left|\left\langle\phi_{j}\middle|a\right\rangle\right|,\left|\left\langle\phi_{j}\middle|b\right\rangle\right|\right)$.
To show this, note that

$\displaystyle\left|\left\langle a\middle|b\right\rangle\right|$
$\displaystyle=\left|\left\langle
a\right|\left(\sum_{j=0}^{d-1}\left|\phi_{j}\middle\rangle\middle\langle\phi_{j}\right|\right)\left|b\right\rangle\right|$
(164) $\displaystyle\leq\sum_{j=0}^{d-1}\left|\left\langle
a\middle|\phi_{j}\right\rangle\left\langle\phi_{j}\middle|b\right\rangle\right|$
(165) $\displaystyle\leq\sum_{j=0}^{d-1}\left|\left\langle
a\middle|\phi_{j}\right\rangle\right|\left|\left\langle\phi_{j}\middle|b\right\rangle\right|,$
(166)

where we have used the fact that
$\sum_{j=0}^{d-1}\left|\phi_{j}\middle\rangle\middle\langle\phi_{j}\right|$ is
a resolution of the identity and the triangle inequality. The largest term in
the sum must be larger than the average, so we have

$\max_{j}\left(\left|\left\langle
a\middle|\phi_{j}\right\rangle\right|\times\left|\left\langle\phi_{j}\middle|b\right\rangle\right|\right)\geq\frac{\left|\left\langle
a\middle|b\right\rangle\right|}{d},$ (167)

but since $\left|\left\langle a\middle|\phi_{j}\right\rangle\right|$ and
$\left|\left\langle\phi_{j}\middle|b\right\rangle\right|$ are between $0$ and
$1$, this means

$\max_{j}\left(\min\left(\left|\left\langle
a\middle|\phi_{j}\right\rangle\right|,\left|\left\langle\phi_{j}\middle|b\right\rangle\right|\right)\right)\geq\frac{\left|\left\langle
a\middle|b\right\rangle\right|}{d},$ (168)

and since $\left[a\right]$ and $\left[b\right]$ are nonorthogonal,
$\left|\left\langle a\middle|b\right\rangle\right|/d>0$, so setting
$\epsilon=\left|\left\langle a\middle|b\right\rangle\right|/d$ gives the
desired result.

Now, we can replace the measure associated with $\left[a\right]$ with

$\mu(\Omega)=\int_{\Lambda_{2}}\mu_{\lambda_{2}}(\Omega_{\lambda_{2}})d\mu_{2}(\lambda_{2}),$
(169)

where as before $\mu_{2}$ is the uniform measure on $\Lambda_{2}$ and
$\Omega_{\lambda_{2}}=\left\\{\left[\lambda_{1}\right]\in\Lambda_{1}\middle|(\left[\lambda_{1}\right],\lambda_{2})\in\Omega\right\\}$,
but now $\mu_{\lambda_{2}}$ is a measure on $\Lambda_{1}$ that depends on
$\lambda_{2}$ via

$\mu_{\lambda_{2}}=\begin{cases}\delta_{a}&\text{if}\,\,\lambda_{2}>\epsilon\\\
\epsilon\mu_{\Omega_{a,b}}&\text{if}\,\,\lambda_{2}\leq\epsilon,\end{cases}$
(170)

where $\mu_{\Omega_{a,b}}$ is an arbitrary measure on $\Omega_{a,b}$ which,
for example, could be the uniform measure. Similarly for $\left[b\right]$,
with $\delta_{a}$ replaced by $\delta_{b}$. Since both measures now agree on
$\Omega_{a,b}$ and assign nonzero probability to it, the model is
$\psi$-epistemic.

This model is only $\psi$-epistemic in a fairly weak sense, since only a
single pair of pure states has any overlap. However, the same idea can be used
as the basis for constructing a model in which every pair of nonorthogonal
pure states has overlap (see §11 for an outline of the construction).

