Correlations constrained
by composite  measurements

Łukasz Czekaj
International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
Ana Belén Sainz
International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
John H. Selby
International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
Michał Horodecki
International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland

How to understand the set of correlations admissible in nature is one outstanding open problem in the core of the foundations of quantum theory. Here we take a complementary viewpoint to the device-independent approach, and explore the correlations that physical theories may feature when restricted by some particular constraints on their measurements. We show that demanding that a theory exhibits a composite measurement imposes a hierarchy of constraints on the structure of its sets of states and effects, which translate to a hierarchy of constraints on the allowed correlations themselves. We moreover focus on the particular case where one demands the existence of an entangled measurement that reads out the parity of local fiducial measurements. By formulating a non-linear Optimisation Problem, and semidefinite relaxations of it, we explore the consequences of the existence of such a parity reading measurement for violations of Bell inequalities. In particular, we show that in certain situations this assumption has surprisingly strong consequences, namely, that Tsirelson's bound can be recovered.

§ INTRODUCTION

Bell nonclassicality is a well-known phenomenon featured by quantum theory, and attests that correlations observed in nature are not always compatible with a classical common cause shared among the distant wings of an experiment [1]. That is, non-classical common causes are necessary to explain our observational data [2]. Bell's theorem not only teaches us a valuable lesson about the foundational aspects of nature, but also underpins a variety of current technological applications. For example, non-classical correlations enable cryptographic applications, such as key distribution [3, 4, 5, 6, 7, 8] and  randomness generation [9, 10, 11, 12], and provide an information-theoretic advantage in other families of so-called non-local games [13, 14, 15].

Understanding quantum correlations – in particular their limitations – is therefore an important open problem within quantum information theory. Research on these lines has recently been carried out within the device-independent formalism, that is, where the only information used to reason about nature are the classical variables that denote measurement choices and their outcomes, together with the observed outcome statistics. Within this paradigm, quantum correlations are studied “from the outside”, by exploring the constraints that physical or information-theoretical principles impose on the observed correlations [16, 17, 18, 19, 20, 21, 22]. In the device-independent framework, hence, such proposed constraints are therefore formulated at the level of the correlations themselves.

In this work we take a complementary viewpoint to the problem of characterising quantum correlations, by examining the possible correlations that may arise when constraints are imposed on the underlying physical theory. From this perspective, hence, one asks how various elements of the physical theory constrain or enable particular correlations. The particular objects we are interested in here are the measurements that the theory may feature. On the one hand, it is well known that the theory known colloquially as `Boxworld' [23], which was formulated in order to realise arbitrary no-signalling correlations, only features local measurements and wirings thereof. That is, Boxworld does not display entangled measurements. This is in contrast to quantum theory, where entangled measurements are ubiquitous – you may for instance think of the so-called Bell measurements. A natural question then arises: is there any relationship between the types of measurements a theory features and the correlations it may produce.
Even seemingly simple at first sight, this question is far from trivial: by enlarging the set of allowed measurements one necessarily needs to shrink down the set of allowed states, since the state space featured by a given physical theory is constrained by the dual space of the effects[The so-called effects in a physical theory may be thought of as its dichotomic measurements.] space. How the set of allowed correlations (measurements on states) changes in consequence is therefore not straightforward.

Progress on this question was made in Ref. [24], where it was shown that demanding the existence of a particular entangled effect would constrain the correlations admissible in a bipartite Bell scenario to those realisable by an entangled pair of qubit quantum systems. That is, by demanding that the theory features a particular entangled measurement, it was shown that the allowed correlations in the so-called Clauser-Horne-Shimony-Holt (CHSH) scenario [25] was indeed the set of quantum ones.

In this paper we explore what types of constraints the existence of bipartite effects impose on the possible correlations that a theory may feature. The framework we use to describe the underlying physical theory is that of General Probabilistic Theories (GPTs) [26, 23, 27, 28, 29, 30, 31, 32, 33, 34] . First, we take a compositional perspective and show how the existence of one (arbitrary) bipartite effect imposes not one but an infinite hierarchy of constraints which must be satisfied by the states and effects of the GPT. This hierarchy of constraints on bipartite states immediately translates to a hierarchy of constraints on the correlations realisable within the theory. Inspired by Ref. [24], we then consider a particular setup where we demand that there exists a measurement in the GPT which can measure the parity of fiducial measurements (or a subset thereof), which we call a (partial) parity reading measurement. We say that the observables which appear in such partial parity reading measurement are parity readable observables.
We then define an Optimisation Problem that computes the maximum violation of a Bell inequality by the corresponding GPT, provided that such a parity reading measurement exists. Such an Optimisation Problem provides a way to characterise the set of correlations allowed by such a GPT. The solution to this optimisation problem is, however, computationally complex, given that the problem itself is polynomial on the optimised variables. We hence present a series of relaxations that upper-bound the solution to the Optimisation problem. We finish by applying our techniques to a variety of Bell inequalities, and discussing the necessity of the Local Tomography assumption.

Inspired by our results, we moreover formulate a conjecture:

Under the assumption of local tomography,
the local observables that are parity readable satisfy Tsirelson's bound, i.e., they cannot violate Bell inequalities better than quantum mechanics does.

Our numerical experiments support
this conjecture. In particular
we have verified that in the simplest case of subsystems which are completely
described by two observables,
the CHSH inequality cannot be violated. For larger systems, we have partial numerical results that did not falsify the above conjecture. Moreover, we have a counterexample demonstrating the necessity of the assumption of local tomography within the conjecture. That is, if we have a GPT which does not satisfy local tomography, then it is possible to create a PR box using observables which are parity readable.

§ DESCRIPTIVE SUMMARY OF THE RESULTS

Suppose that two parties, the ubiquitous Alice and Bob, each have three binary observables, $X$, $Y$, and $Z$, that they can measure on some shared system. Moreover, suppose that there exists some joint measurement that they could perform on their joint system if they were to get together, whose outcome would determine the parity of $XX$ and $ZZ$. That is, the joint measurement does not necessarily reveal the values that they would have obtained had they measured $X$ and $Z$ individually, but just whether or not their $XX$ and $ZZ$ measurements would have been correlated or not.[Such is the case in quantum theory, where $X$ and $Z$ are incompatible measurements, but their parity can nonetheless be measured by performing a suitable coarse graining of a standard Bell measurement.] We say that such observables are parity readable, and illustrate this idea in Fig. <ref>.

Measuring parity of local observables.

Reading parity of two pairs of observables: $XX$ and $ZZ$.

Reading parity of three pairs of observables $XX$, $ZZ$ and $YY$.

The idea of a measurement that reads the parity of the parity-readable observables(denoted by PRM): a composite measurement that measures parities of several pairs of local observables in one go.

In this manuscript, we consider the impact that parity readability has on the correlations that can be generated in a Bell scenario when measuring parity readable observables. For example, is it possible to create PR-box correlations via parity readable observables?

It is clear that, even within the standard quantum mechanical formalism, this imposes a restriction on the correlations which can be observed – not all observables are parity readable, and some correlations can only be achieved by those that are not. However, what about if we go beyond quantum mechanics?

We conjecture that, within this landscape of parity readable observables, quantum theory is always optimal. That is, any correlation that can be generated by parity readable observables, independently of which underlying physical theory they belong to, can also be generated by parity reading observables within quantum theory.

If this conjecture is true, then this would be in stark contrast to the landscape of arbitrary observables, in which there are correlations which cannot be realised within our quantum world. It would therefore show, for the first time, a way in which quantum theory is an optimal physical theory for an information theoretic task.

We explore this conjecture by utilising techniques and insight coming from the field of generalised probabilistic theories. Ultimately, however, we formulate the question as a hierarchy of convex optimisation problems which can be tackled using standard numerical methods. Due to the computational cost of these numerical methods, however, we are only able to test our conjecture within certain simple scenarios.

In particular, we consider scenarios in which each party has at most three observables $X$, $Y$ and $Z$, and in which either two or three of these are parity readable. We then test our conjecture on various Bell inequalities, including the CHSH [25], AMP [35], and AQ [36] inequalities. These all provide evidence either in support of, or at least, not in conflict with, our conjecture. It is clear, however, that further development of both the numerical methods, as well as analytical convex optimisation techniques, are necessary to explore this conjecture further.

The main technique that we develop and apply here, relies on demanding the parity readable observables to yield valid probabilities when applied both to a composite state as well as to products of steered states that can be generated from it (see Figs. <ref>a, <ref>b.). These constraints are indeed phrased as positivity conditions on variables we optimise over. However, to capture the full power of the constraints that these parity readable observables impose, one needs to take into account infinitely many conditions (examples of which are presented in Figs. <ref>c and
<ref>d), described in the article, and which deserve further exploration.

The existence of a measurement that reads the parity of the parity-readable observables, implies an infinite hierarchy of positivity constraints. For instance, the probabilities of outcomes should be positive on a) a single copy of a state b) all pairs of  products of steered states that can be obtained from the original state.

§ WARM-UP: NON-EXISTENCE OF POPESCU-ROHRLICH CORRELATIONS FOR PARITY MEASURABLE $X$ AND $Z$ OBSERVABLES.

While the general problem
of finding a  bound for Bell inequalities  for parity readable observables is a complex one (as we will see further in this paper), one can relatively easily show that
the CHSH inequality cannot achieve its maximal algebraic bound. Namely, we shall show that parity readable observables cannot exhibit so called Popescu-Rohrlich correlations [16].

In this section we shall present such reasoning, as a simple warm-up exercise in anticipation of the
rest of the paper. In this warm-up we will take a device-independent approach, whilst in the remainder of the paper we will utilise the language of Generalised Probabilistic Theories (GPTs)
and, in particular, we will express the question using a formal diagrammatic language. Full justification of some formulas will be found later in the paper. For self-consistency of the main part of the paper, we will repeat there some definitions used here.

§.§ States

The so called PR-box [16] is defined as the set of conditional probabilities:

\begin{equation}
\left \{\begin{array}{lll}
\frac12 &\text{if} & a\oplus b= xy  \\
0 & \text{else} &
\end{array}\right. \,,
\end{equation}

where $x,y\in\{0,1\}$ are inputs, and $a,b\in\{0,1\}$ are outputs. The observables $X,Z$ from the previous section are now encoded by the classical variables $x=0,1$ for Alice, and $y=0,1$ for Bob.
The PR-box is a maximally nonlocal non-signaling box, since it violates the CHSH inequality up to its maximal algebraic value. Moreover, PR-box correlations have the feature that the pairs of observables $XX$, $XZ$, and $ZX$ are perfectly correlated, while the pair $ZZ$ is perfectly anticorrelated.

The principles of Local Tomography and No Signaling allow one to characterise any bipartite state $s$ by means of the following

\begin{equation}
\label{eq:ps-matrixform-warmup}
\mathbf{p_s}=\left[ \begin{array}{ccc}
p_s(00|00)&p_s(00|01)   & p_s^{\mathrm{(1)}}(0|0)   \\
p_s(00|10)& p_s(00|11)  &  p_s^\mathrm{(1)}(0|1)  \\
p_s^\mathrm{(2)}(0|0)& p_s^\mathrm{(2)}(0|1)  & 1
\end{array}
\right]\,,
\end{equation}

where $(i)$ specifies the party (i.e., (1) for Alice and (2) for Bob).

Since a PR box has perfect correlations or anticorrelations for each pair of observables, it is natural that the steered states obtained from it
are all the states that have well defined value for both observables. Thus, we have that, for each party, there are  four steered state, the state $s_1$ with $X=0,Z=0$, the state $s_2$ with   $X=0,Z=1$,
the state $s_3$ with $X=1,Z=0$ and the state $s_4$ with $X=1,Z=1$. These four states are depicted in Fig. <ref>.

Steered states coming from PR box.

We shall present here  the four pairs of steered states which we will use in the proof.
These are products of all four steered states on Bob's site and a single fixed state on Alice's side, namely the state with $X_A=Z_A=1$, as is depicted Fig. <ref>.

Selected pairs of steered states.

We denote these four states as:

\begin{align}
\label{eq:s}
s_{41} = s_4^A \otimes s_1^B, \quad
s_{42} = s_4^A \otimes s_2^B, \quad
s_{43} = s_4^A \otimes s_3^B, \quad
s_{44} = s_4^A \otimes s_4^B.
\end{align}

In the above matrix notation these are characterised by:

\begin{align}
\label{eq:s-matrix}
s_{41}= \left[\begin{array}{ccc}
0& 0 & 0\\
0& 0 & 0\\
1& 1 & 1
\end{array}\right] \, ,
\quad
s_{42}= \left[\begin{array}{ccc}
0& 0 & 0\\
0& 0 & 0\\
1& 0 & 1
\end{array}\right]\, ,
\nonumber \\
s_{43}= \left[\begin{array}{ccc}
0& 0 & 0\\
0& 0 & 0\\
0& 1 & 1
\end{array}\right]\, ,
\quad
s_{44}= \left[\begin{array}{ccc}
0& 0 & 0\\
0& 0 & 0\\
0& 0 & 1
\end{array}\right].
\end{align}

§.§ Parity Reading Measurement.

Now we consider the measurement that measures the parities of $XX$ and $ZZ$ in a single shot, that is, $X$ and $Z$ are parity-readable observables. Specifically, this parity reading measurement (PRM) outputs a pair of bits: the first one reports the parity of $XX$ and the second one reports the parity of $ZZ$.
This is expressed by  the following pair of conditions:

\begin{align}
\label{eq:prm}
R_{XX}(\mathbf{p_s}) &\equiv p_s(00|\PRM) + p_s(01|\PRM) &= p_s(00|00)+ p_s(11|00)\,, \\
R_{ZZ}(\mathbf{p_s}) &\equiv p_s(00|\PRM) + p_s(10|\PRM) &= p_s(00|11)+ p_s(11|11).
\end{align}

where we denote by
$p_s(rq|\PRM)$ the probability of obtaining the pair of outcomes $rq$ when measuring the PRM on state $\mathbf{p_s}$.
Notice that $R_{XX}$ ($R_{ZZ}$) have the interpretation of the probability of $XX$ ($ZZ$) being correlated (i.e., the probability of obtaining the same outcomes by Alice and Bob). In addition, note that the probabilities of the outcomes of the PRM satisfy the normalisation condition:

\begin{equation}
\label{eq:norm}
\end{equation}

We therefore have a system of three linear equations with four unknown quantities
$p_s(rq|\PRM)$. Hence, a solution may be found with one quantity remaining an independent variable, for example the expression:

\begin{align}
\label{eq:c}
\end{align}

Due to local tomography, the probabilities of the outcomes of any measurement for a particular state  can be written as a linear combination of state parameters, $\mathbf{p_s}$. Hence,  as $C$ is a linear combination of such probabilities, it can be computed via:

\begin{equation}
\label{eq:c2}
C(\mathbf{p_s})= \mathcal{C}\cdot \mathbf{p_s} \,,
\end{equation}

where $\cdot$ denotes Frobenius matrix product, and
$\mathcal{C}$ is the matrix of, at the moment, unspecified parameters describing the PRM:

\begin{align}
\label{eq:c-matrix}
\mathcal{C}=
\left[ \begin{array}{ccc}
c_{11}& c_{12} & c_{13}\\
c_{21}& c_{22} & c_{23}\\
c_{31}& c_{32} & c_{33}
\end{array}\right].
\end{align}

Combining  Eqs. (<ref>), (<ref>), (<ref>) and (<ref>), we obtain the formulas  for computing the probabilities of PRM outcomes,  expressed in terms of the state parameters, as well as the free parameter $\mathcal{C}$:

\begin{align}
\label{eq:prm2}
p_s(00|\PRM)&=\frac14\left(2 R_{XX}(\mathbf{p_s}) + 2 R_{ZZ}(\mathbf{p_s}) +\mathcal{C}\cdot \mathbf{p_s} - 1 \right) \,, \nonumber \\
p_s(01|\PRM)&=\frac14\left(2 R_{XX}(\mathbf{p_s}) - 2 R_{ZZ}(\mathbf{p_s}) -\mathcal{C}\cdot \mathbf{p_s} + 1 \right) \,, \nonumber \\
p_s(01|\PRM)&=\frac14\left(-2 R_{XX}(\mathbf{p_s}) + 2 R_{ZZ}(\mathbf{p_s}) -\mathcal{C}\cdot \mathbf{p_s} + 1 \right) \,, \nonumber \\
p_s(11|\PRM)&=\frac14\left(-2 R_{XX}(\mathbf{p_s}) - 2 R_{ZZ}(\mathbf{p_s}) +\mathcal{C}\cdot \mathbf{p_s} + 1 \right) \,.
\end{align}

§.§ Proof of nonexistence of the PR-box.

We shall now argue that, for any arbitrary choice of $\mathcal{C}$, the probability of at least one $\PRM$ output will necessarily be negative on some of the pairs of steered states.
This proves that parity readable observables cannot feature PR-box correlations, and hence cannot violate  the CHSH inequality up to its maximal algebraic bound.

