# The Operational Choi-Jamiołkowski Isomorphism

Emily Adlam 111 Basic Research Community for Physics, Leipzig, Germany;
<EMAIL_ADDRESS>
In this article, I use an operational formulation of the Choi-Jamiołkowski
isomorphism to explore an approach to quantum mechanics in which the state is
not the fundamental object. I first situate this project in the context of
generalized probabilistic theories and argue that this framework may be
understood as a means of drawing conclusions about the intratheoretic causal
structure of quantum mechanics which are independent of any specific
ontological picture. I then give an operational formulation of the Choi-
Jamiołkowski isomorphism and show that in an operational theory which exhibits
this isomorphism, several features of the theory which are usually regarded as
properties of the quantum state can be derived from constraints on non-local
correlations. This demonstrates that there is no need to postulate states to
be the bearers of these properties, since they can be understood as
consequences of a fundamental equivalence between multipartite and temporal
correlations.

##  1 Introduction

In the standard mathematical formulation of quantum mechanics, the quantum
state is the most fundamental object. However, there exists a longstanding
debate about whether we should regard the quantum state as an element of
reality[1, 2]. A recent result known as the PBR theorem purported to settle
this debate once and for all by showing that no interpretation of quantum
mechanics where the quantum state is not an element of reality can reproduce
all of the theory’s empirical predictions[3]. But in order to prove the PBR
theorem it is assumed that information about the past must be carried into the
future by a mediating state, so in fact the setup for the theorem takes it for
granted that the fundamental object in quantum mechanics is some sort of
temporally evolving state, whether or not that state turns out to be the
_quantum_ state.

The assumption that temporal correlations must be mediated by states is seldom
seriously questioned, but it is in fact in tension with certain elements of
modern physics[4], and this suggests that we ought to take seriously the
possibility of re-imagining quantum mechanics with an ontology that does not
employ states at all. Quantum states have certain fundamental temporal
properties (for example, they can’t be broadcasted, they are contextual, they
exhibit interference effects, and they are subject to uncertainty relations)
and a literal reading of the formalism of quantum mechanics would suggest that
the characteristic features of the theory are derived from these properties of
the quantum state. Thus as a first step towards a stateless ontology, we
should determine whether it is necessary to postulate some sort of state to be
the bearer of these properties, or whether they can be accounted for in
another way.

In recent years there have been moves within the quantum foundations community
toward a type of research where quantum mechanics is placed in the context of
a space of generalized probabilistic theories (GPTs), with the aim of showing
that certain features of quantum mechanics can be derived from some chosen
axioms [5, 6, 7, 8, 9, 10, 11, 12, 13]. However, most of these projects use
axioms which employ a concept of ‘state,’ and hence they don’t offer an
obvious route to a stateless ontology. In this paper, I employ the methodology
of the generalized probabilistic theories framework, but I develop an
alternative approach which is explicitly geared towards eliminating the
concept of state.

I begin in section 2 by elaborating on the rationale for studying generalized
probabilistic theories. Then in section 3 I set out an operational form of the
Choi-Jamiołkowski isomorphism, and demonstrate that this operational
isomorphism can be used to derive temporal properties of states from
constraints on the types of non-local correlations that are permitted within
the theory. This demonstrates that in fact we don’t require a state to be the
bearer of the aforementioned temporal properties, and thus paves the way for a
new way of thinking about the intratheoretic causal structure of quantum
mechanics where states are not fundamental objects.

##  2 Intratheoretic Causal Structure

###  2.1 Ontology

The issue of ontology is a vexed one in physics. It is a well-established fact
that ontology is always underdetermined by the empirical results, and
therefore no amount of experimentation can give us certainty about the nature
of the unobservable entities associated with our theory[14]. Furthermore, when
one theory is replaced by another, it is often the case that the theoretical
objects postulated by the original theory do not feature in the new theory,
and thus unless we believe our current theory is the correct final theory of
reality we should probably assume that the theoretical objects posited by our
current theory will not feature in future theories[15]. Considerations such as
these have led many physicists to adopt variants of anti-realism, positivism
or empiricism, holding that physics should be concerned only with empirical
predictions and should be silent on questions of ontology[16, 17].

However, it is important to recognise that the ontology we associated with our
theories affects the way in which science progresses. For example, recall that
classical physics can be written either in the standard Newtonian form, where
we start with initial positions and calculate trajectories from those
positions, or in a Lagrangian form where we define an ’action’ over an entire
history and find the trajectory which optimizes the action [18, 19].
Historically most physicists have considered the Newtonian formulation of
classical physics to be more fundamental than the Lagrangian one, and in this
context it was natural for the founders of quantum mechanics to formulate the
new theory in terms of a state evolving forwards in time, even though we now
know that an alternative Lagrangian formulation in terms of path integrals
exists[18, 20]. Quantum mechanics might look very different (and indeed might
well be less difficult to ’interpret’) had we taken the Lagrangian formulation
seriously from the start.

Moreover, simply becoming an operationalist and refusing to talk about
ontology doesn’t confer immunity to these sorts of biases, because ontological
assumptions are baked into the way in which we think about and do physics[18,
21, 4], and so those who decline to think seriously about ontology are
effectively choosing to uphold the status quo ontological assumptions - which,
in the case of quantum mechanics, entails upholding the assumption of an
ontology based on states.

Thus reconsidering the ontology associated with our theories can be a useful
way of flushing out unjustified assumptions and ultimately coming up with new
physics. However, in adopting this strategy we risk getting caught up in the
details of some specific ontology, which due to the problem of
underdetermination will inevitably not be correct in every particular.
Ideally, what we would like to do is extract the significant _structural_
content of various different ontologies and discard the specifics, which don’t
really contribute to the project of coming up with new physics and are
essentially just window-dressing. This is exactly what the framework of
generalized probabilistic theories allows us to do[5, 6, 7, 8, 9, 10, 11, 12,
13]: it offers a middle way between realism and empiricism, where we impose
hierarchies within our theories by regarding certain features of the theory as
consequences of other features, but do not actually specify the details of the
ontology underlying the hierarchy.

###  2.2 Intratheoretic Causation

Hierarchical structures of this kind occur in many places throughout the
sciences.222I observe in passing that some philosophers have questioned the
coherence of imposing a hierarchy of fundamentality on nature. I will not
engage with this debate here, but will instead observe that whatever its
metaphysical status, the ‘hierarchy of fundamentality’ picture is
instrumentally useful in the actual practice of science as a means of
rethinking the structural assumptions that go into our theories.. They are
most straightforward in theories which come with an ontology attached, because
in such cases a hierarchy of fundamentality can be read directly off the
ontology. For example, the ontology of classical physics postulates objects
which move in inertial reference frames and are subject to certain mechanical
forces, but centrifugal force is not one of those mechanical forces - rather
it is is a ‘fictitious’ force which arises when we try to describe an object
in a reference frame which is accelerating relative to an inertial reference
frame[22]. This ontology encodes a hierarchy where the centrifugal forces are
to be understood as consequences of the more fundamental laws governing motion
relative to inertial frames. Indeed, the hierarchy licences counterfactual
assertions about the relationship between these features: if the laws
governing motion relative to inertial frames had been relevantly different,
the centrifugal forces would also have been different. Thus we observe that
these hierarchies of fundamentality are roughly analogous to causal
structures, with ‘causation’ to be understood in the counterfactual sense: in
a given theory, feature A causes feature B if feature B would not have
obtained if A had not obtained.333There are many subtleties that have to be
dealt with in any counterfactual account of causation; I will not deal with
any of them here, but refer to ref [23] for a general discussion of
counterfactual approaches to causation. It is likely that further subtleties
would arise in the application of this approach to intratheoretic causation; I
will return to these in forthcoming work. Therefore I will henceforth refer to
these hierarchies as ‘intratheoretic causal structures’ though I emphasize
that the word ‘causal’ is here being used in a specialised sense, since the
relata of the putative causal relation are not immanent.

We are already quite familiar with the case where the intratheoretic causal
structure can be read off the ontology. But what if there is no general
consensus regarding the ontology of a theory? Quantum mechanics is one such
case – to this day it is not even clear whether quantum mechanics requires the
existence of the wavefunction[24], let alone other proposed features such as
spontaneous collapses[25], de Broglie-Bohm particles[26] or ‘many-worlds.’[27]
And as a result the intratheoretic causal structure of quantum mechanics is
much less transparent - witness the continuing controversy over the reality of
the quantum state, which can be understood as a discussion about whether we
should adopt an intratheoretic causal structure in which the characteristic
features of quantum mechanics are derived from the properties of the quantum
state, or whether we should take it that the quantum state itself is derived
from some deeper properties.

Obviously it is open to us to resolve these questions by adopting an
interpretation of quantum mechanics and drawing the conclusions implied by its
ontology. For example, the Everett interpretation tells us that the quantum
state is indeed the fundamental object of quantum mechanics[27], whereas
Quantum Bayesianism suggests that the quantum state is simply a description of
an agent’s degrees of belief about the outcomes of measurements[28]. But the
study of generalized probabilistic theories offers an alternative, allowing us
to draw conclusions about intratheoretic causal structure which are
independent of any specific ontological assumptions. Research of this kind
works by placing quantum mechanics in the context of a wider space of GPTs so
that we can meaningfully consider counterfactual questions about how changing
one aspect of a theory might lead to other changes[5, 6, 7, 8, 9, 10, 11, 12,
13]. This often involves proving statements of the form: ‘any generalized
probabilistic theory which has feature $X$ must also have feature $Y$.’ For
example, ref [12] proves that any GPT which obeys no-signalling must also obey
a form of monogamy of correlations, thus demonstrating the existence of a
(putatively causal) relationship between these features which holds regardless
of what the underlying ontology might be. The framework of generalized
probabilistic theories therefore provides exactly the formalism needed for us
to study the counterfactual reasoning associated with intratheoretic causal
structure in a way that does not depend on any particular ontology.