The Pusey–Barrett–Rudolph Theorem implies that the ABCL model must necessarily
fail to satisfy the PIP. To see this, we have to show that, given an ABCL
model for $\mathbb{C}^{d}$, there is some $n$ for which there is no model for
product states on the composite system $\left(\mathbb{C}^{d}\right)^{\otimes
n}$ with which it is compatible. Given that the ABCL construction can be used
to make any single pair of pure states ontologically indistinct, we can apply
it to the states $\left[z+\right]$ and $\left[x+\right]$ in $\mathbb{C}^{2}$
from Example 7.9. Then, compatibility implies that the model for product
states on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ would have to have nonzero
four-way overlap for the states $\left[z+\right]\otimes\left[z+\right]$,
$\left[z+\right]\otimes\left[x+\right]$,
$\left[x+\right]\otimes\left[z+\right]$ and
$\left[x+\right]\otimes\left[x+\right]$, since these are represented by
products of the measures representing $\left[z+\right]$ and $\left[x+\right]$
in the ABCL model, which have nonzero pairwise overlap. However, nonzero
overlap for these four states is ruled out by their antidistinguishability, so
no such model can exist. Similarly, whichever pair of quantum states we choose
to make ontologically indistinct, the Pusey–Barrett–Rudolph Theorem rules out
compatibility with models for the composite system for some $n$.

###  7.6 Criticism of the PIP

Since the PIP is necessary for proving the Pusey–Barrett–Rudolph Theorem, it
is important to settle the question of whether it is reasonable. In this
section, two criticisms of the PIP are discussed, directed at the CPA and NCA
respectively. In §7.6.1, a weakening of the PIP due to Emerson, Serbin,
Sutherland and Veitch (ESSV) [32] is discussed and we show that, under this
weakened assumption, $\psi$-epistemic models can always be constructed. This
is still a useful criticism, as it highlights the fact that giving up the CPA
need not lead to global parameters becoming relevant to local measurements, so
the spirit of separability can perhaps be preserved. However, the weakened PIP
is not really a viable constraint on how systems should compose, as the
construction shows that it can be satisfied trivially. §7.6.2 discusses a
criticism due to Hall [33] that the NCA does not follow from causality
considerations of the type considered in Bell’s Theorem. Whilst this argument
is formally correct, I think it misses the point as the NCA is not motivated
by causality in the first place.

####  7.6.1 Criticism of the CPA

Emerson, Serbin, Sutherland and Veitch (ESSV) [32] have proposed a weakening
of the PIP, aimed at criticizing the CPA, and have shown how a
$\psi$-epistemic model satisfying their weakened PIP can reproduce the
predictions of Example 7.9. Before getting into the details, it is worth
examining what we should expect of a weakened PIP assumption.

The PIP is a constraint on how subsystems should be composed at the
ontological level. The CPA encodes the idea that, when preparing two
independent systems in a product state, there should be no genuinely global
properties that are not reducible to local properties of the individual
systems. It is motivated by the idea that, if this condition were violated,
then it might be necessary to bring the two independent systems together in
order to determine all of the parameters that are relevant even for just local
experiments on one of the systems. Any proposal for weakening the PIP by
introducing global properties should therefore be evaluated on the following
criteria.

  * •

It ought to demonstrate that the motivation for the CPA is misguided by
showing how the existence of global properties need not interfere with local
experiments.

  * •

It ought to provide a viable constraint on how subsystems should be composed
at the ontological level.

In my view, the proposal of ESSV satisfies the first criterion, but not the
second. It does a good job of criticizing the motivation for the CPA, but is
not, in itself, a viable constraint on how subsystems should be composed.
Explanation of what I mean by a viable constraint on subsystem composition is
deferred until after explaining the details of the ESSV proposal.

The basic idea is as follows. Recall that, in order to model direct product
fragments, we can always get away with taking the Cartesian product of ontic
state spaces and using product measures to represent states. Therefore, the
only time global properties would necessarily have to play a role when
modeling product state fragments is when a joint measurement is made, e.g. a
measurement in an entangled basis. Such measurements already involve an
interaction between the two systems at the operational level, so invoking
global properties to explain such experiments does not seem to open up a huge
gap between the structure of the ontological model and the operational
structure of the experiment. The argument that it might be necessary to bring
a system on Earth together with a system on Mars in order to characterize all
the parameters relevant to experiments on Earth does not hold water if the
genuinely global properties do not have any effect on the outcomes of local
measurements. It would still be possible to work with separate systems
completely independently of each another, in blissful ignorance of the global
properties, until we decide to do an experiment that necessarily involves
bringing the systems together. With this in mind, ESSV’s weakened PIP runs as
follows.

