#  $\psi$-epistemic interpretations of quantum theory have a measurement
problem

Joshua B. Ruebeck<EMAIL_ADDRESS>Piers Lillystone  Institute for
Quantum Computing and Department of Physics and Astronomy
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Joseph Emerson
Institute for Quantum Computing and Department of Applied Math
University of Waterloo, Waterloo, Ontario N2L 3G1, Canada  Canadian Institute
for Advanced Research, Toronto, Ontario M5G 1Z8, Canada

(August 27, 2024)

###### Abstract

$\psi$-epistemic interpretations of quantum theory maintain that quantum
states only represent incomplete information about the physical states of the
world. A major motivation for this view is the promise to provide a reasonable
account of state update under measurement by asserting that it is simply a
natural feature of updating incomplete statistical information. Here we
demonstrate that all known $\psi$-epistemic ontological models of quantum
theory in dimension $d\geq 3$, including those designed to evade the
conclusion of the PBR theorem, cannot represent state update correctly.
Conversely, interpretations for which the wavefunction is real evade such
restrictions despite remaining subject to long-standing criticism regarding
physical discontinuity, indeterminism and the ambiguity of the Heisenberg cut.
This revives the possibility of a no-go theorem with no additional
assumptions, and demonstrates that what is usually thought of as a strength of
epistemic interpretations may in fact be a weakness.

##  1 Introduction

There are many interpretations of quantum theory111This is an understatement..
Among the many differences between these interpretations, one that often takes
center stage is the stance that they take towards the wavefunction or quantum
state. Three broad categories have been identified which capture a number of
interpretations. Two of these categories are more commonly juxtaposed:
$\psi$-ontic interpretations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
posit that the quantum state is a part of the _real_ (physical) state of a
system, whereas $\psi$-epistemic interpretations [15, 16, 17, 18, 19, 20, 21,
22, 23] argue that the quantum state is merely a state of _knowledge_ about
the real state of the system. A very thorough review of these two stances can
be found in [24]. A third recently articulated category of $\psi$-doxastic
interpretations [25, 26, 27, 28, 29, 30, 31] argue that the quantum state is a
state of _belief_ , and are distinguished from $\psi$-epistemic
interpretations by the fact that they deny that a system has some ‘real
state.’ While not all interpretations conform to these three descriptors, they
are useful categories insofar as they allow us to qualitatively discuss
certain features separately from the particular interpretation in which they
are embedded.

The $\psi$-epistemic class has garnered attention as a view which provides
very appealing explanations of otherwise paradoxical features of quantum
theory like the state update rule [17], the classical limit under quantum
chaos [20], no cloning [32], and entanglement [33]. For example, through this
viewpoint state update is not a physical ‘collapse’ process and therefore not
subject to paradoxes, indeterminism, and discontinuity; rather it is
understood as analogous to the non-pardoxical ‘collapse’ of a subjective
probability distribution via Bayes’ rule upon consideration of new
information. While several $\psi$-epistemic models have been proposed [16, 34,
35, 36, 37, 22, 38], they generally have undesirable features or are
restricted to a subtheory of full quantum theory. None achieve all of the
features that an optimistic $\psi$-epistemicist would expect.

This suggests the possibility that a fully satisfactory $\psi$-epistemic
interpretation cannot actually explain all of quantum theory despite the
qualitatively compelling features of such a view222As is well-known, the work
of Bell [39] has shown that no $\psi$-epistemic interpretation can evade the
non-locality that manifests trivially in $\psi$-ontic interpretations; this
trivial manifestation of non-locality in $\psi$-ontic interpretations is an
oft-forgotten insight from Einstein [23, 18, 15].. This suspicion has led to a
number of no-go theorems in recent years which establish that, given at least
one additional assumption, any consistent interpretation of quantum theory
cannot be $\psi$-epistemic [40, 36, 41, 42, 43]. These no-go theorems are
generally proven within the ontological models formalism, which describes a
large class of existing interpretations of quantum theory [44, 45, 21].

Within the ontological models formalism333A note on potentially confusing
terminology—an ontological model assumes the reality of _some_ kind of state
(hence “ontological”), but does not assume the reality of the quantum state
specifically (and so is not necessarily $\psi$-ontic)., $\psi$-epistemic
models can be given a precise mathematical definition called the
$\psi$-epistemic criterion [22]. This criterion allows the possibility of
conclusively ruling out this type of model. Outside of this framework, it is
unlikely that $\psi$-epistemic models can be precluded with any kind of
certainty; $\psi$-doxastic interpretations, for example, do not fit neatly
into the ontological models framework and thus are not necessarily ruled out
by these no-go theorems. This is despite the fact that they share many of the
features which make $\psi$-epistemic interpretations appealing. In the present
paper we restrict our attention to the ontological models framework.

The fact that an extra assumption is required to rule out $\psi$-epistemic
theories has purportedly been demonstrated by the existence of
$\psi$-epistemic models which, while being individually unsatisfactory for
various reasons, do satisfy at least the bare minimum requirements of a
$\psi$-epistemic theory [35, 36, 37]. All of these models were specified
within a prepare-measure framework, so they have been proven to reproduce
quantum statistics for all experiments that involve preparing a state and then
measuring it once. In this paper we show that, if we allow sequential
measurements in the operational description, these models cannot reproduce
operational statistics.

Our main contribution in this work is thus to demonstrate that the state
update rule imposes severe constraints on $\psi$-epistemic models. This is in
contrast to the prevailing view that, as articulated by Leifer, “a
straightforward resolution of the collapse of the wavefunction, the
measurement problem, Schrödinger’s cat and friends is one of the main
advantages of $\psi$-epistemic interpretations” [24]. As a consequence, we
revive the possibility of a general no-go theorem for $\psi$-epistemic models
that doesn’t rely on an additional assumption such as the locality assumption
required in [40] which conflicts with the non-locality that is implied by
Bell’s theorem [46, 43]. See Appendix A for further discussion on this point.

Although state update under measurement has been described in a few specific
models [34, 22, 47, 48] and discussed with regards to contextuality [49], it
has yet to be treated generally or in relation to the
$\psi$-epistemic/$\psi$-ontic distinction444Although the Leggett-Garg
inequalities [50, 51] might be construed as a general treatment of state
update, it is more accurate to say that they are about the _absence_ of state
update.. Here we take some preliminary steps in both of these directions, and
argue that $\psi$-epistemic models are the natural arena in which to
investigate interesting behavior of state update under measurement.

In Section 2, we describe the ontological models formalism, adding a
description of state update under measurement and motivating its importance.
Despite this motivation, one might still argue that many distinctly quantum
phenomena (e.g. Bell inequality violations) can be described without reference
to state update; thus, from an operationalist point of view, we shouldn’t need
to consider state update in order to investigate these phenomena. However, we
show in Section 3 that the consideration of state update actually places
nontrivial restrictions on how one can represent even a prepare-and-measure-
once experiment. Thus our results are directly applicable to models which have
only specified behavior for a single measurement. Section 4 reviews a number
of examples of ontological models from the literature; in each case we either
specify its state update rule (in dimension $d=2$, for $\psi$-ontic models,
and for some models of subtheories) or prove its impossibility (for all known
$\psi$-epistemic models in dimension $d\geq 3$). Finally, we discuss the
implications of our results and describe some open questions in Section 5.

##  2 Defining measurement update in ontological models

###  2.1 The ontological models formalism

In the standard treatment, an _operational theory_ [44] is described by a set
of preparations $\mathcal{P}$, a set of transformations $\mathcal{T}$, and a
set of measurements $\mathcal{M}$ along with a probability distribution

$\textup{Pr}(k|M,T,P).$ (1)

This quantity describes the probability of some measurement outcome
$k\in\mathbb{Z}$ given an experimenter’s choice of $P\in\mathcal{P}$,
$T\in\mathcal{T}$, and $M\in\mathcal{M}$. When considering transformations
this is called the _prepare-transform-measure_ operational framework, and when
we omit transformations it is the _prepare-measure_ framework. Often we take
$\mathcal{P}$ to be the set of pure quantum state preparations, $\mathcal{T}$
to be the full set unitary maps on a Hilbert space, and $\mathcal{M}$ to be
all projective measurements on this Hilbert space. In this case, we say we are
describing the full quantum theory555Larger sets can be considered (e.g.
including mixed states, CPTP maps, or non-projective measurements), but all of
the models studied in this paper fit the given definition.; in contrast, a
_subtheory_ is described by taking subsets of
$\mathcal{P},\mathcal{T},\mathcal{M}$ for the full quantum theory. For
example, in quantum information settings we often consider only measurements
in the standard basis.

Note that this standard definition involves a single measurement and a single
measurement outcome despite the fact many important quantum experiments (e.g.
Stern-Gerlach, double slit [52]) and quantum algorithms (e.g. measurement-
based error correction [53]) involve multiple measurements. Thus we will refer
to the usual definition of prepare-measure as prepare-and-measure-once
experiments. In this paper, we are concerned with multiple measurements, so we
will also have to describe probabilities like

$\textup{Pr}(k_{2},k_{1}|M_{2},M_{1},P).$ (2)

Additionally, we note that positive operator valued measures (POVMs) do not
fully specify how a measurement updates a state. Although one can obtain a
POVM $\\{E_{k}\\}$ from a set of generalized measurement operators
$\\{M_{k}\\}$ by the relation $E_{k}=M_{k}^{\dagger}M_{k}$, the decomposition
of $\\{E_{k}\\}$ into $\\{M_{k}\\}$ is not unique. Thus although $\mathcal{M}$
is often described by POVMs, consideration of state update requires that we
specify generalized measurement operators instead. As an example of when this
is important, consider a coarse-graining of the measurement
$\\{M_{k}\\}=\\{{\bm{[}}0{\bm{]}},{\bm{[}}1{\bm{]}},{\bm{[}}2{\bm{]}}\\}$,
where we denote the projector onto a state

${\bm{[}}\psi{\bm{]}}=\outerproduct{\psi}{\psi}$ (3)

as in [24]. We can either coarse-grain coherently, i.e. measure
$\\{M_{k}^{\prime}\\}=\\{{\bm{[}}0{\bm{]}}+{\bm{[}}1{\bm{]}},{\bm{[}}2{\bm{]}}\\}$
or we can coarse-grain decoherently by measuring $\\{M_{k}\\}$ and then
combining outcomes 0 and 1 into a single measurement result and ‘forgetting’
which one actually occurred. While these two processes are represented by the
same POVM
$\\{E_{k}\\}=\\{{\bm{[}}0{\bm{]}}+{\bm{[}}1{\bm{]}},{\bm{[}}2{\bm{]}}\\}$,
their state update behavior is different: if the state
$\frac{1}{\sqrt{2}}\quantity(\ket{0}+\ket{1})$ is measured, it will stay the
same in the coherent case or update to the mixed state
$\frac{1}{2}\quantity({\bm{[}}0{\bm{]}}+{\bm{[}}1{\bm{]}})$ in the decoherent
case.