Of course, showing that certain features of quantum mechanics can be derived
from other features in this way doesn’t actually _prove_ anything about the
intratheoretic causal structure of quantum mechanics, because there exists no
unique axiomatization of quantum mechanics and each different axiomatization
will suggest different intratheoretic causal relations. But that is just to
restate the old problem of underdetermination of theory by data, and a similar
response can be made: the point is not necessarily to know for certain, but to
understand the implications of different possible causal structures as a way
of exploring alternative ways forward for physics. One need not believe that
any particular causal structure is the correct one, or indeed that a unique
correct causal structure even exists, to see the value of the exercise.

###  2.3 Structural Realism

I have suggested that one motivation for research in the space of generalized
probabilistic theories is the idea that we don’t have much hope of coming to
know the true ontology of our theories, so instead of focusing on the details
of specific ontologies we should instead think in terms of intratheoretic
causal structures, which plausibly we could get right even if the true
ontology remains epistemically inaccessible. This suggestion clearly has
common ground with the philosophical view known as structural realism[15],
which proposes that since the specific ontologies of our theories will likely
be discarded when we move to a new theory, we should not be epistemically
committed to specific ontologies but instead commit to the structural
relations implied by the ontologies of the theories, since these relations are
often maintained when we move to a new theory. However, the relations to which
structural realism refers are usually instantiated by specific objects in the
world, whereas the intratheoretic relations that I discuss here hold between
parts of a theory, so the domain of application of this form of structuralism
is importantly different.

##  3 The Choi-Jamiołkowski Isomorphism

In the remainder of this article, I will elaborate on an intratheoretic causal
structure for quantum mechanics where the quantum state is not the fundamental
object. My approach is based on the Choi-Jamiołkowski isomorphism, which
describes the mathematical correspondence between quantum channels and
entangled bipartite states [29]. The Choi-Jamiołkowski isomorphism can be
derived from the standard mathematical formalism of quantum mechanics, and the
fact that it is usually taught in this way encourages physicists to regard it
as a consequence of the properties of the quantum state. In particular, it is
often interpreted by appeal to an operational procedure known as ‘noisy gate
teleportation’ in which agents $A$ and $B$ share entanglement and agent $A$ is
in possession of a system in an unknown state $\rho$ and the aim is for $B$ to
end up with a system in the state $T\rho$ where $T$ is some fixed
transformation; this cannot usually be achieved with perfect certainty, but
insofar as it is possible it works because of the Choi-Jamiołkowski
isomorphism[30, 31]. This presentation implies that the isomorphism is just
one of the many surprising operational consequences which follows from the
properties of the quantum state, so it is of no special foundational interest.

But consider: what the isomorphism actually tells us (roughly speaking), is
that in quantum mechanics, the set of possible multipartite correlations
exhibited by entangled states is equivalent to the set of possible temporal
correlations exhibited by sequences of measurements on a single system over
time. This equivalence has the air of a fact about ontology. Indeed, I would
argue that and indeed, any choice for the ontology of quantum mechanics which
failed to reflect such a striking equivalence would surely be suspect on the
grounds of exhibiting ‘asymmetries which do not appear to be inherent in the
phenomena.’444This phrasing is of course borrowed from ref [32]. Thus in this
paper I will suppose that the Choi-Jamiołkowski isomorphism is indeed a deep
fact about the underlying ontology of quantum mechanics, and therefore
causally prior to most other features of the theory.

In accordance with the discussion of section 2, rather than proposing a
specific ontology which has this feature, I will proceed directly to an
investigation of the consequences of this conjecture for the intratheoretic
causal relations in quantum mechanics. In section 3 I give an operational
formulation of the Choi-Jamiołkowski isomorphism, which relates measurements
on different parts of a multipartite system to sequences of measurements on
the same system and vice versa. In sections 6, 7 and 8 I use it to prove
results of the form ‘Any generalized probabilistic theory which obeys the
operational Choi-Jamiołkowski isomorphism, and which has property A, must also
have property B,’ where A is a constraint on multipartite measurements and B
is a constraint on sequences of measurements, as shown in the table below.
That is, I show that several features of quantum mechanics which are typically
interpreted as temporal properties of the quantum state could in fact be
derived from the properties of some more general entity which is the
substratum for both multipartite states and time evolution.

Property | Implies
---|---
Strong monogamy of correlations and Bell nonlocality | No-broadcasting and quantum interference
No-signalling and Bell nonlocality | Preparation contextuality
Information causality | Fine-grained uncertainty relations

Each of the properties in this table will be given an operational definition
in subsequent sections. Note that the concept of ‘operational state’ will
feature in the definition of some of the properties on the right-hand side of
the table, since they have been chosen specifically as properties which are
normally attached to states. However, crucially, the concept of state is not
used in the operational definitions of the concepts on the left-hand side, or
in the operational Choi-Jamiołkowski isomorphism. Thus as promised, these
results demonstrate features that appear to be properties of states can in
fact be understood as consequences of features of the theory which do not
depend in any way on a concept of state.

###  3.1 Related Work

Although the mainstream literature on quantum mechanics is still very
dominated by state-based approaches, in recent years a variety of interesting
work has been done on non-standard temporal pictures. For example, there has
been growing interest in retrocausal approaches to quantum theory[33, 34, 35],
including a proof by Pusey and Leifer demonstrating that if quantum mechanics
obeys a certain sort of time-symmetry then it must exhibit retrocausality[36]
and a model due to Wharton which suggests a natural resolution to the quantum
reality problem using the “all-at-once”-style analysis of action
principles[37]. Elsewhere, Oreskhov and Cerf[38] have set out the process
matrix formalism, which allows us to generalize the framework of operational
theories in a way that does not depend on a predefined time or causal
structure, thus giving us the mathematical resources to deal with theories
that might contain indefinite causal order, causal loops or other structures
that don’t fit into our familiar notions of time and causality. Similarly,
Shrapnel and Costa have used the process matrix formalism to set out a
generalisation of the ontological models approach which does not assume that
information must be carried through time by a mediating state, and have used
this generalisation to demonstrate that even without mediating states quantum
mechanics must still exhibit a generalized form of contextuality[39]. Thus the
ideas set out here add to a growing body of work on the ways in which quantum
mechanics could be embedded into global and temporally non-local structures.

It has been recognised that the Choi-Jamiolkowski isomorphism is likely to
play an important role in such non-standard temporal pictures - in particular,
the process matrix is defined using the Choi-Jamiolkowski representations of
the relevant CP maps[38]. The process matrix formalism therefore assigns to
the isomorphism an implicit ontological significance, and so the results
presented here can be understood as complementary to that line of research,
since I have made the ontological significance of the isomorphism explicit and
used an operational formulation to explore its consequences for intratheoretic
causal structure. A variety of other authors have also suggested that the
isomorphism should be understood in ontological terms, so for example it is
noted in [40] that the Choi-Jamiołkowski representation for quantum operations
_‘gives a nice way of unifying statics and dynamics in one framework: the
future is entangled (or at least correlated) with the past,’_ and likewise,
ref [41] puts forward a theoretical model in which _‘one particle at N times
is … equivalent to N particles at one time.’_

Finally, a different version of an ‘operational Choi-Jamiołkowski isomorphism’
was put forward in ref [42], where it is shown that in any operational theory
which a) is causal, and b) has the property that every mixed state has a
purification, it is necessarily the case that there exists an isomorphism
between transformations and bipartite states which has the same structural
properties as the Choi-Jamiołkowski isomorphism. This paper is a good example
of what I have described as the study of intratheoretic causal structure, but
the structure proposed is quite different to the one I have suggested here.
First, I have taken the ‘operational Choi-Jamiołkowski isomorphism’ to be
fundamental and derived other features of quantum mechanics from it, whereas
in ref [42] the operational Choi-Jamiołkowski is derived from other features
of the theory, so the intratheoretic causal structure is in fact precisely
reversed. Second, I have specifically avoided using the concept of state in my
formulation of the operational Choi-Jamiołkowski isomorphism and instead
derived properties of the state from the isomorphism together with constraints
on possible correlations, whereas in ref [42] one of the axioms from which the
rest of quantum mehcanics is derived is ‘every mixed state has a
purification,’ so in this approach it seems natural to take the state as a
fundamental object and not an emergent feature.

##  4 Mathematical Background

##  5 Preliminaries

###  5.1 Operational Theories

In attempting to understand some of the special features of quantum mechanics,
it is often helpful to make comparisons between quantum theory and other
possible theories. In this context it is common to employ the language of
generalized probabilistic theories, where theories are defined entirely in
terms of preparation procedures and measurements, eschewing abstract
mathematical constructions. This has been a very active area of study in
recent years, with many interesting results emerging to show how quantum
mechanics relates to the broader field of possible operational theories. For
example, it was shown that quantum mechanics is not the maximally non-local
theory without signalling[10]: there exists a gap between the numerical bound
on the set of non-local correlations which can be produced in a theory limited
only by no-signalling, and the corresponding ‘Tsirelson’ bound on the set of
non-local correlations which can be produced in quantum theory, and much
effort has gone into trying to explain this difference (see for example refs
[43, 44, 45, 11]).