######  Definition 7.16.

A pair of ontological models
$\Theta_{A}=(\Lambda_{A},\Sigma_{A},\Delta_{A},\Xi_{A})$,
$\Theta_{B}=(\Lambda_{B},\Sigma_{B},\Delta_{B},\Xi_{B})$ for fragments
$\mathfrak{F}_{A}=\langle\mathcal{H}_{A},\mathcal{P}_{A},\mathcal{M}_{A}\rangle$
and
$\mathfrak{F}_{B}=\langle\mathcal{H}_{B},\mathcal{P}_{B},\mathcal{M}_{B}\rangle$
are _weakly compatible_ with an ontological model
$\Theta_{AB}=(\Lambda_{AB},\Sigma_{AB},\Delta_{AB},\Xi_{AB})$ for a product
state fragment
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{A}\times\mathcal{P}_{B},\mathcal{M}_{AB}\rangle$
if $\Theta_{AB}$ satisfies

  * •

The Weak Cartesian Product Assumption (WCPA):

$\displaystyle\Lambda_{AB}$
$\displaystyle=\Lambda_{A}\times\Lambda_{B}\times\Lambda_{\text{NL}},$
$\displaystyle\Sigma_{AB}$
$\displaystyle=\Sigma_{A}\otimes\Sigma_{B}\otimes\Sigma_{\text{NL}},$ (171)

where $\Lambda_{\text{NL}}$ represents some global degrees of freedom not
reducible to properties of system $A$ and system $B$ alone, and
$\Sigma_{\text{NL}}$ is a $\sigma$-algebra over $\Lambda_{\text{NL}}$.

  * •

The Weak No-Correlation Assumption (WNCA):
$\forall\rho_{A}\in\mathcal{P}_{A},\rho_{B}\in\mathcal{P}_{B}$, all
$\mu_{AB}\in\Delta_{AB}[\rho_{A}\otimes\rho_{B}]$ satisfy

$\mu_{AB}(\Omega_{AB}\times\Lambda_{\text{NL}})=\mu_{A}\times\mu_{B}(\Omega_{AB}),$
(172)

for all $\Omega_{AB}\in\Sigma_{A}\otimes\Sigma_{B}$ and for some
$\mu_{A}\in\Delta_{A}(\rho_{A})$ and $\mu_{B}\in\Delta_{B}(\rho_{B})$, i.e.
the marginal measure on $\Lambda_{A}\times\Lambda_{B}$ satisfies the NCA.

In general, the ontic state space for a composite system need not break down
neatly into a product of local ontic state spaces and global properties. The
WCPA therefore seems like a relatively mild generalization of the CPA. The
WNCA then says that, if we only have access to $\lambda_{A}$ and
$\lambda_{B}$, then the probability measures corresponding to product states
look just like product measures, even though they may in fact be correlated
via the third variable $\lambda_{\text{NL}}$. Further, not only do they look
like product measures, they look just like the product of measures that the
ontological models $\Theta_{A}$ and $\Theta_{B}$ would assign to the two
systems individually. If we assume that $\Theta_{A}$ and $\Theta_{B}$ are
adequate for modeling $\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$ individually,
it follows that local measurements can be modeled by products of the
conditional probability distributions $\text{Pr}_{A}$ and $\text{Pr}_{B}$
assigned by these models, so the outcomes of local measurements need only
depend on $\lambda_{A}$ and $\lambda_{B}$. Therefore, under local
measurements, this type of model is indistinguishable from one that satisfies
the PIP. It is only under joint measurements that there is a difference, since
the outcomes of these may depend on $\lambda_{\text{NL}}$.