An _ontological model_ [44, 45, 21] supplements this operational point of view
by asserting that a system has a state $\lambda$, called an _ontic state_. To
specify an ontological model, we first choose an _ontic state space_
$\Lambda$. Then, preparations are described by a _preparation distribution_
$\mu(\lambda|P)$, which is the probability of preparing some state
$\lambda\in\Lambda$ given the preparation $P$. Transformations are described
by a _transition matrix_ $\Gamma(\lambda^{\prime}|\lambda,T)$, which is the
probability of preparing a new state $\lambda^{\prime}$ given the previous
state $\lambda$ and the choice $T$ of transformation. Finally, measurements
are represented by a _response function_ $\xi(k|\lambda,M)$ which describes
the probability of an outcome $k$ given the ontic state $\lambda$ and the
choice of measurement $M$. We say that an ontological model successfully
reproduces quantum theory if

$\displaystyle\int_{\Lambda}\differential{\lambda^{\prime}}\int_{\Lambda}\differential{\lambda}\xi(k|\lambda^{\prime},M)\Gamma($
$\displaystyle\lambda^{\prime}|\lambda,T)\mu(\lambda|P)$
$\displaystyle=\textup{Pr}_{Q}(k|M,T,P)$ (4) $\displaystyle\forall$
$\displaystyle P\in\mathcal{P},T\in\mathcal{T},M\in\mathcal{M},$

or, in a prepare-and-measure-once experiment,

$\displaystyle\int_{\Lambda}\differential{\lambda}\xi(k|\lambda,M)\mu(\lambda|P)$
$\displaystyle=\textup{Pr}_{Q}(k|M,P)$ (5) $\displaystyle\forall$
$\displaystyle P\in\mathcal{P},M\in\mathcal{M}.$

In both of these cases, $\textup{Pr}_{Q}$ indicates the outcome probability
calculated by operational quantum theory for the particular experiment under
consideration.

Again, we must supplement this definition in order to model sequential
measurement. In textbook quantum theory, the state updates during a
measurement in a way that depends on the previous state, the measurement
procedure, and the measurement outcome: For a measurement outcome
corresponding to a projector $\Pi_{k}$, a quantum state $\ket{\psi}$ after
measurement will be

$\ket{\psi^{\prime}}=\frac{\Pi_{k}\ket{\psi}}{\expectationvalue{\Pi_{k}}{\psi}}.$
(6)

We allow for dependence on all of these things by choosing to represent this
via a _state update rule_ $\eta(\lambda^{\prime}|k,\lambda,M)$. As with $\mu$,
$\Gamma$, and $\xi$, we require that $\eta$ is normalized. Although this
object looks very similar to the transition matrix for transformations, it is
distinguished by two important features which we emphasize by choosing a new
symbol to represent it.

The first distinction is simple, in that $\eta$ depends on a measurement
outcome $k$, while $\Gamma$ does not; this is analogous to the fact that
generally in quantum theory we can only implement measurement update maps
probabilistically (i.e. by post-selecting on a not-necessarily-deterministic
measurement outcome).

The second distinction is the fact that $\eta(\lambda^{\prime}|k,\lambda,M)$
is not defined for all $\lambda\in\Lambda$. Roughly speaking, it doesn’t make
sense to ask “What is the new state $\lambda^{\prime}$ after measuring state
$\lambda$ and obtaining outcome $k$?” if the outcome $k$ could not have
occurred given the previous state $\lambda$. To express this formally, we
define the _support_ of a distribution

$\textup{Supp}(\xi(k|\dotproduct))=\\{\lambda\in\Lambda:\xi(k|\lambda)>0\\}$
(7)

as the set of ontic states on which it is nonzero. Using this, we can say that
$\eta(\lambda^{\prime}|k,\lambda,M)$ is well-defined only for
$\lambda\in\textup{Supp}(\xi(k|\dotproduct))$. This property of $\eta$ is
analogous to the fact that Eq. 6 is only well-defined for
$\expectationvalue{\Pi_{k}}{\psi}\neq 0$; i.e. only when the measurement
$\Pi_{k}$ could have responded to the previous quantum state $\psi$.

This second distinction is central to the main result of this paper; by
showing that $\eta$ is non-normalizable on some domain, we are able to
conclude that that domain cannot be part of the support of $\xi$.

The consistency condition with quantum theory is given by equations similar to
Eqs. 2.1 and 5; see Appendix B for more formal treatments of the properties of
$\eta$ and other extensions of the ontological models formalism discussed so
far. The rigorous treatment in the appendix shows that, given the pre-existing
assumptions of the ontological models formalism, this is the only way to
represent state update under sequential measurement; this is why we refer to
our results as using ‘no additional assumptions.’

This paper is not explicitly concerned with contextuality [16, 44], but we are
careful to ensure that we do not assume noncontextuality. Under the definition
of generalized contextuality given in [44], an ontological model is
_noncontextual_ if two operationally equivalent preparation, transformation,
or measurement procedures are always represented by equivalent preparation
distributions, transition matrices, or response functions, respectively666This
definition is actually complicated slightly by the inclusion of sequential
measurements, but we do not discuss this here.. It is _contextual_ otherwise.
While this generalized definition may be too broad, accounting for
contextuality under this definition also accounts for the traditional
definition [16], and is therefore more inclusive. Although, as stated above,
we do not assume noncontextuality of any kind, most of our results hold for a
projector independent of its full measurement context. We are explicit about
this when it is the case, and write $\xi(\Pi|\lambda)$ rather than
$\xi(k=0|\lambda,M=\\{\Pi,\ldots\\})$ for notational convenience;
$\eta(\lambda^{\prime}|\lambda,\Pi)$ is defined similarly.

###  2.2 $\psi$-epistemic models

Here we focus on a set of precise criteria for $\psi$-epistemic
interpretations within the ontological models formalism. Consider first the
$\psi$-epistemic criterion, proposed in [23] as a test for whether an
interpretation admits at least some quantum states that are not uniquely
determined by the underlying state of reality. Following [49], we account for
potential preparation contextuality by defining

$\Delta_{\psi}=\bigcup_{P_{\psi}\in\mathcal{P}_{\psi}}\textup{Supp}(\mu(\dotproduct|P_{\psi}))$
(8)

where $\mathcal{P}_{\psi}\subseteq\mathcal{P}$ is the set consisting of every
possible preparation of $\ket{\psi}$. We refer to $\Delta_{\psi}$ as the
support of a state $\ket{\psi}$, to distinguish it from the support of a
particular preparation $P_{\psi}$. Then a pair of states
$\ket{\psi},\ket{\phi}$ is _ontologically distinct_ in a particular model if
$\Delta_{\phi}\cap\Delta_{\psi}=\emptyset$, and _ontologically indistinct_
otherwise777For this purposes of this paper we assume there exists a measure
that is absolutely continuous with respect to all other measures in the
ontological model. Therefore, we can work with probability densities, rather
than the full measure-theoretic treatment. While this assumption is not
strictly true in all of our models, it does not affect our results and
significantly simplifies our presentation.. This leads us to the standard
definition of a $\psi$-epistemic ontological model [23, 24]:

######  Definition 1 ($\psi$-epistemic).

An ontological model is $\psi$-epistemic if there exists a pair of states
$\ket{\psi},\ket{\phi}$ that are ontologically indistinct; i.e.

$\exists\ket{\psi},\ket{\phi}:\Delta_{\phi}\cap\Delta_{\psi}\neq\emptyset.$
(9)

As noted in [24], this definition is highly permissive in the sense that, if
an ontological model were to contain only a single pair of quantum states that
are ontologically indistinct, then it would not achieve the full explanatory
power expected of the $\psi$-epistemic viewpoint; this is exactly the case
with the ABCL0 model discussed in Section 4.2.4.

There are, however, proposals to strengthen the notion of $\psi$-epistemicity,
two of which are relevant to our discussion [24, 54].

######  Definition 2 (Pairwise $\psi$-epistemic).

An ontological model is _pairwise $\psi$-epistemic_ if, for all pairs
$\ket{\psi},\ket{\phi}$ of nonorthogonal quantum states, $\ket{\psi}$ and
$\ket{\phi}$ are ontologically indistinct.

######  Definition 3 (Never $\psi$-ontic).

An ontological model is _never $\psi$-ontic_ if every ontic state
$\lambda\in\Lambda$ is in the support of at least two quantum states:

$\forall\lambda\in\Lambda:\exists\psi,\phi:\lambda\in\Delta_{\psi}\cap\Delta_{\phi}.$
(10)

Note that both of these definitions imply the weaker notion of
$\psi$-epistemicity, but are independent from one another.

###  2.3 Some easy cases of state update rules

There are two cases in which, given a prepare-and-measure-once ontological
model for quantum theory, we can always augment it with a state update rule
for measurement. First, if we only include rank-1 projective measurements in
the subtheory we’re modeling, we can simply re-prepare in the measured
(unique, pure) state:

$\eta(\lambda^{\prime}|k,\lambda,M_{\\{\Pi_{i}\\}})=\mu(\lambda^{\prime}|P_{\Pi_{k}})\text{
for }\tr(\Pi_{k})=1.$ (11)

This is normalized for all $\lambda$ since $\mu$ is normalized, and faithfully
reproduces quantum statistics since $\mu$ does. It is independent of the
previous state $\lambda$, which, besides being unsatisfying, is also not
possible in general (this follows from Section 3).

Second, it is quick to prove, again by construction, that $\psi$-ontic models
can always be given a state update rule. Since there is a unique quantum state
$\ket{\psi_{\lambda}}$ associated with every ontic state $\lambda$, we can
define

$\eta(\lambda^{\prime}|k,\lambda,M)=\mu\quantity(\lambda^{\prime}\bigg{|}P=\frac{M_{k}{\bm{[}}\psi_{\lambda}{\bm{]}}M_{k}^{\dagger}}{\tr(M_{k}{\bm{[}}\psi_{\lambda}{\bm{]}}M_{k}^{\dagger})}).$
(12)

Again, normalization and faithfulness follow because $\mu$ has these
properties. Note that this construction works for any kind of measurement, not
just projective measurements.

These observations together suggest that in order to find anything interesting
involving state update, we ought to examine higher-rank measurements in
$\psi$-epistemic models. This suspicion will be confirmed by the main result
of this paper, which applies to exactly these types of measurements and
models.