In the framework of operational theories, a given theory is specified as a
quadruple $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ where $\mathcal{P}$ is a
set of preparations, $\mathcal{M}$ is a set of measurements, $\mathcal{T}$ is
a set of transformations, and the function $p(M^{x}|P,T)$ specifies, for every
possible combination of preparation $P$, transformation $T$, and measurement
outcome $M^{x}$, the probability of obtaining outcome $M^{x}$ to the
measurement $M$ if it is performed on a system prepared according to $P$ and
then subjected to transformation $T$. [46]

I will sometimes write $p(M^{x}|P)$ as shorthand for $p(M^{x}|P,\mathbb{I})$,
so $p(M^{x}|P)$ denotes the probability of obtaining outcome $M^{x}$ to the
measurement $M$ if it is performed directly on a system prepared according to
$P$. For the operational theories considered in this paper, I stipulate that
performing a transformation followed by a given measurement is equivalent to
simply performing some other measurement, i.e. for any $P,T,\\{M^{x}\\}$,
there exists some $\\{N^{x}\\}$ such that
$p(M^{x}|P,T)=p(N^{x}|P,\mathbb{I})$; for any operational theory where this is
not the case, we can trivially make it the case by expanding the set
$\mathcal{M}$.

###  5.2 Ensemble preparations

The operational formulation of the Choi-Jamiołkowski isomorphism will make use
of the following concept:

######  Definition 5.1.

An _ensemble preparation_ , $P$, specified by a probability distribution
$p(i):i\in\\{1,2...N\\}$ and a set of preparations
$\\{Q_{i}:i\in\\{1,2...N\\}\\}$, is a procedure in which an observer draws a
number $i$ from $\\{1,2...N\\}$ with probability distribution $p(i)$, and then
performs the corresponding preparation $Q_{i}$.

In particular, when the operational theory in question is quantum mechanics,
every possible ensemble preparation can be described by a POVM $\\{M_{i}\\}$
and density operator $\rho$, where $p(i)=Tr(\rho M_{i})$ and $P_{i}$ is a
preparation which produces the quantum state
$\rho_{i}=\frac{\sqrt{\rho}M_{i}\sqrt{\rho}}{Tr(M_{i}\rho)}$ [47].

For brevity, I will also use the following definitions:

######  Definition 5.2.

For any ensemble preparation $P$ for a given system $S$, any set of channels
$T_{2},T_{3},...T_{n}$ which may then be applied to $S$, and any set of
measurements $M_{2},M_{3}...M_{n}$ which can be applied to the distinct
outputs of these channels, I denote by $p_{P;T_{2}...T_{n};M_{2}...M_{n}}$ the
joint probability distribution over the outcome of the random choice in the
ensemble preparation $P$ and the outcome of the measurements
$M_{2},M_{3}...M_{n}$.

$p(3)$$p(2)$$p(1)$$3$$2$$1$$Q_{3}$$Q_{2}$$Q_{1}$$M$$T$$M^{2}$$M^{3}$$M^{1}$
Figure 1: A schematic diagram of an ensemble preparation $P$ with three
possible results for the random number generation step, followed by a
transformation $T$, followed by a measurement $M$ with three possible outcomes
$M^{1},M^{2},M^{3}$. This scenario can be described by the probability
distribution $p_{P;T;M}$.

######  Definition 5.3.

For any joint preparation $P_{12...n}$ for a set of systems
$S_{1},S_{2}...S_{n}$, and any set of measurements $M_{1},M_{2},...M_{n}$
which can be applied separately to systems $S_{1},S_{2},...M_{n}$, I denote by
$p_{P_{12...n};M_{1}...M_{n}}$ the joint probability distribution over the
outcomes of the measurements $M_{1},M_{2},...M_{n}$.

$M_{2}$$M_{1}$$P_{12}$$M_{2}^{1}$$M_{2}^{2}$$M_{2}^{3}$$M_{1}^{1}$$M_{1}^{2}$$M_{1}^{3}$
Figure 2: A schematic diagram of a joint preparation $P_{12}$ on a bipartite
state, followed by followed by measurement $M_{1}$ with possible outcomes
$M_{1}^{1},M_{1}^{2},M_{1}^{3}$ on one system, and measurement $M_{2}$ with
possible outcomes $M_{2}^{1},M_{2}^{2},M_{2}^{3}$ on the other system. This
scenario can be described by the probability distribution
$p_{P_{12};M_{1},M_{2}}$.

###  5.3 The Choi-Jamiołkowski Isomorphism

In its original form, the Choi-Jamiołkowski isomorphism is a one-to-one map
between unnormalized quantum states and completely positive (CP) operators.
Specifically, for a given CP operator $\epsilon_{B|A}$ mapping a Hilbert space
$\mathscr{H}_{A}$ to a Hilbert space $\mathscr{H}_{B}$, the corresponding
Choi-Jamiołkowski state $\rho^{\epsilon}$ is obtained by applying
$\epsilon_{B|A}$ to half of the maximally entangled state
$\rho_{AA^{\prime}}=\frac{1}{d}\sum_{i}\sum_{i^{\prime}}|i\rangle_{A}|i\rangle_{A^{\prime}}\langle
i^{\prime}|_{A}\langle i^{\prime}|_{A^{\prime}}$, where $A^{\prime}$ is a
quantum system with the same Hilbert space as $A$, and $d$ is the dimension of
this Hilbert space: [47, 48]555The formula presented here is for the Choi
version of the isomorphism; the Jamiołkowski version is
$\rho^{\epsilon}_{AB}=(\epsilon_{B|A}\otimes\mathbb{I})(\frac{1}{d}\sum_{i}\sum_{i^{\prime}}|i\rangle_{A}|i\rangle_{A^{\prime}}\langle
i^{\prime}|_{A}\langle i^{\prime}|_{A^{\prime}})^{T_{A}})$, where $T_{A}$
refers to the partial transpose in the basis $\\{|i\rangle\\}$. The key
difference between the two formulations is that the Jamiołkowski version,
unlike the Choi version, is basis-independent. But since I will not henceforth
be working within the quantum formalism, this distinction is not important.

$\rho^{\epsilon}_{AB}=(\epsilon_{B|A}\otimes\mathbb{I})\rho_{AA^{\prime}}$

However, in ref [47] the isomorphism is reformulated as a map between
conditional density operators and completely positive trace-preserving (CPTP)
operators. This formulation, which I paraphrase below, is more transparent in
its physical interpretation and hence I will take it as the starting point for
the operational approach:

######  Lemma 5.4.

For any bipartite state $\rho^{\epsilon}_{AB}$, there exists a CPTP map
$\epsilon$ and a reduced state $\rho_{A}=Tr_{B}(\rho^{\epsilon}_{AB})$ such
that given any two POVMs $M$ and $O$, if $M^{T}$ is obtained by taking the
transpose of all the measurement operators in $M$ with respect to some fixed
basis, then when we perform the ensemble preparation described by the POVM
$M^{T}$ and the density operator $\rho_{A}$, then evolve the state according
to $\epsilon$, then perform the measurement $O$, the probability that state
$\rho_{i}$ is prepared and then the measurement $M$ has outcome $j$ is the
same as the joint probability of obtaining outcomes $M_{i}$ and $O_{j}$ when
the POVM $M$ is performed on system $A$ and the POVM $O$ is performed on
system $B$ for a bipartite system $AB$ in the state $\rho^{\epsilon}_{AB}$.

Conversely, for any pair of a CPTP map and state $\rho$ there exists a
bipartite state $\rho^{\epsilon}_{AB}$ such that the same conditions hold, so
we have defined an isomorphism between bipartite states and pairs
$(\rho_{A},\epsilon^{r})$, where $\epsilon^{r}$ denotes the restriction of the
CPTP map $\epsilon$ to the support of $\rho_{A}$.

###  5.4 Reformulation

Using these concepts, we can define an operational version of the
Choi–Jamiołkowski isomorphism.

######  Definition 5.5.

Operational Choi-Jamiołkowski Isomorphism:

For any joint preparation $P_{123...n}$ on a set of systems
$S,S_{2},...S_{n}$, there exists a set of channels $T_{2},T_{3},...T_{n}$
which may simultaneously be applied to the system $S$, such that for any set
of measurements $M$, $M_{2}$, … $M_{n}$ which may be performed on
$S,S_{2},...S_{n}$, there exists an ensemble preparation $P$ for the system
$S$ such that the distribution $p_{P_{123...n};M,M_{2}...M_{n}}$ is the same
as the distribution $p_{P;T_{2}...T_{n};M_{2}...M_{n}}$.

Conversely, for any set of channels $T_{2},T_{3},...T_{n}$ which may
simultaneously be applied to the system $S$ to produce a set of systems
$S_{2},...S_{n}$, there exists a joint preparation $P_{123...n}$ for systems
$S,S_{2},S_{3}...S_{n}$ such that for any ensemble preparation $P$ which may
be performed for system $S$ and any set of measurements $M_{2},M_{3},...M_{n}$
which may be performed on the products $S_{2},...S_{n}$, there exists a
measurement $M$ on $S$ such that the distribution
$p_{P_{123...n};MM_{2}...M_{n}}$ is the same as the distribution
$p_{P;T_{2}...T_{n};M_{2},...M_{n}}$.

In graphical terms, this says that for any scenario as in figure 1 there
exists a scenario as in figure 2 and vice versa; likewise, for any scenario as
in figure 3 there exists a scenario as in figure 4 and vice versa.

