# Quantum statistics in the minimal scenario

Victor Barizien  Université Paris Saclay, CEA, CNRS, Institut de physique
théorique, 91191 Gif-sur-Yvette, France  Jean-Daniel Bancal  Université Paris
Saclay, CEA, CNRS, Institut de physique théorique, 91191 Gif-sur-Yvette,
France

(August 28, 2024)

###### Abstract

In any given experimental scenario, the rules of quantum theory provide
statistical distributions that the observed outcomes are expected to follow.
The set formed by all these distributions contains the imprint of quantum
theory, capturing some of its core properties. So far, only partial
descriptions have been known for this set, even in the simplest scenarios.
Here, we obtain the analytical description of a complete set of quantum
statistics in terms of extremal points. This is made possible by finding all
bipartite quantum states and pairs of binary measurements which can be self-
tested, i.e. identified from statistics only. Our description provides a
direct insight into the properties and limitations of quantum theory. These
are not expressed in terms of Hilbert spaces, but rather directly in terms of
measurement observation statistics.

Quantum theory is surprising in many regards, one of which being its intrinsic
inability to predict the exact results of experiments, such as the outcome
that is going to be observed when a measurement is performed on a physical
system. Indeed, quantum physics only foretells the statistical distribution of
these outcomes and is, as such, a fundamentally probabilistic theory. Whereas
one might expect this to be a limitation of the theory, it turns out that the
probabilistic predictions of quantum theory often exceed classical and
deterministic ones, leading to new possibilities. The puzzle that a
probabilistic theory could somehow be more powerful than a deterministic one
has stimulated deep questions [1]. Today, the unpredictability of quantum
theory is considered a resource [2] and in light of this new perspective, a
natural question arises: what are the fundamental limits of quantum theory’s
probabilistic predictions?

Addressing this question requires distinguishing probabilistic predictions
that can admit a quantum explanation from those which don’t. Interestingly,
because quantum statistics can violate Bell inequalities [3], determining
whether a given set of probability distributions is compatible with quantum
theory turns out to be a highly nontrivial task. In fact, this problem amounts
to inverting Born’s rule [4], one of quantum theory’s cornerstones. This rule
expresses in simple terms the probability assigned by quantum theory to the
potential outcomes of a measurement as a function of a system’s _realization_
– including its state and measurement operators. While the statistics produced
by a quantum realization is unique, a set of quantum statistics can admit many
quantum realizations simultaneously, rendering their characterization
particularly challenging.

Recently, several efforts have highlighted the wide-ranging consequences of
determining the _quantum set_ – the set of quantum statistical predictions.
Beyond allowing to test whether experimental observations admit a quantum
description or not, it was suggested that by touching upon the limits of
quantum theory itself this description would give access to fundamental
principles satisfied by the theory. Similarly to the constancy of the speed of
light in relativity, such principles could open the way for a fully principle-
based formulation of quantum theory [5]. Several information principle
candidates have been proposed, such as no-signaling, information causality,
macroscopic locality or local orthogonality [6, 7, 8, 9]. So far, none of
these have succeeded in reproducing quantum predictions. Thus, although being
satisfied by the theory, the principles identified until now fall short in
providing a physical explanation for why quantum correlations are limited the
way they are and the search for such a quantum principle remains open.

Knowledge of the quantum set also has direct implications for quantum
applications. Indeed, by inverting Born’s rule without any a priori on the
underlying Hilbert space or on the description of the quantum system at hand,
this approach provides a way of accomplishing tasks that is fundamentally
trustful, in the sense that it is independent of the precise modeling of the
physical devices at hand [10]. This device-independent approach has been shown
to present an interest in entanglement detection and quantification [11, 12]
and is now a standard framework in numerous tasks. The security of adversarial
protocols particularly benefits from assessments that are insensitive to the
implementations details [13, 14, 15]. Since device-independent information
processing analyses rely on observed statistics, they depend directly on the
characterization of the quantum set.

More generally, it has been shown that points in the quantum set are pertinent
to a wide range of topics, including the study of correlations in many-body
systems [16] or of quantum computing advantage [17]. Determining the limits of
this set is thus crucial to identify new possibilities and limitations in
quantum information science.

Considering this problem already in the 1980s, Tsirelson was the first to
obtain bounds on the quantum set [18, 19]. Several results followed, notably
from Landau and Masanes [20, 21], until a major progress was achieved by
Navascués, Pironio and Acín (NPA) in the form of a hierarchy of semidefinite
programming [22, 23]. This hierarchy is now a central tool of quantum
information science [24, 25, 26], with an impact reaching optimization theory
[27]. Concretely, the hierarchy defines a family of problems of increasing
complexity which approximate better and better the quantum set from the
outside and guarantees convergence as the level of the hierarchy goes to
infinity. At a fixed hierarchy level, this technique allows deriving necessary
conditions for the quantum set [28], thus excluding that some behaviors admit
a quantum representation. However, since the NPA technique can generally not
guarantee that specific statistics are quantum, its implications remain
elusive on the boundary of the quantum set.

Recently, fresh insight was gained on the quantum set by analytical studies
which showed that it admits flat nonlocal boundaries [29, 30], as well as
pointy nonlocal extremal points [31, 32]. New curved regions of the set’s
boundaries were also identified analytically [33] and several conjectures were
formulated on the boundary of the quantum set [34, 35].

In this work, we consider the question of identifying the limits of the
quantum set from the perspective of self-testing [36]. Namely, a set of
statistics, or _behavior_ , is said to self-test a quantum realization with
state $\rho$ and measurements $\\{M\\}$ iff it can only be obtained through
Born’s rule by quantum realizations $(\rho^{\prime},\\{M^{\prime}\\})$ which
are related to $(\rho,\\{M\\})$ by local isometries [37]. Self-testing is a
powerful tool to analyze quantum behaviors: it has played an important role in
solving Connes’ embedding problem, a major conjecture of operator algebra
[38], and is a central part of several quantum protocols such as the
certification of quantum devices [39, 40] or delegated quantum computing [41,
42, 43, 44].

So far, numerous families of states have been shown to be self-testable
through some of their behaviors [45, 46]. However, only the statistics
obtained when measuring a maximally entangled state in the minimal scenario,
where two parties each equipped with two binary measurements, have been fully
characterized by self-testing [47]. These behaviors correspond to the boundary
of the quantum set in this scenario with vanishing marginals, which was
described earlier by Tsirelson, Landau and Masanes, see [48].

Here, we obtain all the self-tests in this scenario. This reveals previously
unknown boundaries of the quantum set. In turn, it allows us to identify all
extremal points and their corresponding quantum realizations, thus providing a
complete description of the quantum set in the minimal scenario.