######  Definition 7.17.

A pair of ontological models $\Theta_{A}$ and $\Theta_{B}$ for fragments
$\mathfrak{F}_{A}$ and $\mathfrak{F}_{B}$ that reproduce the quantum
predictions satisfy the _Weak Preparation Independence Postulate (WPIP)_ if,
for all product state fragments $\mathfrak{F}_{AB}$ with $\mathfrak{F}_{A}$
and $\mathfrak{F}_{B}$ as factors, there exists an ontological model
$\Theta_{AB}$ that is weakly compatible with $\Theta_{A}$ and $\Theta_{B}$ and
reproduces the quantum predictions.

Before discussing this further, we prove that the WPIP places no constraints
at all on the local models $\Theta_{A}$ and $\Theta_{B}$. The proof idea is
due to Matt Pusey and Terry Rudolph [114].

######  Theorem 7.18.

Let
$\mathfrak{F}_{A}=\langle\mathcal{H}_{A},\mathcal{P}_{A},\mathcal{M}_{A}\rangle$
and
$\mathfrak{F}_{B}=\langle\mathcal{H}_{B},\mathcal{P}_{B},\mathcal{M}_{B}\rangle$
be PM fragments and let
$\Theta_{A}=(\Lambda_{A},\Sigma_{A},\Delta_{A},\Xi_{A})$ and
$\Theta_{B}=(\Lambda_{B},\Sigma_{B},\Delta_{B},\Xi_{B})$ be any ontological
models of them that reproduce the quantum predictions. Then, $\Theta_{A}$ and
$\Theta_{B}$ satisfy the WPIP.

###### Proof.

Let
$\mathfrak{F}_{AB}=\langle\mathcal{H}_{A}\otimes\mathcal{H}_{B},\mathcal{P}_{A}\times\mathcal{P}_{B},\mathcal{M}_{AB}\rangle$
be a product state fragment with factors $\mathfrak{F}_{A}$ and
$\mathfrak{F}_{B}$. We need to show that there is an ontological model
$\Theta_{AB}$ of $\mathfrak{F}_{AB}$ satisfying the following three
requirements.

  * •

$\Theta_{AB}$ satisfies the WCPA, i.e. it is of the form
$\Theta_{AB}=(\Lambda_{A}\times\Lambda_{B}\times\Lambda_{\text{NL}},\Sigma_{A}\otimes\Sigma_{B}\otimes\Sigma_{NL},\Delta_{AB},\Xi_{AB})$.

  * •

$\Theta_{AB}$ satisfies the WNCA.

  * •

$\Theta_{AB}$ reproduces the quantum predictions.

To satisfy the first requirement, we first need to establish what the
measurable space $(\Lambda_{\text{NL}},\Sigma_{\text{NL}})$ should be. For
this, consider an arbitrary ontological model
$\Theta_{\text{NL}}=(\Lambda_{\text{NL}},\Sigma_{\text{NL}},\Delta_{\text{NL}},\Xi_{\text{NL}})$
of $\mathfrak{F}_{AB}$ that reproduces the quantum predictions. Such an
ontological model always exists. For example, we could use the
Beltrametti–Bugajski model extended to mixed states. Now, just take
$(\Lambda_{\text{NL}},\Sigma_{\text{NL}})$ to be the ontic state space of this
model. With this, we have that the model satisfies the WCPA.

For the second requirement, take the sets
$\Delta_{AB}[\rho_{A}\otimes\rho_{B}]$ to consist of all measures of the form
$\mu_{A}\times\mu_{B}\times\mu_{\text{NL}}$, where
$\mu_{A}\in\Delta_{A}[\rho_{A}]$, $\mu_{B}\in\Delta_{B}[\rho_{B}]$ and
$\mu_{\text{NL}}\in\Delta_{\text{NL}}[\rho_{A}\otimes\rho_{B}]$. This
assignment satisfies the WNCA.

It remains to show that the model can be made to reproduce the quantum
predictions. There are two cases to consider: local measurements for which
$(M_{A},M_{B})\in\mathcal{M}_{A}\times\mathcal{M}_{B}$, and joint measurements
$M_{AB}\in\mathcal{M}_{AB}$ that are not in
$\mathcal{M}_{A}\times\mathcal{M}_{B}$.