$p(3)$$p(2)$$p(1)$$3$$2$$1$$Q_{3}$$Q_{2}$$Q_{1}$$T_{2}$$T_{3}$$M_{2}$$M_{3}$$M_{2}^{3}$$M_{2}^{2}$$M_{2}^{1}$$M_{3}^{1}$$M_{3}^{2}$$M_{3}^{3}$
Figure 3: A schematic diagram of an ensemble $P$ with three possible results
for the random number generation step, followed by two channels $T2$ and
$T_{3}$ each producing a distinct system, followed by a measurement $M_{2}$
with three possible outcomes $M_{2}^{1},M_{2}^{2},M_{2}^{3}$ on one system,
and a measurement $M_{3}$ with three possible outcomes
$M_{3}^{1},M_{3}^{2},M_{3}^{3}$ on the other system. This scenario can be
described by the probability distribution $p_{P;T_{2},T_{3};M_{2},M_{3}}$.
$P_{123}$$M_{1}$$M_{3}$$M_{2}$$M_{1}^{1}$$M_{1}^{2}$$M_{1}^{3}$$M_{3}^{1}$$M_{3}^{2}$$M_{3}^{3}$$M_{2}^{1}$$M_{2}^{2}$$M_{2}^{3}$
Figure 4: A schematic diagram of a joint preparation $P_{12}$ on a bipartite
state, followed by followed by a measurement $M_{1}$ with possible outcomes
$M_{1}^{1},M_{1}^{2},M_{1}^{3}$ on one system, and measurement $M_{2}$ with
possible outcomes $M_{2}^{1},M_{2}^{2},M_{2}^{3}$ on the other system. This
scenario can be described by the probability distribution
$p_{P_{123};M_{1},M_{2},M_{3}}$.

####  5.4.1 Relation to original Choi-Jamiołkowski isomorphism

This operational definition of the isomorphism differs from the original in
several respects. It is of course to be expected that an operational
formulation of the isomorphism will not be exactly equivalent to the original
quantum-mechanical statement of it, since the quantum-mechanical version is
heavily theory-dependent and it is not possible to jettison these theory-
dependent elements without losing some content. I believe nonetheless that
this formulation captures much of what is important about the original Choi-
Jamiołkowski isomorphism.

First, it should be noted that the operational formulation does not actually
postulate the existence of an isomorphism: it makes a statement of the form
‘for each x there exists a y, and for each y there exists an x,’ but this does
not imply that there is an isomorphism between the sets $\\{x\\}$, $\\{y\\}$,
since I do not insist that each distinct $x$ should be mapped to a different
$y$. One could of course make this further stipulation and it might turn out
that further interesting results would arise from the correspondingly stronger
constraint, but it was not necessary to do this for any of the results
obtained here and I thought it best to employ the weakest possible version of
the constraint that was still sufficient to derive these results. I have
continued to use the term ‘isomorphism’ for consistency with the established
terminology.

Second, the operational formulation applies to all multipartite states,
whereas the quantum-mechanical Choi-Jamiołkowski isomorphism applies only to
bipartite states. This change has been made in order to make it possible to
derive the no-cloning result in a particularly simple and transparent way.
This particular generalization has in fact been considered before: it was used
to demonstrate the connection between no-cloning/no-broadcasting theorems and
the monogamy of entanglement in ref [49]. Furthermore, recall that the
motivation for putting forward this operational Choi-Jamiołkowski isomorphism
is the intuition that the quantum isomorphism expresses a deep fact about the
underlying ontology of the theory - channels behave like preparations because
they are in fact the same sort of thing at the level of the underlying
ontology. If this is the case, then we would naturally expect the constraint
to apply to all multipartite states rather than just bipartite ones, since
bipartite states are simply a special case of multipartite states.

##  6 No-broadcasting theorem and the monogamy of entanglement

###  6.1 Background

It is a particularly distinctive feature of quantum information that quantum
states cannot be cloned - that is, there is no quantum operation which makes
perfect copies of two non-orthogonal quantum states. In fact, the
impossibility of cloning quantum states is a special case of the impossibility
of broadcasting quantum states:

######  Definition 6.1.

A map $M$ from the Hilbert space $\mathscr{H}$ to the the Hilbert space
$\mathscr{H}_{A}\otimes\mathscr{H}_{B}$ broadcasts the set of states $S$ iff
for any state $\rho\in S$, $Tr_{A}(M(\rho))=\rho$ and $Tr_{B}(M(\rho))=\rho$.

######  Definition 6.2.

A universal broadcasting map broadcasts any set of states.

######  theorem 1.

No-Broadcasting Theorem: A set of quantum states can be broadcasted if and
only if they commute pairwise [50]666A pair of quantum states
$\rho_{0},\rho_{1}$ are said to commute iff
$\rho_{0}\rho_{1}-\rho_{1}\rho_{2}=0$[50]

######  corollary 6.3.

There is no universal broadcasting map in quantum mechanics

###  6.2 Operational Formulations

In ref [49], it is shown that corollary 6.3 can be derived directly from the
monogamy of entanglement using the Choi-Jamiołkowski isomorphism:

######  theorem 2.

Supposing the existence of a universal broadcasting map is equivalent to
supposing the existence of a tripartite state of a system $A,B,C$ where both
the bipartite reduced state of $A,B$ and the bipartite reduced state of $B,C$
are pure and maximally entangled. 777Ref [49] also derives the no-broadcasting
theorem in full generality from the monogamy of entanglement, but I have not
pursued that result here, because the proof relies on facts about the fixed
point set of TPCP maps, and in the operational framework facts concerning the
detailed structure of quantum operations are not available to us.

Here I will derive a similar result, but my derivation will be presented in
the context of general operational theories without presupposing the Hilbert
space structure of quantum mechanics. Therefore before embarking on the proof,
I need to provide definitions of broadcasting and monogamy which are suited to
the operational context.

In order to define ‘broadcasting’ we will use the concept of as an
‘operational state’; note, again, that this concept will be used only to
define broadcasting, and will not appear in the definitions of the features
from which no-broadcasting will be derived.

######  Definition 6.4.

Given two preparation procedures $P_{a},P_{b}$ which appear in an operational
theory, I say that these procedures produce the same _operational state_ iff
when a system is prepared using one of these procedures, there is no
subsequent measurement or sequence of measurements which can give us any
information about whether the system was prepared using $P_{a}$ or $P_{b}$.

######  Definition 6.5.

A transformation $T$ broadcasts the set of operational states $S$ iff when $T$
is applied to a system whose operational state is $\rho\in S$, together with
one or more ancilla systems, the output includes two distinct systems both
having the operational state $\rho$.

######  Definition 6.6.

A universal broadcasting map broadcasts any set of operational states.

Defining the relevant monogamy property for an operational context is less
straightforward. In quantum mechanics, the monogamy of entanglement refers to
the fact that the amount of entanglement a quantum system has with a given
system limits the amount of entanglement it can have with any other system -
in the most extreme case, a quantum system which is maximally entangled with a
given quantum system cannot be entangled at all with any other quantum system.
But entanglement is a property of quantum states, usually quantified using
state-dependent measures like the ‘concurrence’ (or ‘tangle’)[51], and these
measures do not have a straightforward physical interpretation, making it
difficult to translate them to the framework of operational theories.

Therefore it is more helpful for us to focus on the ‘monogamy of
correlations,’ which is a property of certain theories in which there are
limits on how strongly a system can be correlated with other systems. The
monogamy of correlations is defined purely in terms of observable statistics,
and is therefore much more easily cast in an operational framework.

In order to facilitate comparison to the quantum case, I will define monogamy
for a general operational theory using the CHSH quantity:

######  Definition 6.7.

CHSH quantity: For any joint preparation $P_{AB}$ of two systems, and any pair
of measurements $M_{A}^{0},M_{A}^{1}$ which may be performed on system $A$,
and any pair of measurements $M_{B}^{0},M_{B}^{1}$ which may be performed on
system $B$, the CHSH quantity
$\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})$ for this
combination of preparation and measurements is defined by the following sum of
expectation values:

$\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1}):=\\\ \langle
AB\rangle_{P_{AB},M_{A}^{0},M_{B}^{0}}+\langle
AB\rangle_{P_{AB},M_{A}^{0},M_{B}^{1}}+\langle
AB\rangle_{P_{AB},M_{A}^{1},M_{B}^{0}}-\langle
AB\rangle_{P_{AB},M_{A}^{1},M_{B}^{1}}\\\ $ (1)

Note that is also possible to define a related CHSH quantity
$\mathscr{B}_{AB}(P^{0},P^{1},T,M_{B}^{0},M_{B}^{1})$ for the scenario where
an experimenter applies one of two ensemble preparations $P^{0},P^{1}$ to a
system, then applies a fixed transformation $T$, then performs one of two
measurements $M_{B}^{0},M_{B}^{1}$. This quantity is defined just as in
equation LABEL:Belloperator, where the result of the measurement $M_{A}$ is
replaced with the result of the probabilistic choice in the ensemble
preparation. This definition ensures that if for some transformation $T$, some
set of preparations $P_{AB}$, $P^{0},P^{1}$ and some set of measurements
$M^{0}_{1},M^{1}_{1}$, $M^{0}_{2},M^{1}_{2}$, for any $x,y\in\\{0,1\\}$ the
distributions $p_{P_{AB};M^{x}_{1}M^{y}_{2}}$ and $p_{P^{x};T;M^{y}_{2}}$ are
the same, then
$\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})=\mathscr{B}_{AB}(P^{0},P^{1},T,M_{B}^{0},M_{B}^{1})$.