##  3 The consequences of state update

We now prove our central claim that consideration of a rule for state update
under measurement has consequences for the response function of an ontological
model, so that consistent state update puts restrictions on how one may
represent even a prepare-and-measure-once experiment. We begin with a lemma
that articulates a general property of the update rule $\eta$, and then
examine its consequences for response functions $\xi$.

######  Lemma 1.

Suppose we have an ontological model with ontic space $\Lambda$, preparation
distributions $\mu(\lambda|P)$, indicator functions $\xi(k|\lambda,M)$, and
state update maps $\eta(\lambda^{\prime}|k,\lambda,M)$. For a particular ontic
state $\lambda$ and measurement projector $\Pi$, we define the set
$S_{\lambda,\Pi}$ of quantum states that one could obtain after measurement of
any quantum state consistent with $\lambda$:

$S_{\lambda,\Pi}=\left\\{\frac{\Pi\ket{\phi}}{\sqrt{\expectationvalue{\Pi}{\phi}}}\>\middle|\>\forall\ket{\phi}:\lambda\in\Delta_{\phi}\right\\}.$
(13)

It is then true that, independently of the measurement context of $\Pi$,

$\textup{Supp}(\eta(\dotproduct|\lambda,\Pi))\subseteq\bigcap_{\ket{\psi}\in
S_{\lambda,\Pi}}\Delta_{\psi}.$ (14)

###### Proof.

Suppose that measuring a state $\ket{\phi}$ with a measurement $M$ results in
the updated state $\ket{\psi}$ when we get outcome $k$, where $\Pi$ is the
$k$th projector in $M$. Then let $P_{M,k,P_{\phi}}\in\mathcal{P}_{\psi}$ be
the preparation procedure associated with post-selection of this measurement
outcome after a particular preparation $P_{\phi}$. It must be normalized on
$\Delta_{\psi}$:

$\displaystyle 1$
$\displaystyle=\int_{\Delta_{\psi}}\differential{\lambda^{\prime}}\mu(\lambda^{\prime}|P_{M,k,P_{\phi}})$
$\displaystyle=\int_{\Delta_{\psi}}\differential{\lambda^{\prime}}\int_{\Delta_{\phi}}\differential{\lambda}\eta(\lambda^{\prime}|k,\lambda,M)\mu(\lambda|P_{\phi})$
$\displaystyle=\int_{\Delta_{\phi}}\differential{\lambda}\mu(\lambda|P_{\phi})\int_{\Delta_{\psi}}\differential{\lambda^{\prime}}\eta(\lambda^{\prime}|k,\lambda,M)$

Since $\eta$ is always positive, normalization of $\mu(\lambda|P_{\phi})$ then
implies that

$\int_{\Delta_{\psi}}\differential{\lambda^{\prime}}\eta(\lambda^{\prime}|k,\lambda,M)=1\qquad\forall\lambda\in\Delta_{\phi}.$

If $\eta$ is normalized on a region, its support must be contained in that
region. Thus for all $\lambda$ that are consistent with some preparation
$\ket{\phi}$ that could result in the post-measurement state $\ket{\psi}$,

$\textup{Supp}(\eta(\dotproduct|k,\lambda,M))\subseteq\Delta_{\psi}.$

The fact that this is true for all $\ket{\psi}$ that could result from the
measurement leads to Eq. 42. ∎

We note that Lemma 1 is easy to account for in $\psi$-ontic theories and for
rank-1 measurements, since in both cases $S_{\lambda,\Pi}$ has a single
element. This is why we were able to write down update rules for these
situations in Section 2.3. Outside of these trivial cases, Eq. (42) is a very
restrictive condition; depending on the structure of $S_{\lambda,\Pi}$, the
intersection may be a very small set. In particular, if any two of the post-
selected quantum states are orthogonal, then $S_{\lambda,\Pi}$ is empty.

######  Theorem 1 (Main theorem).

Suppose that a projector $\Pi$ maps two states $\ket{\alpha},\ket{\beta}$ to
ontologically distinct states $\Pi\ket{\alpha},\Pi\ket{\beta}$. Then the
response function for $\Pi$ cannot have support on the overlap of
$\ket{\alpha},\ket{\beta}$ for any measurement context of $\Pi$; i.e.

$\Delta_{\Pi\ket{\alpha}}\cap\Delta_{\Pi\ket{\beta}}=\emptyset\implies\xi(\Pi|\lambda)=0\
\ \forall\lambda\in\Delta_{\alpha}\cap\Delta_{\beta}.$ (15)

###### Proof.

Pick some $\lambda\in\Delta_{\alpha}\cap\Delta_{\beta}$. By Lemma 1,
$\textup{Supp}(\eta(\dotproduct|\lambda,\Pi))=\emptyset$ so
$\eta(\lambda^{\prime}|\lambda,\Pi)$ is not normalizable. As discussed in
Section 2.1, this is only allowable if $\xi(\Pi|\lambda)=0$. ∎

Both the lemma and the theorem hold for non-projective measurements as well.
We emphasize that this result does not say anything _directly_ about the
overlap of the supports of quantum states, just their overlap within the
support of a particular response function. In the following section, we deploy
this theorem by showing that, in every known $\psi$-epistemic model for $d\geq
3$, measurements of this type exist _and_ have support on the relevant
overlaps, leading to contradiction and demonstrating that these models cannot
reproduce state update under measurement.

##  4 Examples of state update under measurement (or its impossibility)

We provide a number of examples of ontological models from the literature,
illustrating some of the properties described in the previous sections. For
each model, we either specify its state update rule or prove that it cannot
reproduce state update. Although many of these models can be easily defined
for arbitrary types of measurements, we only consider projective measurements
for simplicity and notational consistency.

(a) Kochen-Specker model

(b) Montina model

(c) Beltrametti-Bugajski model

(d) Bell Model

Figure 1: Visualizations of the state space, preparation distributions, and
response functions for (a) the Kochen-Specker model, (b) Montina’s model, (c)
the Beltrametti-Bugajski model for $d=2$, and (d) Bell’s model for $d=2$. Blue
represents the support of preparations, and green the support of the response
functions, where possible. Black objects are generic elements of the state
space.

###  4.1 $\psi$-epistemic models of a qubit

The only $\psi$-epistemic models that are able to represent state update are
those that restrict to modeling a qubit. This corresponds to the fact that the
only nontrivial projective measurements in a qubit are rank-1, which, as
described earlier, do not result in restrictions from state update.

####  4.1.1 Kochen-Specker model

The Kochen-Specker model of a qubit [16, 24] is an exemplar of what we look
for in a $\psi$-epistemic theory, with the unfortunate feature that it only
works in $d=2$ dimensions. It is both pairwise $\psi$-epistemic and never
$\psi$-ontic, and provides a very intuitively pleasing interpretation of the
statistical nature of quantum theory. We take the ontic space to be the unit
sphere $S^{2}$, and denote by $\vec{\psi}$ the Bloch vector corresponding to
$\ket{\psi}$ under the usual mapping. Preparations and measurement outcomes
are represented by distributions over hemispheres, with response functions
uniform and preparation distributions peaked towards the center (Fig. 1(a)).
Unitary transformations are represented by rotations of the sphere. Since the
only nontrivial measurements on a qubit are rank-1 measurements, this is a
case where we can use the state update rule described in Eq. 11 and just re-
prepare the measured state for our update rule:

$\displaystyle\Lambda$ $\displaystyle=S^{2}$
$\displaystyle\mu(\vec{\lambda}|P_{\psi})$
$\displaystyle=\frac{1}{\pi}\Theta(\vec{\psi}\cdot\vec{\lambda})\vec{\psi}\cdot\vec{\lambda}$
$\displaystyle\Gamma(\vec{\lambda}^{\prime}|\vec{\lambda},T_{U})$
$\displaystyle=\delta(\vec{\lambda}^{\prime}-R_{U}\vec{\lambda})$
$\displaystyle\xi(k|\vec{\lambda},M_{\phi})$
$\displaystyle=\Theta(k\vec{\phi}\cdot\vec{\lambda})$
$\displaystyle\eta(\vec{\lambda}^{\prime}|k,\vec{\lambda},M_{\phi})$
$\displaystyle=\frac{1}{\pi}\Theta(k\vec{\phi}\cdot\vec{\lambda}^{\prime})k\vec{\phi}\cdot\vec{\lambda}^{\prime}$
(16)

Here $\Theta$ is the Heaviside step function, $R_{U}$ is the rotation of the
Bloch sphere corresponding to a unitary $U$, and $k\in\\{+1,-1\\}$. This
particular state update rule is not particularly satisfactory in an
explanatory sense, since it is independent of the previous ontic state.

####  4.1.2 Montina model

In [34], Montina introduces an ontological model based on the Kochen-Specker
model. The model was constructed to show that state update in a qubit can be
successfully modeled by only updating a finite amount of information in the
ontic state. To do so, Montina extends the ontic space of the Kochen-Specker
model by taking two vectors $\vec{x}_{+1},\vec{x}_{-1}$ on the Bloch sphere
and adding two bits, labeled $r$ and $s$, such that the vectors on the Bloch
sphere are dynamic under transformations but remain static under state update
(Fig. 1(b)). The bit $r\in\\{-1,+1\\}$ acts as an index which decides which of
the two Bloch vectors is ‘active;’ $s\in\\{-1,+1\\}$ stores the result of a
hypothetical standard basis measurement on the state. We take the standard
basis to be defined by a special vector $\vec{n}$ pointing along the $z$-axis.

As in the Kochen-Specker model, unitary transformations act by rotating the
Bloch vectors; additionally, if the vector $\vec{x}_{r}$ (i.e. the ‘active’
Bloch vector) crosses the horizontal equator of the sphere during this
transformation, then the bit $s$ flips to $-s$. $r$ does not change during a
transformation.

A measurement in the standard basis simply reveals the value of $s$, and then
updates $r$ so that the active vector is the one which was more closely
aligned with $\vec{n}$ at the time of measurement. For any other basis, we
apply the unitary that maps our desired measurement basis to the standard
basis, measure, and then rotate back—this whole process has been wrapped into
our definitions of $\eta$ and $\xi$ below. In either case, the vectors
$\vec{x}_{+1},\vec{x}_{-1}$ do not change during measurement.