For the local measurements, the sets $\Xi_{AB}[M_{A},M_{B}]$ are chosen to
consist of all conditional probability distributions of the form

$\displaystyle\text{Pr}_{AB}(E,F|M_{A},M_{B},\lambda_{A},\lambda_{B},\lambda_{\text{NL}})$
$\displaystyle\quad=\text{Pr}_{A}(E|M_{A},\lambda_{A})\text{Pr}_{B}(F|M_{B},\lambda_{B}),$
(173)

where $E\in M_{A}$ and $F\in M_{B}$, and where
$\text{Pr}_{A}\in\Xi_{A}[M_{A}]$ and $\text{Pr}_{B}\in\Xi_{B}[M_{B}]$. In
other words, the nonlocal variable is ignored and we just use the conditional
probabilities from the two local ontological models. This reproduces the
quantum predictions by virtue of the fact that $\Theta_{A}$ and $\Theta_{B}$
do.

For the nonlocal measurements, the sets $\Xi_{AB}[M_{AB}]$ are chosen to
consist of all conditional probability distributions of the form

$\text{Pr}_{AB}(E|M_{AB},\lambda_{A},\lambda_{B},\lambda_{\text{NL}})=\text{Pr}_{\text{NL}}(E|M_{AB},\lambda_{\text{NL}}),$
(174)

where $E\in M_{AB}$, and where
$\text{Pr}_{\text{NL}}\in\Xi_{\text{NL}}[M_{AB}]$. This reproduces the quantum
predictions by virtue of the fact that $\Theta_{\text{NL}}$ does. ∎

If the PIP is replaced with the WPIP, the implications for $\psi$-ontology are
as follows. Consider the situation in which the factors $\mathfrak{F}_{A}$ and
$\mathfrak{F}_{B}$ consist of all pure states and measurements in orthonormal
bases on a finite dimensional Hilbert space of dimension $d$. The ABCL
construction shows that $\psi$-epistemic models always exist for these
factors. Since any pair of ontological models satisfy the WPIP, ABCL models do
in particular, so the WPIP cannot be used to prove a $\psi$-ontology theorem.
One might object that we should demand that the weakly compatible model of
$\mathfrak{F}_{AB}$ should also be $\psi$-epistemic, although that was not
part of the definition of the WPIP. However, if we restrict attention to
fragments $\mathfrak{F}_{AB}$ that consist of measurements in orthonormal
bases then this can be achieved by using an ABCL model for dimension $d^{2}$
as $\Theta_{\text{NL}}$. Since we can make an arbitrary pair of states
ontologically indistinct in an ABCL model, if $\left[\psi_{0}\right]_{A}$ and
$\left[\psi_{1}\right]_{A}$ are ontologically indistinct in $\Theta_{A}$ and
$\left[\phi_{0}\right]_{B}$ and $\left[\phi_{1}\right]_{B}$ are ontologically
indistinct in $\Theta_{B}$ then we can choose
$\left[\psi_{0}\right]_{A}\otimes\left[\phi_{0}\right]_{B}$ and
$\left[\psi_{1}\right]_{A}\otimes\left[\phi_{1}\right]_{B}$ to be
ontologically indistinct in $\Theta_{\text{NL}}$ and then they will remain so
in $\Theta_{AB}$. One might make the further objection that all four states
$\left[\psi_{0}\right]_{A}\otimes\left[\phi_{0}\right]_{B}$,
$\left[\psi_{0}\right]_{A}\otimes\left[\phi_{1}\right]_{B}$,
$\left[\psi_{1}\right]_{A}\otimes\left[\phi_{0}\right]_{B}$ and
$\left[\psi_{1}\right]_{A}\otimes\left[\phi_{1}\right]_{B}$ ought to be
pairwise ontologically indistinct, as they would be in the model of the direct
product fragment discussed in §7.1, but this can also be arranged by using the
extension of the ABCL model to be discussed in §11, in which every pair of
pure states can be made ontologically indistinct. Finally, it should be
evident that the construction used in proving Theorem 7.18 can be iterated to
larger numbers of subsystems. We simply introduce a new $\Theta_{\text{NL}}$
for the joint system every time a new subsystem is added.