With a definition for the CHSH quantity in hand, we may now define two types
of monogamy which might be obeyed by an operational theory:

######  Definition 6.8.

Non-signalling monogamy of correlations: An operational theory obeys no-
signalling monogamy of correlations iff for any joint preparation of three
systems $S_{A},S_{B},S_{C}$, for any choice of measurements
$M_{A}^{0},M_{A}^{1}$ on $S_{A}$, any choice of measurements
$M_{B}^{0},M_{B}^{1}$ on $S_{B}$, and any choice of measurements
$M_{C}^{0},M_{C}^{1}$ on $S_{C}$, the associated CHSH quantities satisfy:

$\mathscr{B}_{AB}(P_{ABC},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})+\mathscr{B}_{BC}(P_{ABC},M_{B}^{0},M_{B}^{1},M_{C}^{0},M_{C}^{1})\leq
4$

It can be shown that this bound is necessarily obeyed by any operational
theory which does not allow signalling (see section )[12].

######  Definition 6.9.

Strong monogamy of correlations: An operational theory obeys strong monogamy
of correlations iff for any joint preparation of three systems
$S_{A},S_{B},S_{C}$, for any choice of measurements $M_{A}^{0},M_{A}^{1}$ on
$S_{A}$, any choice of measurements $M_{B}^{0},M_{B}^{1}$ on $S_{B}$, and any
choice of measurements $M_{C}^{0},M_{C}^{1}$ on $S_{C}$, the associated CHSH
quantities satisfy:

$\mathscr{B}_{AB}(P_{ABC},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})^{2}+\mathscr{B}_{BC}(P_{ABC},M_{B}^{0},M_{B}^{1},M_{C}^{0},M_{C}^{1})^{2}\leq
8$

Strong monogamy of correlations need not be obeyed by all non-signalling
operational theories, but it _is_ obeyed by quantum mechanics[52].

###  6.3 Bell nonlocality

We will also require the notion of Bell nonlocality, which is again defined by
analogy with the quantum case:

######  Definition 6.10.

An operational theory exhibits Bell nonlocality iff there exists at least one
joint preparation $P_{AB}$ for two systems $S_{A}$ and $S_{B}$, such that for
some pair of measurements $M_{A}^{0},M_{A}^{1}$ on $S_{A}$ and some pair of
measurements $M_{B}^{0},M_{B}^{1}$ on $S_{B}$, the CHSH quantity satisfies:

$\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})>2$

###  6.4 theorem

Using these concepts, we have the following theorem (see appendix .1 for the
proof).

######  theorem 3.

In an operational theory which obeys the operational Choi-Jamiolkowksi
isomorphism and exhibits Bell nonlocality, the existence of a universal
broadcasting map implies that the theory violates strong monogamy of
correlations

Note that if we impose the ontological Choi-Jamiołkowski isomorphism as a
fundamental constraint on an operational theory, it follows by modus tollens
that the strong monogamy of correlations together with the existence of Bell
nonlocality implies the absence of broadcasting, as advertised in table 3.

###  6.5 Quantum interference

The results of section 6.4 may also be used to derive the existence of quantum
interference from the monogamy of correlations together with the existence of
Bell non-locality.

To see this, let us limit ourselves to considering GPTs whose observables have
the algebraic structure of the self-adjoint operators in a $C^{*}$ algebra.
Not all GPTs meet this criterion888See ref [53] for a discussion of the
general conditions under which a theory can be represented as a $C^{*}$
algebra, but nonetheless it is a reasonable heuristic framework since a large
class of important GPTs, including classical mechanics and quantum mechanics,
can indeed be thus represented. Then observe that for any theory represented
as a $C^{*}$-algebra, it can be shown that if the $C^{*}$ algebra is
commutative then the theory has a univeral broadcasting map [54, 13]. Thus if
a GPT does not have a universal broadcasting map, any $C^{*}$ algebra that
represents it must be non-commutative. Finally, note that theories represented
by commutative $C^{*}$ algebras are classical phase space theories, whereas
theories represented by non-commutative $C^{*}$ algebras have observables
which cannot be simultaneously measured and hence exhibit superposition and
interference[54].

Thus, if we limit ourselves to GPTs which have the algebraic structure of the
self-adjoint operators in a $C^{*}$ algebra and which obey the operational
Choi-Jamiołkowski isomorphism, it follows from theorem 3 that any GPT which
exhibits strong monogamy of correlations and Bell non-locality must also
exhibit superposition and interference. This means that superposition and
interference need not be regarded as properties of the quantum state - with
the help of the operational Choi-Jamiołkowski isomorphism they can be derived
directly from properties of non-local correlations, with no need to appeal to
a concept of state at all.

##  7 Preparation Contextuality and No-Signalling

###  7.1 Background

One of the most puzzling feature of quantum theory is the fact that it is
_contextual_ : that is, it is not possible to interpret the outcomes of
quantum-mechanical measurements as if they simply reveal the pre-existing
value of some property of the system being measured.

The original formulation of contextuality[55] tied the concept of
contextuality to the concept of determinism, but the notion was subsequently
generalized by Spekkens[56]. Spekkens’ form of contextuality is most easily
defined within the ontological models framework, where one supposes that every
system has a single real ‘ontic state,’ which determines the probabilities for
the outcomes of any measurement on that system999The use of this faramework
does not necessarily imply the endorsement of a description of quantum reality
in terms of underlying states; the ontological models approach could simply be
regarded as a helpful language in which I may express mathematical facts about
the structure of quantum theory, such as its contextual character.. An
ontological model thus consists of a space $\Lambda$ of ontic states
$\lambda$, a set of probability distributions $\mu^{P}(\lambda)$ giving the
probability that the system ends up in the state $\lambda$ when we perform the
preparation procedure $P$, a set of response functions $\xi_{M,X}(\lambda)$
giving the probability that we obtain outcome $M^{x}$ when we perform
measurement $M$ on a system whose ontic state is $\lambda$, and a set of
column-stochastic matrices $T^{O}$ representing the way in which the ontic
state is transformed when some operation $O$ is applied to the system. A valid
ontological model must satisfy the following positivity and normalization
conditions:

  1. 1.

$\forall P,\lambda\quad\mu_{P}(\lambda)\in[0,1]$

  2. 2.

$\forall P\quad\sum_{\lambda}\mu_{P}(\lambda)=1$

  3. 3.

$\forall M,X\quad\vec{\xi}_{M,X}(\lambda)\in[0,1]$

  4. 4.

$\forall M\quad\sum_{X}\vec{\xi}_{M,X}(\lambda)=1$

  5. 5.

$\forall O\quad T_{0}$ is a column-stochastic matrix. 101010A column-
stochastic matrix is a matrix whose entries are all non-negative and whose
columns sum to one. Left-multiplication by a column-stochastic matrix is the
most general possibility for the representation of operations in an
ontological model, since such an operation must map ontic state $\lambda_{i}$
to $\lambda_{j}$ with some probability, which is specified by entry $(i,j)$ in
the transformation matrix. Given that the entries are all probabilities, they
must be nonnegative, and given that each transformation must map $\lambda_{i}$
to some state, the sum of the entries in a column must be one.

An ontological model is _preparation contextual_ if it does _not_ represent
every quantum state by a unique probability distribution $\mu(\lambda)$;
_transformation contextual_ iff it does _not_ represent every possible quantum
operation $O$ by a unique transformation matrix $T_{O}$; and _measurement
contextual_ if it does _not_ represent every possible quantum measurement
element $M^{x}$ by a unique set of response functions $\xi_{M,X}(\lambda)$.
Spekkens proved[56] that any ontological model of quantum mechanics which
reproduces all the correct measurement statistics must exhibit preparation
contextuality, but examples such as the Kochen-Specker model [57] show that
this is not true for measurement contextuality. 111111Transformation
contextuality has not been studied so thoroughly, and most existing explicit
ontological models omit transformations, so to my knowledge the question of
whether or not quantum mechanics necessarily exhibits transformation
contextuality remains unresolved.

###  7.2 Operational Formulations

I now generalize the definition of preparation contextuality to the framework
of operational theories. (Similar generalizations can be made for measurement
and transformation contextuality, but I will not need these concepts here).

######  Definition 7.1.

We say that an operational theory is _preparation contextual_ iff it is not
possible to represent the theory by a valid ontological model in which every
operational state is represented by a unique probability distribution over
ontic states.

I will also employ a definition of the no-signalling principle suited for an
operational framework. This generalization is a straightforward one, since it
is just the operational version of the no-signalling principle as it is used
in the context of relativistic cryptography[58]:

######  Definition 7.2.

Operational no-signalling principle: In a process involving a set of non-
communicating devices $\\{D_{i}\\}:i\in\\{1\ldots N\\}$ such that device
$D_{i}$ accepts an input variable $N_{i}$ and produces an output variable
$O_{i}$, let $J$ be any subset of $\\{1\ldots N\\}$, let $O_{J}$ be the set of
variables $\\{O_{j}:j\in J\\}$, let $N_{J}$ be the set of variables
$\\{N_{j}:j\in J\\}$; then if the inputs $\\{N_{i}\\}$ are uncorrelated, the
outcomes satisfy $p(O_{J}|N_{1},\ldots,N_{n},)=p(O_{J}|N_{J})$.

###  7.3 theorem

Thus we have the following theorem (see appendix .1 for the proof).

######  theorem 4.

Given an operational theory which obeys the operational Choi-Jamiołkowski
isomorphism and the no-signalling principle, if the theory is preparation non-
contextual, it does not exhibit Bell nonlocality.

Note that if we impose the ontological Choi-Jamiołkowski isomorphism as a
fundamental constraint on an operational theory, it follows by modus tollens
that if the theory exhibits Bell nonlocality but obeys the no-signalling
principle, it must be preparation contextual; that is, preparation
contextuality is a necessary feature of any such operational theory describing
a world which is non-local but also non-signalling.