Finally, we prepare a state $\vec{\psi}$ by measuring in the basis
$\\{\vec{\psi},-\vec{\psi}\\}$ and applying a rotation that maps
$-\vec{\psi}\to\vec{\psi}$ if we measured $-\vec{\psi}$. Summarizing these
constructions, we can write

$\displaystyle\Lambda$ $\displaystyle=S^{2}\times
S^{2}\times\\{-1,+1\\}\times\\{-1,+1\\}$ $\displaystyle\lambda$
$\displaystyle=(\vec{x}_{+1},\vec{x}_{-1},r,s)$
$\displaystyle\mu(\lambda|P_{\psi})$
$\displaystyle=\frac{1}{(4\pi)^{2}}\Theta\left[s\left(\vec{x}_{r}\cdot\vec{\psi}\right)\left(\vec{x}_{r}\cdot\vec{n}\right)\right]$
$\displaystyle\quad\cdot\Theta\left[r\left[\left(\vec{x}_{+1}\cdot\vec{\psi}\right)^{2}-\left(\vec{x}_{-1}\cdot\vec{\psi}\right)^{2}\right]\right]$
$\displaystyle\Gamma(\lambda^{\prime}|\lambda,T_{U})$
$\displaystyle=\delta\left(\vec{x}_{+1}^{\prime}-R_{U}\vec{x}_{+1}\right)\delta\left(\vec{x}_{-1}^{\prime}-R_{U}\vec{x}_{-1}\right)$
$\displaystyle\quad\cdot\Theta\left[ss^{\prime}\left(\vec{x}_{r}^{\prime}\cdot\vec{n}\right)\left(\vec{x}_{r}\cdot\vec{n}\right)\right]\Theta[rr^{\prime}]$
$\displaystyle\xi(k|\lambda,M_{\phi})$
$\displaystyle=\Theta\left[ks\left(\vec{x}_{r}\cdot\vec{n}\right)\left(\vec{x}_{r}\cdot\vec{\phi}\right)\right]$
$\displaystyle\eta(\lambda^{\prime}|k,\lambda,M_{\phi})$
$\displaystyle=\delta\left(\vec{x}_{+1}^{\prime}-\vec{x}_{+1}\right)\delta\left(\vec{x}_{-1}^{\prime}-\vec{x}_{-1}\right)$
$\displaystyle\quad\cdot\Theta\left[ss^{\prime}\quantity(\vec{x}_{r}\cdot\vec{n})\quantity(\vec{x}_{r}\cdot\vec{\phi})\quantity(\vec{x}_{r^{\prime}}\cdot\vec{n})\quantity(\vec{x}_{r^{\prime}}\cdot\vec{\phi})\right]$
$\displaystyle\quad\cdot\Theta\left[r^{\prime}\left[\left(\vec{x}_{+1}\cdot\vec{\phi}\right)^{2}-\left(\vec{x}_{-1}\cdot\vec{\phi}\right)^{2}\right]\right]$
(17)

The original presentation is not stated in terms of the ontological models
formalism, and only explicitly models measurements in the standard basis. This
lead to the claim that state update under measurement is accounted for by
updating a single bit, but it is clear from the form of $\eta$ above that by
including all measurements in our subtheory we have caused both bits to be
updated during measurement.

We include this model here for two reasons. First, it is one of the few
ontological models in the literature that has explicitly considered state
update under measurement. Second, it demonstrates that the generic rank-1
update (Eq. 11) that we used for the Kochen-Specker model is not the only
possibility; even though all measurements in this model are rank-1, $\eta$ has
nontrivial dependence on the previous ontic state $\lambda$. Thus just because
we _can_ construct a trivial update rule in some cases does not mean that
there is then nothing interesting to investigate. It also includes these
features while remaining pairwise $\psi$-epistemic and never $\psi$-ontic.

###  4.2 Models of full quantum theory for arbitrary dimension

####  4.2.1 Beltrametti-Bugajski model

The Beltrametti-Bugajski model [9, 24] is perhaps the simplest ontological
model that describes a system of arbitrary dimension. Although it is
$\psi$-ontic, it is the starting point for the construction of the next three
models in this section. For a $d$-dimensional quantum system, we take the
ontic space to be the quantum state space, which we denote
$\mathcal{PH}^{d-1}$ (the projective Hilbert space of dimension $d-1$).
Preparations, transformations, measurements, and state update rules then
follow directly from the usual quantum rules:

$\displaystyle\Lambda$ $\displaystyle=\mathcal{PH}^{d-1}$
$\displaystyle\mu(\lambda|P_{\psi})$
$\displaystyle=\delta(\ket{\lambda}-\ket{\psi})$
$\displaystyle\Gamma(\lambda^{\prime}|\lambda,T_{U})$
$\displaystyle=\delta(\ket{\lambda^{\prime}}-U\ket{\lambda})$
$\displaystyle\xi(k|\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\bra{\lambda}\Pi_{k}\ket{\lambda}$
$\displaystyle\eta(\lambda^{\prime}|k,\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\delta\quantity(\ket{\lambda^{\prime}}-\frac{\Pi_{k}\ket{\lambda}}{\sqrt{\bra{\lambda}\Pi_{k}\ket{\lambda}}})$
(18)

This provides an example of the generic update-rule for $\psi$-ontic models
(Eq. 12), and is depicted in Fig. 1(c).

####  4.2.2 Bell’s model

Lewis et al. [35] extended a model of a qubit orignally proposed by Bell [39]
to arbitrary dimension, which can be seen as a modification of the
Beltrametti-Bugajski model [24]. The ontic space is the Cartesian product of
the projective Hilbert space with the unit interval $[0,1]$. Now we write
$\lambda$ as an ordered pair $\lambda=(\ket{\lambda},p_{\lambda})$ where
$\ket{\lambda}\in\mathcal{PH}^{d-1}$, as in the Beltrametti-Bugajski model,
and $p_{\lambda}\in[0,1]$. Preparations remain essentially the same, becoming
a product distribution of a delta function on the quantum state space with a
uniform distribution over the unit interval. The response functions divide up
the unit interval into lengths corresponding to probabilities of measuring
each outcome, and respond with outcome $k$ when $p_{\lambda}$ is in the
corresponding interval (Fig. 1(d)). This has the effect of making the model
outcome deterministic.

$\displaystyle\Lambda$ $\displaystyle=\mathcal{PH}^{d-1}\times[0,1]$
$\displaystyle\mu(\lambda|P_{\psi})$
$\displaystyle=\delta(\ket{\lambda}-\ket{\psi})$
$\displaystyle\Gamma(\lambda^{\prime}|\lambda,T_{U})$
$\displaystyle=\delta(\ket{\lambda^{\prime}}-U\ket{\lambda})$
$\displaystyle\xi(k|\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\Theta\quantity[p_{\lambda}-\sum_{j=0}^{k-1}\tr(\Pi_{j}{\bm{[}}\lambda{\bm{]}})]$
$\displaystyle\quad\cdot\Theta\quantity[-p_{\lambda}+\sum_{j=0}^{k}\tr(\Pi_{j}{\bm{[}}\lambda{\bm{]}})]$
$\displaystyle\eta(\lambda^{\prime}|k,\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\delta\quantity(\ket{\lambda^{\prime}}-\frac{\Pi_{k}\ket{\lambda}}{\sqrt{\bra{\lambda}\Pi_{k}\ket{\lambda}}})$
(19)

Since this model is still $\psi$-ontic, we once again use the generic state
update rule for $\psi$-ontic models. In this case, we can also see that this
works because of the structure of the preparations as product distributions.
Since every state has a uniform distribution over $p_{\lambda}$, and this is
uncorrelated with $\ket{\lambda}$, we can just update $\ket{\lambda}$
according to the Beltrametti-Bugajski update rule and then re-randomize
uniformly over $p_{\lambda}$.

####  4.2.3 LJBR model

In [35], Lewis et al. define a $\psi$-epistemic model based on their
generalization of Bell’s model. Referred to here as the LJBR model, it is
motivated by the observation that the order of segments in the response
function of Bell’s model does not matter: a re-ordering of these segments
allows arbitrary modification of preparation distributions within a subset of
the ontic space, so they can be made to overlap. We present here a brief
description of the ‘most epistemic’ version of this model, and refer the
reader to [35] for a more thorough construction and motivation.

The LJBR model has the same ontic space as the Bell model, so we again write
ontic states as $\lambda=(\ket{\lambda},p_{\lambda})$. It is constructed in a
preferred basis $\\{\ket{j}\\}$, which we use in defining two helper
functions. First,

$z_{j}(\ket{\lambda})=\inf_{\ket{\phi}:\tr({\bm{[}}j{\bm{]}}{\bm{[}}\phi{\bm{]}})\geq
1/d}\tr({\bm{[}}\lambda{\bm{]}}{\bm{[}}\phi{\bm{]}}).$ (20)

Note that $z_{j}(\ket{\lambda})>0$ if and only if
$\tr({\bm{[}}j{\bm{]}}{\bm{[}}\lambda{\bm{]}})>\frac{d-1}{d}$, so
$z_{j}(\ket{\lambda})$ is nonzero for at most a single element of the
preferred basis; we denote this unique vector as $\ket{j_{\lambda}}$. Second,
we define a permutation $\pi_{M,\lambda}$ for each measurement $M$ and ontic
state $\lambda$:

$\displaystyle\tr(M_{\pi_{M,\lambda}(0)}{\bm{[}}j_{\lambda}{\bm{]}})$
$\displaystyle\geq\tr(M_{\pi_{M,\lambda}(1)}{\bm{[}}j_{\lambda}{\bm{]}})\geq\cdots$
$\displaystyle\cdots$
$\displaystyle\geq\tr(M_{\pi_{M,\lambda}(\quantity|M|-1)}{\bm{[}}j_{\lambda}{\bm{]}}).$
(21)

If there is no $\ket{j_{\lambda}}$, i.e. $z_{j}(\ket{\lambda})=0$ for all $j$,
then we take $\pi_{M,\lambda}$ to be the identity permutation. The final
element we need before defining the model itself is a set

$\mathcal{E}_{j}=\\{\lambda:z_{j}(\ket{\lambda})>0\\}$ (22)

defined for each basis vector. Without further ado, the full specification of
the model:

$\displaystyle\Lambda$ $\displaystyle=\mathcal{PH}^{d-1}\times[0,1]$
$\displaystyle\lambda$ $\displaystyle=(\ket{\lambda},p_{\lambda})$
$\displaystyle\mu(\lambda|P_{\psi})$
$\displaystyle=\delta(\ket{\lambda}-\ket{\psi})\prod_{j}\Theta[p_{\lambda}-z_{j}(\ket{\psi})]$
$\displaystyle\quad+\sum_{j}z_{j}(\ket{\psi})\mu_{\mathcal{E}_{j}}(\lambda)$
$\displaystyle\xi(k|\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\Theta\quantity[p_{\lambda}-\sum_{l=0}^{k-1}\tr(\Pi_{\pi_{M,\lambda}(l)}{\bm{[}}\lambda{\bm{]}})]$
$\displaystyle\quad\cdot\Theta\quantity[-p_{\lambda}+\sum_{l=0}^{k}\tr(\Pi_{\pi_{M,\lambda}(l)}{\bm{[}}\lambda{\bm{]}})]$
(23)

where $\mu_{\mathcal{E}_{j}}(\lambda)$ is the uniform distribution over
$\mathcal{E}_{j}$.