We shall first impose positivity of $p_s(qr|\PRM)$
for each of the four states in Eq.  (<ref>).
Note that, for the state $s_{44}$, by definition, both $XX$ and $ZZ$ are perfectly correlated, since $X_A=X_B=1$ and $Z_A=Z_B=1$. Hence, $R_{XX}(s_{44})=R_{ZZ}(s_{44})=1$ for the state $s_{44}$. An analogous reasoning for the remaining three states yields:

\begin{align}
R_{XX}(s_{41})&=0, \quad  R_{ZZ}(s_{41})=0, \nonumber \\
R_{XX}(s_{42})&=0, \quad  R_{ZZ}(s_{42})=1, \nonumber \\
R_{XX}(s_{43})&=1, \quad  R_{ZZ}(s_{43})=0, \nonumber \\
R_{XX}(s_{44})&=1, \quad  R_{ZZ}(s_{44})=1.
\end{align}

Of course, we might alternatively compute these values from the matrix form of the states, and the definition of $R_{XX}$, $R_{ZZ}$. Inserting them into Eq.(<ref>) for each of the four states, yields the following conditions for the positivity of PRM probabilities:

\begin{align}
p_{s_{41}}(qr|\PRM) &\geq 0 \quad \Leftrightarrow \quad \mathcal{C}\cdot s_{14} =1 \,, \nonumber \\
p_{s_{42}}(qr|\PRM) &\geq 0 \quad \Leftrightarrow \quad \mathcal{C}\cdot s_{24} =-1 \,, \nonumber \\
p_{s_{43}}(qr|\PRM) &\geq 0 \quad \Leftrightarrow \quad \mathcal{C}\cdot s_{34} =-1 \,, \nonumber \\
p_{s_{44}}(qr|\PRM) &\geq 0 \quad \Leftrightarrow \quad \mathcal{C}\cdot s_{44} =1.
\end{align}

We then use the matrix form of our states of Eq. (<ref>) and the form of $\mathcal{C}$ of Eq. (<ref>) and rewrite these conditions as:

\begin{align}
c_{31}+c_{32}+c_{33}=1,\quad c_{31}+c_{33}=-1, \quad c_{32}+c_{33}=-1, \quad c_{33}=1.
\end{align}

We see that this set of equalities does not have any real solutions. We therefore conclude that there does not exist a Parity Reading Measurement whose outcomes  give legitimate probabilities for the above four pairs of steered states. Since the existence of a PR-box allows for such steered states, we conclude that existence of Parity Reading Measurement excludes PR-box correlations.

The remainder of this work explores this curious property in more detail. In particular: on the one hand, can we go beyond simply ruling out the PR-box and see how PRMs constrain the set of correlations; and, on the other hand, can we go beyond the assumptions involved in this derivation, namely, that of local tomography and that there are simply two observables per party.

§ GENERALISED PROBABILISTIC THEORIES

In order to explore the previously presented questions, we will work within the framework of
generalised probabilistic theories (GPTs) [26, 23]. This framework was developed in order to be able to describe essentially arbitrary conceivable theories of nature – taking quantum and classical theory as just two particular points within a broad landscape of potential physical theories. The GPT framework is based on the idea that, ultimately, the way that we characterise physical devices is by the probabilities that they give rise to in experiments. From this simple observation, one can build a rich mathematical structure which any GPT must have.

GPTs have already proved a vital tool in the study of computation [37, 38, 39, 40, 41, 42, 43, 44, 45] and cryptography [46, 47, 48, 49, 50, 23, 51] beyond quantum theory. Moreover, recently tools from convex optimisation theory have been used to gain new insight into GPTs [47, 46, 52, 53, 54, 48]. In particular, the generalisation of quantum theory to GPTs  is  analogous to the generalisation  from semi-definite to conic programmes. These optimisation tools will be vital for developing a complete understanding of how the structures of the GPT impact on the realisable correlations of the GPT.

For simplicity of the presentation, in this manuscript we will focus on a particular class of GPTs, namely, those that satisfy the principle of Local Tomography. The majority of our results, such as the formulation of the hierarchy of constraints, however, do not require this principle to hold, hence we will highlight the instances where the assumption is indeed necessary.

In this paper we take a categorical approach to tomographically local GPTs. This is an intrinsically compositional approach, which allows us to describe arbitrary experimental scenarios. Moreover, the diagrammatic representation in terms of string diagrams, which comes from this approach, provides an intuitive way to reason about these complex situations. We provide a brief technical introduction to the formalism in Appendix <ref>, and refer the reader  to Refs. [34, 33, 32, 55, 56, 57] for more extensive introductions to these tools.

§.§ Constraints on states and effects

We can see how the state and effect spaces constrain one another when demanding that scalars are probabilities.
For some system $V$ (see the Appendix for details on notation), the states can be thought of as vectors $s\in\Omega_V$ living inside $V$, and effects as linear functionals, $e\in \mathcal{E}_V$ living in the dual space $V^*$. Any pair of an effect and a state must satisfy:

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[style=none] (1) at (0, -1.5) ;
[style=none] (2) at (0.75, -0.5) ;
[style=none] (3) at (0, -0.5) ;
[style=none] (4) at (0, 0.5) ;
[style=none] (5) at (-0.75, 0.5) ;
[style=none] (6) at (0, 1.5) ;
[style=none] (7) at (0.75, 0.5) ;
[style=point] (8) at (0, -1) $s$;
[style=copoint] (9) at (0, 1) $e$;
[style=right label] (10) at (0, -0) $V$;

[qWire] (8.center) to (9.center);

∈[0,1]  .
The geometric consequences of this for local and composite states are presented in App. <ref>.

It was noted in Ref. [58], however, that even if a pair of state and effect spaces satisfy the standard constraints discussed in App. <ref> (i.e., Eqs. (<ref>) and (<ref>)), it is not straightforward that they actually define a valid GPT, at least when the No-restriction hypothesis is not assumed (i.e., when it is not required that $\Omega = \mathcal{E}^*$). An important condition that must also be checked, as shown in Theorem 9 of Ref. [58], is that the steered states are also valid states within the theory. That is, any bipartite state $s$ must satisfy:

Diagrams/steerV  ∈Ω_V
and  Diagrams/steerW  ∈Ω_W
for all $e_v\in\mathcal{E}_V$ and $e_w\in\mathcal{E}_W$. This constraint can be interpreted in many forms:

* as a constraint on the bipartite state space, namely, that a bipartite state must lead to valid steered states,

* as a constraint on the local state spaces, namely, it forces them to include all of these steered states as valid local states,

* as a constraint on the local effect spaces, namely, an effect is only allowed if it leads to valid steered states when composed with any bipartite state.

However, we believe that the constraints of Eq. (<ref>) are probably best viewed not from any of these individual perspectives, but instead just as a compatibility condition between local states and effects, and bipartite states.

Similarly, we can consider bipartite effects $e$ and note that these have a similar compatibility condition together with local states and effects:
Diagrams/steerVeffect  ∈ℰ_V
and  Diagrams/steerWeffect  ∈ℰ_W
for all $s_v\in\Omega_V$ and $s_w\in\Omega_W$.

One may be inclined to think that these constraints on bipartite states/effects, steered states, and steered effects, are sufficient to characterise a valid GPT. However, there are only the tip of the iceberg – a whole plethora of further compatibility constraints lie underneath the surface. For example, consider a normalised bipartite state $s$ and a bipartite effect $e$. By taking two copies of each, one should be capable of wiring them as follows and obtain a valid probability:

Diagrams/hier1 ∈[0,1].
In addition, if one takes two copies of $s$ and one of $e$, one should be capable of wiring them as follows and obtain a valid bipartite state:

Other types of compatibility constraints include diagrams like the following, which arise due to symmetry in the special case when all the local systems have the same type:

Diagrams/hiersym∈[0,1],  Diagrams/hiersym2∈[0,1].
One can readily see how these belong to an infinite family of constraints, each featuring the same (but arbitrary) number of normalised bipartite states $s$ and bipartite effects $e$ being connected in this “braided” fashion.

Even if the bipartite states consist of local systems of the same type, it is not necessary that they are symmetric under a swap operation of the local systems. Hence, a different hierarchy of braided-type constraints will arise by requiring consistent probability assignments to diagrams of the form:

\begin{align}\label{eq:braid2}
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\node [style=none] (39) at (6, 2) {$e$};
\node [style=none] (40) at (3.75, 1.5) {};
\node [style=right label] (41) at (5, 1) {$V$};
\node [style=none] (42) at (5, 1.5) {};
\node [style=none] (43) at (3, -1.5) {};
\node [style=none] (44) at (4.25, 1.5) {};
\node [style=none] (45) at (3.75, -1.5) {};
\node [style=right label] (46) at (3, -1.25) {$V$};
\node [style=right label] (47) at (3, 1) {$V$};
\node [style=none] (48) at (3, 1.5) {};
\node [style=none] (49) at (5, -1.5) {};
\node [style=right label] (50) at (5, -1.25) {$V$};
\node [style=none] (51) at (4.25, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [qWire, in=-90, out=90] (1.center) to (0.center);
\draw (5.center) to (7.center);
\draw (7.center) to (6.center);
\draw (6.center) to (5.center);
\draw (11.center) to (10.center);
\draw (10.center) to (8.center);
\draw (8.center) to (11.center);
\draw [qWire, in=90, out=-90, looseness=0.50] (16.center) to (12.center);
\draw [qWire, in=90, out=-90, looseness=0.50] (15.center) to (24.center);
\draw [qWire, in=90, out=-90] (13.center) to (19.center);
\draw (20.center) to (21.center);
\draw (21.center) to (22.center);
\draw (22.center) to (20.center);
\draw (29.center) to (30.center);
\draw (30.center) to (31.center);
\draw (31.center) to (29.center);
\draw (34.center) to (35.center);
\draw (35.center) to (37.center);
\draw (37.center) to (34.center);
\draw (32.center) to (38.center);
\draw (38.center) to (36.center);
\draw (36.center) to (32.center);
\draw [qWire, in=-90, out=90] (43.center) to (42.center);
\draw [qWire, in=90, out=-90] (48.center) to (49.center);
\end{pgfonlayer}
\end{tikzpicture}
& \in [0,1].
\end{align}

In Section <ref> we will see how to formalise these types of hierarchies, and how they can be used to constrain the potential correlations in a GPT.

§.§ Correlations in a GPT

To describe correlations in a GPT we must first introduce classical systems. Here we describe a classical system by classical random variable, which can take values from a set, such as $X,Y,A,B$. We denote these classical systems by thin gray wires (to distinguish them from GPT wires). Correlations, in this formalism, are then viewed as no-signalling stochastic maps, $\mathsf{N}:X\times Y \to A\times B$, between these random variables. Diagrammatically, we denote these no-signalling boxes as:
Diagrams/nsBox ,
which must satisfy the no-signalling constraints:

Diagrams/nsBox1  =  Diagrams/nsBox4     and    Diagrams/nsBox3  =  Diagrams/nsBox2 .
This is equivalent to the standard view of correlations [59] as being described by a conditional probability distribution $\mathsf{Pr}(A,B|X,Y) = \{p(ab|xy)\}_{\{a\in A,b\in B, x\in X, y\in Y\}}$, which can be seen by defining:
p(ab|xy) :=
[style=none] (0) at (-1, -0.5) ;
[style=none] (1) at (-1, -1.75) ;
[style=none] (2) at (-1, 0.5) ;
[style=none] (3) at (-1, 1.25) ;
[style=none] (4) at (1, 1.5) ;
[style=none] (5) at (1, 0.5) ;
[style=none] (6) at (1, -1.75) ;
[style=none] (7) at (1, -0.5) ;
[style=right label] (8) at (-1, -1) $X$;
[style=right label] (9) at (1, -1) $Y$;
[style=right label] (10) at (-1, 0.75) $A$;
[style=right label] (11) at (1, 0.75) $B$;
[style=none] (12) at (-1.5, 0.5) ;
[style=none] (13) at (-1.5, -0.5) ;
[style=none] (14) at (0, 0) $\mathsf{N}$;
[style=none] (15) at (1.5, 0.5) ;
[style=none] (16) at (1.5, -0.5) ;
[style=copoint] (17) at (-1, 1.5) $a$;
[style=copoint] (18) at (1, 1.5) $b$;
[style=point] (19) at (-1, -1.75) $x$;
[style=point] (20) at (1, -1.75) $y$;

[cWire, in=90, out=-90] (0.center) to (1.center);
[cWire, in=-90, out=90] (2.center) to (3.center);
[cWire, in=90, out=-90] (7.center) to (6.center);
[cWire, in=-90, out=90] (5.center) to (4.center);
(13.center) to (12.center);
(12.center) to (15.center);
(15.center) to (16.center);
(16.center) to (13.center);

and checking that the no-signalling conditions of Eqs. (<ref>) are equivalent to the standard no-signalling conditions for the conditional probability distribution. To do so it is useful to note that, for example:

[style=none] (0) at (0, -1.5) ;
[style=none] (1) at (0, -0.25) ;
[style=none] (2) at (0, -1.5) ;
[style=none] (3) at (0, -0.25) ;
[style=upground] (4) at (0, -0.0) ;
[style=right label] (5) at (0, -1.25) $X$;

[cWire] (3.center) to (2.center);

:=   ∑_x∈X

[style=none] (0) at (0, -1.5) ;
[style=none] (1) at (0, -0.25) ;
[style=none] (2) at (0, -1.5) ;
[style=none] (3) at (0, -0.25) ;
[style=copoint] (4) at (0, 0.0) $x$;
[style=right label] (5) at (0, -1.25) $X$;

[cWire] (3.center) to (2.center);

Then, in order to understand the possible correlations in a GPT, it is useful to describe measurements as transformations from a GPT to a classical system, where the choice of measurement is controlled by another classical system. These controlled measurements must satisfy the constraint:

[style=right label] (0) at (1.25, -1.5) $V$;
[style=none] (1) at (0.5, -0.5) ;
[style=none] (2) at (1.25, -1.75) ;
[style=none] (3) at (0.5, 0.5) ;
[style=none] (4) at (-0.75, 0.5) ;
[style=none] (5) at (-0.75, -0.5) ;
[style=none] (6) at (0, -0) $M$;
[style=none] (7) at (-0.25, -0.5) ;
[style=none] (8) at (-0.25, -1.75) ;
[style=none] (9) at (-0.25, 0.5) ;
[style=none] (10) at (-0.25, 1.75) ;
[style=right label] (11) at (-0.25, -1.5) $X$;
[style=right label] (12) at (-0.25, 1.25) $A$;
[style=none] (13) at (1, -0.5) ;
[style=upground] (14) at (-0.25, 2) ;

[qWire, in=90, out=-90, looseness=1.00] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
[cWire, in=90, out=-90, looseness=1.00] (7.center) to (8.center);
[cWire, in=-90, out=90, looseness=1.00] (9.center) to (10.center);
(5.center) to (13.center);
(13.center) to (3.center);

[style=right label] (0) at (0.75, -0.75) $V$;
[style=none] (1) at (0.75, 0.25) ;
[style=none] (2) at (0.75, -1) ;
[style=none] (3) at (-0.75, 0.25) ;
[style=none] (4) at (-0.75, -1) ;
[style=right label] (5) at (-0.75, -0.75) $X$;
[style=upground] (6) at (-0.75, 0.5) ;
[style=upground] (7) at (0.75, 0.5) ;

[qWire, in=90, out=-90, looseness=1.00] (1.center) to (2.center);
[cWire, in=90, out=-90, looseness=1.00] (3.center) to (4.center);

Correlations that can be generated in a Bell experiment are hence of the form:

where the local controlled measurements $M_A$ and $M_B$ (for Alice and Bob respectively) are performed on a bipartite system on state $s$, with local system types $V,W$ in the GPT.
These local measurements are controlled on the input classical variable and have an outcome recorded in the output classical variable.