Figure 1: Quantum measurements on a bipartite system produces outcomes which
follow a set of probability distributions that depend on the quantum state
$\rho_{AB}$ and on the measurement operators $A_{x},B_{y}$, as dictated by
Born’s rule. Here, we consider the set $\mathcal{Q}$ of all probability
distributions which can be obtained according to quantum theory in the
celebrated Clauser-Horne-Shimony-Holt (CHSH) scenario, where both parties can
choose between two possible measurement settings, i.e. $x,y=0,1$, and where
the outcomes of the measurements $\mathsf{a},\mathsf{b}=\pm 1$ are binary.
This includes all measurements on every possible quantum states. By
identifying all the extremal quantum behaviors, we obtain a description of
quantum theory in this setting.

We derive our results in the bipartite setting depicted in Fig. 1, in which
two users, Alice and Bob, each have access to a shared quantum state
$\rho_{AB}\in L(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$, where
$\mathcal{H}_{A/B}$ denotes their respective Hilbert space111Here,
$L(\mathcal{H})$ refers to the space of linear applications on the Hilbert
space $\mathcal{H}$.. In the scenario we consider, each party can perform one
of two possible measurements, denoted by $x,y=0,1$, each with two possible
outcomes, denoted $\mathsf{a},\mathsf{b}=\pm 1$, and described by the POVM
elements $P_{\mathsf{a},x}^{A}$, $P_{\mathsf{b},y}^{B}$ respectively. To each
POVM, one can associate hermitian operators, $A_{x}\in L(\mathcal{H}_{A})$ and
$B_{y}\in L(\mathcal{H}_{B})$, with eigenvalues in $[-1,1]$, defined as
$A_{x}=P_{1,x}^{A}-P_{-1,x}^{A}$ and $B_{y}=P_{1,y}^{B}-P_{-1,y}^{B}$. In this
setting, a behavior is fully determined according to Born’s rule by the eight
real parameters

$\displaystyle\langle A_{x}\rangle$ $\displaystyle=\tr(\rho_{AB}A_{x}\otimes\
\mathds{1}\ ),$ (1) $\displaystyle\langle B_{y}\rangle$
$\displaystyle=\tr(\rho_{AB}\ \mathds{1}\ \otimes B_{y}),$ (2)
$\displaystyle\langle A_{x}B_{y}\rangle$
$\displaystyle=\tr(\rho_{AB}A_{x}\otimes B_{y}),$ (3)

which encode a vector, or _point_ , $\vec{P}\in\mathbb{R}^{8}$. The set of all
such vectors $\vec{P}$ form the quantum set $\mathcal{Q}$ of interest. It
should be stressed that all states and measurements in arbitrary Hilbert
spaces $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, including infinite dimensional
Hilbert spaces, can be considered here. Hence, this set encodes all statistics
that can be achieved in this setting within the framework of quantum theory.

The fact that the Hilbert space dimension is unbounded allows simplifying the
analysis of $\mathcal{Q}$ in two ways. First, it ensures that the quantum set
is convex and therefore that it is fully described as the convex hull of its
extremal points: $\mathcal{Q}=\text{Conv}(\text{Ext}(\mathcal{Q}))$ [49]. We
can thus focus on the description of these extremal points, which are known to
be infinitely many. Second, it ensures by Naimark’s dilation theorem that
points of $\mathcal{Q}$ can be realized in terms of projective measurements
acting on pure states. The description of extremal points can then be further
simplified by Jordan’s lemma, which guarantees that they can be realized on
Hilbert spaces of dimension two whenever the parties hold two binary
measurements, i.e. $\text{Ext}(\mathcal{Q})\subset\mathcal{Q}_{2}$. A recent
result by Mikos-Nuszkiewicz and Kaniewski [35] further guarantees that such
realizations can be parametrized in the simple real form

$\displaystyle\ket{\phi_{\theta}}$
$\displaystyle=\cos{\theta}\ket{00}+\sin\theta\ket{11},$ (4a) $\displaystyle
A_{x}$ $\displaystyle=\cos a_{x}\sigma_{z}+\sin a_{x}\sigma_{x},$ (4b)
$\displaystyle B_{y}$ $\displaystyle=\cos b_{y}\sigma_{z}+\sin
b_{y}\sigma_{x}.$ (4c)

Considering the symmetries of the set $\mathcal{Q}$ allows us to finally
restrict our attention to parameters in the range

$\theta\in[0,\pi),\ 0\leq a_{0}\leq b_{0}\leq b_{1}<\pi,\ a_{0}\leq
a_{1}<\pi,$ (5)

see Appendix A for more details on these restriction steps.

Figure 2: Action of the non-linear steering map $T_{\theta}$ on Alice’s
measurements $A_{0}$ and $A_{1}$ when $\theta\in(0,\pi/4)$. Bob’s measurement
direction $B_{0}$ and $B_{1}$ are also represented in the same plane. The map,
defined as
$T_{\theta}:\ket{u}\mapsto\frac{\sqrt{2}(\innerproduct{\phi_{\theta}}{u})\ket{\phi^{+}}}{||\innerproduct{\phi_{\theta}}{u}||_{2}}$,
describes the steering of vectors in $\mathbb{C}^{2}$ through the state
$\ket{\phi_{\theta}}$. By construction, this transformation preserves the
normalization so that $T_{\theta}(\ket{u})T_{\theta}(\ket{u})^{\dagger}$ is a
unit projector for every normalized $\ket{u}$. However, the map does not
preserve the scalar product: in general,
$T_{\theta}(\ket{u})^{\dagger}T_{\theta}(\ket{u^{\prime}})\neq\innerproduct{u}{u^{\prime}}$.
For this reason, we associate to every projective measurement of Alice
$A_{x}=\outerproduct{u_{+1,x}}{u_{+1,x}}-\outerproduct{u_{-1,x}}{u_{-1,x}}$
two image measurements through this map, one for each projector:
$\tilde{A}^{\mathsf{a}}_{x}=\mathsf{a}(2T_{\theta}(\ket{u_{\mathsf{a},x}})T_{\theta}(\ket{u_{\mathsf{a},x}})^{\dagger}-\mathds{1})$.
When the initial measurement $A_{x}$ has an angle $a_{x}$ with respect to
$\sigma_{z}$ in the ($\sigma_{z},\sigma_{x}$)-plane, the modified measurements
still lie on this plane, but with angles
$\tilde{a}^{\mathsf{a}}_{x}=2\operatorname{atan}\left(\tan(\frac{a_{x}}{2})\tan(\theta)^{\mathsf{a}}\right)$
with respect to $\sigma_{z}$, as illustrated above. We say that Alice’s and
Bob’s measurement fully alternate if the measurement angles verify
$0\leq[\tilde{a}_{0}^{\mathsf{s}}]_{\pi}\leq
b_{0}\leq[\tilde{a}_{1}^{\mathsf{t}}]_{\pi}\leq b_{1}<\pi$ for all
$\mathsf{s},\mathsf{t}\in\\{-1,1\\}$, where
$[\alpha]_{\pi}:\equiv\alpha[\pi]$. In Appendix B, we show that the
expectation value of these modified operators on the maximally entangled state
$\ket{\phi^{+}}$ relate to the initial statistics as
$\bra{\phi^{+}}\tilde{A}^{\mathsf{a}}_{x}B_{y}\ket{\phi^{+}}=\frac{\langle
A_{x}B_{y}\rangle+\mathsf{a}\langle B_{y}\rangle}{1+\mathsf{a}\langle
A_{x}\rangle}$.