##  8 Uncertainty Relations

###  8.1 Background

’Heisenberg’s Uncertainty Principle’ is one of the most well-known features of
quantum physics, not least because the fact that there exist incompatible
measurements whose results cannot be simultaneously predicted with certainty
seems strongly in conflict with our classical intuitions. This provides a
clear imperative to seek an explanation for the existence of such apparently
arbitrary limitations on the properties that quantum systems can possess.

To be more precise, there are in fact a number of different uncertainty
relations which characterize tradeoffs between the precision with which the
outcomes of various sets of measurements can be predicted. Most of these
relations were originally formulated in terms of standard deviations and
commutators, but this approach has been criticized on the grounds that the use
of commutators makes the relations state-dependent, which should not be the
case given that in quantum mechanics there always exists a dynamical evolution
which transforms any initial pure state into any any other pure state[59].
Thus more recently there has been a move towards expressing uncertainty
relations as a constraint on the Shannon entropies corresponding to the
probability distributions over outcomes associated with various possible
measurements[59, 60, 61]. However these entropic uncertainty relations have
their own limitations - in particular, they are not capable of distinguishing
between different possible combinations of outcomes[62] \- and therefore in
certain cases it is more useful to consider _fine-grained_ uncertainty
relations, which directly constrain particular sets of outcomes of different
measurements.

The derivation of this section is inspired by the work of ref [62], where it
is shown that it is possible to derive the Tsirelson bound from the following
fine-grained entropic uncertainty relation, which applies to any pair of
orthogonal two-outcome measurements $M$ and $M^{\prime}$ on a two-dimensional
quantum system (such as the binary spin-half observables $X$ and $Z$):

$\forall m\forall
n\forall\rho\in\\{0,1\\}\;p(m|M)_{\rho}+p(n|M^{\prime})_{\rho}\leq
1+\frac{1}{\sqrt{2}}$ (2)

Here, $p(m|M)_{\rho}$ denotes the probability that I obtain outcome
$m\in\\{0,1\\}$ when I perform measurement $M$ on a system in quantum state
$\rho$.

In the operational framework, equation 2 becomes a constraint applying to all
preparations in the operational theory which prepare systems of dimension two,
according to the operational definition of ‘dimension’ which I provide in
section 8.2. I also reverse the argument of ref [62], in order to derive the
uncertainty relation from the Tsirelson bound. I then note that it was shown
in ref [11] that the Tsirelson bound can be derived from the principle of
information causality:

######  Definition 8.1.

Information causality: if Alice and Bob pre-share a set of devices which
exhibit nonlocal correlations, and Alice receives a bit string
$N_{0}N_{1}...N_{n}$ and sends Bob a classical message $M$ of $m$ bits, and
Bob performs a measurement with some setting $k$ and obtains outcome $O$, then
$\sum_{r}I(MO:N_{r}|k=r)>m$ [11]

Thus, if we take it that information causality is a fundamental constraint on
quantum theory, as suggested in [11], the fact that quantum states obey this
particular fine-grained uncertainty relation in quantum theory may be
understood as a consequence of the fundamental constraint. Moreover, in
quantum mechanics, the inequality in equation 2 is tight (for example, the
inequality is saturated if we choose measurements $X$, $Z$ and prepare the
system to be measured in an eigenstate of $\frac{1}{\sqrt{2}}(X+Z)$ or
$\frac{1}{\sqrt{2}}(X-Z)$) and thus argue in this particular case information
causality may be understood as setting the exact limits of quantum
uncertainty.

###  8.2 Operational Formulations

Since the fine-grained uncertainty relation of equation 2 applies only for
systems of dimension two, we need an operational way to single out systems of
dimension two. To do so, we must first add some structure to the usual
framework of operational theories. Thus far, we have assumed that for a given
operational theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ it is possible to
perform any preparation followed by any transformation followed by any
measurement, but in a theory with a meaningful concept of dimension this is
not so, because we can only perform measurements which match the dimension of
the system I have prepared. Thus in this section I consider an operational
theory to be a collection of subtheories
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$, such that within any given
subtheory it is possible to perform any preparation followed by any
transformation followed by any measurement, but it is not possible to perform
a preparation from one subtheory followed by a transformation or measurement
from another subtheory. I then define the dimension of a sub-theory as
follows.

######  Definition 8.2.

A sub-theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ of an operational
theory is d-dimensional iff $d$ is the smallest number such that there exists
a set of $d^{2}-1$ continuous parameters in $[0,1]$ with the following
properties121212This is a fairly weak notion of ‘dimension,’ in that I have
not ruled out the possibility that there could be two or more disjoint
subtheories classified as two-dimensional under this definition. In other
contexts it might be helpful to strengthen the definition by requiring that
the subtheory of dimension two should be unique, but this is not necessary for
the derivation I give here and therefore for brevity I employ this simpler and
weaker notion of dimension. :

  1. 1.

Specifying the values of all $d^{2}-1$ parameters for any preparation
$P\in\mathcal{P}$ fully determines the probabilities $p(M^{x}|P)$ for every
outcome $M^{x}$ of every measurement $M$ in $\mathcal{M}$.

  2. 2.

For every possible set of values of the $d^{2}-1$ parameters, there exists a
preparation $P\in\mathcal{P}$ described by those parameters. 131313This
continuity assumption is stronger than I strictly need for the argument: it
would suffice to make some kind of symmetry assumption, such as the assumption
that if there exists a preparation described by a set of parameters $\vec{x}$
with $x_{1}=p$, then there exists a preparation described by the same set of
parameters except with $x_{1}=1-p$. I have chosen to use the stronger
assumption here as it is simpler and seems more natural; it is clear that both
assumptions are satisfied by quantum mechanics.

Obviously, the choice to associate $d$ dimensions with $d^{2}-1$ parameters is
inspired by quantum theory: a $d$-dimensional quantum state is described by a
set of $d$ complex parameters, or $d^{2}$ continuous real parameters, minus
$1$ to account for normalization. But this particular mapping is a naming
convention only - I could equally well have chosen to say that a subtheory
described by a set of $d$ parameters is $d$-dimensional, which might have
seemed the natural choice if it were not for the example of quantum theory.
And therefore this definition does not take for granted any particular
structure for the state space of the subtheory, other than the continuity
implied by the second property.

The fine-grained uncertainty relation also applies only for pairs of
orthogonal measurements, and therefore we require a notion of ‘orthogonal
measurement’ for the operational framework:

######  Definition 8.3.

In some subtheory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ of an operational
theory, two measurements $M_{1},M_{2}\in\mathcal{M}$ are orthogonal iff given
an arbitrary unknown preparation $P$, the set of probabilities
$\\{p(M^{x}_{1}|P)\\}$ and $\\{p(M^{x}_{2}|P)\\}$ are independent.

This definition implies that given a preparation $P$, and any set of $q$
orthogonal measurements $\\{M_{i}\\}$ each having two outcomes
$M_{i}^{0},M_{i}^{1}$, we need $q$ independent parameters to specify the set
of $q$ probabilities $p(M_{i}^{0}|M_{i},P)$. Therefore I can set $q$ out of
the $d^{2}-1$ parameters needed to specify the full set of outcome
probabilities to be equal to the $q$ probabilities $p(M_{i}^{0}|M_{i},P)$. In
particular, for a two-dimensional system, given any set of three orthogonal
measurements $\\{M_{1},M_{2},M_{3}\\}$ each having two outcomes, specifying
the three probabilities
$p(M_{1}^{0}|M_{1},P),p(M_{2}^{0}|M_{2},P),p(M_{3}^{0}|M_{3},P)$ is sufficient
to fix the probabilities for any other measurement which may be performed
after the preparation $P$.

###  8.3 theorem

Using these concepts, we have the following theorem (see appendix .3 for the
proof).

######  theorem 5.

If an operational theory obeys the operational Choi-Jamiołkowski isomorphism
and information causality, then given any subtheory
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$ of dimension two, for any
preparation $P\in\mathcal{P}$ and any pair of orthogonal measurements
$M_{1},M_{2}\in\mathcal{M}$, and any two outcomes $M_{1}^{m}$, $M_{2}^{n}$ of
the measurements $M_{1},M_{2}$, we must have:

$p(M_{1}^{m}|M)+p(M_{2}^{n}|M^{\prime})\leq 1+\frac{1}{\sqrt{2}}$

##  9 Discussion

I have suggested that the study of intratheoretic causal structures offers us
a way of rethinking the structural assumptions built into our theories in a
way that is independent of any particular ontology. In particular, I have
focused here on a particularly stubborn ontological assumption which is deeply
woven into both classical and quantum physics: the idea that information about
the past must be carried into the future by a mediating state.

Although it is still very common to assume that temporal correlations must be
mediated, we do not usually suppose that the non-local multipartite
correlations in quantum mechanics are mediated by an intervening process, not
least because any such process would either have to travel faster than light
or go backwards in time. Thus by considering an intratheoretic causal
structure for quantum mechanics in which the equivalence between multipartite
correlations and temporal correlations is treated as fundamental, we are
moving towards a picture of quantum mechanics where temporal correlations are
_not_ mediated. Therefore the structure set up here offers a way of thinking
about quantum mechanics where neither the quantum state nor any other sort of
state is a fundamental object.