In any GPT, such a diagram corresponds to a no-signalling stochastic map:

[style=none] (0) at (1, 0.75) ;
[style=none] (1) at (1, -0.5) ;
[style=right label] (2) at (1, -0.25) $W$;
[style=right label] (3) at (-1, -0.25) $V$;
[style=none] (4) at (0, -1) $s$;
[style=none] (5) at (1.75, -0.5) ;
[style=none] (6) at (-1.75, -0.5) ;
[style=none] (7) at (0, -1.75) ;
[style=none] (8) at (-1, 0.75) ;
[style=none] (9) at (-1, -0.5) ;
[style=none] (10) at (-0.5, 0.75) ;
[style=none] (11) at (-1, 1.75) ;
[style=none] (12) at (-2.5, 1.75) ;
[style=none] (13) at (-2.5, 0.75) ;
[style=none] (14) at (-1.75, 1.25) $M_A$;
[style=none] (15) at (-1.75, -2) ;
[style=none] (16) at (-1.75, -3) ;
[style=none] (17) at (-1.75, 1.75) ;
[style=none] (18) at (-1.75, 3) ;
[style=none] (19) at (0.5, 0.75) ;
[style=none] (20) at (1, 1.75) ;
[style=none] (21) at (1.75, 3) ;
[style=none] (22) at (1.75, 1.75) ;
[style=none] (23) at (1.75, 1.25) $M_B$;
[style=none] (24) at (2.5, 0.75) ;
[style=none] (25) at (2.5, 1.75) ;
[style=none] (26) at (1.75, -3) ;
[style=none] (27) at (1.75, -2) ;
[style=right label] (28) at (-1.75, -2.75) $X$;
[style=right label] (29) at (1.75, -2.75) $Y$;
[style=right label] (30) at (-1.75, 3) $A$;
[style=right label] (31) at (1.75, 3) $B$;
[style=none] (32) at (-2.75, 2.25) ;
[style=none] (33) at (-2.75, -2) ;
[style=none] (34) at (2.75, -2) ;
[style=none] (35) at (2.75, 2.25) ;
[style=none] (36) at (-2, 0.75) ;
[style=none] (37) at (2, 0.75) ;

[qWire, in=-90, out=90] (1.center) to (0.center);
(5.center) to (7.center);
(7.center) to (6.center);
(6.center) to (5.center);
[qWire] (8.center) to (9.center);
(13.center) to (10.center);
(10.center) to (11.center);
(11.center) to (12.center);
(12.center) to (13.center);
[cWire, in=90, out=-90] (15.center) to (16.center);
[cWire, in=-90, out=90] (17.center) to (18.center);
(24.center) to (19.center);
(19.center) to (20.center);
(20.center) to (25.center);
(25.center) to (24.center);
[cWire, in=90, out=-90] (27.center) to (26.center);
[cWire, in=-90, out=90] (22.center) to (21.center);
[thick gray dashed edge] (32.center)
to (35.center)
to (34.center)
to (33.center)
to cycle;
[cWire, in=90, out=-90] (36.center) to (15.center);
[cWire, in=90, out=-90] (37.center) to (27.center);

=:  Diagrams/nsBox
It can then be shown that the constraint on measurements of Eq. <ref> immediately implies the relevant no-signalling conditions, for example:
Diagrams/nsBox1  =
[style=none] (0) at (1, 0.75) ;
[style=none] (1) at (1, -0.5) ;
[style=right label] (2) at (1, -0.25) $W$;
[style=right label] (3) at (-1, -0.25) $V$;
[style=none] (4) at (0, -1) $s$;
[style=none] (5) at (1.75, -0.5) ;
[style=none] (6) at (-1.75, -0.5) ;
[style=none] (7) at (0, -1.75) ;
[style=none] (8) at (-1, 0.75) ;
[style=none] (9) at (-1, -0.5) ;
[style=none] (10) at (-0.5, 0.75) ;
[style=none] (11) at (-1, 1.75) ;
[style=none] (12) at (-2.5, 1.75) ;
[style=none] (13) at (-2.5, 0.75) ;
[style=none] (14) at (-1.75, 1.25) $M_A$;
[style=none] (15) at (-1.75, -2) ;
[style=none] (16) at (-1.75, -3) ;
[style=none] (17) at (-1.75, 1.75) ;
[style=none] (18) at (-1.75, 3) ;
[style=none] (19) at (0.5, 0.75) ;
[style=none] (20) at (1, 1.75) ;
[style=none] (21) at (1.75, 3) ;
[style=none] (22) at (1.75, 1.75) ;
[style=none] (23) at (1.75, 1.25) $M_B$;
[style=none] (24) at (2.5, 0.75) ;
[style=none] (25) at (2.5, 1.75) ;
[style=none] (26) at (1.75, -3) ;
[style=none] (27) at (1.75, -2) ;
[style=right label] (28) at (-1.75, -2.75) $X$;
[style=right label] (29) at (1.75, -2.75) $Y$;
[style=right label] (30) at (-1.75, 3) $A$;
[style=right label] (31) at (1.75, 2.5) $B$;
[style=none] (32) at (-2.75, 2.25) ;
[style=none] (33) at (-2.75, -2) ;
[style=none] (34) at (2.75, -2) ;
[style=none] (35) at (2.75, 2.25) ;
[style=none] (36) at (-2, 0.75) ;
[style=none] (37) at (2, 0.75) ;
[style=upground] (38) at (1.75, 3.25) ;

[qWire, in=-90, out=90] (1.center) to (0.center);
(5.center) to (7.center);
(7.center) to (6.center);
(6.center) to (5.center);
[qWire] (8.center) to (9.center);
(13.center) to (10.center);
(10.center) to (11.center);
(11.center) to (12.center);
(12.center) to (13.center);
[cWire, in=90, out=-90] (15.center) to (16.center);
[cWire, in=-90, out=90] (17.center) to (18.center);
(24.center) to (19.center);
(19.center) to (20.center);
(20.center) to (25.center);
(25.center) to (24.center);
[cWire, in=90, out=-90] (27.center) to (26.center);
[cWire, in=-90, out=90] (22.center) to (21.center);
[thick gray dashed edge] (32.center)
to (35.center)
to (34.center)
to (33.center)
to cycle;
[cWire, in=90, out=-90] (36.center) to (15.center);
[cWire, in=90, out=-90] (37.center) to (27.center);

[style=none] (0) at (1, 0.75) ;
[style=none] (1) at (1, 0) ;
[style=right label] (2) at (1, 0.25) $W$;
[style=right label] (3) at (-1, 0.25) $V$;
[style=none] (4) at (0, -0.5) $s$;
[style=none] (5) at (1.75, 0) ;
[style=none] (6) at (-1.75, 0) ;
[style=none] (7) at (0, -1.25) ;
[style=none] (8) at (-1, 0.75) ;
[style=none] (9) at (-1, 0) ;
[style=none] (10) at (-0.5, 0.75) ;
[style=none] (11) at (-1, 1.75) ;
[style=none] (12) at (-2.5, 1.75) ;
[style=none] (13) at (-2.5, 0.75) ;
[style=none] (14) at (-1.75, 1.25) $M_A$;
[style=none] (15) at (-1.75, -1.5) ;
[style=none] (16) at (-1.75, -3.25) ;
[style=none] (17) at (-1.75, 1.75) ;
[style=none] (18) at (-1.75, 3) ;
[style=none] (26) at (3, -3.25) ;
[style=none] (27) at (3, -2.5) ;
[style=right label] (28) at (-1.75, -3) $X$;
[style=right label] (29) at (3, -3) $Y$;
[style=right label] (30) at (-1.75, 3) $A$;
[style=none] (32) at (-3, 2.25) ;
[style=none] (33) at (-3, -1.5) ;
[style=none] (34) at (2, -1.5) ;
[style=none] (35) at (2, 2.25) ;
[style=none] (36) at (-2, 0.75) ;
[style=upground] (38) at (1, 1) ;
[style=upground] (39) at (3, -2.25) ;

[qWire, in=-90, out=90] (1.center) to (0.center);
(5.center) to (7.center);
(7.center) to (6.center);
(6.center) to (5.center);
[qWire] (8.center) to (9.center);
(13.center) to (10.center);
(10.center) to (11.center);
(11.center) to (12.center);
(12.center) to (13.center);
[cWire, in=90, out=-90] (15.center) to (16.center);
[cWire, in=-90, out=90] (17.center) to (18.center);
[cWire, in=90, out=-90] (27.center) to (26.center);
[thick gray dashed edge] (32.center)
to (35.center)
to (34.center)
to (33.center)
to cycle;
[cWire, in=90, out=-90] (36.center) to (15.center);

=:  Diagrams/nsBox4 .
A more general version of this proof first appeared in Ref. [60] and was generalised to arbitrary causal structures in Ref. [61].

Bell inequalities [1, 59] are then a particular class of linear functionals from this space of stochastic maps to the reals. A linear functional corresponding to a Bell inequality, hereon denoted by $\mathcal{I}$, can be diagrammatically denoted as:
Diagrams/InequalityBell2    ::   Diagrams/nsBox  ↦  Diagrams/InequalityBell3 ∈R  .
Note that $\mathcal{I}$ should not be interpreted as a process within the GPT –  $\mathcal{I}$ is simply some linear functional, and can lead to negative values.

The value of a Bell inequality on a stochastic map $\mathsf{N}$ – realised within the GPT as per Eq. (<ref>) – is given by:
Diagrams/InequalityBell1 .

The maximal value of a Bell inequality $\mathcal{I}$ achievable by correlations within a given GPT $\mathcal{G}$, is therefore given by the following optimisation problem:

ℐ_max:=𝗌𝗎𝗉{  Diagrams/InequalityBell1  |  V,W, M_A,M_B,s ∈𝒢  } .
Notice that the optimisation is carried over the types of systems $V$ and $W$ present in $\mathcal{G}$, as well as over the local measurements $M_A$ and $M_B$, and bipartite states $s$.
The solution to this optimisation problem will of course depend on the properties of the GPT being studied. However, we readily see that when maximising over states and measurements from the theory, the compatibility constraints we discussed in the previous section will play a crucial role. Indeed, if the GPT admits some bipartite effect $e$, then the above mentioned hierarchy of constraints (see Eqs. (<ref>), (<ref>), (<ref>), and (<ref>)) will restrict the sets of states that the value of $\mathcal{I}$ is optimised over. In other words, the existence of bipartite effects $e$ within the GPT will impose a hierarchy of constraints on the correlations that such GPT may feature. In the next sections we elaborate on this fact with a concrete example.

§ PARITY READING MEASUREMENT

In this section we will explore the constraints on the correlations that a GPT may feature, given that bipartite effects associated to a particular measurement – which we call a Parity reading measurement (PRM) – exist within the GPT.

Suppose we have a controlled measurement, $M$, for a system $V$, with a setting variable labeled by the set $\nSet:=\{0,...,n-1\}$ such that $|\nSet|=n$, and a binary outcome variable $\bSet:=\{0,1\}$ as an outcome.  This is diagrammatically denoted  by:
Diagrams/ControlledMeasurement .
Recall that, as this is a measurement, it must satisfy:

[style=right label] (0) at (0.25, -0.5) $V$;
[style=none] (1) at (0.25, -0) ;
[style=none] (2) at (0.25, -0.75) ;
[style=none] (3) at (-1, -0) ;
[style=none] (4) at (0.75, -0) ;
[style=none] (5) at (0.75, 1) ;
[style=none] (6) at (-0.75, 1) ;
[style=none] (7) at (0, 0.5) $M$;
[style=none] (8) at (-0.5, -0) ;
[style=none] (9) at (-0.5, -0.75) ;
[style=none] (10) at (0, 1) ;
[style=none] (11) at (0, 1.75) ;
[style=right label] (12) at (-0.5, -0.5) $\nSet$;
[style=right label] (13) at (0, 1.375) $\bSet$;
[style=upground] (14) at (0, 2) ;

[qWire] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
(5.center) to (6.center);
(6.center) to (3.center);
[cWire] (11.center) to (10.center);
[cWire] (8.center) to (9.center);

[style=right label] (0) at (0.75, -0.75) $V$;
[style=none] (1) at (0.75, 0.25) ;
[style=none] (2) at (0.75, -1) ;
[style=none] (3) at (-0.75, 0.25) ;
[style=none] (4) at (-0.75, -1) ;
[style=right label] (5) at (-0.75, -0.75) $X$;
[style=upground] (6) at (-0.75, 0.5) ;
[style=upground] (7) at (0.75, 0.5) ;

[qWire, in=90, out=-90, looseness=1.00] (1.center) to (2.center);
[cWire, in=90, out=-90, looseness=1.00] (3.center) to (4.center);

which ensures that the correlations it can generate are no-signalling.

Then we can define a measurement $\mathcal{P}[M]$ which reads out the parity of such a measurement $M$ as follows:

Parity Reading Measurement.–

A parity reading measurement for $M$, denoted by $\mathcal{P}[M]$, is a bipartite measurement on $V\otimes V$ with $n$ binary variables as outputs:
Diagrams/ParityReading ,
such that tracing out all but the $i$-th outcome gives the parity of the $i$-th setting for $M$:
∀i∈  ,  Diagrams/ParityReading2 =   Diagrams/ParityReading1 .

We will also be interested in situations in which we have a measurement which can only read the parity of a certain subset of the setting variable ${\iSet} \subseteq \nSet$:
Partial Parity Reading Measurement.–

A partial parity reading measurement for the settings  ${\iSet} \subseteq \nSet$ of measurement $M$, denoted by $\mathcal{P}[M]_{\iSet}$ is simply a parity reading measurement for the measurement:

[style=right label] (0) at (0.25, -0.5) $V$;
[style=none] (1) at (0.25, -0) ;
[style=none] (2) at (0.25, -0.75) ;
[style=none] (3) at (-1, -0) ;
[style=none] (4) at (0.75, -0) ;
[style=none] (5) at (0.75, 1) ;
[style=none] (6) at (-0.75, 1) ;
[style=none] (7) at (0, 0.5) $M_{\iSet}$;
[style=none] (8) at (-0.5, -0) ;
[style=none] (9) at (-0.5, -0.75) ;
[style=none] (10) at (0, 1) ;
[style=none] (11) at (0, 1.75) ;
[style=right label] (12) at (-0.5, -0.5) $\iSet$;
[style=right label] (13) at (0, 1.5) $\bSet$;

[qWire] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
(5.center) to (6.center);
(6.center) to (3.center);
[cWire] (11.center) to (10.center);
[cWire] (8.center) to (9.center);

[style=right label] (0) at (0.25, -1.75) $V$;
[style=none] (1) at (0.25, -0) ;
[style=none] (2) at (0.25, -2) ;
[style=none] (3) at (-1, -0) ;
[style=none] (4) at (0.75, -0) ;
[style=none] (5) at (0.75, 1) ;
[style=none] (6) at (-0.75, 1) ;
[style=none] (7) at (0, 0.5) $M$;
[style=none] (8) at (-0.5, -0) ;
[style=none] (9) at (-0.5, -1) ;
[style=none] (10) at (0, 1) ;
[style=none] (11) at (0, 1.75) ;
[style=right label] (12) at (-0.5, -0.5) $\nSet$;
[style=right label] (13) at (0, 1.5) $\bSet$;
[style=none] (14) at (-0.5, -2) ;
[style=none] (15) at (-0.5, -1) ;
[style=right label] (16) at (-0.5, -1.75) $\iSet$;
[style=black dot] (17) at (-0.5, -1) ;

[qWire] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
(5.center) to (6.center);
(6.center) to (3.center);
[cWire] (11.center) to (10.center);
[cWire] (8.center) to (9.center);
[cWire] (15.center) to (14.center);

where $\bullet$ is the canonical embedding of $\iSet$ into $\nSet$[That is, it maps $\iSet$ viewed as a set in its own right into $\iSet$ viewed as a subset of $\nSet$]. Using this notation we can succinctly define these partial partity reading measurements by:𝒫[M]_:=𝒫[M_] .
Note that when $\iSet=\nSet$ then we recover the notion of a PRM.

In order to see how the existence of a (partial) PRM constraints the correlations that the GPT may feature, we will fist discuss the concepts of a Fiducial Measurement and Fiducial effects.

Let $n$ be the affine dimension of the normalised state space, that is, $n = |V|-1$. A fiducial measurement, $\mathcal{F}$, is a controlled measurement with $n$ settings (described by the set $\nSet$) and binary outcomes (described by the set $\bSet$):
$\mathcal{F}$ is called a fiducial measurement if all of the fiducial effects can be obtained from such a measurement. Fiducial effects, in turn, form a (minimal) spanning set for the effect space of the GPT. As an example, consider the case where $n=2$: here $\mathcal{F}$ will have a binary input system, and the three fiducial effects will be given by
{ Diagrams/FiducialEffect1 ,  Diagrams/FiducialEffect2 ,  Diagrams/FiducialEffect3 = Diagrams/FiducialEffect5 = Diagrams/FiducialEffect4} ,
where the equality for the third effect comes from the fact that $\mathcal{F}$ is a valid controlled measurement and hence satisfies:

[style=right label] (0) at (0.25, -1.5) $V$;
[style=none] (1) at (0.25, -0.5) ;
[style=none] (2) at (0.25, -1.75) ;
[style=none] (3) at (0.75, 0.5) ;
[style=none] (4) at (-0.75, 0.5) ;
[style=none] (5) at (-1, -0.5) ;
[style=none] (6) at (0, -0) $\mathcal{F}$;
[style=none] (7) at (-0.5, -0.5) ;
[style=none] (8) at (-0.5, -1.75) ;
[style=none] (9) at (0, 0.5) ;
[style=none] (10) at (0, 1.25) ;
[style=right label] (11) at (-0.5, -1.5) $\nSet$;
[style=right label] (12) at (0, 0.875) $\bSet$;
[style=none] (13) at (0.75, -0.5) ;
[style=upground] (14) at (0, 1.5) ;

[qWire, in=90, out=-90, looseness=1.00] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
[cWire, in=90, out=-90, looseness=1.00] (7.center) to (8.center);
[cWire, in=-90, out=90, looseness=1.00] (9.center) to (10.center);
(5.center) to (13.center);
(13.center) to (3.center);

[style=right label] (0) at (0.75, -0.75) $V$;
[style=none] (1) at (0.75, 0.25) ;
[style=none] (2) at (0.75, -1) ;
[style=none] (3) at (-0.75, 0.25) ;
[style=none] (4) at (-0.75, -1) ;
[style=right label] (5) at (-0.75, -0.75) $\nSet$;
[style=upground] (6) at (-0.75, 0.5) ;
[style=upground] (7) at (0.75, 0.5) ;

[qWire, in=90, out=-90, looseness=1.00] (1.center) to (2.center);
[cWire, in=90, out=-90, looseness=1.00] (3.center) to (4.center);

Coming back to the case of an arbitrary $n$, notice that the fact that fiducial effects,
[style=copoint] (0) at (0, .5) $e_j$;
[style=none] (1) at (0, -1) ;
[style=right label] (2) at (0, -0.5) $V$;

[qWire] (1.center) to (0);

}_j=0:nspan the corresponding vector space, means that any state $s$ can be uniquely characterised by the vector of probabilities:

𝐩_s  :=   (
[style=copoint] (0) at (0, 1) $e_0$;
[style=point] (1) at (0, -1) $s$;
[style=right label] (2) at (0, -0.25) $V$;

[qWire] (1.center) to (0);

,  … ,
[style=copoint] (0) at (0, 1) $e_{n}$;
[style=point] (1) at (0, -1) $s$;
[style=right label] (2) at (0, -0.25) $V$;

[qWire](1.center) to (0);

)^T .
For example, going back to the case where $n=2$, this vector of probabilities will be given by:

𝐩_s  :=  (Diagrams/FiducialProb2 ,  Diagrams/FiducialProb3 ,  Diagrams/FiducialProb1 )^T.