Our first result concerns the set $\mathcal{Q}_{2}$ of statistics that can be
obtained by measuring quantum systems of dimension 2. Due to the dimension
restriction, this set is non-convex [50], which makes it hard to characterize.
Defining the correlators $[\tilde{A}_{x}^{\mathsf{a}}B_{y}]=\frac{\langle
A_{x}B_{y}\rangle+\mathsf{a}\langle B_{y}\rangle}{1+\mathsf{a}\langle
A_{x}\rangle}$ for all $\mathsf{a}\in\\{\pm 1\\}$, $x,y\in\\{0,1\\}$, we have
the following necessary condition for pure realizations in $\mathcal{Q}_{2}$.

######  Proposition 1 (Necessary condition for pure projective realizations of
local dimension 2).

Any quantum point obtained from projective measurements on a pure entangled
two-qubit state verifies

$\begin{array}[]{r}-\pi\leq-\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{0}]-\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{0}]-\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[\tilde{A}_{0}^{\mathsf{s}}B_{1}]-\operatorname{asin}[\tilde{A}_{1}^{\mathsf{t}}B_{1}]\leq\pi\end{array}$
(6)

for all $\mathsf{s},\mathsf{t}\in\\{\pm 1\\}$.

The main idea of the proof consists in considering the quantum state
$\ket{\phi_{\theta}}$ as a non-linear map $T_{\theta}$ that brings an
arbitrary pure entangled qubit behavior in relation with 4 distributions
issued from the maximally entangled state, see Fig. 2. Constraints on
maximally entangled statistics derived by Masanes [21] then imply the
necessary constraints on the original statistics issued from a non-maximally
entangled state that are given here, see Section B.4 for more details.

Note that the correlators $[\tilde{A}_{x}^{\mathsf{a}}B_{y}]$ are well-
defined, except in the case where some of Alice’s marginals are equal to $\pm
1$. This can only happen when $\theta\equiv 0[\pi/2]$, which gives rise to
local points. Note also that the labeling of the subsystems is arbitrary here,
so additional necessary conditions can be obtained by exchanging Alice and
Bob.

One easily verifies that when the marginal of both parties are zero, the
inequalities above reduce to the 8 Masanes inequalities. These inequalities
are convex and identify the boundary of the quantum set in this restricted
CHSH scenario [48]. In the general case, however, the inequalities Eq. 48 do
not describe a convex set. Therefore, they cannot describe the set
$\mathcal{Q}$. Indeed, measurements on a high-dimensional state or on a mixed
two-qubit state can result in quantum points that do not satisfy the above
inequalities.

It has been shown that the saturation of the Masanes inequalities can only be
achieved by self-testing behaviors, which identify a singlet state and real
unitary measurements [47]. It is thus natural to ask whether the points
saturating some of the above conditions could lead to self-testing as well. It
turns out that saturating a single inequality is not sufficient. Similarly,
one can find non-extremal points saturating two inequalities. However,
whenever three of the conditions are met for different values of $(s,t)$, the
fourth one is also. Indeed, out of the above 32 conditions, only 24 are
linearly independent. In this special case, we show that the resulting
statistics self-test a qubit realization.

######  Lemma 1.

Any nonlocal behavior which satisfies

asin [~A_0^sB_0] +asin [~A_1^tB_0] - asin [~A_0^sB_1]+asin[~A_1^tB_1] =π

(7)

for all four $(\mathsf{s},\mathsf{t})\in\\{\pm 1\\}^{2}$, self-tests a quantum
realization of local dimension 2.

In particular, all realizations in the range Eq. 5 for which the measurement
angles fully alternates, i.e. verify

$0\leq[\tilde{a}_{0}^{\mathsf{s}}]_{\pi}\leq
b_{0}\leq[\tilde{a}_{1}^{\mathsf{t}}]_{\pi}\leq b_{1}<\pi$ (8)

satisfy, up to relabelling, the condition of 1. As such, those realizations
are self-tested by their associated quantum point, which is therefore extremal
in $\mathcal{Q}$ [29]. Note that conversely, satisfying Eq. 7 for all
$(\mathsf{s},\mathsf{t})\in\\{\pm 1\\}^{2}$ self-tests realizations that fully
alternate (up to relabeling).

The proof of 1, given in Appendix C, strongly relies on the steering
transformation. This time, the nonlinear transformation is used to map self-
testing statements on a maximally entangled state into a self-testing
statement on a partially entangled one. This is made possible by strong
geometrical relations between the vectors
$\tilde{A}_{x}^{\mathsf{a}}\ket{\phi^{+}}$ and $B_{y}\ket{\phi^{+}}$ issued
from the steered realization when Eq. 7 holds.

Note that the condition of nonlocal behavior implies that the marginals cannot
take the value $\pm 1$, which guarantees that the correlators
$[\tilde{A}_{x}^{\mathsf{s}}B_{y}]$ are well-defined. Indeed, any vector with
a single marginal probability equal to $\pm 1$ would have more than two zeros
of probabilities on the same line or column of the probability table, ensuring
that the quantum point is local [30].

1 identifies many extremal points of the quantum set and provides self-
testings for all qubit partially-entangled states with a large family of
measurement settings. However, it does not say whether points which do not
satisfy the equality case are extremal in $\mathcal{Q}$. In order to address
this, we consider the complementary question of showing that realizations
which do not satisfy the full alternation condition Eq. 8 lead to behaviors
that are non-extremal. As a first step in this direction, we prove that in the
absence of full alternation, the statistics are non-exposed, i.e. that they do
not uniquely reach the Tsirelson bound of any Bell expression. Note that since
non-exposed points include both decomposable points and some extremal points,
this result in itself does not allow concluding yet.