In particular, I have demonstrated that several of the characteristic
properties of the quantum state can in fact be derived from features of non-
local correlations without appeal to any concept of state. For example,
theorem 4 shows that using the operational Choi-Jamiołkowski isomorphism, the
fact that quantum states exhibit preparation contextuality can be derived from
the fact that we live in a world which is non-local but non-signalling.
Intuitions will vary, of course, but to me non-locality and no-signalling seem
like very fundamental features of reality, and therefore I find it plausible
to regard preparation contextuality as a consequence of non-locality and no-
signalling. Thus preparation contextuality need not be regarded as a
fundamental property of some entity known as the quantum state - it can be
thought of as a behaviour that is in fact caused by very general facts about
space and time together with the equivalence between multipartite correlations
and temporal correlations.

The same goes for the other derivations presented here. There is no need to
postulate a quantum state to be the bearer of the properties of ‘no-
broadcasting’, ’interference,’ and ‘obeying uncertainty relations,’ since
these features can be regarded as consequences of constraints on non-local
correlations. Thus these derivations point towards a picture of quantum
mechanics where the state is not a fundamental entity but instead emerges from
deeper properties of the theory. There do exist several specific proposals for
the ontology of quantum mechanics which eliminate states from the picture -
particularly those approaches where the ontology consists entirely of events,
such as the GRW flash ontology [63] [64] or Kent’s solution to the Lorentzian
quantum reality problem [65] \- but by couching the discussion in terms of
intratheoretic causal structure it is possible to understand some general
consequences of the elimination of states without committing to specific
ontological details.

I will finish by reinforcing that the possibility of a stateless ontology for
quantum mechanics has significant consequences for physics outside quantum
foundations, because the attitude that we take toward the quantum state has a
strong influence on the way we are likely to proceed when working on
extensions of the theory, such as quantum field theory and quantum gravity. If
we think of the quantum state as the fundamental object of quantum theory, it
is natural to deal with gravity by writing down quantum states of spacetime -
that is, by ’quantizing gravity,’ as is done in most of the mainstream
approaches to quantum gravity, including covariant quantum gravity, canonical
quantum gravity and string theory. On the other hand, if we think that the
quantum state is actually just a codification of relations between events or
some other ontological substratum, it seems more natural to regard spacetime
itself as emerging from the underlying substratrum, as in causal set theory.
Which direction we take will depend to a large extent on the beliefs we hold,
implicitly or explicitly, about the underlying intratheoretic causal structure
of quantum mechanics, and so exploring new structures where states are not
fundamental may shed new light on outstanding problems in quantum field theory
and quantum gravity.

###  .1 Proof of theorem 3

###### Proof.

Consider any operational theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$
which obeys the operational Choi-Jamiołkowski isomorphism. Suppose the theory
exhibits Bell-nonlocality. Then for some joint preparation $P_{AB}$ and some
pair of measurements $M_{A}^{0},M_{A}^{1}$ on $S_{A}$ and some pair of
measurements $M_{B}^{0},M_{B}^{1}$ on $S_{B}$, the CHSH quantity satisfies:

$\mathscr{B}_{AB}(P_{AB},M_{A}^{0},M_{A}^{1},M_{B}^{0},M_{B}^{1})>2$

From the operational Choi-Jamiołkowski isomorphism, there exist ensemble
preparations $P^{0},P^{1}$ and a transformation $T$ which give rise to the
same probability distributions, and therefore

$\mathscr{B}_{AB}(P^{0},P^{1},T,M_{B}^{0},M_{B}^{1})>2$

By stipulation (recall section 5.1), I can always combine the transformation
$T$ with the measurements $P^{0},P^{1}$ to produce new measurements
$N^{0},N^{1}$, thus there exist measurements $N^{0},N^{1}$ such that
$\mathscr{B}_{AB}(P^{0},P^{1},\mathbb{I},N^{0},N^{1})>2$.

Now suppose the theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ has a
universal broadcasting map, i.e. a channel which takes any given operational
state to two perfect copies of the same operational state. This channel may be
represented as two copies of the identity transformation $\mathbb{I}$ acting
on the input state.

Let us apply this universal broadcasting map to the result of an ensemble
preparation $P$, producing two copies of the input state on two systems
$S_{B},S_{C}$. I may then apply measurements $N_{B},N_{C}$, giving rise to a
joint distribution $p_{P;\mathbb{I};N_{B}}$ over the outcome of the ensemble
preparation and the outcome of the measurement $N_{B}$, and another joint
distribution $p_{P;\mathbb{I};N_{C}}$ over the outcome of the ensemble
preparation and the outcome of the measurement $N_{C}$, and hence a joint
distribution $p_{P;N_{B},N_{C}}$ over the outcome of the ensemble preparation
and the two outcomes of the two measurements.

From the operational Choi-Jamiołkowski isomorphism, it follows that there
exists a joint preparation $P_{ABC}$ of three systems $S_{A},S_{B},S_{C}$ and
a measurement $N_{A}$ on system $S_{A}$ such that if I perform $P_{ABC}$ and
then the measurements $N_{A},N_{B},N_{C}$ on the three systems respectively,
the joint distribution $p_{P_{ABC};N_{A}N_{B}N_{C}}$ over the measurement
outcomes is identical to the distribution $p_{P;N_{B},N_{C}}$.

Thus consider applying the universal broadcasting map to either of the two
ensemble preparations $P^{0},P^{1}$ and then choosing from the set of
measurements $N^{0},N^{1}$ for the systems $S_{B},S_{C}$. I have that:

$\mathscr{B}_{AB}(P^{0},P^{1},\mathbb{I},N^{0},N^{1})>2$
$\mathscr{B}_{AC}(P^{0},P^{1},\mathbb{I},N^{0},N^{1})>2$

It follows that there exists a preparation $P^{\prime}_{ABC}$ such that if I
perform $P_{ABC}^{\prime}$ and then choose from the measurements $N^{0},N^{1}$
for the systems $S_{A},S_{B},S_{C}$, I have:

$\mathscr{B}_{AB}(P_{ABC}^{\prime},N_{A}^{0},N_{A}^{1},N_{B}^{0},N_{B}^{1})>2$
$\mathscr{B}_{AC}(P_{ABC}^{\prime},\mathbb{I},N_{A}^{0},N_{A}^{1},N_{C}^{0},N_{C}^{1})>2$

And therefore:

$\mathscr{B}_{AB}(P_{ABC}^{\prime},N_{A}^{0},N_{A}^{1},N^{0},N^{1})^{2}+\mathscr{B}_{AC}(P_{ABC}^{\prime},\mathbb{I},N_{A}^{0},N_{A}^{1},N^{0},N^{1})^{2}>8$

Thus the monogamy of correlations is necessarily violated in the theory
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$.

∎

###  .2 Proof of theorem 4

###### Proof.

Consider an operational theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$ which
obeys the operational Choi-Jamiołkowski isomorphism and the no-signalling
principle.

Within such a theory, suppose we perform a joint preparation $P_{AB}$ for two
distinct systems $A,B$, then perform one measurement from a set of $n$
measurements $\\{M_{i}\\}$ on system $A$, and a fixed measurement $F_{b}$ on
system $B$.

It follows from the operational Choi-Jamiołkowski isomorphism that there
exists a set of $n$ ensemble preparations $\\{P_{i}\\}$ on a system $S$ and a
channel $T$ such that if we perform the preparation $P_{i}$ on $S$, then input
$S$ to the channel $T$, then perform measurement $F_{b}$ on the product of the
channel, each of the resulting distributions $p_{P_{i};T;F_{b}}$ is identical
to the corresponding distribution $p_{P_{AB};M_{i}F_{b}}$. Thus each $P_{i}$
is associated with a probability distribution $p^{i}(x)$ such that
$p^{i}(x)=p^{A}(M^{x}_{i}|M_{i},P_{AB})$.

Since the theory is no-signalling, there is no choice of measurement $F_{b}$
on system $B$ which gives any information about which measurement $M_{i}$ was
performed on system $A$ in the former scenario, and therefore there is no
choice of measurement $F_{b}$ which gives any information about which ensemble
preparation $P_{i}$ was chosen in the latter scenario. This means that any one
of the preparation procedures in $\\{P_{i}\\}$, followed by the transformation
$T_{AB}$, gives rise to the same operational state, which I denote by $\chi$.
I will henceforth denote by $P_{i}^{\prime}$ the composite preparation
procedure obtained by performing the procedure $P_{i}$ followed by the
transformation $T_{AB}$.

Let us now write down an ontological model for this operational theory.