Now we can briefly state the case we will explore in this section: GPTs that have a  PRM for a Fiducial measurement, and where a bipartite Bell experiment is carried by Alice and Bob performing this controlled fiducial measurement in each wing.

A PRM $\mathcal{P[F]}$ for the fiducial measurement $\mathcal{F}$ will satisfy the following constraints:

Diagrams/ParFid1  =   Diagrams/ParFid2     ∀i∈ .
It is worth mentioning that a PRM $\mathcal{P[F]}$ is not necessarily uniquely singled out by these constraints – more than one PRM may qualify as potential candidates for the role. We denote by $\mathsf{ParMeas}[\mathcal{F}]$ the set of all PRM $\mathcal{P[F]}$ that satisfy Eq. (<ref>) for the given $\mathcal{F}$.

§.§ Examples

Let us conclude this discussion with an example from quantum theory. Consider the case of qubit systems, where the affine local dimension is $n=3$. A fiducial measurement corresponds to measuring the three Pauli observables $X$, $Y$, and $Z$. A PRM is given by (a suitable post-processing of) the Bell measurement
{|ϕ^+⟩⟨ϕ^+|, |ϕ^-⟩⟨ϕ^-|, |ψ^+⟩⟨ψ^+|, |ψ^-⟩⟨ψ^-|},
with $\ket{\phi^\pm} = \frac{\ket{00} \pm \ket{11}}{\sqrt{2}}$ and $\ket{\psi^\pm} = \frac{\ket{01} \pm \ket{10}}{\sqrt{2}}$.  Denoting the four element set of outcomes of the Bell measurement as $\mathbf{B}:=\{0,1,2,3\}$, and the qubit system by $Q_2$, we can diagrammatically represent this as:

[style=none] (0) at (0, 1.5) ;
[style=none] (4) at (0, 0.5) ;
[style=right label] (5) at (0, 1.25) $\mathbf{B}$;
[style=none] (18) at (-0.75, 0.5) ;
[style=none] (19) at (0.75, 0.5) ;
[style=none] (20) at (1.25, -0.5) ;
[style=none] (21) at (-1.25, -0.5) ;
[style=none] (22) at (-0.75, -0.5) ;
[style=none] (23) at (-0.75, -1.5) ;
[style=right label] (24) at (-0.75, -1.25) $Q_2$;
[style=none] (25) at (0.75, -0.5) ;
[style=none] (26) at (0.75, -1.5) ;
[style=right label] (27) at (0.75, -1.25) $Q_2$;
[style=none] (28) at (0, 0) Bell;

[cWire] (0.center) to (4.center);
(18.center) to (21.center);
(21.center) to (20.center);
(20.center) to (19.center);
(19.center) to (18.center);
[qWire] (22.center) to (23.center);
[qWire] (25.center) to (26.center);

To see that a post-processing is necessary for this measurement to fit into our definition of a PRM is easy: the measurement of Eq. (<ref>) is a four outcome measurement, but, a PRM for $X$, $Y$, and $Z$ should have three binary outcomes. The required post-processing can be described diagrammatically as:

[style=white dot] (0) at (0, -1) ;
[style=none] (1) at (-1.75, 0.25) ;
[style=none] (2) at (0, 0.25) ;
[style=none] (3) at (1.75, 0.25) ;
[style=none] (4) at (0, -2.25) ;
[style=right label] (5) at (0, -2) $\mathbf{B}$;
[style=right label] (6) at (1.75, 0) $\mathbf{B}$;
[style=right label] (7) at (0, 0) $\mathbf{B}$;
[style=right label] (8) at (-1.75, 0) $\mathbf{B}$;
[style=small box] (9) at (-1.75, 1) $C_X$;
[style=small box] (10) at (0, 1) $C_Y$;
[style=small box] (11) at (1.75, 1) $C_Z$;
[style=none] (12) at (-1.75, 2.25) ;
[style=none] (13) at (0, 2.25) ;
[style=none] (14) at (1.75, 2.25) ;
[style=right label] (15) at (-1.75, 2) $\bSet$;
[style=right label] (16) at (0, 2) $\bSet$;
[style=right label] (17) at (1.75, 2) $\bSet$;

[cWire, in=150, out=-90] (1.center) to (0);
[cWire] (0) to (2.center);
[cWire, in=-90, out=30] (0) to (3.center);
[cWire] (0) to (4.center);
[cWire] (12.center) to (9);
[cWire] (9) to (1.center);
[cWire] (13.center) to (10);
[cWire] (10) to (2.center);
[cWire] (14.center) to (11);
[cWire] (11) to (3.center);

where the white dot first makes three copies of the outcome, and the processes $C_X$, $C_Y$ and $C_Z$ correspond to the three different  equal bipartitions of $\mathbf{B}$. For example:

[style=none] (1) at (0, -1) ;
[style=right label] (8) at (0, -1) $\mathbf{B}$;
[style=small box] (9) at (0, 0) $C_Z$;
[style=none] (12) at (0, 1) ;
[style=right label] (15) at (0, 0.75) $\bSet$;

[cWire] (12.center) to (9);
[cWire] (9) to (1.center);

[style=none] (1) at (0, -1.75) ;
[style=right label] (8) at (0, -1.5) $\mathbf{B}$;
[style=copoint] (9) at (0, -0.75) $0$;
[style=point] (18) at (0, 0.75) $0$;
[style=none] (19) at (0, 1.75) ;
[style=right label] (20) at (0, 1.5) $\bSet$;

[cWire] (9) to (1.center);
[cWire] (19.center) to (18);

[style=none] (1) at (0, -1.75) ;
[style=right label] (8) at (0, -1.5) $\mathbf{B}$;
[style=copoint] (9) at (0, -0.75) $1$;
[style=point] (18) at (0, 0.75) $0$;
[style=none] (19) at (0, 1.75) ;
[style=right label] (20) at (0, 1.5) $\bSet$;

[cWire] (9) to (1.center);
[cWire] (19.center) to (18);

[style=none] (1) at (0, -1.75) ;
[style=right label] (8) at (0, -1.5) $\mathbf{B}$;
[style=copoint] (9) at (0, -0.75) $2$;
[style=point] (18) at (0, 0.75) $1$;
[style=none] (19) at (0, 1.75) ;
[style=right label] (20) at (0, 1.5) $\bSet$;

[cWire] (9) to (1.center);
[cWire] (19.center) to (18);

[style=none] (1) at (0, -1.75) ;
[style=right label] (8) at (0, -1.5) $\mathbf{B}$;
[style=copoint] (9) at (0, -0.75) $3$;
[style=point] (18) at (0, 0.75) $1$;
[style=none] (19) at (0, 1.75) ;
[style=right label] (20) at (0, 1.5) $\bSet$;

[cWire] (9) to (1.center);
[cWire] (19.center) to (18);

To read the parity of observable $ZZ$,
we hence apply post-processing $\{0,1\}\to 0$ and $\{2,3\}\to 1$ depicted in Eq. (<ref>), which quantum mechanically comes from

\begin{eqnarray}
&& |\phi_+\rangle\<\phi_+| + |\phi_-\rangle\<\phi_-|=|00\rangle\<00|+
|11\rangle\<11|,\nonumber \\
&& |\psi_+\rangle\<\psi_+| + |\psi_-\rangle\<\psi_-|=|01\rangle\<01|+
|10\rangle\<10|,  \\
&& \sigma_z \otimes \sigma_z =
|11\rangle\<11| - (|01\rangle\<01|+
\end{eqnarray}

All three required post-processings, and the observables whose parity they read, are presented below:

\begin{eqnarray}
\label{eq:Bell-postproc}
\begin{array}{l|c}
\text{post-processing}& \text{parity} \\
\hline
C_X:\quad \{\phi_+,\psi_+\}\to 0, \{\phi_-,\psi_-\}\to 1 & XX \\
\{\phi_-,\psi_+\}\to 0, \{\phi_+,\psi_-\}\to 1
& YY \\
\{\phi_+,\phi_-\}\to 0, \{\psi_+,\psi_-\}\to 1
& ZZ
\end{array}
\end{eqnarray}

With this we conclude the argument for why the measurement given by

[style=white dot] (0) at (0, 0.25) ;
[style=none] (1) at (-1.75, 1.5) ;
[style=none] (2) at (0, 1.5) ;
[style=none] (3) at (1.75, 1.5) ;
[style=none] (4) at (0, -0.75) ;
[style=right label] (5) at (0, -0.5) $\mathbf{B}$;
[style=right label] (6) at (1.75, 1.25) $\mathbf{B}$;
[style=right label] (7) at (0, 1.25) $\mathbf{B}$;
[style=right label] (8) at (-1.75, 1.25) $\mathbf{B}$;
[style=small box] (9) at (-1.75, 2.25) $C_X$;
[style=small box] (10) at (0, 2.25) $C_Y$;
[style=small box] (11) at (1.75, 2.25) $C_Z$;
[style=none] (12) at (-1.75, 3.5) ;
[style=none] (13) at (0, 3.5) ;
[style=none] (14) at (1.75, 3.5) ;
[style=right label] (15) at (-1.75, 3.25) $\bSet$;
[style=right label] (16) at (0, 3.25) $\bSet$;
[style=right label] (17) at (1.75, 3.25) $\bSet$;
[style=none] (18) at (-0.75, -0.75) ;
[style=none] (19) at (0.75, -0.75) ;
[style=none] (20) at (1.25, -1.75) ;
[style=none] (21) at (-1.25, -1.75) ;
[style=none] (22) at (-0.75, -1.75) ;
[style=none] (23) at (-0.75, -2.75) ;
[style=right label] (24) at (-0.75, -2.5) $Q_2$;
[style=none] (25) at (0.75, -1.75) ;
[style=none] (26) at (0.75, -2.75) ;
[style=right label] (27) at (0.75, -2.5) $Q_2$;
[style=none] (28) at (0, -1.25) Bell;

[cWire, in=150, out=-90] (1.center) to (0);
[cWire] (0) to (2.center);
[cWire, in=-90, out=30] (0) to (3.center);
[cWire] (0) to (4.center);
[cWire] (12.center) to (9);
[cWire] (9) to (1.center);
[cWire] (13.center) to (10);
[cWire] (10) to (2.center);
[cWire] (14.center) to (11);
[cWire] (11) to (3.center);
(18.center) to (21.center);
(21.center) to (20.center);
(20.center) to (19.center);
(19.center) to (18.center);
[qWire] (22.center) to (23.center);
[qWire] (25.center) to (26.center);

is a parity reading measurement for the $X$, $Y$ and $Z$ observables.

Mirror quantum correlations.
Such correlations have been considered in Ref. [62].
To introduce them, let us first notice a feature of the Bell measurement:

* the effect corresponding to $\psi_-$
appears only in the bipartitions of $\mathbf{B}$ that give rise to anticorrelations, i.e., a value of $1$ for the classical system $\bSet$,

* the other three effects appear each only once in a bipartition that measures anticorrelations.

The following table summarises this feature, where we specify, for each observable whose parity we want to read, whether each effect belongs to the correlation (0) or anticorrelation (1) bipartitions:

\begin{equation}\label{eq:tableqt}
\begin{array}{c|ccc}
\text{effect} & \text{XX} &\text{YY} &\text{ZZ} \\
\hline
\phi_+ & 0 & 1 & 0  \\
\phi_- & 1 & 0 & 0  \\
\psi_+ & 0 & 0 & 1  \\
\psi_- & 1 & 1 & 1  \\
\end{array}
\end{equation}

We then say that quantum parity reading measurement has signature

\begin{align}
\{(+,-,+),(-,+,+),(+,+,-),(-,-,-)\}.
\end{align}

In the case of mirror quantum mechanics [62] the table in Eq. (<ref>) does not hold anymore, and instead the following are satisfied:

\begin{equation}
\label{eq:tablemirror}
\begin{array}{c|ccc}
\text{effect} & \text{XX} &\text{YY} &\text{ZZ} \\
\hline
(\phi_+)^{PT} & 1 & 0 & 1  \\
(\phi_-)^{PT} & 0 & 1 & 1  \\
(\psi_+)^{PT} & 1 & 1 & 0  \\
(\psi_-)^{PT} & 0 & 0 & 0  \\
\end{array}
\end{equation}

where PT stands for partial transposition[In mirror quantum theory, these effects are partially transposed effects of Bell measurement.].

Thus in mirror quantum case we have signature:

\begin{align}
\{(-,+,-),(+,-,-),(-,-,+),(+,+,+)\}.
\end{align}

This case then corresponds to the following post-processing of the measurement outcomes:

\begin{eqnarray}
\label{eq:Mirror-postproc}
\begin{array}{l|c}
\text{post-processing}& \text{parity} \\
\hline
C_X:\quad \{0,2\}\to 0, \{1,3\}\to 1 & XX \\
\{0,3\}\to 0, \{1,2\}\to 1
& YY \\
\{2,3\}\to 0, \{0,1\}\to 1
& ZZ
\end{array}
\end{eqnarray}

Classical theory.
In the quantum case, the parity reading measurement was entangled. It had to be so, because it measured parities of observables that are not jointly measurable. In classical theory we can consider three bit system, described by $X,Y,Z$ which are now jointly measurable. Then the PRM just amounts to post-process the joint measurement of all the 6 observables (three per party).

§.§ $\mathcal{P[F]}$ and maximal violations of Bell inequalities for fiducial measurements $\mathcal{F}$

So, how does the existence of a $\mathcal{P[F]}$ in the GPT constrain the correlations we may observe in a Bell test?
More precisely, what are the constraints on the correlations that a bipartite system of a certain type can produce when there exists a PRM for the fiducial measurements on that system type?
In this section we aim at optimising the value of a Bell inequality $\mathcal{I}$ when the measurements that the parties perform are given by $\mathcal{F}$ on a system of type $V$.

Notice that, in general, the cardinality of the input settings for the Bell test need not coincide with the number of settings for the fiducial measurement $\mathcal{F}$. That is, $|X|=|\nSet|=|Y|$ does not necessarily hold. In this manuscript we will work with the case where $|X|\leq|\nSet|$ and $|Y|\leq|\nSet|$, and hence only some of the settings of the controlled measurement $\mathcal{F}$ might be used for the Bell test. In such a case, then, we will focus on the constraints that a partial PRM imposes when its existence is demanded on the settings of $\mathcal{F}$ used in the Bell test. For simplicity in the discussion, in this section we will present the case where $|X|=|\nSet|=|Y|$, but the most general case follows similarly. We will return to the optimisation problems for partial PRM later on in the manuscript.