######  Lemma 2.

Consider a quantum realization with state $\ket{\phi_{\theta}}$ and
measurements satisfying Eq. 5. If the following series of inequalities does
not hold:

$\forall\mathsf{s},\mathsf{t},\
0\leq[\tilde{a}_{0}^{\mathsf{s}}]_{\pi}<b_{0}<[\tilde{a}_{1}^{\mathsf{t}}]_{\pi}<b_{1}<\pi,$
(9)

where
$\tilde{a}^{\mathsf{a}}_{x}=\operatorname{atan}(\tan(\frac{a_{x}}{2})\tan(\theta)^{\mathsf{a}})$,
and $[\alpha]_{\pi}:\equiv\alpha[\pi]$, then the corresponding point is non-
exposed in $\mathcal{Q}$.

The idea that we employ in Appendix D to prove that these points are not
exposed is to consider all possible Bell expressions, and to show for each of
them that the value achieved by the considered quantum point can be also
achieved by a distinct quantum point. In order to prove this, we first find
necessary conditions that Bell expressions maximized by the considered
statistics must satisfy. This allows reducing the dimension of the search
space for Bell expressions from 8 to 3. For all remaining Bell expressions, we
then provide a local point which gives the same Bell value as the considered
quantum point.

Equipped with Lemmas 1 and 2, let us consider a qubit realization of the form
Eqs. 4 and 5. If the measurement angles satisfy the alternating condition Eq.
8, then 1 ensures that the quantum realization is self-tested and therefore
the point is extremal. This shows that the alternating property Eq. 8 is
constitutive of these points: no point obtained with non-alternating settings
can be obtained with other, alternating settings. Using this property together
with 2 ensures that all realizations that do not verify Eq. 8 correspond to
points on the interior of the set of non-exposed points of $\mathcal{Q}$.
However, Straszewicz’s theorem states that all extremal points are limits of
exposed points [49]. This ensures that non-alternating realizations lead to
non-extremal points of $\mathcal{Q}$. This can be summarized in the following
theorem.

######  Theorem 1 (Characterization of $\text{Ext}(\mathcal{Q})$).

  1. 1.

A nonlocal point in the CHSH scenario is extremal in $\mathcal{Q}$ iff

$\forall\,\mathsf{u}\in\\{\pm
1\\}^{2},\sum_{x,y}\epsilon_{xy}\asin[\tilde{A}_{x}^{\mathsf{u}_{x}}B_{y}]=\pi$
(10)

for some $\epsilon_{xy}\in\\{\pm 1\\}$ such that
$\prod_{x,y}\epsilon_{xy}=-1$.

  2. 2.

A quantum realization leads to a nonlocal extremal point iff it can be mapped
by local channels and relabelings to a quantum realization on the entangled
state $\ket{\phi_{\theta}}$ with measurements satisfying Eq. 5 s.t.

$\forall(\mathsf{s},\mathsf{t})\in\\{\pm
1\\}^{2},0\leq[\tilde{a}_{0}^{\mathsf{s}}]_{\pi}\leq
b_{0}\leq[\tilde{a}_{1}^{\mathsf{t}}]_{\pi}\leq b_{1}<\pi.$ (11)

(a)  (b)  (c)  (d)

Figure 3: Geometrical representation of the realizations leading to extremal
points of the quantum set in the CHSH scenario. Up to local isometries and
relabelings, these realizations correspond to measurements on a partially
entangled two-qubit state $\ket{\phi_{\theta}}$ taking place in the
$(\sigma_{z},\sigma_{x})$-plane and verifying the fully alternating condition
Eq. 8. If the initial measurement angles alternate, i.e. verify $0\leq
a_{0}\leq b_{0}\leq a_{1}\leq b_{1}<\pi$, then the corresponding points are
extremal for all entanglement parameters $\theta$ in
$[\theta_{\text{min}},\pi/4]$, where the threshold entanglement value is given
by $\theta_{\min}=\max\\{\theta\leq\pi/4\text{ s.t.
}\tilde{a}_{x}^{\mathsf{a}}=b_{y}\\}$. As such, for $\theta=\pi/4$, pictured
in Fig. 3a, the point is extremal. For $\theta_{\min}<\theta<\pi/4$, pictured
in Fig. 3b, the orange areas and black arrows still alternate, so the point is
extremal. For $\theta=\theta_{\min}$, see Fig. 3c, the point is still extremal
but non-exposed in $\mathcal{Q}$. Last, for $\theta<\theta_{\min}$, like in
Fig. 3d, the black arrow of $B_{0}$ is inside an orange area: the fully
alternating property is broken, and the point admits a decomposition.

Note that the full alternance condition is equivalent whether the steering
transformation is applied on Alice’s measurements (as done here) or on Bob’s
ones, see Section B.3. Therefore, exchanging the role of Alice and Bob in
conditions Eqs. 10 and 11, with modified angles $\tilde{b}_{y}^{\mathsf{b}}$
and correlators $[A_{x}\tilde{B}_{y}^{\mathsf{b}}]$, identifies the same
extremal point.

Together with the 8 deterministic strategies [3], which are local extremal
points of $\mathcal{Q}$, 1 gives a complete characterization of the extremal
point of the quantum set in the minimal scenario. It also describes the
quantum realizations behind them in terms of measurements on two-qubit states,
as illustrated in Fig. 3. This description implies that all nonlocal extremal
points in this scenario self-test a two-qubit realization.

Note that the analytical description of the quantum set given by 1 provides
new insight on its geometry. Indeed, it implies that the 8-dimensional convex
set is generated by a 5-dimensional sub-manifold of extremal points. Indeed,
as discussed above, whenever Alice’s marginals are not zero, three out of the
four equations above are linearly independent. Furthermore, by 2, extremal
points admitting a probability equal to zero verify
$[\tilde{A}_{x}^{\mathsf{a}}B_{y}]=\pm 1$ for some $\mathsf{a},x,y$ and are
thus non-exposed.

###### Acknowledgements.

We thank Valerio Scarani for feedback on the manuscript.