Roughly, all quantum states $\ket{\psi}$ with
$\tr({\bm{[}}j{\bm{]}}{\bm{[}}\psi{\bm{]}})>\frac{d-1}{d}$ will have support
on $\mathcal{E}_{j}$, and so will all overlap with each other. The permutation
included in the definition of the measurements is constructed so that this
shared support does not affect the prepare-and-measure-once statistics: the
measurement ordered first by the permutation has a support which entirely
contains $\mathcal{E}_{j}$. This model is not pairwise $\psi$-epistemic, nor
is it never $\psi$-ontic. Note additionally that it was originally only
defined for rank-1 projective measurements, but it works just as well for
higher-rank projective measurements without modification.

This model is the first to fall to Theorem 1:

######  Theorem 2.

The LJBR model cannot represent state update under measurement in dimension
$d\geq 3$.

###### Proof.

The general idea of the proof is to find a measurement which maps any two
nonidentical states $\ket{\alpha},\ket{\beta}$ to two again nonidentical
states $\Pi\ket{\alpha},\Pi\ket{\beta}$ which both have no support on any of
the $\mathcal{E}_{j}$, and so are ontologically distinct.

Consider the preferred basis $\ket{j}$ of the LJBR model and the generalized
$x$-basis defined by

$\ket{X_{k}}=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}\omega^{jk}\ket{j},\qquad\omega=e^{2\pi
i/d}.$ (24)

These $x$-basis states have the property
$\tr({\bm{[}}X_{j}{\bm{]}}{\bm{[}}k{\bm{]}})=\frac{1}{d}$ for all $j,k$. There
must exist two elements $\ket{X_{k_{1}}},\ket{X_{k_{2}}}$ of the $x$-basis
such that $\ket{\alpha},\ket{\beta}$ differ on that two-dimensional subspace
or else $\ket{\alpha},\ket{\beta}$ would be identical. Pick two such elements,
and consider the projector
$\Pi={\bm{[}}X_{k_{1}}{\bm{]}}+{\bm{[}}X_{k_{2}}{\bm{]}}$. The quantum overlap
of the post-measurement state $\Pi\ket{\alpha}$ with any basis vector
$\ket{j}$ is, using the submultiplicativity of the trace,

$\frac{\tr({\bm{[}}j{\bm{]}}\Pi{\bm{[}}\alpha{\bm{]}}\Pi)}{\tr(\Pi{\bm{[}}\alpha{\bm{]}})}\leq\tr({\bm{[}}j{\bm{]}}\Pi)=\frac{2}{d}\leq\frac{d-1}{d}$
(25)

and the same is true for $\Pi\ket{\beta}$. As described above, only states
with $\tr({\bm{[}}j{\bm{]}}{\bm{[}}\psi{\bm{]}})>\frac{d-1}{d}$ have overlap
with any other states in the LJBR model, so the post-measurement states are
ontologically distinct; by Theorem 1, $\xi(\Pi|\lambda)=0$ for all
$\lambda\in\mathcal{E}_{j}$ for all $j$ since $\ket{\alpha}$ and $\ket{\beta}$
were arbitrary.

However, when measured in the context of the rest of the rank-1 $x$-basis
projectors, $\Pi$ will be ordered first by $\pi_{M,\lambda}$ for all $\lambda$
since

$\tr(\Pi{\bm{[}}j{\bm{]}})=\frac{2}{d}>\frac{1}{d}$ (26)

for all $j$. Thus, by the construction of the LJBR model, $\xi(\Pi|\lambda)=1$
for all $\lambda\in\mathcal{E}_{j}$, resulting in a contradiction. ∎

####  4.2.4 ABCL models

In [36], Aaronson et. al. construct two $\psi$-epistemic models. The first,
which we will call ABCL0 , is very closely related to the LJBR model but is
not identical; rather than continuous regions of quantum states which overlap,
this model has exactly one pair of quantum states which are ontologically
indistinct. However, it gains the feature that _any_ two nonorthogonal quantum
states can be chosen as the single pair that overlaps. With malice
aforethought, we will call this defining pair $\ket{\alpha},\ket{\beta}$.

The second, ABCL1 , is a convex mixture (to be defined) of the ABCL0 model
constructed for all $\ket{\alpha},\ket{\beta}$ and is intended to demonstrate
the possibility of a pairwise $\psi$-epistemic model. This is the only known
example of a pairwise $\psi$-epistemic model in $d\geq 3$, but it still is not
never $\psi$-ontic [24, 54]. These models have come under criticism for their
“unnaturalness,” but we show here that their problems go deeper due to an
inability to represent state-update.

We begin with ABCL0 , defining a couple of helper functions like in the LJBR
model. Rather than ordering measurements with respect to traces with a
preferred basis, we use the defining states $\ket{\alpha},\ket{\beta}$ and a
function

$g_{\alpha\beta}(\Pi)=\min\\{\tr(\Pi{\bm{[}}\alpha{\bm{]}}),\tr(\Pi{\bm{[}}\beta{\bm{]}})\\}.$
(27)

We now define a new permutation $\sigma_{M}$888To be precise, we should label
this with $\alpha,\beta$ as well to emphasize that it belongs to the model
defined by that particular pair of states. for each measurement $M$ [24]:

$g_{\alpha\beta}(M_{\sigma_{M}(0)})\geq
g_{\alpha\beta}(M_{\sigma_{M}(1)})\geq\cdots\geq
g_{\alpha\beta}(M_{\sigma_{M}(|M|-1)}).$ (28)

With this, we can specify the ABCL0 model.

$\displaystyle\qquad\Lambda=\mathcal{PH}^{d-1}\times[0,1]$
$\displaystyle\qquad\lambda=(\ket{\lambda},p_{\lambda})$
$\displaystyle\mu(\lambda|P_{\psi})=\begin{cases}\begin{aligned}
&\bigg{[}\Theta(p_{\lambda}-\varepsilon)\delta(\ket{\lambda}-\ket{\psi})\\\
&+\frac{1}{2}\Theta(\varepsilon-
p_{\lambda})\quantity[\delta(\ket{\lambda}-\ket{\alpha})+\delta(\ket{\lambda}-\ket{\beta})]\bigg{]}\end{aligned}\\\
\hfill\text{if }\ket{\psi}=\ket{\alpha},\ket{\beta}\\\ \\\
\delta(\ket{\lambda}-\ket{\psi})\hfill\text{otherwise}\end{cases}$
$\displaystyle\xi(k|\lambda,M_{\\{\Pi_{i}\\}})=\Theta\quantity[p_{\lambda}-\sum_{j=0}^{k-1}\tr(\Pi_{\sigma_{M}(j)}{\bm{[}}\lambda{\bm{]}})]$
$\displaystyle\qquad\qquad\qquad\cdot\Theta\quantity[-p_{\lambda}+\sum_{j=0}^{k}\tr(\Pi_{\sigma_{M}(j)}{\bm{[}}\lambda{\bm{]}})]$
(29)

for $\varepsilon\leq\frac{|\innerproduct{\alpha}{\beta}|}{d}$. Now the
preparation distributions for $\ket{\alpha}$ and $\ket{\beta}$ overlap on
$\\{\ket{\alpha},\ket{\beta}\\}\times[0,\varepsilon]$; as in the LJBR model,
the permutation in $\xi$ ensures that the preparation change doesn’t affect
the prepare-and-measure-once statistics by making sure measurements whose
support must contain this overlap region are ordered first. Once again, this
model fails to meet the conditions required in order to faithfully represent
state update:

######  Theorem 3.

The ABCL0 model cannot represent state update under measurement in dimension
$d\geq 3$.

###### Proof.

Call the two states defining the model $\ket{\alpha},\ket{\beta}$. Let
$\Pi={\bm{[}}\alpha{\bm{]}}+{\bm{[}}\gamma{\bm{]}}$, where $\ket{\gamma}$ is
some state such that

$\innerproduct{\alpha}{\gamma}=0\quad\text{and}\quad
0<\quantity|\innerproduct{\gamma}{\beta}|^{2}<1-\quantity|\innerproduct{\alpha}{\beta}|^{2}.$
(30)

Under this measurement, $\ket{\alpha}$ maps to $\ket{\alpha}$ and
$\ket{\beta}$ does not get mapped to either $\ket{\alpha}$ or $\ket{\beta}$.
Thus the post-measurement states are ontologically distinct, so by Theorem 1,
$\xi(\Pi|\lambda)=0$ for all $\lambda\in\Delta_{\alpha}\cap\Delta_{\beta}$.

For the other half of the contradiction, note that
$\tr(\Pi{\bm{[}}\alpha{\bm{]}})=1$ means
$g_{\alpha\beta}(\Pi)=\tr(\Pi{\bm{[}}\beta{\bm{]}})>0$ (since
$\ket{\alpha},\ket{\beta}$ are nonorthogonal) and
$g_{\alpha\beta}(\mathbb{I}-\Pi)=1-\tr(\Pi{\bm{[}}\alpha{\bm{]}})=0$, so $\Pi$
is ordered first by $\sigma_{M}$ when measured in the context
$M=\\{\Pi,\mathbb{I}-\Pi\\}$. Thus $\xi(\Pi|\lambda)=1$ for all
$\lambda\in\Delta_{\alpha}\cap\Delta_{\beta}$, resulting in a contradiction. ∎

We outline the ABCL1 model schematically and refer the reader to [24, 36] for
details. Given two ontological models specified by
$\Lambda_{1},\mu_{1},\xi_{1}$ and $\Lambda_{2},\mu_{2},\xi_{2}$ respectively,
the authors define a convex combination of these models as
$\Lambda_{3},\mu_{3},\xi_{3}$ such that

$\displaystyle\Lambda_{3}$ $\displaystyle=\Lambda_{1}\oplus\Lambda_{2}$
$\displaystyle b\mu_{3}$ $\displaystyle=p\mu_{1}+(1-p)\mu_{2}$
$\displaystyle\xi_{3}$ $\displaystyle=\xi_{1}+\xi_{2}$ (31)

Here $p\in(0,1)$ is some mixing parameter. If there’s overlap between two
states in either of models 1 or 2, then model 3 has overlap on these states.
The ABCL1 model is then defined essentially as a convex mixture of the ABCL0
models for _all_ pairs $\ket{\alpha},\ket{\beta}$, taking care with respect to
the uncountable size of this set.