However, the proof of Theorem 7.18 also illustrates why I do not consider the
WPIP to be a viable constraint on subsystem composition. All we have done is
to load an ontological model $\Theta_{\text{NL}}$ for the joint system, that
could potentially reproduce the quantum predictions for both local and joint
measurements all by itself, onto the nonlocal variable $\lambda_{\text{NL}}$.
When we make a local measurement, this model is completely ignored and we use
the local models $\Theta_{A}$ and $\Theta_{B}$ instead, but when we make a
joint measurement we do the opposite, using the nonlocal model
$\Theta_{\text{NL}}$ and completely ignoring $\Theta_{A}$ and $\Theta_{B}$.
This is fully compatible with the WPIP, but it is not clear why this model is
better than just using $\Theta_{\text{NL}}$ on its own in the first place.

I accept the criticism that dropping the CPA need not necessarily give rise to
global parameters becoming relevant to local experiments. However, arguably,
the form of the local ontological models should place some constraints on what
the model for the joint system should look like. The PIP, in its original
form, does this, but the WPIP does not, since it is compatible with just
assigning an ontological model to the nonlocal variable that is completely
independent of the local ontological models. In response to this, I think one
should either drop the PIP entirely, or come up with something of intermediate
strength that places nontrivial constraints on the ontological model for the
joint system. Without that, giving examples of $\psi$-epistemic models that
satisfy the WPIP does not, in my opinion, strengthen the case against
$\psi$-ontology.

Of course, ESSV do not have in mind the trivial sort of model constructed
here. In their model of Example 7.9, the nonlocal variable does not contain
enough information to reproduce the quantum predictions all on its own, and
the probability measures over
$\Lambda_{A}\times\Lambda_{B}\times\Lambda_{\text{NL}}$ are nontrivially
correlated. The point is that ESSV have not done enough to articulate the way
in which their model differs from the trivial construction presented here. On
this point, note that both the ESSV model and the construction given here
share the feature that the amount of overlap between states is lower than it
would be in the model of the direct product fragment discussed in §7.1, due to
the probability measure on the extra factor $\Lambda_{\text{NL}}$. One might
therefore impose the additional requirement that the reduction in overlap
introduced by the nonlocal variable ought to be as small as possible, which
could potentially rule out the construction presented here. One might also
demand that there should be no redundancy in the model, in the sense that it
should not be possible to reproduce the quantum predictions for local
measurements using the nonlocal variable on its own. Further work along these
lines is needed to determine whether there is a weakened version of the PIP
that can serve as a reasonable constraint on subsystem composition.

####  7.6.2 Criticism of the NCA

The PIP has also been criticized on other grounds by Hall [33] (this criticism
is also mentioned by Schlosshauer and Fine [38]). Hall objects to the NCA on
the grounds that, even if they are space-like separated, the events
corresponding to preparing two systems have a common past, so their lack of
correlation cannot be derived from causality by the same sort of reasoning
that Bell used to motivate his locality condition. More explicitly, suppose we
have a model that does satisfy the CPA, so that we can associate separate
ontic states $\lambda_{A}$ and $\lambda_{B}$ to Alice and Bob’s systems.
Suppose that the intersection of the past lightcones of Alice and Bob’s
preparation events also has some physical properties described its own ontic
state $\lambda_{\text{past}}$. For concreteness, suppose that all the ontic
state spaces are finite, i.e. the ontic state spaces of Alice’s system, Bob’s
system, and their common past are all finite. The argument does not depend on
this, but it makes things conceptually simpler.