## References

  * Einstein  _et al._ [1935] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?, Physical Review 47, 777 (1935).
  * Pironio  _et al._ [2010] S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, Random numbers certified by Bell’s theorem, Nature 464, 1021 (2010).
  * Brunner  _et al._ [2014] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics 86, 419 (2014).
  * Born [1926] M. Born, Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 38, 803 (1926).
  * Scarani [2012] V. Scarani, The device-independent outlook on quantum physics, Acta Physica Slovaca 62, 347 (2012).
  * Popescu and Rohrlich [1992] S. Popescu and D. Rohrlich, Generic quantum nonlocality, Physics Letters A 166, 293–297 (1992).
  * Pawłowski  _et al._ [2009] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, Information causality as a physical principle, nature 461, 1101 (2009).
  * Miguel and Harald [2010] N. Miguel and W. Harald, A glance beyond the quantum model, Proc. R. Soc. A 466, 881–890 (2010).
  * Fritz  _et al._ [2013] T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, Local orthogonality as a multipartite principle for quantum correlations, Nature Communications 4, 2263 (2013).
  * Acín  _et al._ [2007] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-independent security of quantum cryptography against collective attacks, Physical Review Letters 98, 230501 (2007).
  * Barreiro  _et al._ [2013] J. T. Barreiro, J.-D. Bancal, P. Schindler, D. Nigg, M. Hennrich, T. Monz, N. Gisin, and R. Blatt, Demonstration of genuine multipartite entanglement with device-independent witnesses, Nature Physics 9, 559–562 (2013).
  * Moroder  _et al._ [2013] T. Moroder, J.-D. Bancal, Y.-C. Liang, M. Hofmann, and O. Gühne, Device-independent entanglement quantification and related applications, Physical Review Letters 111, 030501 (2013).
  * Nadlinger  _et al._ [2022] D. P. Nadlinger, P. Drmota, B. C. Nichol, G. Araneda, D. Main, R. Srinivas, D. M. Lucas, C. J. Ballance, K. Ivanov, E. Y.-Z. Tan, P. Sekatski, R. L. Urbanke, R. Renner, N. Sangouard, and J.-D. Bancal, Experimental quantum key distribution certified by Bell’s theorem, Nature 607, 682 (2022).
  * Zhang  _et al._ [2022] W. Zhang, T. van Leent, K. Redeker, R. Garthoff, R. Schwonnek, F. Fertig, S. Eppelt, W. Rosenfeld, V. Scarani, C. C.-W. Lim, and H. Weinfurter, A device-independent quantum key distribution system for distant users, Nature 607, 687 (2022).
  * Liu  _et al._ [2022] W.-Z. Liu, Y.-Z. Zhang, Y.-Z. Zhen, M.-H. Li, Y. Liu, J. Fan, F. Xu, Q. Zhang, and J.-W. Pan, Toward a Photonic Demonstration of Device-Independent Quantum Key Distribution, Physical Review Letters 129, 050502 (2022).
  * Tura  _et al._ [2014] J. Tura, R. Augusiak, A. B. Sainz, T. Vértesi, M. Lewenstein, and A. Acín, Detecting nonlocality in many-body quantum states, Science 344, 1256 (2014).
  * Bravyi  _et al._ [2018] S. Bravyi, D. Gosset, and R. König, Quantum advantage with shallow circuits, Science 362, 308 (2018).
  * Cirel’son [1980] B. S. Cirel’son, Quantum generalizations of Bell’s inequality, Letters in Mathematical Physics 4, 93 (1980).
  * Tsirelson [1993] B. S. Tsirelson, Some results and problems on quantum bell-type inequalities, Hadronic Journal Supplement 8, 329–345 (1993).
  * Landau [1988] L. J. Landau, Empirical two-point correlation functions, Foundations of Physics 18, 449 (1988).
  * Masanes [2003] L. Masanes, Necessary and sufficient condition for quantum-generated correlations (2003), arxiv:quant-ph/0309137 .
  * Navascués  _et al._ [2007] M. Navascués, S. Pironio, and A. Acín, Bounding the Set of Quantum Correlations, Physical Review Letters 98, 010401 (2007).
  * Navascués  _et al._ [2008] M. Navascués, S. Pironio, and A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations, New Journal of Physics 10, 073013 (2008).
  * Skrzypczyk and Cavalcanti [2023] P. Skrzypczyk and D. Cavalcanti,  _Semidefinite Programming in Quantum Information Science_ (IOP Publishing, 2023).
  * Mironowicz [2024] P. Mironowicz, Semi-definite programming and quantum information, Journal of Physics A: Mathematical and Theoretical 57, 163002 (2024).
  * Tavakoli  _et al._ [2024] A. Tavakoli, A. Pozas-Kerstjens, P. Brown, and M. Araújo, Semidefinite programming relaxations for quantum correlations 10.48550/arXiv.2307.02551 (2024), arXiv:2307.02551 [quant-ph].
  * Bhardwaj  _et al._ [2021] A. Bhardwaj, I. Klep, and V. Magron, Noncommutative polynomial optimization (2021), arXiv:2109.00786 [math.OC] .
  * de Vicente [2015] J. I. de Vicente, Simple conditions constraining the set of quantum correlations, Physical Review A 92, 032103 (2015).
  * Goh  _et al._ [2018] K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai, Y.-C. Liang, and V. Scarani, Geometry of the set of quantum correlations, Physical Review A 97, 022104 (2018).
  * Chen  _et al._ [2023] K.-S. Chen, G. N. M. Tabia, J. Chellasamy, S. Mal, J.-Y. Wu, and Y.-C. Liang, Quantum correlations on the no-signaling boundary: self-testing and more, Quantum 7, 1054 (2023).
  * Barizien  _et al._ [2024] V. Barizien, P. Sekatski, and J.-D. Bancal, Custom bell inequalities from formal sums of squares, Quantum 8, 1333 (2024).
  * Barizien and Bancal [2024] V. Barizien and J.-D. Bancal, Extremal tsirelson inequalities (2024), arXiv:2401.12791 [quant-ph] .
  * Liu  _et al._ [2024] Y. Liu, H. Y. Chung, and R. Ramanathan, (almost-)quantum bell inequalities and device-independent applications (2024), arXiv:2309.06304 [quant-ph] .
  * Ishizaka [2018] S. Ishizaka, Necessary and sufficient criterion for extremal quantum correlations in the simplest Bell scenario, Physical Review A 97, 050102 (2018).
  * Mikos-Nuszkiewicz and Kaniewski [2023] A. Mikos-Nuszkiewicz and J. Kaniewski, Extremal points of the quantum set in the chsh scenario: conjectured analytical solution (2023), arXiv:2302.10658 [quant-ph] .
  * Mayers and Yao [2004] D. Mayers and A. Yao, Self testing quantum apparatus (2004), arxiv:quant-ph/0307205 .
  * Šupić and Bowles [2020] I. Šupić and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020).
  * Ji  _et al._ [2021] Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, Mip* = re, Communications of the ACM 64, 131–138 (2021).
  * Magniez  _et al._ [2006] F. Magniez, D. Mayers, M. Mosca, and H. Ollivier, Self-testing of quantum circuits, in  _Automata, Languages and Programming_, Lecture Notes in Computer Science, edited by M. Bugliesi, B. Preneel, V. Sassone, and I. Wegener (Springer, Berlin, Heidelberg, 2006) p. 72–83.
  * Sekatski  _et al._ [2018] P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Certifying the building blocks of quantum computers from bell’s theorem, Phys. Rev. Lett. 121, 180505 (2018).
  * Reichardt  _et al._ [2013] B. W. Reichardt, F. Unger, and U. Vazirani, Classical command of quantum systems, Nature 496, 456–460 (2013).
  * Gheorghiu  _et al._ [2015] A. Gheorghiu, E. Kashefi, and P. Wallden, Robustness and device independence of verifiable blind quantum computing, New Journal of Physics 17, 083040 (2015).
  * Hajdušek  _et al._ [2015] M. Hajdušek, C. A. Pérez-Delgado, and J. F. Fitzsimons, Device-independent verifiable blind quantum computation 10.48550/arXiv.1502.02563 (2015), arXiv:1502.02563 [quant-ph].
  * McKague [2016] M. McKague, Interactive proofs for $\mathsf{BQP}$ via self-tested graph states, Theory of Computing 12, 1–42 (2016).
  * Coladangelo  _et al._ [2017] A. Coladangelo, K. T. Goh, and V. Scarani, All pure bipartite entangled states can be self-tested, Nature Communications 8, 15485 (2017).
  * Šupić  _et al._ [2018] I. Šupić, A. Coladangelo, R. Augusiak, and A. Acín, Self-testing multipartite entangled states through projections onto two systems, New Journal of Physics 20, 083041 (2018).
  * Wang  _et al._ [2016] Y. Wang, X. Wu, and V. Scarani, All the self-testings of the singlet for two binary measurements, New Journal of Physics 18, 025021 (2016).
  * Le  _et al._ [2023] T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, Quantum Correlations in the Minimal Scenario, Quantum 7, 947 (2023).
  * Rockafellar [1970] R. T. Rockafellar,  _Convex Analysis_ (Princeton University Press, 1970).
  * Donohue and Wolfe [2015] J. M. Donohue and E. Wolfe, Identifying nonconvexity in the sets of limited-dimension quantum correlations, Physical Review A 92, 10.1103/physreva.92.062120 (2015).
  * Masanes [2005] L. Masanes, Extremal quantum correlations for n parties with two dichotomic observables per site (2005), arXiv:quant-ph/0512100 [quant-ph] .
  * Harris and Pandey [2016] S. J. Harris and S. K. Pandey, Entanglement and non-locality pmath 990/qic 890 (2016).