The optimisation problem that we focus on then reads:
ℐ_max:=𝗌𝗎𝗉{  Diagrams/OptimisationProblem  |  s ∈𝒢  } ,
where $X,Y,A,B$ are all binary variables, and we are using the shorthand notation:

[style=right label] (0) at (-0.5, -1) $V$;
[style=none] (1) at (-0.5, -0.5) ;
[style=none] (2) at (-0.5, -1.25) ;
[style=none] (3) at (0.75, -0.5) ;
[style=none] (4) at (-1, -0.5) ;
[style=none] (5) at (-1, 0.5) ;
[style=none] (6) at (0.5, 0.5) ;
[style=none] (7) at (-0.25, 0) $\mathcal{F}$;
[style=none] (8) at (0.25, -0.5) ;
[style=none] (9) at (0.25, -1.25) ;
[style=none] (10) at (-0.25, 0.5) ;
[style=none] (11) at (-0.25, 1.25) ;
[style=right label] (12) at (0.25, -1) $\nSet$;
[style=right label] (13) at (-0.25, 1) $\bSet$;

[qWire] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
(5.center) to (6.center);
(6.center) to (3.center);
[cWire] (11.center) to (10.center);
[cWire] (8.center) to (9.center);

[style=left label] (0) at (-0.5, -1.25) $V$;
[style=none] (1) at (0.25, 0) ;
[style=none] (2) at (-0.5, -1.5) ;
[style=none] (3) at (-1, 0) ;
[style=none] (4) at (0.75, 0) ;
[style=none] (5) at (0.75, 1) ;
[style=none] (6) at (-0.75, 1) ;
[style=none] (7) at (0, 0.5) $\mathcal{F}$;
[style=none] (8) at (-0.5, 0) ;
[style=none] (9) at (0.25, -1.5) ;
[style=none] (10) at (0, 1) ;
[style=none] (11) at (0, 1.75) ;
[style=right label] (12) at (0.25, -1.25) $\nSet$;
[style=right label] (13) at (0, 1.5) $\bSet$;

[qWire, in=90, out=-90] (1.center) to (2.center);
(3.center) to (4.center);
(4.center) to (5.center);
(5.center) to (6.center);
(6.center) to (3.center);
[cWire] (11.center) to (10.center);
[cWire, in=90, out=-90] (8.center) to (9.center);

Given a particular GPT $\mathcal{G}$ – and, in particular, given a specification of its state and effect spaces  –  this optimisation problem reduces to a type of cone program which has been explored in recent literature [46, 47, 63] regarding their relationship to GPTs. That is, there is some convex spanning cone of states $K_{V\otimes V}\subset V\otimes V$, which $s$ belongs to, and some normalisation constraint on $s$, $u_V\otimes u_V(s)=1$ so the above problem can be rewritten as:

ℐ_max:=𝗌𝗎𝗉{  Diagrams/OptimisationProblem  |  s ∈K_V⊗V, u_V⊗u_V(s)=1  }.
Here, however, we do not consider a particular GPT $\mathcal{G}$ which we optimise over. What we carry out here is an optimisation over the space of GPTs which have the relevant structure – those which admit a PRM for the fiducial measurement. This optimisation problem is much more complex than that of Eq. (<ref>), as we will now explain.

In Section <ref> we elaborated on the types of compatibility constraints between states and effects that a GPT must feature. Here, we will demand that the GPT admits the fiducial local measurement $\mathcal{F}$ and a PRM $\mathcal{P} \in \mathsf{ParMeas}[\mathcal{F}]$. By imposing the compatibility constraints motivated in Section <ref>, we hence restrict the possible cones of states $K[\mathcal{P}]$ that such GPT could feature. If we have a characterisation of $K[\mathcal{P}] \subset  V\otimes V$, then the optimisation problem becomes:

ℐ_max:=𝗌𝗎𝗉{  Diagrams/OptimisationProblem  |   𝒫∈𝖯𝖺𝗋𝖬𝖾𝖺𝗌[ℱ], u_V⊗u_V(s)=1, s ∈K[𝒫]  } .
This turns out to be a non-linear optimisation problem, as we will show next.

§.§ The cone $K$ of bipartite states

The key question here is: how to characterise the cones of states $K[\mathcal{P}]$? Here we will take the types of diagrammatic constraints motivated in Section <ref>, and define a systematic hierarchy of conditions that the existence of $\mathcal{P[F]}$ imposes on the cone $K$.
This hierarchy of conditions will be specified in terms of the number of copies of the bipartite state $s$ featured in the diagram. Framing these constraints in the form of a hierarchy is useful because, as we will see, interesting results can be obtained without needing to impose all of the constraints. For example, in our case we will be interested in  possible violations of a given Bell inequality,  and, we can obtain upper bounds on this by simply working at the second level of the hierarchy.

Given the fiducial measurement $\mathcal{F}$ and the PRM $\mathcal{P}\in\mathsf{ParMeas}[\mathcal{F}]$, the normalised bipartite states $s \in K[\mathcal{P}]$ must satisfy:

\begin{align}
&\tikzfig{Diagrams/const1} \geq 0 \quad \text{,} \label{eq:const1}\\
&\tikzfig{Diagrams/const2} \geq 0 \quad \text{,} \label{eq:singsin} \\
&\tikzfig{Diagrams/const3}\geq 0    \quad \text{,} \label{eq:singswap}
\end{align}

where by $\geq 0$ we mean that every matrix element is non-negative.

Notice that the constraints that come from the deterministic effect $u$ – in particular, the normalisation condition $u_V\otimes u_V(s)=1$ – together with these positivity constraints, ensures that diagrams in Eqs. <ref>, <ref>, and <ref> are stochastic maps.

Notice moreover that the constraint of Eq. (<ref>) has the same structure as that of Eq. (<ref>) but applied instead to the swapped state:

[style=none] (0) at (-1, 1) ;
[style=none] (1) at (1, -0.25) ;
[style=left label] (2) at (-1, 0.75) $V$;
[style=right label] (3) at (1, 0.75) $V$;
[style=none] (4) at (0, -0.75) $s$;
[style=none] (5) at (1.75, -0.25) ;
[style=none] (6) at (-1.75, -0.25) ;
[style=none] (7) at (0, -1.5) ;
[style=none] (8) at (1, 1) ;
[style=none] (9) at (-1, -0.25) ;
[style=none] (21) at (-1, 1) ;

[qWire, in=270, out=90] (1.center) to (0.center);
(5.center) to (7.center);
(7.center) to (6.center);
(6.center) to (5.center);
[qWire, in=90, out=-90] (8.center) to (9.center);

We see then that there is a certain structure emerging: (i) there are two layers – one corresponding to the state $s$ and one to the measurements $\mathcal{F}$ and $\mathcal{P}$ –, and (ii) we can vary the order in which the output wires of the state are plugged into the measurements of the second layer. This motivates the definition for the remaining levels of the hierarchy, which relies on the concept of a wiring, which we explain next.

A process which describes how a collection of input systems are connected to a collection of output systems is here referred to as a wiring, and denoted usually by $W$.

When all the input systems are of the same type – a case we focus on here –  wirings reduce to permutations of the systems, e.g.:

[style=none] (0) at (5, 1) ;
[style=none] (1) at (5, 2.25) ;
[style=right label] (2) at (5, 1.25) $V$;
[style=none] (3) at (-3, -1) ;
[style=none] (4) at (-3, -2.25) ;
[style=right label] (5) at (-3, -2) $V$;
[style=right label] (6) at (-5, -2) $V$;
[style=none] (7) at (-5, -1) ;
[style=none] (8) at (-5, -2.25) ;
[style=none] (9) at (-3.25, 2.25) ;
[style=none] (10) at (-3.25, 1) ;
[style=right label] (11) at (-3.25, 1.25) $V$;
[style=right label] (12) at (-5, 1.25) $V$;
[style=none] (13) at (-5, 2.25) ;
[style=none] (14) at (-5, 1) ;
[style=none] (15) at (-1.25, 1) ;
[style=right label] (16) at (-1.25, 1.25) $V$;
[style=none] (17) at (-1.25, 2.25) ;
[style=none] (18) at (5, -1) ;
[style=none] (19) at (5, -2.25) ;
[style=right label] (20) at (5, -2) $V$;
[style=right label] (21) at (3, -2) $V$;
[style=none] (22) at (3, -1) ;
[style=none] (23) at (3, -2.25) ;
[style=none] (24) at (1, -1) ;
[style=none] (25) at (1, -2.25) ;
[style=right label] (26) at (1, -2) $V$;
[style=right label] (27) at (-1, -2) $V$;
[style=none] (28) at (-1, -1) ;
[style=none] (29) at (-1, -2.25) ;
[style=none] (30) at (1, 2.25) ;
[style=none] (31) at (1, 1) ;
[style=right label] (32) at (1, 1.25) $V$;
[style=none] (33) at (3, 1) ;
[style=right label] (34) at (3, 1.25) $V$;
[style=none] (35) at (3, 2.25) ;
[style=none] (36) at (-5.5, 1) ;
[style=none] (37) at (-5.5, -1) ;
[style=none] (38) at (5.5, -1) ;
[style=none] (39) at (5.5, 1) ;
[style=none] (40) at (0, -0) $W$;

[qWire, in=-90, out=90, looseness=1.00] (0.center) to (1.center);
[qWire, in=-90, out=90, looseness=1.00] (4.center) to (3.center);
[qWire] (7.center) to (8.center);
[qWire, in=-90, out=90, looseness=1.00] (10.center) to (9.center);
[qWire] (13.center) to (14.center);
[qWire, in=90, out=-90, looseness=1.00] (17.center) to (15.center);
[qWire, in=-90, out=90, looseness=1.00] (19.center) to (18.center);
[qWire] (22.center) to (23.center);
[qWire, in=-90, out=90, looseness=1.00] (25.center) to (24.center);
[qWire] (28.center) to (29.center);
[qWire, in=-90, out=90, looseness=1.00] (31.center) to (30.center);
[qWire, in=90, out=-90, looseness=1.00] (35.center) to (33.center);
(36.center) to (39.center);
(39.center) to (38.center);
(38.center) to (37.center);
(37.center) to (36.center);

[style=none] (0) at (3, 1) ;
[style=none] (1) at (3, 2.25) ;
[style=right label] (2) at (3, 1.25) $V$;
[style=none] (3) at (-3, -1) ;
[style=none] (4) at (-3, -2.25) ;
[style=right label] (5) at (-3, -2) $V$;
[style=right label] (6) at (-5, -2) $V$;
[style=none] (7) at (-5, -1) ;
[style=none] (8) at (-5, -2.25) ;
[style=none] (9) at (-1, 2.25) ;
[style=none] (10) at (-1, 1) ;
[style=right label] (11) at (-1, 1.25) $V$;
[style=right label] (12) at (-5, 1.25) $V$;
[style=none] (13) at (-5, 2.25) ;
[style=none] (14) at (-5, 1) ;
[style=none] (15) at (1, 1) ;
[style=right label] (16) at (1, 1.25) $V$;
[style=none] (17) at (1, 2.25) ;
[style=none] (18) at (5, -1) ;
[style=none] (19) at (5, -2.25) ;
[style=right label] (20) at (5, -2) $V$;
[style=right label] (21) at (3, -2) $V$;
[style=none] (22) at (3, -1) ;
[style=none] (23) at (3, -2.25) ;
[style=none] (24) at (1, -1) ;
[style=none] (25) at (1, -2.25) ;
[style=right label] (26) at (1, -2) $V$;
[style=right label] (27) at (-1, -2) $V$;
[style=none] (28) at (-1, -1) ;
[style=none] (29) at (-1, -2.25) ;
[style=none] (30) at (-3, 2.25) ;
[style=none] (31) at (-3, 1) ;
[style=right label] (32) at (-3, 1.25) $V$;
[style=none] (33) at (5, 1) ;
[style=right label] (34) at (5, 1.25) $V$;
[style=none] (35) at (5, 2.25) ;
[style=none] (36) at (-5.5, 1) ;
[style=none] (37) at (-5.5, -1) ;
[style=none] (38) at (5.5, -1) ;
[style=none] (39) at (5.5, 1) ;

[qWire, in=-90, out=90, looseness=1.00] (0.center) to (1.center);
[qWire, in=-90, out=90, looseness=1.00] (4.center) to (3.center);
[qWire] (7.center) to (8.center);
[qWire, in=-90, out=90, looseness=1.00] (10.center) to (9.center);
[qWire] (13.center) to (14.center);
[qWire, in=90, out=-90, looseness=1.00] (17.center) to (15.center);
[qWire, in=-90, out=90, looseness=1.00] (19.center) to (18.center);
[qWire] (22.center) to (23.center);
[qWire, in=-90, out=90, looseness=1.00] (25.center) to (24.center);
[qWire] (28.center) to (29.center);
[qWire, in=-90, out=90, looseness=1.00] (31.center) to (30.center);
[qWire, in=90, out=-90, looseness=1.00] (35.center) to (33.center);
[thick gray dashed edge] (36.center) to (39.center);
[thick gray dashed edge] (39.center) to (38.center);
[thick gray dashed edge] (38.center) to (37.center);
[thick gray dashed edge] (37.center) to (36.center);
[qWire, in=90, out=-90, looseness=1.00] (14.center) to (7.center);
[qWire, in=90, out=-90, looseness=1.00] (10.center) to (3.center);
[qWire, in=90, out=-90, looseness=1.00] (15.center) to (28.center);
[qWire, in=90, out=-90, looseness=0.75] (31.center) to (24.center);
[qWire, in=90, out=-90, looseness=1.00] (33.center) to (22.center);
[qWire, in=90, out=-90, looseness=1.00] (0.center) to (18.center);

Given the fiducial measurement $\mathcal{F}$ and the PRM $\mathcal{P}\in\mathsf{ParMeas}[\mathcal{F}]$, the normalised bipartite states $s \in K[\mathcal{P}]$ must satisfy:

\begin{align}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (12) at (4.5, 5.5) {};
\node [style=none] (13) at (4.5, 4.25) {};
\node [style=right label] (15) at (4.5, 5) {$\bSet$};
\node [style=none] (17) at (4.5, 2) {};
\node [style=none] (31) at (4.5, 3.25) {};
\node [style=right label] (32) at (4.5, 2.25) {$V$};
\node [style=none] (40) at (4.75, 3.75) {$\mathcal{F}$};
\node [style=none] (45) at (3.75, 4.25) {};
\node [style=none] (46) at (3.75, 3.25) {};
\node [style=none] (47) at (5.75, 3.25) {};
\node [style=none] (48) at (5.25, 4.25) {};
\node [style=none] (52) at (6.25, 1.25) {};
\node [style=right label] (53) at (6.25, 1.25) {$\nSet$};
\node [style=none] (54) at (5.25, 3.25) {};
\node [style=none] (55) at (-6.5, 0) {};
\node [style=none] (56) at (-6.5, -1.25) {};
\node [style=right label] (57) at (-6.5, -1) {$V$};
\node [style=right label] (58) at (-8.5, -1) {$V$};
\node [style=none] (59) at (-7.5, -1.75) {$s$};
\node [style=none] (60) at (-5.75, -1.25) {};
\node [style=none] (61) at (-9.25, -1.25) {};
\node [style=none] (62) at (-7.5, -2.5) {};
\node [style=none] (63) at (-8.5, 0) {};
\node [style=none] (64) at (-8.5, -1.25) {};
\node [style=none] (93) at (-6.75, 3.25) {};
\node [style=none] (94) at (-6.75, 2) {};
\node [style=right label] (95) at (-6.75, 2.25) {$V$};
\node [style=right label] (96) at (-8.5, 2.25) {$V$};
\node [style=none] (101) at (-8.5, 3.25) {};
\node [style=none] (102) at (-8.5, 2) {};
\node [style=none] (103) at (-8.5, 4.25) {};
\node [style=none] (104) at (-8.5, 5.5) {};
\node [style=right label] (105) at (-8.5, 5) {$\bSet$};
\node [style=none] (106) at (-6.75, 4.25) {};
\node [style=none] (107) at (-7.25, 3.25) {};
\node [style=none] (108) at (-4.75, 4.25) {};
\node [style=none] (110) at (-7.25, 4.25) {};
\node [style=none] (111) at (-4.25, 3.25) {};
\node [style=none] (112) at (-5.75, 3.75) {$\mathcal{P}$};
\node [style=none] (113) at (-4.25, 3.25) {};
\node [style=none] (114) at (-4.25, 4.25) {};
\node [style=none] (115) at (-4.75, 2) {};
\node [style=none] (116) at (-4.75, 5.5) {};
\node [style=none] (117) at (-6.75, 5.5) {};
\node [style=right label] (118) at (-6.75, 5) {$\bSet$};
\node [style=right label] (119) at (-4.75, 2.25) {$V$};
\node [style=right label] (120) at (-4.75, 5) {$\bSet$};
\node [style=none] (121) at (-7.25, 4.25) {};
\node [style=none] (122) at (-4.75, 3.25) {};
\node [style=none] (123) at (-8.75, 3.75) {$\mathcal{F}$};
\node [style=none] (124) at (-7.75, 4.25) {};
\node [style=none] (125) at (-7.75, 3.25) {};
\node [style=none] (126) at (-9.75, 3.25) {};
\node [style=none] (127) at (-9.25, 4.25) {};
\node [style=none] (128) at (-10.25, 1.25) {};
\node [style=right label] (129) at (-10.25, 1.25) {$\nSet$};
\node [style=none] (130) at (-9.25, 3.25) {};
\node [style=none] (131) at (4.5, 0) {};
\node [style=none] (132) at (4.5, -1.25) {};
\node [style=right label] (133) at (4.5, -1) {$V$};
\node [style=right label] (134) at (2.5, -1) {$V$};
\node [style=none] (135) at (3.5, -1.75) {$s$};
\node [style=none] (136) at (5.25, -1.25) {};
\node [style=none] (137) at (1.75, -1.25) {};
\node [style=none] (138) at (3.5, -2.5) {};
\node [style=none] (139) at (2.5, 0) {};
\node [style=none] (140) at (2.5, -1.25) {};
\node [style=none] (141) at (0, -1.25) {$\cdots$};
\node [style=none] (142) at (-8, -3) {};
\node [style=none] (143) at (4, -3) {};
\node [style=none] (144) at (-2, -3.75) {$k$};
\node [style=none] (145) at (-2.5, 0) {};
\node [style=none] (146) at (-2.5, -1.25) {};
\node [style=right label] (147) at (-2.5, -1) {$V$};
\node [style=right label] (148) at (-4.5, -1) {$V$};
\node [style=none] (149) at (-3.5, -1.75) {$s$};
\node [style=none] (150) at (-1.75, -1.25) {};
\node [style=none] (151) at (-5.25, -1.25) {};
\node [style=none] (152) at (-3.5, -2.5) {};
\node [style=none] (153) at (-4.5, 0) {};
\node [style=none] (154) at (-4.5, -1.25) {};
\node [style=none] (155) at (0.75, 3.25) {};
\node [style=none] (156) at (0.75, 2) {};
\node [style=right label] (157) at (0.75, 2.25) {$V$};
\node [style=none] (158) at (0.75, 4.25) {};
\node [style=none] (159) at (0.25, 3.25) {};
\node [style=none] (160) at (2.75, 4.25) {};
\node [style=none] (161) at (0.25, 4.25) {};
\node [style=none] (162) at (3.25, 3.25) {};
\node [style=none] (163) at (1.75, 3.75) {$\mathcal{P}$};
\node [style=none] (164) at (3.25, 3.25) {};
\node [style=none] (165) at (3.25, 4.25) {};
\node [style=none] (166) at (2.75, 2) {};
\node [style=none] (167) at (2.75, 5.5) {};
\node [style=none] (168) at (0.75, 5.5) {};
\node [style=right label] (169) at (0.75, 5) {$\bSet$};
\node [style=right label] (170) at (2.75, 2.25) {$V$};
\node [style=right label] (171) at (2.75, 5) {$\bSet$};
\node [style=none] (172) at (0.25, 4.25) {};
\node [style=none] (173) at (2.75, 3.25) {};
\node [style=none] (174) at (-2, 3.75) {$\cdots$};
\node [style=none] (175) at (3.5, 6.5) {};
\node [style=none] (176) at (-7.5, 6.5) {};
\node [style=none] (177) at (-2, 7.25) {$k-1$};
\node [style=none] (178) at (-9, 2) {};
\node [style=none] (179) at (-9, 0) {};
\node [style=none] (180) at (5, 0) {};
\node [style=none] (181) at (5, 2) {};
\node [style=none] (182) at (-2, 1) {$W$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [cWire, in=-90, out=90] (13.center) to (12.center);
\draw [qWire, in=-90, out=90] (17.center) to (31.center);
\draw (45.center) to (48.center);
\draw (48.center) to (47.center);
\draw (47.center) to (46.center);
\draw (46.center) to (45.center);
\draw [cWire, in=-90, out=90] (52.center) to (54.center);
\draw [qWire, in=-90, out=90] (56.center) to (55.center);
\draw (60.center) to (62.center);
\draw (62.center) to (61.center);
\draw (61.center) to (60.center);
\draw [qWire] (63.center) to (64.center);
\draw [qWire, in=-90, out=90] (94.center) to (93.center);
\draw [qWire] (101.center) to (102.center);
\draw [cWire, in=-90, out=90] (103.center) to (104.center);
\draw [qWire, in=90, out=-90] (122.center) to (115.center);
\draw (107.center) to (111.center);
\draw (121.center) to (107.center);
\draw [cWire, in=-90, out=90] (106.center) to (117.center);
\draw (110.center) to (114.center);
\draw (114.center) to (113.center);
\draw [cWire, in=-90, out=90] (108.center) to (116.center);
\draw (127.center) to (124.center);
\draw (124.center) to (125.center);
\draw (125.center) to (126.center);
\draw (126.center) to (127.center);
\draw [cWire, in=-90, out=90] (128.center) to (130.center);
\draw [qWire, in=-90, out=90] (132.center) to (131.center);
\draw (136.center) to (138.center);
\draw (138.center) to (137.center);
\draw (137.center) to (136.center);
\draw [qWire] (139.center) to (140.center);
\draw [braceedge] (143.center) to (142.center);
\draw [qWire, in=-90, out=90] (146.center) to (145.center);
\draw (150.center) to (152.center);
\draw (152.center) to (151.center);
\draw (151.center) to (150.center);
\draw [qWire] (153.center) to (154.center);
\draw [qWire, in=-90, out=90] (156.center) to (155.center);
\draw [qWire, in=90, out=-90] (173.center) to (166.center);
\draw (159.center) to (162.center);
\draw (172.center) to (159.center);
\draw [cWire, in=-90, out=90] (158.center) to (168.center);
\draw (161.center) to (165.center);
\draw (165.center) to (164.center);
\draw [cWire, in=-90, out=90] (160.center) to (167.center);
\draw [braceedge] (176.center) to (175.center);
\draw (178.center) to (181.center);
\draw (181.center) to (180.center);
\draw (180.center) to (179.center);
\draw (179.center) to (178.center);
\end{pgfonlayer}
\end{tikzpicture} &\geq 0
\quad \text{,} \quad  \label{eq:hie1}\\
\begin{tikzpicture}[baseline=(current bounding box.center)]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (55) at (-4, 0) {};
\node [style=none] (56) at (-4, -1.25) {};
\node [style=right label] (57) at (-4, -1) {$V$};
\node [style=right label] (58) at (-6, -1) {$V$};
\node [style=none] (59) at (-5, -1.75) {$s$};
\node [style=none] (60) at (-3.25, -1.25) {};
\node [style=none] (61) at (-6.75, -1.25) {};
\node [style=none] (62) at (-5, -2.5) {};
\node [style=none] (63) at (-6, 0) {};
\node [style=none] (64) at (-6, -1.25) {};
\node [style=none] (93) at (-6, 3.25) {};
\node [style=none] (94) at (-6, 2) {};
\node [style=right label] (95) at (-6, 2.25) {$V$};
\node [style=none] (106) at (-6, 4.25) {};
\node [style=none] (107) at (-6.5, 3.25) {};
\node [style=none] (108) at (-4, 4.25) {};
\node [style=none] (110) at (-6.5, 4.25) {};
\node [style=none] (111) at (-3.5, 3.25) {};
\node [style=none] (112) at (-5, 3.75) {$\mathcal{P}$};
\node [style=none] (113) at (-3.5, 3.25) {};
\node [style=none] (114) at (-3.5, 4.25) {};
\node [style=none] (115) at (-4, 2) {};
\node [style=none] (116) at (-4, 5.5) {};
\node [style=none] (117) at (-6, 5.5) {};
\node [style=right label] (118) at (-6, 5) {$\bSet$};
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\node [style=none] (139) at (0, 0) {};
\node [style=none] (140) at (0, -1.25) {};
\node [style=none] (141) at (-2, -1.25) {$\cdots$};
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\node [style=none] (144) at (-2, -3.75) {$k$};
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\node [style=none] (163) at (1, 3.75) {$\mathcal{P}$};
\node [style=none] (164) at (2.5, 3.25) {};
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\draw (60.center) to (62.center);
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\draw [qWire] (63.center) to (64.center);
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\end{tikzpicture} &\geq 0
\quad \text{,} \label{eq:hie2}
\end{align}

for all distinct totally connected wirings $W$ and $W'$.

Here, two wirings are distinct if they do not give the same diagram under some permutation of the external wires. A wiring is totally connected if the diagram it gives rise to does not factorise.

The constraint that the wirings are distinct ensures that there is no redundancy within a particular level in the hierarchy. In addition, the condition that the wirings are totally connected ensures that the constraints they impose do not reduce to constraints at lower levels in the hierarchy. Enumerating and finding a simple description of the distinct totally connected wirings is left as an interesting open problem.

We see that the constraints that each level $k$ imposes are then of two types: (i) one where the process in the second layer is the product of $k$ copies of $\mathcal{P}$ – Eq. (<ref>) –, and (ii) one where the process in the second layer is the product of $k-1$ copies of $\mathcal{P}$ and two copies of $\mathcal{F}$ – Eq. (<ref>). Note that if we had more copies of $\mathcal{F}$ (and so fewer of $\mathcal{P}$) then it would be impossible to have a totally connected wiring, hence we need consider at most two copies of $\mathcal{F}$.

One particular example of a of constraint imposed by the hierarchy is

\begin{align}
&\tikzfig{Diagrams/const6} \geq 0\,. \label{eq:steestatmeas}
\end{align}

This condition, imposed first in Level 2, can be given the following interpretation: the PRM $\mathcal{P}$ must give valid probabilities on products of two steered states, each constructed from $s$.

§ THE OPTIMISATION PROBLEMS

The hierarchy of constraints presented in the previous subsection ultimately defines some convex cone. To see this, suppose $\sigma_1$ and $\sigma_2$ satisfy the compatibility constraints of Eqs. (<ref>) and (<ref>) for all $k$. Then, $r_1\sigma_1 + r_2\sigma_2$ will satisfy the constraints for all $r_1,r_2 \in \mathds{R}^+$.

The optimisation problem of Eq. (<ref>) is therefore carried out over $\Sigma$, the union of the cones $K[\mathcal{P}]$:
s ∈⋃_𝒫∈𝖯𝖺𝗋𝖬𝖾𝖺𝗌[ℱ]K[𝒫]  =:  Σ.

This allows us to cast the optimization problem in a deceptively simple form:

ℐ_max:=𝗌𝗎𝗉{  Diagrams/OptimisationProblem  |  s ∈Σ, u_V⊗u_V(s)=1}.

While this form of the optimisation may appear simple, determining membership of the set $\Sigma$ is computationally extremely difficult. Indeed, $\Sigma$ is defined as the union of a (potentially infinite) set of cones, each of which is defined by an infinite hierarchy of constraints. In the remaining of the paper we will see how to relax these constraints, to make the optimisation problem computationally tractable, and by so compute upper bounds to $\mathcal{I}_{max}$. Note that, since the objective function is linear, we can make this a convex optimisation problem by optimising over the convex closure of $\Sigma$.

In addition, one may wish to optimise the value of a Bell inequality where the cardinality of the input variables in the Bell test does not coincide with the number of settings in the fiducial measurement, i.e., $|X| \leq |\nSet|$ and/or $|Y| \leq |\nSet|$. In this case, only a subset $\iSet \subset \nSet$ of the control settings are of interest, and the relevant constraint is the existence of a partial PRM for $\iSet$. In this case, the Optimisation problem becomes:
\beginprob'op:orig
ℐ_max:=𝗌𝗎𝗉{  Diagrams/OptPartial  |  s ∈Σ_, u_V⊗u_V(s)=1},
where $\Sigma_{\iSet}$ is the set of potential states compatible with the existence of a partial PRM $\mathcal{P[F]}_{\iSet}$ for the settings  $\iSet \subseteq \nSet$.
\endprob'

§.§ A relaxation to Optimisation Problem <ref>

In this section we will specify a particular subset of constraints imposed by the hierarchy that defines $K[\mathcal{P}]$. We will focus on a particular set of minimum requirements to demand to the GPT, which are colloquially stated as:

* Valid states under local fiducial measurements – Eq. (<ref>),

* $\mathcal{P[F]}$ is a valid measurement on generic composite states – Eq. (<ref>),

* $\mathcal{P[F]}$ is a valid measurement on products of steered states – Eq. (<ref>),

* $\mathcal{P[F]}$ is a PRM – Eq. (<ref>).

Implementing these conditions is already computationally demanding, since we are optimising over the possible PRM $\mathcal{P[F]}$ and the possible bipartite states $s$ of the GPT.

The optimisation problem to solve therefore reads as follows:

\begin{align}
\mathcal{I}^\mathrm{R}_{max} \quad &= \quad \stackrel[\{s, \mathcal{P}\}]{}{\mathsf{sup}}\,\, \tikzfig{Diagrams/OptimisationProblem} \,\,,\\
&\textrm{s.t.} \begin{cases} \nonumber
\quad \tikzfig{Diagrams/const1} \geq 0\,,\\
\quad \tikzfig{Diagrams/const2New} \geq 0 \,,\\
\quad \tikzfig{Diagrams/const6New} \geq 0 \,,\\
\quad \tikzfig{Diagrams/ParFid1} = \tikzfig{Diagrams/ParFid2} \qquad \forall i\in\nSet\,.
\end{cases}
\end{align}

It is readily seen how the Optimisation Problem <ref> is a relaxation of the Optimisation Problem <ref> – a solution $\mathcal{I}^\mathrm{R}_{max}$ to the former will yield an upper bound to the solution $\mathcal{I}_{max}$ of the latter.

To be able to implement this sort of optimisation problem on a computer, we must switch from the high level diagrammatic description of the processes in a GPT, to a lower level tensorial representation. The method for doing this is presented in App. <ref>.

The constraints that appear in Optimisation Problem <ref> can then be recast by means of the tensor representation as constraints on real vectors. This lower level form of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), will be used when coding the scripts to carry out the numerical calculations of the next section.

Example 1.–

Consider the case where $n=2$. Here, as we discussed in Section <ref>, a normalised state $s$ can be fully parametrised as in Eq. (<ref>) by the vector of probabilities:

\begin{align}
\mathbf{p}_s = \left( p_s(0|0)\,,\, p_s(0|1)\,,\, 1 \right)^T\,,
\end{align}

where $p_s(a|x)$ is the probability that outcome $a$ is obtained when the fiducial measurement $x$ is performed on a system on state $s$. In a locally tomographic GPT, a bipartite system can be parametrised as follows:

\begin{align}
\label{eq:LTparam-ex1}
\mathbf{p}_s = \left( p_s(00|00)\,,\, p_s(00|10)\,,\, p^\mathrm{(2)}_s(0|0) \,,\, p_s(00|01)\,,\, p_s(00|11)\,,\, p^\mathrm{(2)}_s(0|1) \,,\, p^\mathrm{(1)}_s(0|0)\,,\, p^\mathrm{(1)}_s(0|1)\,,\, 1 \right)^T\,,
\end{align}

where $p^\mathrm{(j)}_s(a|x)$ denotes the marginal conditional probability of subsystem $j$, and $p_s(ab|xy)$ denotes the joint conditional probabilities. Note that these parameters are also precisely those required to characterise a no-signalling box with binary inputs and outputs. This is not a coincidence – indeed, this is precisely the no-signalling box that we will obtain when we measure this state with the fiducial measurement $\mathcal{F}$ on both systems. Hence, this parameterisation of the bipartite state and the form of $\mathcal{F}$ ensure that the observed correlations are no-signalling.

Using the tensorial notation described in App. <ref> and, in particular, the above parameterisation of the composite state (eq. (<ref>)), the constraints in Optimisation Problem <ref> can be recast as follows.

The first one, i.e., Eq. (<ref>), reads:

\begin{align}
p_s(ab|xy) \geq 0 \,\, \forall \,  a,b,x,y \in \{0,1\} \,, \quad
\sum_{a,b=0:1} p_s(ab|xy) = 1 \,\, \forall \, x,y \in \{0,1\} \,,
\end{align}

\begin{align}
\sum_{a=0:1} p_s(ab|xy) = p_s(b|y)\,\, \quad
\sum_{b=0:1} p_s(ab|xy) = p_s(a|y)\,\,
\quad
\forall \, a,b,x,y \in \{0,1\} \,.
\end{align}

That is, the outcome statistics of fiducial measurements on a state $s$ are a well-defined no-signalling normalised conditional probability distribution.

The second constraint, i.e., Eq. (<ref>), reads

\begin{align}
\sum_{vw} \mathcal{P}^{qr}_{vw} \, s^{vw} \geq 0 \quad \forall q,r \in \{0,1\}\,,
\end{align}

where $v$ and $w$ are the indices associated to the two GPT vector spaces $V$, and $q$ and $r$ are the indices associated to the classical outcomes of the parity reading measurement, that is, each corresponds to one of the parities which is being read.
Since $s^{vw}$ is represented by a $9$-dimensional probability vector $\mathbf{p}_s$, $\mathcal{P}^{qr}_{vw}$ may be represented by a $4 \times 9$ matrix, $[\mathcal{P}]$.

The third constraint, i.e., Eq. (<ref>), reads:

\begin{align}
\sum_{v_1,w_1,v_2,w_2} \mathcal{F}_{xv_1}^{a} \, \mathcal{P}^{rq}_{w_1 v_2} \, \mathcal{F}_{yw_2}^{b} \, s^{v_1w_2} \, s^{v_2w_2} \geq 0 \quad \forall x,y,q,r,a,b \in \{0,1\} \,,
\end{align}

where the indices $x$ and $y$ are the indices associated to the measurement settings of the two fiducial measurements, and the indices $a$ and $b$ to their outcomes.
Notice that the tensors $\mathcal{F}_{xv_1}^{a}$ and $\mathcal{F}_{yw_2}^{b}$  correspond to the definition of the fiducial effects for system of type $V$. Specifically, we can write that:
e_a|x = (ℱ^a_x0,ℱ^a_x1,ℱ^a_x2)  and  e_b|y = (ℱ^b_y0,ℱ^b_y1,ℱ^b_y2)  .