In order to include state update in a convex combination of ontological
models, the most obvious (and perhaps only) option is to specify

$\eta_{3}=\eta_{1}+\eta_{2}.$ (32)

The failure of the ABCL1 model to reproduce state update follows directly from
the failure of the ABCL0 model.

######  Theorem 4.

The ABCL1 model cannot represent state update under measurement in dimension
$d\geq 3$

###### Proof.

When we take a convex combination of models, we see that

$\displaystyle\textup{Supp}(\xi_{3}(k|\dotproduct,M))=$
$\displaystyle\textup{Supp}(\xi_{1}(k|\dotproduct,M))\cup\textup{Supp}(\xi_{2}(k|\dotproduct,M))$
$\displaystyle\textup{Supp}(\eta_{3}(\dotproduct|k,\lambda,M))$
$\displaystyle=\textup{Supp}(\eta_{1}(\dotproduct|k,\lambda,M))$
$\displaystyle\qquad\qquad\cup\textup{Supp}(\eta_{2}(\dotproduct|k,\lambda,M))$

Thus if either of models 1 or 2 violates Theorem 1, model 3 must violate it as
well. Since all of the ABCL0 models being mixed violate Theorem 1, so must
ABCL1 . ∎

####  4.2.5 A note on transformations

As Leifer notes, transformations also play a role in restricting the structure
of $\psi$-epistemic ontological models [24, Section 8.1]. If an ontological
model successfully represents all unitary transformations, then
$\left|\innerproduct{\psi}{\phi}\right|=\left|\innerproduct{\psi^{\prime}}{\phi^{\prime}}\right|$
implies that $\ket{\psi},\ket{\phi}$ are ontologically distinct if and only if
$\ket{\psi^{\prime}},\ket{\phi^{\prime}}$ are ontologically distinct. If the
model also includes all CPTP maps, then
$\left|\innerproduct{\psi}{\phi}\right|\geq\left|\innerproduct{\psi^{\prime}}{\phi^{\prime}}\right|$
implies that $\ket{\psi^{\prime}},\ket{\phi^{\prime}}$ are ontologically
distinct if $\ket{\psi},\ket{\phi}$ are.

It immediately follows that the LJBR and ABCL0 models cannot faithfully
represent unitary transformations. In each model there exist quantum states
which are ontologically distinct from every other state; pick one of these
states, and it is easy to find examples of pairs of ontologically distinct
states with any inner product.

However, transformations cannot necessarily rule out the ABCL1 model: since it
is pairwise $\psi$-epistemic, the unitary condition could _in principle_ be
satisfied. That said, the transformation rule would be complicated because it
would have to map _between_ models that are mixed together, so it is certainly
an open question whether this is actually possible.

###  4.3 Models of subtheories

Although we have dealt so far with models that include the full quantum set of
preparations, transformations, and measurements, there is the possibility that
we can retain $\psi$-epistemic models of subtheories. It turns out that
although the stabilizer subtheory can be represented by a $\psi$-epistemic
model, the more general Kitchen Sink model which models any finite subtheory
cannot in general represent state update under measurement.

####  4.3.1 Kitchen Sink model

The Kitchen Sink model is a $\psi$-epistemic ontological model for any finite
subtheory of quantum theory [37, Section IIIC]. Given a finite set of
projective measurements $\mathcal{M}=\\{M^{(i)}\\}$, we choose our ontic
states to be a list of measurement outcomes
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{\left|\mathcal{M}\right|})$.
That is, $\lambda_{i}=k$ means that if $M^{(i)}=\\{\Pi^{(i)}_{j}\\}$ is
measured on the ontic state $\lambda$, the outcome $\Pi_{k}$ will occur with
certainty. For a system of dimension $d$, the maximum number of projectors in
any given measurement is $d$, so we pad all of our measurements with $0$s
until they have $d$ elements. The Kitchen Sink model is then defined by

$\displaystyle\Lambda$
$\displaystyle=\mathbb{Z}_{d}^{\left|\mathcal{M}\right|}$
$\displaystyle\mu(\lambda|\psi)$
$\displaystyle=\prod_{i=1}^{\left|\mathcal{M}\right|}\tr(\Pi^{(i)}_{\lambda_{i}}{\bm{[}}\psi{\bm{]}})$
$\displaystyle\xi(k|\lambda,M^{(i)})$ $\displaystyle=\delta(k,\lambda_{i})$
(33)

The Kitchen Sink is pairwise $\psi$-epistemic for any subtheory, and also
never $\psi$-ontic if we include all pure states in our subtheory.
Transformations can additionally be modeled under the assumption of a closed
subtheory.

We can only rule out the Kitchen Sink model for certain subtheories, as it is
easy to construct subtheories with trivial update rules (e.g. by only
including rank-1 measurements). That said, our requirements are few and are
satisfied by the multi-qu$p$it stabilizer subtheory, arguably the most
important subtheory of quantum theory. Specifically, we only need to include
two states $\ket{\alpha},\ket{\beta}$ and two measurements
$M^{(1)}=\\{\Pi,\mathbb{I}-\Pi\\},M^{(2)}$ satisfying

$\displaystyle\innerproduct{\alpha}{\beta}$ $\displaystyle\neq 0$ (34)
$\displaystyle\expectationvalue{\Pi}{\alpha}$ $\displaystyle\neq 0$ (35)
$\displaystyle\expectationvalue{\Pi}{\beta}$ $\displaystyle\neq 0$ (36)
$\displaystyle M^{(2)}\text{ distinguishes }$
$\displaystyle\Pi\ket{\alpha}\text{ and }\Pi\ket{\beta}$ (37)

The first is required because we don’t expect orthogonal states to be
ontologically indistinguishable. The next two stipulate that there is a
nonzero chance of obtaining an outcome $\Pi$ when measuring $\ket{\alpha}$ and
$\ket{\beta}$, so that its support overlaps with their support. The last
condition implies that the post-measurement states are ontologically distinct
[24].

######  Theorem 5.

For any finite subtheory containing states and measurements satisfying the
conditions given in Eqs. 34–37, the Kitchen sink model cannot model state
update under measurement.

###### Proof.

Theorem 1 implies that, if $\Pi$ maps $\ket{\alpha},\ket{\beta}$ to
ontologically distinct states, then

$\int_{\Lambda}\differential{\lambda}\xi(\Pi|\lambda)\mu(\lambda|\alpha)\mu(\lambda|\beta)=0.$
(38)

We evaluate this quantity for the Kitchen Sink’s response functions and
preparation distributions, using the states and measurement $M^{(1)}$
satisfying Eqs. 34–37:

$\displaystyle\int_{\Lambda}$
$\displaystyle\differential{\lambda}\xi(k=0|\lambda,M^{(1)})\mu(\lambda|\alpha)\mu(\lambda|\beta)$
$\displaystyle=\sum_{\lambda\in\mathbb{Z}_{r}^{\left|\mathcal{M}\right|}}\delta(0,\lambda_{1})\prod_{j=1}^{\left|\mathcal{M}\right|}\tr(M^{(j)}_{\lambda_{j}}{\bm{[}}\alpha{\bm{]}})\tr(M^{(j)}_{\lambda_{j}}{\bm{[}}\beta{\bm{]}})$
$\displaystyle=\sum_{\lambda_{1}\in\mathbb{Z}_{r}}\delta(0,\lambda_{1})\tr(M^{(1)}_{\lambda_{1}}{\bm{[}}\alpha{\bm{]}})\tr(M^{(1)}_{\lambda_{1}}{\bm{[}}\beta{\bm{]}})$
$\displaystyle\qquad\qquad\cdot\prod_{j=2}^{\left|\mathcal{M}\right|}\sum_{l\in\mathbb{Z}_{r}}\tr(M^{(j)}_{l}{\bm{[}}\alpha{\bm{]}})\tr(M^{(j)}_{l}{\bm{[}}\beta{\bm{]}})$
$\displaystyle=\tr(M^{(1)}_{0}{\bm{[}}\alpha{\bm{]}})\tr(M^{(1)}_{0}{\bm{[}}\beta{\bm{]}})$
$\displaystyle\qquad\qquad\cdot\prod_{j=2}^{\left|\mathcal{M}\right|}\sum_{l\in\mathbb{Z}_{r}}\tr(M^{(j)}_{l}{\bm{[}}\alpha{\bm{]}})\tr(M^{(j)}_{l}{\bm{[}}\beta{\bm{]}})$
(39)

This final expression will be zero if and only if at least one of its factors
is 0. The first two factors are nonzero by Eqs. 35 and 36. The rest of the
factors are nonzero due to Eq. 34 and the completeness condition on the
measurements. Thus Eq. 39 is nonzero and we have a contradiction, so state
update cannot be represented faithfully. ∎

This demonstrates that Theorem 1 can create trouble even in subtheories. In
particular, the stabilizer subtheory satisfies the requirements in Eqs. 34–37,
so it cannot be modeled by the Kitchen Sink. That said, we can show that the
stabilizer subtheory still supports a $\psi$-epistemic interpretation using
other models.

####  4.3.2 Qu$p$it stabilizer subtheory

We begin with the straightforward case of $n$ $p$-dimensional systems, for $p$
an odd prime. In this case, the stabilizer subtheory has an ontological model
given by the discrete Wigner function [55, 33] (see Appendix C for definitions
of the stabilizer subtheory and the phase-point operaters $A_{\lambda}$):

$\displaystyle\Lambda$
$\displaystyle=\mathbb{Z}_{p}^{n}\times\mathbb{Z}_{p}^{n}$
$\displaystyle\mu(\lambda|P_{\psi})$
$\displaystyle=\frac{1}{p^{n}}\tr(A_{\lambda}{\bm{[}}\psi{\bm{]}})$
$\displaystyle\xi(k|\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\tr(\Pi_{k}A_{\lambda})$ (40)

As the Wigner function is a quasi-probability distribution [56], it
necessarily takes on negative values if we try to model the full quantum
theory. If, however, we restrict to modeling preparations, transformations,
and measurements in the qu$p$it stabilizer subtheory, then the representation
is positive and it forms a well-defined ontological model [57, 33]. It is both
pairwise $\psi$-epistemic and never $\psi$-ontic.