## SUPPLEMENTAL MATERIAL

In this supplemental material, we explain the technical details underlying the
results presented in the paper. In particular, Appendix A focuses on the
parametrization of the extremal points in the CHSH scenario, Appendix B on the
steering transformation $T_{\theta}$, while Appendices C and D provide
detailed proofs of Lemma 1 and 2 respectively.

##  Appendix A Sufficient parametrization for the extremal points of the
quantum set in the CHSH scenario

Quantum correlations are defined as the statistics that can be observed upon
performing arbitrary measurements on an arbitrary quantum state. As such,
describing the quantum set exhaustively requires in principle considering all
possible realizations (states and measurements) in every Hilbert space.
Thankfully, it is known that several realizations give rise to identical
statistics, and therefore a subset of all quantum realizations is sufficient
to reproduce all quantum correlations in a given Bell scenario. In turn,
realizations can be even further restricted when considering extremal quantum
correlations only. In this appendix, we discuss these restrictions in detail.
This leads us to the explicit parametrization given in A.2, which is
sufficient to reproduce all extremal points in the CHSH scenario.

This appendix is organised as follows. First, we discuss up to A.3 the way
that restrictions can be made on measurement operators without affecting the
set of generated statistics. Then, we incorporate restrictions on the state as
well as those due to discrete symmetries of the Bell scenario.

While we focus on the case of the CHSH scenario, several of the results
described here also apply to general Bell scenarios. Also, most of the steps
described in this appendix were already discussed in earlier works, see e.g.
[51, 35], however some of our proofs are more concise. We include a proof of
every restriction step for completeness.

######  Definition A.1.

A _binary measurement_ – a quantum measurement with two possible outcomes – is
described by an operator $A\in L(\mathcal{H})$ such that $A^{\dagger}=A$ and
$-\mathds{1}\preceq A\preceq\mathds{1}$. Furthermore, the measurement is
_projective_ iff $A^{2}=\mathds{1}$, in which case the only possible
eigenvalues of $A$ are $\pm 1$ and $A$ is unitary, i.e.
$A^{\dagger}A=AA^{\dagger}=\mathds{1}$.

######  Definition A.2.

_Quantum correlations in the CHSH scenario_ are the correlation vectors
$\vec{P}\in\mathbb{R}^{8}$ with components

$\displaystyle\langle A_{x}\rangle$ $\displaystyle=\tr(\rho_{AB}A_{x}\otimes\
\mathds{1}\ ),$ (12) $\displaystyle\langle B_{y}\rangle$
$\displaystyle=\tr(\rho_{AB}\ \mathds{1}\ \otimes B_{y}),$ (13)
$\displaystyle\langle A_{x}B_{y}\rangle$
$\displaystyle=\tr(\rho_{AB}A_{x}\otimes B_{y}),$ (14)

where $\rho_{AB}\in L(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$ is a quantum
state (satisfying $\rho_{AB}\succeq 0$, $\Tr(\rho_{AB})=1$) and $A_{x}\in
L(\mathcal{H}_{A})$, $B_{y}\in L(\mathcal{H}_{B})$ with $x,y\in{0,1}$ are
binary measurement operators. Each such vector encodes the statistics
corresponding to measuring the quantum state $\rho_{AB}$ with the
corresponding two pairs of binary measurements.

######  Remark.

It is convenient to arrange the components of $\vec{P}$ in a table with
elements indexed by $x,y\in\\{-1,0,1\\}$. Namely, defining
$A_{-1}=\mathds{1}$, $B_{-1}=\mathds{1}$, we can write

$\\{\vec{P}\\}_{xy}=\Tr(\rho_{AB}(A_{x}\otimes B_{y})),$ (15)

where the first element of this table,
$\\{\vec{P}\\}_{-1,-1}=\Tr(\rho_{AB})=1$, is a constant.