Suppose that Alice decides to prepare the quantum state
$\left[\psi\right]_{A}$ and Bob decides to prepare the quantum state
$\left[\phi\right]_{B}$. In general, the resulting $\lambda_{A}$ and
$\lambda_{B}$ might depend on both Alice and Bob’s choices of preparation and
the ontic state $\lambda_{\text{past}}$ of their common past, so the
preparation procedure would be specified by conditional probabilities
$\text{Pr}(\lambda_{A},\lambda_{B}|\left[\psi\right]_{A},\left[\phi\right]_{B},\lambda_{\text{past}})$.
Given that the preparation procedures might occur at space-like separation, it
is not unreasonable to impose a factorization of probabilities akin to Bell
locality, which would read

$\displaystyle\text{Pr}(\lambda_{A},\lambda_{B}|\left[\psi\right]_{A},\left[\phi\right]_{B},\lambda_{\text{past}})$
$\displaystyle=\text{Pr}(\lambda_{A}|\left[\psi\right]_{A},\lambda_{\text{past}})\text{Pr}(\lambda_{B}|\left[\phi\right]_{B},\lambda_{\text{past}}),$
(175)

However, this is not enough to entail the NCA because the measure $\mu$
corresponding to the preparation of
$\left[\psi\right]_{A}\otimes\left[\phi\right]_{B}$ would then be,

$\displaystyle\mu(\lambda_{A},\lambda_{B})=\sum_{\lambda_{\text{past}}}\text{Pr}(\lambda_{A}|\left[\psi\right]_{A},\lambda_{\text{past}})$
$\displaystyle\quad\times\text{Pr}(\lambda_{B}|\left[\phi\right]_{B},\lambda_{\text{past}})\text{Pr}(\lambda_{\text{past}}),$
(176)

where $\text{Pr}(\lambda_{\text{past}})$ is the prior distribution over the
variables in the common past. This can induce correlations between
$\lambda_{A}$ and $\lambda_{B}$ due to their common dependence on
$\lambda_{\text{past}}$.

This argument is correct, but all it shows is that the NCA cannot be regarded
as a causality assumption akin to Bell locality. It does not imply that the
NCA is necessarily unreasonable. To infer that, one would have to believe that
the _only_ reasonable type of assumption to make in a no-go theorem is one
that follows from Bell-type locality. In contrast, §7.3 discussed a more
general strategy for positing reasonable assumptions, which is to look at the
operational structure of quantum theory and try to impose similar structure at
the ontological level. In the case of the NCA, we noted that a product state
displays no correlations in any quantum measurement, so it makes sense to
posit that no such correlations exist at the ontological level either. It is
this, and not Bell locality, that is the motivation for the NCA. Dropping the
NCA in light of the Pusey–Barrett–Rudolph Theorem is not an obviously crazy
thing to do, but it is the Pusey–Barrett–Rudolph Theorem itself rather than
causality arguments that should motivate this move. The same reasoning applies
to all other no-go theorems that are not motivated by locality. For example, I
think it is fairly clear that proofs of preparation contextuality are a good
reason for dropping the assumption of preparation noncontextuality, but the
fact that the latter cannot be derived from Bell locality is a much less
compelling reason for doing so.

###  7.7 Other Criticisms of the Pusey–Barrett–Rudolph Theorem

Apart from criticism of the PIP, several other criticisms of the
Pusey–Barrett–Rudolph Theorem have been raised. §7.7.1 discusses a criticism
due to Drezet [30, 31] and Schlosshauer and Fine [38] based on the idea that
the conditional probability distributions representing measurements should
depend on the quantum state in addition to the ontic state. I argue that this
criticism is simply a misunderstanding of what is meant by the term “ontic
state” in the ontological models framework. I then discuss two further
criticisms due to Schlosshauer and Fine [38], the second of which has also
been made by Dutta et. al. [115]. The first criticism, discussed in §7.7.2, is
a claim that the $\psi$-ontic/$\psi$-epistemic distinction is merely
conventional because one kind of model can be converted into a structurally
equivalent model of the other kind. The second criticism, discussed in §7.7.3,
is based on the idea that modeling detector inefficiencies offers a way out of
the dilemma posed by the Pusey–Barrett–Rudolph Theorem. Whilst this is
strictly speaking true, it only concerns the practical implementation of
Pusey–Barrett–Rudolph-type experiments and has no impact on the
Pusey–Barrett–Rudolph Theorem as a structural result about quantum theory
itself, assuming ideal experiments. If one accepts the PIP, error analysis can
be used to constrain this escape route in more practical experiments. Finally,
in §7.7.4, I discuss criticisms that reject some aspect of the ontological
models framework outright. These criticisms are mostly based on a neo-
Copenhagen point of view, and thus are easy to deal with as the
Pusey–Barrett–Rudolph Theorem was never intended to rule out such approaches.