Each ensemble preparation $P_{i}$ is associated with a set of preparations
$P_{i}^{x}$ each being chosen with probability $p^{i}(x)$. Each preparation
$P_{i}^{x}$ is associated with a probability distribution
$\mu(\lambda|P_{i}^{x})$ in the ontological model, and the result of
performing a preparation $P^{i}_{x}$ and then applying the transformation $T$
is also associated with a probability distribution,
$\mu(\lambda|T,P^{i}_{x})$141414Note that I do not assume that there exists a
unique column-stochastic matrix in the ontological model which represents the
transformation $T$ \- if there is more than one such transformation, I simply
use any one which produces the correct statistics for this particular scenario
- and therefore I need not assume that the theory is transformation
contextual.. Thus each of the ensemble preparations $P_{i}\in\\{P_{i}\\}$
gives rise to a representation of $\chi$ as a probability distribution over
ontic states:

$p(\lambda|\chi)=\sum_{x}p^{i}(x)\mu(\lambda|T,P^{i}_{x})$

$\hskip
36.98857pt=\sum_{x}p^{A}(M^{x}_{i}|M_{i},P_{AB})\mu(\lambda|T,P^{i}_{x})$

For any $M_{i}\in\\{M_{i}\\}$, the joint probability distribution
$p_{P_{AB};M_{i};F_{b}}$ over the outcomes of the measurements on the systems
$A,B$ after the preparation $P_{AB}$ can be written as follows151515Note that
I do not assume that there exists a unique set of response functions
$p(F^{y}_{b}|F_{b},\lambda)$ for the measurement $F_{b}$ \- if there is more
than one such response function, we can simply use any one which produces the
correct statistics for this particular scenario - and therefore I need not
assume that the theory is measurement non-contextual.:

$p(M_{i}^{x},F_{b}^{y})_{P_{AB},M_{i},F_{b}}=p^{A}(M^{x}_{i}|M_{i},P_{AB})p(F^{y}_{b}|M^{x}_{i},M_{i},F_{b},P_{AB})\\\
\hskip 93.89409pt=p^{A}(M^{x}_{i}|M_{i},P_{AB})p(F^{y}_{b}|T,P^{i}_{x})\\\
\hskip
93.89409pt=p^{A}(M^{x}_{i}|M_{i},P_{AB})\sum_{\lambda}\xi(F^{y}_{b}|F_{b},\lambda)p(\lambda|T,P^{i}_{x})\\\
$ (3)

Now suppose the operational theory $(\mathcal{P},\mathcal{M},\mathcal{T},p)$
is not preparation contextual. This means that it is possible to choose the
ontological model in such a way that all representations of $\chi$ are
identical, so there exists a unique probability distribution $p(\lambda)$ such
that for any $i\in\\{1,2,...n\\}$ I have:

$\sum_{x}p^{A}(M^{x}_{i}|M_{i},P_{AB})p(\lambda|T,P^{i}_{x})=p(\lambda)$

Now for any $i\in\\{1,2...n\\}$ let us define:

$\xi(M^{x}_{i}|M_{i},\lambda):=\frac{p^{A}(M^{x}_{i}|M_{i},P_{AB})p(\lambda|T,P^{i}_{x})}{p(\lambda)}$

Note that $\forall
i\quad\sum_{x}\xi(M^{x}_{i}|M_{i}\lambda)=\sum_{x}\frac{p^{A}(M^{x}_{i}|M_{i},P_{AB})p(\lambda|T,P^{i}_{x})}{p(\lambda)}=\frac{p(\lambda)}{p(\lambda)}=1$,
and since every $\xi(M^{x}_{i}|M_{i},\lambda)$ is nonnegative, it follows that
$\forall i,x\quad\xi(M^{x}_{i}|M_{i},\lambda)\in[0,1]$. These are the
sufficient conditions for a function to be a valid response function of an
ontological model, so $\xi(M^{x}_{c}|M_{a},\lambda)$ is a valid response
function.

The joint probability distribution over the outcomes of the measurements on
the systems $A,B$ after the preparation $P_{AB}$ can then be written as
follows:

$p(M_{i}^{x},F_{b}^{y})_{P_{AB},M_{i},F_{b}}=\sum_{\lambda}\xi(F^{y}_{b}|F_{b},\lambda)\xi(M^{x}_{i}|M_{i},\lambda)p(\lambda)\\\
$ (4)

Thus any joint probability distributions within the theory
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$ are factorizable, and it is a well-
known result[66] that measurements described by factorizable probability
distributions cannot satisfy $\mathscr{B}_{AB}>2$, so the theory
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$ cannot exhibit Bell nonlocality.

∎

###  .3 Proof of theorem 5

###### Proof.

Let $M_{1},M_{2},M_{3}\in\mathcal{P}$ be a set of three orthogonal
measurements for the two-dimensional subtheory
$(\mathcal{P},\mathcal{M},\mathcal{T},p)$, each having two outcomes which I
label by $0,1$.

Let $P_{a}\in\mathcal{P}$ be a preparation which maximizes
$p(0|M_{1},P_{a})+p(0|M_{2},P_{a})$, with the maximization performed over all
preparations in $\mathcal{P}$. Let us say that $p(0|M_{1},P_{a})=i$,
$p(0|M_{2},P_{a})=j$ and $p(0|M_{3},P_{a})=k$ Since I can choose
$p(0|M_{1},P),p(0|M_{2},P),p(0|M_{2},P)$ to be the set of three parameters
which suffice to fix the measurement outcomes for any other measurement
performed after $P$, and by definition, for any valid set of values of these
parameters, there exists a preparation $P\in\mathcal{P}$ described by those
parameters, it follows that there exist preparations $P_{b},P_{c},P_{d}$ with
outcome probabilities as follows:

| $P_{a}$ | $P_{b}$ | $P_{c}$ | $P_{d}$
---|---|---|---|---
$p(0|M_{1})$ | $i$ | $1-i$ | $i$ | $1-i$
$p(0|M_{2})$ | $j$ | $1-j$ | $1-j$ | $j$
$p(0|M_{3})$ | $k$ | $k$ | $k$ | $k$

Suppose Alice performs either preparation $P_{a}$ or $P_{b}$, chosen uniformly
at random, and then sends the resulting system to Bob. If Bob then performs
the measurement $M_{1}$, he will obtain the outcome $0$ with probability
$\frac{1}{2}$ and if Bob then performs the measurement $M_{2}$, he will obtain
the outcome $0$ with probability $\frac{1}{2}$. Likewise, if Alice performs
$P_{c}$ or $P_{d}$ chosen uniformly at random and then sends the system to
Bob, if Bob performs the measurement $M_{1}$ Bob will obtain the outcome $0$
with probability $\frac{1}{2}$ and if Bob then performs the measurement
$M_{2}$, he will obtain the outcome $0$ with probability $\frac{1}{2}$.
Clearly the probabilities for the outcomes of measurement $M_{3}$ are also the
same for both decompositions. Since the probabilities
$p(0|M_{1},P),p(0|M_{2},P),p(0|M_{2},P)$ suffice to fix the measurement
outcomes for any other measurement performed after $P$, this implies that
these two probabilistic preparations give rise to the same measurement
statistics for any subsequent measurements, and hence both decompositions
produce the same operational state, which I denote by $\chi$.

It follows from the operational Choi-Jamiołkowski isomorphism that there
exists a preparation $P_{AB}$ and a measurement $S_{0}$ such that when $S_{0}$
is performed on one subsystem $A$ of a bipartite system $AB$ jointly prepared
using preparation $P_{AB}$ with probability $\frac{1}{2}$ I obtain outcome $0$
and the resulting operational state of subsystem $B$ is the same as the
operational state produced by preparation $P_{a}$, and with probability
$\frac{1}{2}$ I obtain outcome $1$ and the resulting operational state of
subsystem $B$ is the same as the operational state produced by preparation
$P_{b}$. Likewise, there exists a measurement $S_{1}$ defined similarly for
preparations $P_{c},P_{d}$.

Now consider a CHSH game in which Alice and Bob are each given input bits
$c,d$ and are required to produce outcomes $a,b$ such that $a\oplus b=cd$.
Alice and Bob may employ the following strategy:

  1. 1.

Before the start of the game, Alice and Bob perform the preparation $P_{AB}$,
and then Alice takes subsystem $A$ and Bob takes subsystem $B$.

  2. 2.

When Alice is given input $c\in\\{0,1\\}$, she performs measurement $S_{c}$
and then returns her measurement outcome.

  3. 3.

When Bob is given input $d\in\\{0,1\\}$, he performs measurement $M_{d+1}$ and
then returns his measurement outcome.

If the inputs are chosen uniformly at random, the probability that Alice and
Bob win the game using this strategy is:

$\frac{1}{4}\sum_{c,d,a}p(a|c)p(x(acd)|acd)$

where $p(a|s)$ the probability that Alice obtains outcome $a$ when she
performs measurement $s$, $x(acd)$ is defined such that for any $a,c,d$,
$a\oplus x(a,c,d)=cd$, and $p(q|acd)$ is the probability that Bob obtains
outcome $q$ when Alice performs measurement $S_{c}$ and obtains outcome $a$
and Bob performs measurement $M_{d+1}$.

$S_{0}$ and $S_{1}$ are defined such that $\forall a\ \forall
s\;p(a|s)=\frac{1}{2}$. Moreover, by the definition of $S_{0}$, if Alice
performs measurement $S_{0}$ and obtains outcome $0$, the resulting
probabilities $p(q|0,0,d)$ for the outcomes of Bob’s measurement $M_{d+1}$ are
the same as if Alice were to simply perform preparation $P_{a}$ and hand the
prepared system over to Bob. That is, $p(0|0,0,0)=i$ and $p(0|0,0,1)=j$.
Similar relationships hold for other possible values of $a$ and $c$, and using
the table above, it is straightforward to obtain:

$\forall a,c\quad\sum_{d=1}^{2}p(x(a,c,d)|a,c,d)=i+j$

Hence the probability that Alice and Bob win the game using this strategy is
$\frac{1}{2}(i+j)$.

Ex hypothesi the operational theory satisfies information causality. It is
known that information causality implies the Tsirelson bound[11], and the
Tsirelson bound implies that the maximum probability of winning this CHSH game
is $1+\frac{1}{\sqrt{2}}$; hence $i+j\leq 1+\frac{1}{\sqrt{2}}$.

Since we chose $P_{a}$ to be the preparation which maximizes
$p(0|M_{1},P_{a})+p(0|M_{2},P_{a})$, it follows that for any preparation $P$,
$p(0|M_{1},P)+p(0|M_{2},P)\leq 1+\frac{1}{\sqrt{2}}$.

Then note that for any $m\in\\{0,1\\},n\in\\{0,1\\}$ I can make a similar
argument by starting from the requirement that $P_{a}$ is the state which
maximizes $p(m|0)+p(n|1)$, and hence:

∎

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