######  Remark.

This definition is equivalent to the definition of quantum correlations in
terms of full probabilities, which identifies distributions of the form
$p(\mathsf{a}\mathsf{b}|xy)=\Tr(\rho_{AB}(P_{\mathsf{a}|x}^{A}\otimes
P_{\mathsf{b}|y}^{B}))$, where $\rho_{AB}$ is the measured quantum state and
$P_{\mathsf{a}|x}^{A},P_{\mathsf{b}|y}^{B}$ are POVM elements on subsystems
$A$ and $B$, corresponding to the measurement choices $x$ and $y$ and outcomes
$\mathsf{a}$ and $\mathsf{b}$. Indeed, any binary outcome POVM
$P_{\mathsf{s}|t}$ verifying $P_{\mathsf{s}|t}\succeq 0$ and
$P_{0|t}+P_{1|t}=\mathds{1}$ is in one-to-one correspondence with a binary
measurement operator $X_{t}=P_{0|t}-P_{1|t}$ verifying $-\mathds{1}\preceq
X_{t}\preceq\mathds{1}$. Therefore, it is possible to express the statistics
in terms of the components of the correlation vector $\vec{P}$ as

$p(\mathsf{a}\mathsf{b}|xy)=\frac{1+\mathsf{a}\langle
A_{x}\rangle+\mathsf{b}\langle B_{y}\rangle+\mathsf{a}\mathsf{b}\langle
A_{x}B_{y}\rangle}{4}.$ (16)

Note that the measurement operator $X_{t}$ is unitary iff the corresponding
POVM is a projector.

######  Lemma A.1 (Naimark’s dilation theorem for binary measurement
operators).

For any binary measurement operator $A$ acting on a Hilbert space
$\mathcal{H}$, there exists a Hilbert space $\tilde{\mathcal{H}}$, an isometry
$V:\mathcal{H}\to\tilde{\mathcal{H}}$ and a unitary binary measurement
operator $\tilde{A}$ s.t.

$A=V^{\dagger}\tilde{A}V.$ (17)

###### Proof.

Consider the tensor product Hilbert space
$\tilde{\mathcal{H}}=\mathcal{H}\otimes\mathbb{C}^{2}$ and introduce the
following isometry and unitary binary measurement operator:

$\displaystyle
V=\sum_{i=0}^{1}\sqrt{\frac{1+(-1)^{i}A}{2}}\otimes\ket{i}_{\mathbb{C}^{2}},\quad\tilde{A}=\mathds{1}_{\mathcal{H}}\otimes\sigma_{z}.$
(18)

On can easily verify that the first term is well-defined as both $\frac{1\pm
A}{2}$ are positive operators, and $V^{\dagger}V=\mathds{1}_{\mathcal{H}}$ so
$V$ is an isometry, that $\tilde{A}^{2}=\mathds{1}_{\tilde{\mathcal{H}}}$ and
that property Eq. 17 is satisfied. ∎

######  Remark.

It is well known that Naimark’s dilation theorem also applies to measurements
involving an arbitrary number of measurements outcomes $k\geq 1$, see e.g.
[52, Proposition 9.6].

######  Corollary A.1 (Naimark’s dilation for two measurements).

For any two binary measurement described by operators $A_{0}$, $A_{1}$ acting
on a Hilbert space $\mathcal{H}$, there exists a Hilbert space
$\tilde{\mathcal{H}}$, an isometry $V:\mathcal{H}\to\tilde{\mathcal{H}}$ and
unitary binary measurement operators $\tilde{A}_{x}$ s.t.

$\forall x\in\\{0,1\\},\ A_{x}=V^{\dagger}\tilde{A}_{x}V.$ (19)

###### Proof.

We can apply Naimark’s dilation for the operator $A_{0}$. Therefore, there
exists a Hilbert space $\mathcal{H}_{1}$, an isometry
$V_{1}:\mathcal{H}\to\mathcal{H}_{1}$ and a unitary operator $A_{0}^{\prime}$
s.t. $A_{0}=V_{1}^{\dagger}A_{0}^{\prime}V_{1}$. Then one can define a
measurement operator $A_{1}^{\prime}=V_{1}A_{1}V_{1}^{\dagger}$ on
$\mathcal{H}_{1}$, verifying $-\mathds{1}\preceq
A_{1}^{\prime}\preceq\mathds{1}$. One can thus apply Naimark’s dilation to
$A_{1}^{\prime}$. This guarantees the existence of a Hilbert space
$\mathcal{H}_{2}$, an isometry $V_{2}:\mathcal{H}_{1}\to\mathcal{H}_{2}$ and a
unitary operator $\tilde{A}_{1}$ s.t.
$A_{1}^{\prime}=V_{2}^{\dagger}\tilde{A}_{1}V_{2}$. Let’s define
$\tilde{A}_{0}=V_{2}A_{0}^{\prime}V_{2}^{\dagger}+V_{2}V_{2}^{\dagger}-\mathds{1}_{\mathcal{H}_{2}}$.
Since $V_{2}^{\dagger}V_{2}=\mathds{1}_{\mathcal{H}_{1}}$, we have
$\tilde{A}_{0}^{2}=\mathds{1}_{\mathcal{H}_{2}}$, so $\tilde{A}_{0}$ is a
unitary operator. Now, we can define $V:=V_{2}\circ V_{1}$, verifying

$\forall x\in\\{0,1\\},\ A_{x}=V^{\dagger}\tilde{A}_{x}V$ (20)

where $\tilde{A}_{x}$ describe projective measurements. ∎

######  Remark.

Naimark’s dilation can be further extended to an arbitrary family of $m\geq 1$
measurements by repeating the above construction iteratively, see e.g. [52,
Theorem 9.8].

######  Corollary A.2 (Naimark’s dilation for the CHSH scenario).

Any quantum correlation in the CHSH scenario can be realized by _projective_
measurements, i.e. for all $\vec{P}\in\mathcal{Q}$, there exists a quantum
state $\rho_{AB}$ and unitary binary measurement operators
$(\tilde{A}_{x},\tilde{B}_{y})$ such that
$\vec{P}=\\{\Tr(\rho_{AB}(\tilde{A}_{x}\otimes\tilde{B}_{y}))\\}_{xy}$.