The stabilizer subtheory presents a challenge in that measuring a single
qu$p$it is described by a rank-$p^{n-1}$ measurement, which may run into
trouble due to Theorem 1. In particular, there are many examples in the
stabilizer subtheory of the type of measurements that broke the Kitchen Sink
model. Nonetheless, we can specify the update rule

$\displaystyle\eta(\lambda^{\prime}|k,\lambda,M_{\\{\Pi_{i}\\}})$
$\displaystyle=\frac{1}{p^{n}}\frac{\tr(A_{\lambda}\Pi_{k}A_{\lambda^{\prime}}\Pi_{k})}{\tr(\Pi_{k}A_{\lambda})}.$
(41)

Note the normalization factor in $\eta$ which makes clear that $\eta$ is only
defined in the support of $\xi$. The fact that this successfully reproduces
quantum statistics follows from the fact that post-selected measurement is a
completely positive map, and this is how completely positive maps are
represented in quasi-probability representations [56]. $\eta$ is always
positive for the stabilizer subtheory, though we do not include the proof
here.

For the case $p=2$, Lillystone and Emerson construct a $\psi$-epistemic model
of the $n$-qubit stabilizer formalism that successfully represents state
update under measurement [48]. This model starts from the Kitchen Sink model
and augments $\Lambda$ so that the problematic overlaps of the Kitchen sink
are removed. This model is not pairwise $\psi$-epistemic, but a modified
version (see appendix of [48]) is never-$\psi$-ontic. We don’t present the
construction here because it is significantly more convoluted than the model
above for $p\geq 3$. This reflects the often-observed ill-behaved nature of
the qubit stabilizer subtheory.

Tangentially, if we extend the Wigner function to the full quantum theory, we
get negatively represented state update, as expected. One consequence of this
is that some state updates can’t be normalized, so the Wigner function state
update must include a renormalization step not allowed in ontological models
or quasiprobability representations. Although further discussion of state
update under measurement in quasi-probability representations is beyond the
scope of this paper, we note that Theorem 1 does not hold for quasi-
probability representations so this could be one potential direction for
related future work.

##  5 Discussion

We have demonstrated that state update under measurement poses a serious
challenge to $\psi$-epistemic interpretations of quantum theory in the
ontological models framework: all currently known $\psi$-epistemic models for
full quantum theory in $d\geq 3$ cannot faithfully represent state update.
This runs in direct contrast to the prevailing view that $\psi$-epistemic
models provide a compelling explanation of state update.

There are a number of remaining open questions. Most pressingly, we have re-
opened the possibility of proving a general $\psi$-onticity result without
additional assumptions—will the methods of this paper be useful in doing so?

On the one hand, the proofs above do not rule out the possibility of extending
the ontology of the broken models in order to represent state update under
measurement while still retaining the epistemicity of the model. This is
exactly the route taken in [48] for the $n$-qubit stabilizer subtheory.
Granted, the $n$-qubit stabilizer subtheory was brought within an inch of
$\psi$-onticity by this process, so it seems unlikely that a similar technique
will work for the full quantum theory.

In the other direction, we’ve shown that consideration of state update puts
powerful constraints on the structure of $\psi$-epistemic models. These
restrictions would ideally lead to a categorical statement like
“$\psi$-epistemic models cannot represent state-update,” but there are
challenges to achieving this conclusion. In particular, we note that all of
the $\psi$-epistemic models that we considered share the property of outcome
determinism, which means that Theorem 1 may be less trouble in non-outcome-
deterministic models. At the very least, any no-go theorem will have to
include measurements, states, and/or transformations from outside the
stabilizer subtheory since we have shown that $\psi$-epistemic models for this
subtheory exist.

What import does our result have for the general interpretational project of
quantum theory? First of all, we have demonstrated that $\psi$-epistemicists
have yet another challenge to overcome: a successful explanation of state-
update. This is in contrast with the usual claim that this arena is one where
epistemic interpretations have an advantage over ontic interpretations. As we
emphasized in the introduction, our results only strictly apply to
interpretations that can be described by the ontological models formalism, but
there may be a qualitative message for epistemic and doxastic interpretations
that are outside this formalism as well.

##  6 Acknowledgements

This research was supported by the Canada First Research Excellence Fund and
the NSERC Discovery program. J.B.R. acknowledges funding from an Ontario
Graduate Scholarship. The authors would like to thank Matt Leifer and David
Schmid for productive discussions. We would also like to thank two anonymous
reviewers and our editor at Quantum, whose remarks motivated the discussion in
Appendix A.

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##  Appendix A Is state update an additional assumption?

Throughout this paper, we have used the phrase ‘no additional assumptions’ in
order to contrast our results with those of existing no-go theorems. By this
we don’t mean to raise an argument over the semantics of exactly what one
means by ‘assumption,’ but rather we mean to say that the consideration of
state update, which we propose may be able to categorically rule out
$\psi$-epistemic models, is very different from the ‘additional assumptions’
used in existing no-go theorems.

To make this precise, we can break up our consideration of state update into
two parts. The first part is simply the fact that we are considering a broader
set of empirical situations than considered previously: experiments involving
sequential measurements. The ontological models formalism, as originally
defined, does not have the tools to model this empirical situation, so we are
forced to extend the formalism to do so. This leads to the second part, which
is an assumption about _how_ we ought to model this empirical situation.
While, in principle, there are many ways one can do this, we show in Appendix
B that there is only one way to do this that is consistent with the broader
conceptual underpinnings (i.e. pre-existing assumptions) of the ontological
models formalism. By examining the core assumptions of the ontological models
formalism rather than its strict mathematical formulation, we show that all of
the structure associated with state update follows directly from the pre-
existing assumptions of the ontological models formalism.

This is in contrast to the assumptions used in previously published no-go
theorems. First of all, they start from empirical situations that can be
described by the vanilla ontological models formalism. Then, they impose a
particular restriction on the form of the ontological models that happens
directly at the level of the ontic state; for example, the PBR theorem [40]
assumes the preparation independence postulate (PIP) which says that says
that, for two subsystems $A$ and $B$, if the quantum state is a product state,
then the preparation distribution $\mu$ is a product distribution:

$\displaystyle\Lambda_{AB}$ $\displaystyle=\Lambda_{A}\times\Lambda_{B}$ (42)
and $\displaystyle\Psi_{AB}=\Psi_{A}\otimes\Psi_{B}\quad$
$\displaystyle\implies\mu(\lambda_{A},\lambda_{B}|\Psi_{AB})$
$\displaystyle=\mu(\lambda_{A}|\Psi_{A})\cdot\mu(\lambda_{B}|\Psi_{B}).$ (43)

We can see from this symbolic expression that this is indeed a direct
assumption about the structure of the ontological model, rather than an
empirical consideration. More to the point, it imposes structure on the
ontological model that is not required by the bare formalism.

As another example, the Colbeck-Renner argument [41] claims to not make any
assumptions of the kind described in the PIP; they do, however, assume a
spacetime structure which is, again, additional to the ontological models
formalism. Spacetime and locality arguments are also what motivate the PIP.
These relativistically-motivated considerations are interesting in their own
right, but since Bell’s theorem already rules out $\psi$-epistemic models
based on reasonable spacetime structure (i.e. the locality assumption), the
goal of our paper is to arrive at a similar conclusion _without_ imposing
spacetime structure. In particular, a $\psi$-epistemicity no-go theorem would
directly imply the results of Bell’s theorem and is thus a stronger result
[24].

Whether _operational_ state update, i.e. the Lüders rule, is considered an
assumption of some kind or a consequence of other operational axioms is
irrelevant to our analysis. The key point for our analysis is that the state-
update rule is an experimentally validated operational feature/prediction of
QM theory, much like entanglement or unitary dynamics, and therefore is an
empirical feature of the QM framework that the framework of ontological models
ought to explain. So just like the discovery from Bell that entangled states
lead to new constraints/insights into the necessary features of
($\psi$-epistemic) ontological models, similarly our insight is that the
dynamics associated with state-update also leads to new constraints/insights
into the necessary features of ($\psi$-epistemic) ontological models.

##  Appendix B Justifying the form of the state update rule: ontological
models as hidden Markov models of stochastic channels

In the context of state update under measurement, it is illuminating to
motivate the definition of an ontological model from the point of view of the
hidden Markov models (HMM) literature. We do this in order to (a) provide a
rigorous treatment of multiple-measurement scenarios presented informally in
Section 2.1 and (b) clarify the assumptions that define the ontological models
framework. In the process, we use these assumptions to show that the form of
our state update rule is the only one consistent with the already-defined
components of the ontological models formalism.

We picture a quantum circuit as a memoryful stochastic channel (Fig. 2). The
channel that we often discuss with regards to a quantum circuit is the
(quantum) channel that takes the input quantum state and maps it to the output
quantum state. For present purposes, we will instead think of it as a channel
from the experimenter to individual measurement outcomes used repeatedly at
each time step. Pictorially, one might think of this as ‘rotating the channel
ninety degrees’ in a circuit diagram.

@C=.5cm @R=.2em |  |  |  |  k [1] |  |  k’ [3] |
---|---|---|---|---|---|---|---
|  5P |  3U |  |  1M |  |  |
|  P |  U |  |  M |  3U” |  |
|  P |  U |  |  |  U” |  3M’ |
|  P |  U |  2U’ |  |  U” |  M’ |
|  P |  |  U’ |  |  U” |  M’ |

$k_{0}=0$$a_{0}=P$$k_{1}=0$$a_{1}=U$$k_{2}=0$$a_{2}=U^{\prime}$$k_{3}=k$$a_{3}=M$$k_{4}=0$$a_{4}=U^{\prime\prime}$$k_{5}=k^{\prime}$$a_{5}=M^{\prime}$
channel

Figure 2: A quantum circuit can be pictured as a stochastic channel, as
described in the text. The inputs to the channel $a_{t}$ are the choice of
operation, and the outputs $k_{t}$ report the results of measurements.