Hence, this equation can be further written as:

\begin{align}
[\mathcal{P}] \circ \left( (e_{a|x} \otimes \mathds{1}_V) \circ s \right) \otimes \left( (\mathds{1}_V \otimes e_{b|y}) \circ s \right) \geq 0 \quad \forall x,y,a,b \in \{0,1\} \,,
\end{align}

where by $\geq 0$ we mean that every element of the matrix must be $\geq 0$. This equivalent form of Eq. (<ref>) makes it clear to see that it indeed imposes that $\mathcal{P[F]}$ is a valid measurement on products of steered states, which are steered by fiducial measurements.
We can denote the (subnormalised) steered states explicitly by
\begin{align}
\label{eq:steered-ex1}
s^{(1)}_{b|y}:=(\mathds{1}_V \otimes e_{b|y}) \circ s \\
s^{(2)}_{a|x}:=(e_{a|x} \otimes \mathds{1}_V) \circ s.
\end{align}

Then, the condition (<ref>)
can be finally written as

\begin{align}
[\mathcal{P}] \circ  s^{(1)}_{b|x} \otimes
s^{(2)}_{a|x} \geq 0
\quad \forall x,y,a,b \in \{0,1\} \,,
\end{align}

that is, the parity reading measurement must give valid probabilities on products of steered states.

The last constraint, given by Eq. (<ref>), can be recast in the $n=2$ case as follows:

\begin{align}
\label{eq:PRMfidcond-2}
\sum_{q=0:1}  \mathcal{P}_{vw}^{qr} = \mathcal{F}^{0}_{1v} \mathcal{F}^{r}_{1w} + \mathcal{F}^{1}_{1v} \mathcal{F}^{1 \oplus r}_{1w} \quad \forall \, r\in\{0,1\} \,, \\
\sum_{r=0:1}  \mathcal{P}_{vw}^{qr} = \mathcal{F}^{0}_{0v} \mathcal{F}^{q}_{0w} + \mathcal{F}^{1}_{0v} \mathcal{F}^{1 \oplus q}_{0w} \quad \forall \, q\in\{0,1\}\,,
\end{align}

where $\oplus$ denotes sum mod 2. These tensorial equations indeed correspond to equality constraints between $9$-dimensional covectors:

\begin{align}
&u_{\boldsymbol{\beta}} \otimes \vec{r} \circ [\mathcal{P}] = e_{0|1} \otimes e_{r|1} + ( u_V - e_{0|1}) \otimes (u_V - e_{r|1}) \quad \forall \, r\in\{0,1\} \,, \\
& \vec{q} \otimes u_{\boldsymbol{\beta}} \circ [\mathcal{P}] = e_{0|0} \otimes e_{q|0} + ( u_V - e_{0|0}) \otimes (u_V - e_{q|0}) \quad \forall \, q\in\{0,1\} \,.
\end{align}

which can be straightforwardly verified by noting that:
u_ = (1,1) ,   0⃗ = (1,0) ,   1⃗ = (0,1) ,
and that
u_V=(ℱ_x0^0+ℱ_x0^1, ℱ_x1^0+ℱ_x1^1, ℱ_x2^0+ℱ_x2^1)  ∀ x∈{0,1}.

Optimisation Problem <ref>, despite being a relaxation of Optimisation Problem <ref> ,  still shares a common feature with the latter: they are both nonlinear optimisation problems. Indeed, we can see clearly in the formulation of Optimisation Problem <ref> how the constraints feature products of the variables being optimised over. Solutions to such polynomial optimisation problems may be approximated by standard techniques in the literature. Here, we will consider the hierarchy of semidefinite relaxations to polynomial optimisation problems given by Lasserre [64]. Each level of such hierarchy will give an upper bound to the solution $\mathcal{I}^\mathrm{R}_{max}$ of Optimisation Problem <ref>.

Optimisation Problem <ref> is formulated for the situations where the cardinality of the input variables in the Bell test match the number of settings in the fiducial measurement, i.e., $|X|=|Y|=|\boldsymbol{\eta}|$. However, as we mentioned in Section <ref>, this is not always necessarily the case. We will therefore next reformulate Optimisation Problem <ref> to encompass the case of partial PRMs.
This adjusted version of the optimisation problem will come in handy when exploring quantum correlations, since for example it allows the study of Bell inequalities with two measurement settings per wing (see, e.g., the CHSH scenario in which $|X|=|Y|=2$) on qubits (whose affine dimension is $3$ rather than $2$).
In addition, importantly, this adjusted version of the optimisation problem might allow us to make device-independent studies of the results, since do not require full knowledge of the dimension of the local systems to impose the constraint of existence of partial PRMs. Namely, we hope that the constraints for correlations
obeyed by set of local observables imposed by existence of PRM persists, regardless of the dimension of the system.

\beginprob'op:rel

Let $\iSet \subset \nSet$.

\begin{align}
\mathcal{I}^\mathrm{RP}_{max} \quad &= \quad \stackrel[\{s, \mathcal{P_\iSet}\}]{}{\mathsf{sup}}\,\, \tikzfig{Diagrams/OptQuant1} \,\,,\\
&\textrm{s.t.} \begin{cases} \nonumber
\quad \tikzfig{Diagrams/const1} \geq 0\,,\\
\quad \tikzfig{Diagrams/OptQuant2} \geq 0 \,,\\
\quad \tikzfig{Diagrams/OptQuant3} \geq 0 \,,\\
\quad \tikzfig{Diagrams/OptQuant4} = \tikzfig{Diagrams/OptQuant5} \qquad \forall i\in\iSet\subseteq\nSet\,.
\end{cases}
\end{align}

\endprob'
Notice finally that Optimisation Problem <ref> is indeed a relaxation of Optimisation Problem <ref>, in the same way that Optimisation Problem <ref> is a relaxation of Optimisation Problem <ref>.

Example 2.–

Consider the case where $n=3$. Here, as we discussed in Section <ref>, a normalised state $s$ can be fully parameterised as in Eq. (<ref>) by the vector of probabilities:

\begin{align}
\mathbf{p}_s = \left( p_s(0|0)\,,\, p_s(0|1)\,,\,p_s(0|2)\,,\, 1 \right)^T\,,
\end{align}

where $p_s(a|x)$ is the probability that outcome $a$ is obtained when the fiducial measurement $x$ is performed on a system on state $s$. In a locally tomographic GPT, a bipartite system can be parameterised as follows:

\begin{align} \nonumber
\mathbf{p}_s = \Big( &p_s(00|00)\,,\, p_s(00|10)\,,\,p_s(00|20)\,,\, p^\mathrm{(2)}_s(0|0) \,,\, p_s(00|01)\,,\, p_s(00|11)\,,\,p_s(00|21)\,,\, p^\mathrm{(2)}_s(0|1) \,,\\ &p_s(00|02)\,,\, p_s(00|12)\,,\,p_s(00|22)\,,\, p^\mathrm{(2)}_s(0|2) \,,\, p^\mathrm{(1)}_s(0|0)\,,\, p^\mathrm{(1)}_s(0|1)\,,\,p^\mathrm{(1)}_s(0|2)\,,\, 1 \Big)^T\,,
\end{align}

where $p^\mathrm{(j)}_s(a|x)$ denotes the marginal probability of subsystem $j$.

We will furthermore consider the case where $\iSet=\{0,1\}\subset \{0,1,2\}=\nSet$.
With this choice, using the tensorial notation of App. <ref>, the constraints in Optimisation Problem <ref> can be recast as follows. The first one, i.e., Eq. (<ref>),  reads:

\begin{align}
&p_s(ab|xy) \geq 0 \,\, \forall \,  a,b \in \{0,1\},\ x,y \in \{0,1,2\} \,, \quad
\sum_{a,b=0:1} p_s(ab|xy) = 1 \,\, \forall \, x,y \in \{0,1,2\} \,,\\
&\sum_{a=0:1} p_s(ab|xy) = p_s(b|y)\,\, \quad
\sum_{b=0:1} p_s(ab|xy) = p_s(a|y)\,\,
\quad
\forall \, a,b \in \{0,1\},x,y \in \{0,1,2\} \,.
\end{align}

That is, the outcome statistics of fiducial measurements on a state $s$ are a well-defined no-signalling normalised conditional probability distribution.

The second constraint, i.e., Eq. (<ref>), reads

\begin{align}
\sum_{vw} {\mathcal{P}_{\iSet}}^{qr}_{vw} \, s^{vw} \geq 0 \quad \forall q,r \in \{0,1\}\,.
\end{align}

Since $s^{ij}$ is represented by a $16$-dimensional probability vector $\mathbf{p}_s$, ${\mathcal{P}_{\iSet}}^{qr}_{vw}$ may be represented by a $4 \times 16$ matrix. Notice that the fact that ${\mathcal{P}_{\iSet}}$ is a partial PRM is captured by the fact that it only has two output systems – hence its matrix representation has four rows.

The third constraint, i.e., Eq. (<ref>), reads:

\begin{align}
\sum_{v_1,w_1,v_2,w_2} \mathcal{F}_{xv_1}^{a} \, {\mathcal{P}_{\iSet}}^{rq}_{w_1 v_2} \, \mathcal{F}_{yw_2}^{b} \, s^{v_1w_2} \, s^{v_2w_2} \geq 0 \quad \forall a,b,q,r \in \{0,1\},\ x,y\in\{0,1,2\} \,.
\end{align}

Notice that the tensors $\mathcal{F}_{xv_1}^{a}$ and $\mathcal{F}_{yw_2}^{b}$ actually correspond to the definition of the fiducial effects for system of type $V$. If we represent ${\mathcal{P}_{\iSet}}^{qr}_{vw}$ by a $4 \times 16$ matrix $[\mathcal{P}_{\iSet}]$,  hence, this equation can be further written as:

\begin{align}
[\mathcal{P}_{\iSet}] \circ \left( (e_{a|x} \otimes \mathds{1}_V) \circ s \right) \otimes \left( (\mathds{1}_V \otimes e_{b|y}) \circ s \right) \geq 0 \quad \forall a,b \in \{0,1\},\ x,y \in \{0,1,2\} \,,
\end{align}

where by $\geq 0$ we mean that every element of the matrix must be $\geq 0$. This equivalent form of Eq. (<ref>) makes it clear to see that it indeed imposes that $\mathcal{P}_{\iSet}$ is a valid measurement on products of steered states, which are steered by fiducial measurements.

The last constraint, given by Eq. (<ref>), can be recast in the $n=3$ case as follows:

\begin{align}
\sum_{q=0:1}  {\mathcal{P}_{\iSet}}_{vw}^{qr} = \mathcal{F}^{0}_{1v} \mathcal{F}^{r}_{1w} + \mathcal{F}^{1}_{1v} \mathcal{F}^{1 \oplus r}_{1w} \quad \forall \, r\in\{0,1\} \,, \\
\sum_{r=0:1}  \mathcal{P_{\iSet}}_{vw}^{qr} = \mathcal{F}^{0}_{0v} \mathcal{F}^{q}_{0w} + \mathcal{F}^{1}_{0v} \mathcal{F}^{1 \oplus q}_{0w} \quad \forall \, q\in\{0,1\}\,,
\end{align}

where $\oplus$ denotes sum mod 2. These tensorial equations indeed correspond to equality constraints between $16$-dimensional covectors:

\begin{align}
&u_{\boldsymbol{\beta}} \otimes \vec{r} \circ [\mathcal{P}] = e_{0|1} \otimes e_{r|1} + ( u_V - e_{0|1}) \otimes (u_V - e_{r|1}) \quad \forall \, r\in\{0,1\} \,, \\
& \vec{q} \otimes u_{\boldsymbol{\beta}} \circ [\mathcal{P}] = e_{0|0} \otimes e_{q|0} + ( u_V - e_{0|0}) \otimes (u_V - e_{q|0}) \quad \forall \, q\in\{0,1\} \,.
\end{align}

§ MAIN CONJECTURE, AND NUMERICAL EXPLORATION

In Ref. [24] it was shown that existence of the Bell measurement for two systems, each with three observables (i.e., for $|\nSet|=3$) imposes that there are no post-quantum correlations.
Here we want to pose the more general problem of whether parity readable observables can lead to post-quantum correlations.

Firstly, let us restate (colloquially)  the aforementioned

Suppose that a bipartite system satisfies local tomography and no-signaling.
Moreover, suppose that product of states is a valid state. Then the observables for which there exists parity reading measurement
do not violate any Bell inequality more than quantum mechanics does.

Formally, the conjecture, if true, means that for any Bell inequality the Optimization Problem <ref> returns at most the quantum  bound. Note, however, that we do not always expect OP<ref> to actually reach the quantum bound, as the maximal quantum value is not necessarily achieved by observables which are parity readable within quantum theory.

In this section we shall summarize our numerical results which support (or at least do not falsify) the conjecture.
Moreover, we will provide an analytical proof that GPTs which violate local tomography admit PR-box correlations under the constraints of OP<ref> – that is, OP<ref> may yield a value of $\frac{1}{2}$ for the CHSH inequality (in the notation of Eq. (<ref>)) within non-tomographically local GPTs.
This, however, does not necessarily imply that non-tomographically local GPTs violate Conjecture <ref>, since the actual optimisation problem to be solved – OP<ref> – imposes additional constraints to those appearing in OP<ref>.

Our numerical explorations focus on the relaxed problems OP<ref> and  OP<ref>, depending on the cardinalities of $\bSet$ and $\nSet$.
As mentioned in the previous section, we will approximate the solution to OP<ref> by means of a hierarchy of semidefinite relaxations formulated by  Lasserre [64], using mostly the Lasserre hierarchy levels 1+AB and 2.
We will now provide a brief explanation of what these two levels mean, and refer the reader to Ref. [64] for a thorough exposition.
Each Lasserre hierarchy level is related to the semidefiniteness of a matrix (whose definition we will not give here), and the rows and columns of this matrix have particular labels depending on what level we are focusing on. Let $\Upsilon$ be the set that contains the variables we are optimising over plus the element $1$. In the first level of the Lasserre hierarchy, the matrix under study has row and columns labelled by the elements of $\Upsilon$. In the second level of the hierarchy, however, the matrix under study is of much larger size, and its rows and columns are labelled by the elements of $\Upsilon \times \Upsilon$, where $\times$ denotes the Cartesian product of sets. The so-called 1+AB level lies in between the first and the second – the matrix corresponding to 1+AB is a sub-matrix of that of level 2, and  the matrix corresponding to level 1 is a sub-matrix of that of 1+AB. In particular, the rows and columns of the matrix corresponding to level 1+AB are labelled by the elements of the set $\Upsilon \times \Upsilon \setminus \{{(\upsilon, \upsilon)} | \upsilon \in \Upsilon\}$.

All numerical computations was performed with Python 3.7. SDP relaxation of polynomial programming was calculated with package Ncpol2sdpa [65]. SDP problem was solved using SDPA [66]

§.§ CHSH inequality

In a bipartite Bell scenario featuring two dichotomic measurements per party, the most studied inequality is the  Clauser, Horne,  Shimony, Holt (CHSH) inequality  [25], which, using the notation of
Eq. (<ref>), is:

\begin{align}
\label{eq:chsh}
I_{\text{CHSH}}(\mathbf{p}_s)=-p_s^{(1)}(0|0)-p_s^{(2)}(0|0)+p_s(00|00)+p_s(00|10)+p_s(00|01)-p_s(00|11)\, .
\end{align}

This inequality is bounded from above, and the corresponding classical, quantum, and non-signalling bounds are:

\begin{align}
\beta_{\text{CHSH}}^\text{C}=0\,,\quad \beta_{\text{CHSH}}^\text{Q}=\frac{\sqrt{2}-1}{2}\sim 0.2071\,, \text{ {and}}\quad \beta_{\text{CHSH}}^\text{NS}=\frac{1}{2}\,.
\end{align}

Note that in this section we shall make a slight abuse of notation, and denote
Alice and Bob's observables  by $X,Y,Z$, which is not to be confused with the use of
$X$ and $Y$ to denote the sets of inputs.

The numerical results presented in this subsection are summarized in Fig. <ref>.