Since the criticisms in this section are directed against the ontological
models framework in general, they are not specific to Pusey–Barrett–Rudolph
but could also be directed against Hardy’s Theorem, the Colbeck–Renner Theorem
and the results discussed in Part III. I discuss them here because they were
made as responses to the Pusey–Barrett–Rudolph Theorem and not these other
results, but this is simply because the Pusey–Barrett–Rudolph Theorem was the
first and is still the most prominent $\psi$-ontology theorem.

####  7.7.1 The conditional probabilities for measurements should depend on
the quantum state

Several authors have pointed out that, in the ontological models framework,
the conditional probability distribution $\text{Pr}(E|M,\lambda)$ representing
a measurement is assumed to be independent of the quantum state $\rho$ that is
prepared [30, 31, 38]. If a dependence $\text{Pr}(E|\rho,M,\lambda)$ is
allowed then the theorem can be trivially evaded. The conditional
probabilities can simply be made independent of the ontic state and and can
just return the quantum probabilities by setting
$\text{Pr}(E|\rho,M,\lambda)=\text{Tr}\left(E\rho\right)$, since we then have,
for any probability measure $\mu$,

$\displaystyle\int_{\Lambda}\text{Pr}(E|\rho,M,\lambda)d\mu(\lambda)$
$\displaystyle=\text{Tr}\left(E\rho\right)\int_{\Lambda}d\mu(\lambda)$ (177)
$\displaystyle=\text{Tr}\left(E\rho\right).$ (178)

The ontic state space and probability measures can then be anything at all, so
the model can trivially be made to satisfy the PIP. For example,
$\mathbb{C}^{d}$ could be associated with the ontic state space
$\Lambda_{d}=\\{1,2,\ldots,d\\}$ and
$\mathbb{C}^{d}\otimes\mathbb{C}^{d^{\prime}}$ with the Cartesian product
$\Lambda_{d}\times\Lambda_{d^{\prime}}$, which is isomorphic to
$\Lambda_{dd^{\prime}}$. Then, the uniform measure can be used to represent
all quantum states, which makes the model trivially $\psi$-epistemic.

Of course, proponents of this view do not have this sort of model in mind as a
realistic candidate for describing quantum theory. They think the conditional
probabilities should depend on the ontic states in some way in addition to the
quantum state. The above model is just intended to show how trivial the
$\psi$-ontic/$\psi$-epistemic distinction becomes when the conditional
probabilities are allowed to depend on the quantum state.

I think there are two intuitions behind this sort of objection. The first is
based on elementary probability theory and the second on a misunderstanding of
how conventional hidden variable theories, such as de Broglie–Bohm theory, are
meant to fit into the ontological models framework. First of all, putting
aside everything we know about the ontological models framework for the
moment, suppose we are interested in some PM fragment
$\mathfrak{F}=\langle\mathcal{H},\mathcal{P},\mathcal{M}\rangle$ and we have a
theory for reproducing its predictions that involves some sort of additional
variable $\lambda$ that takes values in a set $\Lambda$. For the purposes of
this argument, assume that $\Lambda$, $\mathcal{P}$ and $\mathcal{M}$ are
finite, so that elementary probability theory can be used. This restriction
can easily be lifted, but dealing with the measure theoretic complications
would obscure the argument.

A quite general way that such a theory could be formulated is in terms of a
joint probability distribution $\text{Pr}(E,\rho,M,\lambda)$ over all the
variables involved, where $\text{Pr}(E,\rho,M,\lambda)$ is specified for every
$\rho\in\mathcal{P}$, $M\in\mathcal{M}$, $E\in M$, $\lambda\in\Lambda$. In
order to determine whether such a model reproduces the quantum predictions, we
need to determine the conditional probabilities $\text{Pr}(E|\rho,M)$, i.e.
the probability for an outcome given the choice of state and measurement,
because quantum theory tells us this should equal