The input string $\overleftrightarrow{a_{0}}=\ldots
a_{-2}a_{-1}a_{0}a_{1}a_{2}\ldots$ of the channel is the experimenter’s choice
of action, which we take to be a sequence of operational preparation,
transformation, or measurement procedures. The output string
$\overleftrightarrow{k_{0}}$ of the channel reports either the results of
measurements or a trivial output for preparations and transformations. The
subscript $0$ indicates in both cases the time that we take as an
origin/reference point. The channel is then described by the conditional
probability distribution
$\textup{Pr}(\overleftrightarrow{k_{0}}|\overleftrightarrow{a_{0}})$.
Following [58], we denote substrings with $a_{t:t+L}=a_{t}a_{t+1}\ldots
a_{t+L-1}$, and also define the past $\overleftarrow{a_{t}}=a_{-\infty:t}$ and
future $\overrightarrow{a_{t}}=a_{t:\infty}$. We can now define two properties
of stochastic channels:

######  Definition 4 (Stationary).

A _stationary_ channel is one that has time-translation symmetry, so
statistics are not affected by our choice of time-origin:

$\displaystyle\textup{Pr}(k_{t:t+L}|\overleftrightarrow{a_{t}})=\textup{Pr}(k_{0:L}|\overleftrightarrow{a_{o}})\qquad\text{and}$
$\displaystyle\textup{Pr}(\overleftrightarrow{k_{t}}|\overleftrightarrow{a_{t}})=\textup{Pr}(\overleftrightarrow{k_{0}}|\overleftrightarrow{a_{0}})\qquad\forall
t,L,\overleftrightarrow{a}.$ (44)

######  Definition 5 (Causal).

A _causal_ channel is one for which a finite output substring depends only on
input symbols in its past:

$\textup{Pr}(k_{t:t+L}|\overleftrightarrow{a})=\textup{Pr}(k_{t:t+L}|\overleftarrow{a}_{t+L}).\qquad\forall
t,L,\overleftarrow{a}$ (45)

It is shown in [58] that a channel satisfying these two properties can be
specified entirely by the single-symbol recurrence relation

$\textup{Pr}(k_{0}|a_{0},\overleftarrow{a_{0}},\overleftarrow{k_{0}}).$ (46)

Note that this does not imply a Markov process, since it depends in general on
the entire histories $\overleftarrow{a_{0}}$ and $\overleftarrow{k_{0}}$. All
we mean by single-symbol is that we are not specifying the probabilities over
the whole future, just a single output symbol. We now construct an HMM as
follows:

######  Definition 6 (Hidden Markov Model).

A _hidden Markov model_ (HMM) of a stationary, causal channel is specified by
an additional random variable $\lambda$ taking values in a state space
$\Lambda$. It is given a joint probability distribution over
$\overleftrightarrow{\lambda_{0}},\overleftrightarrow{a_{0}},\overleftrightarrow{k_{0}}$
so that the recurrence relation above (Eq. 46) becomes

$\textup{Pr}(k_{0},\lambda_{1}|a_{0},\lambda_{0},\overleftarrow{a}_{0},\overleftarrow{k}_{0},\overleftarrow{\lambda}_{0})=\textup{Pr}(k_{0},\lambda_{1}|a_{0},\lambda_{0}).$
(47)

In other words, the state $\lambda$ renders the future conditionally
independent of the past and induces a Markov process over the state space
$\Lambda$ that mediates the channel statistics. This is the property of the
ontological models formalism called “$\lambda$-mediation” in [59]. An
influence diagram [60] of a stationary, causal channel is shown in Figure 3
before and after the specification of an HMM.

To see that specification of an HMM as in Eq. 47 is equivalent to the
definition of an ontological model given in Section 2.1, we first note that
generally we don’t think of preparations and transformations having output; to
account for this, we stipulate that they give a trivial, deterministic output
$k_{0}=0$. We then factor the probability distribution from Eq. 47 and look
separately at the cases where $a_{0}$ is a preparation, transformation, or
measurement:

Pr $\displaystyle(k_{0},\lambda_{1}|\lambda_{0},a_{0})$
$\displaystyle=\textup{Pr}(\lambda_{1}|k_{0},\lambda_{0},a_{0})\textup{Pr}(k_{0}|\lambda_{0},a_{0})$
$\displaystyle=\begin{cases}\mu(\lambda_{1}|P)\delta_{k_{0},0}&a_{0}=P\in\mathcal{P}\\\
\Gamma(\lambda_{1}|\lambda_{0},T)\delta_{k_{0},0}&a_{0}=T\in\mathcal{T}\\\
\eta(\lambda_{1}|k_{0},\lambda_{0},M)\xi(k_{0}|\lambda_{0},M)&a_{0}=M\in\mathcal{M}\end{cases}$
(48)

where $\delta$ is the Kronecker delta. This can be seen as the rigorous
definition of $\eta$ which emerges naturally from this recognition that
ontological models are equivalent to hidden Markov models. In particular, this
gives a mathematical reason why $\eta$ is only defined in the support of
$\xi$. If we take the joint distribution
$\textup{Pr}(k_{0},\lambda_{1}|\lambda_{0},a_{0})$ to be the more fundamental
object, then it is clear we can obtain $\xi$ directly by marginalization

$\xi(k_{0}|\lambda_{0},M)=\int_{\Lambda}\differential{\lambda_{1}}\textup{Pr}(k_{0},\lambda_{1}|\lambda_{0},M)$
(49)

which is always well defined, and then find $\eta$ by rearranging Eq. 48:

$\eta(\lambda_{1}|k_{0},\lambda_{0},M)=\frac{\textup{Pr}(k_{0},\lambda_{1}|\lambda_{0},M)}{\xi(k_{0}|\lambda_{0},M)}$
(50)

Thus clearly $\eta(\lambda_{1}|k_{0},\lambda_{0},M)$ is only well-defined when
$\xi(k_{0}|\lambda_{0},M)\neq 0$, which is how we defined its support.

$k_{2}$ $k_{1}$ $k_{0}$ $k_{-1}$ $a_{2}$ $a_{1}$ $a_{0}$ $a_{-1}$ …… (a) A
causal and stationary channel

$k_{2}$ $\lambda_{2}$ $a_{2}$ $k_{1}$ $\lambda_{1}$ $a_{1}$ $k_{0}$
$\lambda_{0}$ $a_{0}$ $k_{-1}$ $a_{-1}$ …… (b) A hidden Markov model of the
above channel

Figure 3: Influence diagrams [60] for stochastic channels. The boxes represent
choices made by the experimenter, circles represent random variables, and
arrows represent a possible causal influence. Note that no arrows point
backwards in time, and that in the hidden Markov model the state $\lambda_{t}$
mediates all causal influences through time.

Definitions 4–6 constitute an equivalent formulation of the ontological models
formalism. The assumptions of this construction can be broken down as follows:
(a) quantum theory is described by a stochastic channel, (b) this channel is
stationary, (c) it is causal, and (d) we assign the system a state which acts
as an HMM of the channel. The authors of [59] identify (c) and (d), calling
them non-retrocausality and $\lambda$-mediation, respectively. Assumptions (a)
and (b) were implicitly present but not explicitly identified. Since our
formulation of the state update rule follows directly from these assumptions,
we see that it is not ‘additional’ to the ontological models formalism, but a
unique extension to describe a more general empirical scenario.

##  Appendix C A brief introduction to the stabilizer subtheory

We focus here on the stabilizer subtheory for $n$ qu$p$its, where $p$ is a
prime. For a more detailed exposition, we refer the reader to [33, 55].

##### Mathematical objects

The generalizations of the $X$ and $Z$ operators to a single qu$p$it are
defined by their action on the standard basis $\\{\ket{j}\\}$ for
$j=0,1,\ldots,p-1$:

$\displaystyle X\ket{j}$ $\displaystyle=\ket{j+1}$ (51) $\displaystyle
Z\ket{j}$ $\displaystyle=\omega^{j}\ket{j}$ (52) $\displaystyle\omega$
$\displaystyle=e^{2\pi i/p}$ (53)

All integer arithmetic is done $\bmod p$. Then the full set of generalized
Pauli operators on $n$ qu$p$its is given by

$\displaystyle T_{(\mathbf{x},\mathbf{z})}$
$\displaystyle=\begin{cases}\bigotimes_{j=0}^{n-1}X^{x_{j}}Z^{z_{j}}&p=2\\\
\bigotimes_{j=0}^{n-1}\omega^{x_{j}z_{j}/2}X^{x_{j}}Z^{z_{j}}&p>2\end{cases}$
(54) $\displaystyle\text{for }\mathbf{x}$
$\displaystyle=(x_{0},x_{1},\ldots,x_{n-1})\in\mathbb{Z}_{p}^{n}$
$\displaystyle\text{and }\mathbf{z}$
$\displaystyle=(z_{0},z_{1},\ldots,z_{n-1})\in\mathbb{Z}_{p}^{n}$

We also define the symplectic inner product as

$\left[(\mathbf{x},\mathbf{z}),(\mathbf{x}^{\prime},\mathbf{z}^{\prime})\right]=\mathbf{z}\dotproduct\mathbf{x}^{\prime}-\mathbf{x}\dotproduct\mathbf{z}^{\prime}.$
(55)

Finally, the phase-point operators $A_{\lambda}$, for
$\lambda=(\mathbf{x},\mathbf{z})\in\mathbb{Z}_{p}^{n}\times\mathbb{Z}_{p}^{n}$,
are a symplectic Fourier transform of the Pauli operators:

$A_{\lambda}=\frac{1}{p^{n}}\sum_{\lambda^{\prime}\in\mathbb{Z}_{p}^{n}\times\mathbb{Z}_{p}^{n}}\omega^{[\lambda,\lambda^{\prime}]}T_{\lambda^{\prime}}$
(56)

##### The stabilizer subtheory

A stabilizer group $S$ is a set of $p^{n}$ mutually commuting Pauli operators,
which can be specified by a set of $n$ generators. There is a unique state (up
to global phase) which is an eigenvector of all of these operators with
eigenvalue $+1$; we say that $S$ _stabilizes_ this state. For example, for two
qubits, the Bell state

$\ket{\Psi}=\ket{00}+\ket{11}$ (57)

is stabilized by

$\displaystyle S_{\Psi}$ $\displaystyle=\langle Z_{1}Z_{2},X_{1}X_{2}\rangle$
(58) $\displaystyle=\\{\mathbb{I},Z_{1}Z_{2},X_{1}X_{2},-Y_{1}Y_{2}\\}.$ (59)

Here a subscript indicates on which qubit the operator is acting, e.g.
$Z_{1}=Z\otimes\mathbb{I}$ describes $Z$ acting on the first qubit. A
stabilizer state, then, is a state which is stabilized by a group of $p^{n}$
Pauli operators.

Stabilizer measurements are simply measurements of the Pauli operators. Note
that since each Pauli operator has $p$ eigenvalues, these amount to a
measurement of $p$ projectors, each with rank $p^{n-1}$. Lower-rank projectors
can be constructed by performing commuting Pauli measurements sequentially.

Finally, the transformations of the stabilizer subtheory are called Clifford
transformations. These are the transformations that map the set of Pauli
operators to itself, up to a global phase. In other words, it is the
normalizer of the Pauli group.

The stabilizer subtheory thus consists of preparations corresponding the set
of stabilizer states, measurements of Pauli observables, and Clifford
transformations, along with convex combinations thereof.