Case $|\iSet|=|\nSet|=2$. This is the simplest possible problem, where there are only two observables per party (that is, Alice has two observables $X,Z$, and the same for Bob) and the PRM measures the two parities $XX$ and $ZZ$.
We approximated the solution of OP<ref> applied to the CHSH inequality, by applying a Lasserre SDP relaxation with hierarchy level `a bit lower than 1+AB'. We will specify shortly what this means, but will first elaborate on the specific parameterisation we chose for OP<ref>.
The state is described by 8 parameters coming from local tomography (see Eq. (<ref>)) which we recall here in a more compact, matrix notation:

\begin{equation}
\label{eq:ps-matrixform}
\mathbf{p_s}=\left[ \begin{array}{ccc}
p_s(00|00)&p_s(00|01)   & p_s^{\mathrm{(1)}}(0|0)   \\
p_s(00|10)& p_s(00|11)  &  p_s^\mathrm{(1)}(0|1)  \\
p_s^\mathrm{(2)}(0|0)& p_s^\mathrm{(2)}(0|1)  & 1
\end{array}
\right].
\end{equation}

The (unnormalized) Alice states steered by Bob
given by Eq. (<ref>) are expressed as:

\begin{align}
s^\mathrm{(1)}_{0|0} &= (p_s(00|00), p_s(00|10), p^\mathrm{(2)}(0|0))^T \,, \nonumber \\
s^\mathrm{(1)}_{1|0} &= (p^\mathrm{(1)}(0|0) - p_s(00|00), p^\mathrm{(1)}(0|1) - p_s(00|10), 1 - p^\mathrm{(2)}(0|0))^T \,,
\nonumber \\
s^\mathrm{(1)}_{0|1} &= (p_s(00|01), p_s(00|11),p^\mathrm{(2)}(0|1))^T \,, \text{ and}
\nonumber \\
s^\mathrm{(1)}_{1|1} &= (p^\mathrm{(1)}(0|0) - p_s(00|01), p^\mathrm{(1)}(0|1) - p_s(00|11), 1 - p^\mathrm{(2)}(0|1))^T \,. \label{eq:st}
\end{align}

Bob's states steered by Alice have the same form as in Eq. (<ref>) but exchanging $XZ\leftrightarrow ZX$  and $\mathrm{(1)}\leftrightarrow \mathrm{(2)}$.
Now, the constraint that PRM measures parity
(given by Eq. (<ref>))

\begin{align}
\mathcal{P}^{00}+  \mathcal{P}^{01}
=\left[ \begin{array}{ccc}
2 & 0 & -1  \\
0 & 0 & 0 \\
-1 & 0 & 1
\end{array} \right]\,,
\nonumber \\
\mathcal{P}^{10}+  \mathcal{P}^{11}
=\left[ \begin{array}{ccc}
0 & 0 &  0  \\
0 & 2 & -1 \\
0 & -1 & 1
\end{array} \right]\,.
\end{align}

This can be obtained from Eq. (<ref>) as follows. First we have, for example,

\begin{equation}
(\mathcal{P}^{00}+  \mathcal{P}^{01}) \cdot \mathbf{p}_s=
\end{equation}

where $\cdot$ represents the Frobenius inner product of the two matrices.
Then using

\begin{align}
p_s(00|00)+p_s(01|00) &= p^{(1)}(0|0)\,, \nonumber \\ p_s(11|00)+p_s(01|00) &= p^{(2)}(0|0) \,, \nonumber \\
p_s(00|00)+  p_s(01|00)+ p_s(10|00)+ p_s(11|00) &= 1\,,
\end{align}

we get

\begin{equation}
(\mathcal{P}^{00}+  \mathcal{P}^{01}) \cdot \mathbf{p}_s=
1+2 p_s(00|00) - p^{(1)}(0|0)-p^{(2)}(0|0)\,,
\end{equation}

which leads to the form above.
Preserving probability by PRM reads as

\begin{align}
\label{eq:id}
\mathcal{P}^{00}+  \mathcal{P}^{01} +
\mathcal{P}^{10}+  \mathcal{P}^{11}=
\left[ \begin{array}{ccc}
\end{array}
\right]=\Id\,.
\end{align}

Thus, the condition that $\mathcal{P}$ is a PRM is captured by the following free parameters:

\begin{align}
\label{eq:free}    \mathcal{P}^{00}-  \mathcal{P}^{01} -
\mathcal{P}^{10}+  \mathcal{P}^{11}
=\left[ \begin{array}{ccc}
c_{11}& c_{12} & c_{13}\\
c_{21}& c_{22} & c_{23}\\
c_{31}& c_{32} & c_{33}
\end{array}
\right]\equiv \cal C.
\end{align}

Finally, the constraints that we still need to impose are the positivity of PRM effects
both on the state, as well as on tensor products on all pairs of
steered states that can be obtained from it.
We see then that OP<ref> requires us to optimise over the free parameters (state $s$ given by Eq. (<ref>)
and $\cal C$), under the positivity constraints of the previous sentence.

Now, to approximate the solution to OP<ref>, we apply a particular level of the Lasserre hierarchy, which is slightly lower than the previously described 1+AB, and which we will denote by 1+AB${}^*$. Let $\Upsilon_p$ denote the set of free parameters given by given by Eq. (<ref>), and  $\Upsilon_{\mathcal{C}}$ that given by the free parameters in $\cal C$. Here, $\Upsilon = \Upsilon_p \bigcup \Upsilon_{\mathcal{C}} \bigcup \{1\}$. However, the matrix under study in the level we consider here has rows and columns labeled by the elements of $\left(\Upsilon \times \Upsilon\right) \setminus \left( \Upsilon_p \times \Upsilon_p \right) \setminus \left( \Upsilon_{\mathcal{C}} \times \Upsilon_{\mathcal{C}} \right)  $ – that is, it is a submatrix of that considered in level 1+AB.

The upper bound to OP<ref> given by the 1+AB${}^*$ level of the Lasserre hierarchy, gives a value of $\sim 0.2071$, which implies that $\mathcal{I}^\mathrm{R}_{max} = \frac{\sqrt{2}-1}{2}$. This equality follows from recalling that the Tsirelson's bound value can be achieved within Quantum theory by a Bell measurement, and hence yields a lower bound to $\mathcal{I}^\mathrm{R}_{max}$. In other words, here we recover Tsirelson's bound for the CHSH inequality.
To corroborate this numerical result, we approximated the solution to OP<ref> by the second level of the Lasserre hierarchy, and reached the same conclusion.

Case $|\iSet|=2$, $|\nSet|=3$.
Here we still assume that the PRM measures just two parities (i.e., those of $XX$ and $ZZ$), but now Alice and Bob have one more additional observable (i.e., Y). This is an important case, as it allows for the possibility that the constraints imposed by the existence of a PRM are sensitive to the dimension of the local systems – if this is not the case, then we are within a device-independent setup.

Our numerical results show that in this case Tsirelson's bound is also not violated. The results are computed exactly as in the previous case with $|\nSet|=2$: we upperbound the value of $\mathcal{I}^\mathrm{RP}_{max}$ via the 1+AB${}^*$ level of the Lasserre hierarchy, which yields Tsierlson's bound for the CHSH inequality as the solution to OP<ref>.

Case $|\iSet|=3$, $|\nSet|=3$.
Here, both parties have three observables, and the PRM measures the parity of all three of them.
Unfortunately, the number of parameters to be optimised over for solving OP<ref> is too large, and the complexity of the 1st level of the Lasserre hierarchy was already too high for our computational resources.
We have thus decided to provide partial numerical evidence for our conjecture, by considering special cases which involve additional restrictions on the PRMs. That is, instead of only demanding the existence of a PRM, we will further request that these PRM satisfy extra properties, which will ultimately reduce the number of free parameters in the optimisation. We will here consider two cases: requesting the PRMs to (i) only have four linearly independent outcomes, or (ii) depend only on  $XX$, $ZZ$, $YY$ and $\Id$.
What we mean precisely by these two cases will be explained below, and we will see how neither falsifies our conjecture.

Ad (i) Here we focus on the case in which we are no longer considering arbitrary PRMs, but, instead, only those that have a particular form (motivated by quantum theory). Specifically, we consider PRMs of eight outcomes (with outcome set $\bSet\times\bSet\times\bSet$) which arise from the classical post-processing of some four outcome measurement (with outcome set $\mathbf{B}$) – just like in the case of the Bell measurements previously discussed. These post-processings have the particular form

[style=white dot] (0) at (0, -1) ;
[style=none] (1) at (-1.75, 0.25) ;
[style=none] (2) at (0, 0.25) ;
[style=none] (3) at (1.75, 0.25) ;
[style=none] (4) at (0, -2.25) ;
[style=right label] (5) at (0, -2) $\mathbf{B}$;
[style=right label] (6) at (1.75, 0) $\mathbf{B}$;
[style=right label] (7) at (0, 0) $\mathbf{B}$;
[style=right label] (8) at (-1.75, 0) $\mathbf{B}$;
[style=small box] (9) at (-1.75, 1) $C_X$;
[style=small box] (10) at (0, 1) $C_Y$;
[style=small box] (11) at (1.75, 1) $C_Z$;
[style=none] (12) at (-1.75, 2.25) ;
[style=none] (13) at (0, 2.25) ;
[style=none] (14) at (1.75, 2.25) ;
[style=right label] (15) at (-1.75, 2) $\bSet$;
[style=right label] (16) at (0, 2) $\bSet$;
[style=right label] (17) at (1.75, 2) $\bSet$;

[cWire, in=150, out=-90] (1.center) to (0);
[cWire] (0) to (2.center);
[cWire, in=-90, out=30] (0) to (3.center);
[cWire] (0) to (4.center);
[cWire] (12.center) to (9);
[cWire] (9) to (1.center);
[cWire] (13.center) to (10);
[cWire] (10) to (2.center);
[cWire] (14.center) to (11);
[cWire] (11) to (3.center);

in which $\mathcal{C}_X$, $\mathcal{C}_Y$, and $\mathcal{C}_Z$ correspond to simple bipartitions of $\mathbf{B}$.

The quantum and mirror quantum examples describe two possible choices for these bipartitions, as presented in Eqs. (<ref>) and (<ref>) respectively.

Note that, at least for a fixed choice of post-processing, this substantially reduces the computational cost, as the number of effects which we must optimise over is reduced from $8$ down to just $4$.

Coming back to CHSH, we here approximate the solution to OP<ref> by solving the 1+AB level of the Lasserre hierarchy that relaxes it, with the additional constraint that the PRMs need to satisfy the above mentioned 4-outcome property. We focused on two types of post-processings, one given by “quantum” and one by “mirror quantum”, as presented in  Eqs. (<ref>) and (<ref>). Up to numerical precision, we found that $\mathcal{I}^\mathrm{R}_{max} (\text{4-outcome}) \leq \frac{\sqrt{2}-1}{2}$. Since there is a quantum realisation of the Tsirelson's bound that satisfies all these constraints, it follows that $\mathcal{I}^\mathrm{R}_{max} (\text{4-outcome}) = \frac{\sqrt{2}-1}{2}$.

Finally, we also explored other post-processings beyond those of `quantum' and `mirror quantum'.
They are determined by the following signatures:

\begin{align}
\{(-,-,-),(-,-,+),(+,+,-),(+,+,+)\}&,
\nonumber \\
\{(-,-,-),(-,+,-),(+,-,+),(+,+,+)\}&,
\nonumber \\
\{(-,-,-),(-,+,+),(+,-,-),(+,+,+)\}&,
\nonumber \\
\{(-,-,-),(-,+,+),(+,-,+),(+,+,-)\}&, \quad \text{(quantum)}
\nonumber \\
\{(-,-,+),(-,+,-),(+,-,-),(+,+,+)\}&, \quad\text{(mirror quantum)}
\nonumber \\
\{(-,-,+),(-,+,-),(+,-,+),(+,+,-)\}&,
\nonumber \\
\{(-,-,+),(-,+,+),(+,-,-),(+,+,-)\}&,
\nonumber \\
\{(-,+,-),(-,+,+),(+,-,-),(+,-,+)\}&.
\end{align}

Ad (ii)
Here we focus on another family of PRMs, which we denote as “those that only depend on the observables $XX$, $ZZ$, $YY$ and $I$". So let us begin by explaining what this means.

The expectation values of the observables $XX$, $ZZ$, $YY$ and $I$ are expressed in terms of the correlations (i.e., free parameters of the state vector) as:

\begin{eqnarray}
&& XX\cdot \mathbf{p}_s = E_{XX}^s=p_s(00|00)+p_s(11|00) -  p_s(01|00)-p_s(10|00)\,,
\nonumber \\
&&YY\cdot \mathbf{p}_s = E_{YY}^s=p_s(00|11)+p_s(11|11) -  p_s(01|11)-p_s(10|11)\,,  \nonumber \\
&&ZZ\cdot \mathbf{p}_s = E_{ZZ}^s=p_s(00|22)+p_s(11|22) -  p_s(01|22)+p_s(10|22)\,.
\end{eqnarray}

Given that

\begin{equation}
p_s(00|00)+p_s(11|00)=2 p(00|00) +1-p_s^{(1)}(0|0)-
\end{equation}

\begin{equation}
p_s(00|00)+p_s(11|00) +  p_s(01|00)+p_s(10|00) = 1\,,
\end{equation}

these expectation values can be expressed as:

\begin{eqnarray}
\label{eq:correlators}
&&E_{XX}^s = 4 p_s(00|00) - 2p_s^{(1)}(0|0)
-2 p_s^{(1)}(0|0) +1 \,,
\nonumber \\
&&E_{YY}^s = 4 p_s(00|11) - 2p_s^{(1)}(0|1)
-2 p_s^{(1)}(0|1) +1 \,,
\nonumber \\
&&E_{ZZ}^s = 4 p_s(00|22) - 2p_s^{(1)}(0|2)
-2 p_s^{(1)}(0|2) +1\,.
\end{eqnarray}

We can therefore associate to these product observables the following matrix representation:

\begin{align}
\label{eq:xxyyzz}
\left[
\begin{array}{cccc}
4 & 0 &0 & -2  \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
-2 & 0 & 0 & 1
\end{array}
\right],\,
\begin{array}{cccc}
0 & 0 &0 & 0  \\
0 & 4 & 0 & -2 \\
0 & 0 & 0 & 0 \\
0 & -2 & 0 & 1
\end{array}
\right], \,
\begin{array}{cccc}
0 & 0 & 0 & 0  \\
0 & 0 & 0 & 0 \\
0 & 0 & 4 & -2 \\
0 & 0 & -2 & 1
\end{array}
\right].
\end{align}

The constraint that we now impose on the PRMs is that each of their operators $\mathcal{P}^{ij}$
must be a linear combination of four matrices: the three from Eq. (<ref>), and the one from Eq. (<ref>) corresponding to $\Id$.
This means that the number of free parameters in the optimisation problem is reduced by $12$, and, hence, becomes tractable with the computational resources we had available.

Coming back to the CHSH inequality, we here approximate the solution to OP<ref> by solving the 1+AB level of the Lasserre hierarchy that relaxes it, with the additional constraint that the PRMs need to satisfy the linear combination constraint specified in the previous paragraph.
Up to numerical precision, we found that $\mathcal{I}^\mathrm{R}_{max} (\text{XX-YY-ZZ}) \leq \frac{\sqrt{2}-1}{2}$. Since there is a quantum realisation of the Tsirelson's bound that satisfies all these constraints, it follows that $\mathcal{I}^\mathrm{R}_{max} (\text{XX-YY-ZZ}) = \frac{\sqrt{2}-1}{2}$, which supports our conjecture.

§.§ AMP inequalities

In a bipartite Bell scenario featuring two dichotomic measurements per party, a relevant family of inequalities was defined by Acín, Massar, and Pironio (AMP) [35]. These correspond to tilted CHSH inequalities, and have been found to be useful for randomness `generation' [35]. In traditional language, the value assigned to the linear functional associated to the inequality reads:

\begin{align}
I_{\alpha,\gamma} = \langle \gamma \, A_0 + \alpha \, {A_0B_0} + \, \alpha {A_0B_1} + {A_1B_0} - {A_1B_1} \rangle \,,
\end{align}

where the parameters $\alpha$ and $\gamma$ satisfy: $\alpha \geq 1$, $\gamma \geq 0$, and $\gamma < 2$.
In our notation, that is, in terms of the probabilities, they are equivalently written as:

\begin{align} \nonumber
I_{\alpha,\gamma} = &-(2\alpha+\gamma) \, p_s^{(1)}(0|0) - (1+\alpha) \, p_s^{(2)}(0|0) + 2\alpha \, p_s(00|00)+ 2 \, p_s(00|01)  \\
&+ (1-\alpha) \, p_s^{(2)}(0|1) + 2\alpha \, \text{ and} p_s(00|10)  -2 \, p_s(00|11)\,.
\end{align}

These inequalities are bounded from above, and their corresponding classical, quantum, and non-signalling bounds when $\alpha \gamma \leq 2$ are:

\begin{align}
\beta_{\text{AMP}}^{\text{C}}(\alpha,\gamma) &= 2\alpha + \gamma,\\
\beta_{\text{AMP}}^{\text{Q}}(\alpha,\gamma) &= 2\sqrt{(1+\alpha^2)\left(1+\frac{\gamma^2}{4}\right)}\,, \text{ and}\\
\beta_{\text{AMP}}^{\text{NS}}(\alpha,\gamma) &= 2+2\alpha\,.
\end{align}

Here, we have only considered the case of
$|\iSet|=|\nSet|=2$. We considered values for $\alpha$ taken from the set
and then, for each $\alpha$, considered six different values for
$\gamma$ (see Sec. <ref> in the
We further focused on the optimisation problem OP<ref>, and implemented levels 1+AB and 2 of the Lasserre hierarchy.

In most cases, the optimisation
returns an upper bound for $\mathcal{I}^\mathrm{R}_{max}$ that is smaller than the inequality's Tsirelson's bound, which supports out conjecture. In addition, the remaining cases do not pose a disproof of the conjecture, since the relation between the values obtained for level 1+AB, together with the possible ones for level 2, suggest that higher levels of the Lasserre hierarchy may yield values within the desired range. In these cases, further numerical studies with increased computational capacity are required to make more definitive statements.
The numerical results are presented in Fig. <ref>.

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