###### Proof.

Consider a vector $\vec{P}\in\mathcal{Q}$. It admits a realization involving a
shared state $\sigma_{AB}$ and binary measurement operators $(A_{x},B_{y})$.
Using A.1, one can define isometries
$V_{A(B)}:\mathcal{H}_{A(B)}\to\tilde{\mathcal{H}}_{A(B)}$ such that for all
$x,y$, $A_{x}=V_{A}^{\dagger}\tilde{A}_{x}V_{A}$ and
$B_{y}=V_{B}^{\dagger}\tilde{B}_{y}V_{B}$ with $\tilde{A}_{x}$ and
$\tilde{B}_{y}$ unitary binary measurement operators. Then the coefficients of
$\vec{P}$ are given by

$\begin{split}\\{\vec{P}\\}_{xy}&=\Tr(\sigma_{AB}(A_{x}\otimes B_{y}))\\\
&=\Tr(\sigma_{AB}(V_{A}^{\dagger}\tilde{A}_{x}V_{A}\otimes
V_{B}^{\dagger}\tilde{B}_{y}V_{B}))\\\ &=\Tr((V_{A}\otimes
V_{B})\sigma_{AB}(V_{A}^{\dagger}\otimes
V_{B}^{\dagger})(\tilde{A}_{x}\otimes\tilde{B}_{y}))\\\
&=\Tr(\rho_{AB}(\tilde{A}_{x}\otimes\tilde{B}_{y})),\end{split}$ (21)

where we defined the quantum state $\rho_{AB}:=(V_{A}\otimes
V_{B})\sigma_{AB}(V_{A}\otimes V_{B})^{\dagger}$ on the Hilbert space
$\tilde{\mathcal{H}}_{A}\otimes\tilde{\mathcal{H}}_{B}$. ∎

######  Remark.

Due to the previous extension, A.2 can be expanded to an arbitrary Bell
scenario, in which $n$ parties each have a $m_{i}$ measurement settings and
each measurement has $k_{i,j_{i}}$ possible outcomes, where $i=1,\ldots,n$ and
$j_{i}=1,\ldots,m_{i}$.

######  Remark.

At this stage, one could also apply a dilation to the state $\rho_{AB}$ to
show that any point $\vec{P}\in\mathcal{Q}$ can be realized by measuring a
_pure_ state with projective measurements.

######  Lemma A.2 (Jordan’s lemma).

For any pair of projective binary measurements $A_{0},A_{1}$ acting on a
Hilbert space $\mathcal{H}$, there exists an orthogonal basis of $\mathcal{H}$
such that $A_{0}$ and $A_{1}$ are block-diagonal with blocks of size at most
two. Furthermore, in this basis, $A_{0}$ and $A_{1}$ are real operators.

###### Proof.

Consider the hermitian operator $R=A_{0}+A_{1}$. For any eigenvector
$\ket{\psi}$ of $R$ of eigenvalue $\lambda\in\mathbb{R}$, one can consider the
subspace $V_{\psi}=\text{Vect}\langle\ket{\psi},A_{0}\ket{\psi}\rangle$. Then,
one of the two following cases happens:

  1. Case 1 :

$\dim V_{\psi}=1$. Then there exist $\mu\in\mathbb{C}$ s.t.
$A_{0}\ket{\psi}=\mu\ket{\psi}$. As $A_{0}$ is hermitian, $\mu$ has to be
real. Then,
$A_{1}\ket{\psi}=R\ket{\psi}-A_{0}\ket{\psi}=(\lambda-\mu)\ket{\psi}$. So
$\ket{\psi}$ is an eigenvector of both $A_{0}$ and $A_{1}$ of real eigenvalue
(block of size $1$).

  2. Case 2 :

$\dim V_{\psi}=2$. Then, since $A_{0}^{2}=\mathds{1}$, $V_{\psi}$ is
stabilized by $A_{0}$ and in this subspace $A_{0}|_{V_{\psi}}=\sigma_{x}$.
Furthermore, $A_{1}\ket{\psi}=\lambda\ket{\psi}-A_{0}\ket{\psi}$ belongs to
$V_{\psi}$ and since $A_{1}^{2}=\mathds{1}$, we have that
$A_{1}(A_{0}\ket{\psi})=\lambda A_{1}\ket{\psi}-\ket{\psi}$ also belongs to
$V_{\psi}$. Finally, $V_{\psi}$ is stabilized by both $A_{0}$ and $A_{1}$
(block of size 2) and all have real coefficients in the sub-basis.

The rest of the proof goes on finite recursion on the eigenvectors of $R$. At
each step, one has to find an eigenvector $\ket{\psi^{\prime}}$ orthogonal to
all previous subspaces, which is always possible since $R$ is hermitian. ∎

######  Corollary A.3 (Jordan’s lemma for the CHSH scenario).

Every quantum correlations in the CHSH scenario can be obtained as a convex
combination of realizations involving local Hilbert spaces of dimension $2$
and real projective measurements.

###### Proof.

For $\vec{P}\in\mathcal{Q}$, we consider a projective realization thanks to
A.2. Furthermore, one can assume, up to trivial dilation, that the dimensions
of $\mathcal{H}_{A(B)}$ are even. We can then apply Jordan’s lemma for each
party and assume that all block are of size 2. Thus, there exists local
projections $\Pi_{A}^{i}$, $\Pi_{B}^{j}$ on $\mathcal{H}_{A(B)}$ respectively
such that $A_{x}^{(i)}:=\Pi_{A}^{i}A_{x}\Pi_{A}^{i}$ are (2,2)-real matrices
and $A_{x}=\bigoplus_{i}A_{x}^{(i)}$, likewise for Bob. As such

$\begin{split}\\{\vec{P}\\}_{xy}&=\Tr(\rho_{AB}(A_{x}\otimes B_{y}))\\\
&=\Tr(\rho_{AB}(\bigoplus_{i}A_{x}^{(i)}\otimes\bigoplus_{j}B_{y}^{j}))\\\
&=\sum_{ij}\Tr((\Pi_{A}^{i}\otimes\Pi_{B}^{j})\rho_{AB}(\Pi_{A}^{i}\otimes\Pi_{B}^{j})(A_{x}^{(i)}\otimes
B_{y}^{j}))\\\ &=\sum_{ij}p^{ij}\\{\vec{P}_{ij}\\}_{xy},\end{split}$ (22)

where we defined
$p^{ij}:=\Tr((\Pi_{A}^{i}\otimes\Pi_{B}^{j})\rho_{AB}(\Pi_{A}^{i}\otimes\Pi_{B}^{j}))$,
$\\{\vec{P}_{ij}\\}_{xy}=\Tr(\rho_{AB}^{ij}(A_{x}^{(i)}\otimes B_{y}^{j}))$
and
$\rho_{AB}^{ij}:=(\Pi_{A}^{i}\otimes\Pi_{B}^{j})\rho_{AB}(\Pi_{A}^{i}\otimes\Pi_{B}^{j})/p^{ij}$,
which verify $\Tr(\rho_{AB}^{ij})=1$, $\rho_{AB}^{ij}\succeq 0$, $p^{ij}\geq
0$ and $\sum_{ij}p^{ij}=1$. ∎

######  Remark.

A.3 generalizes straightforwardly to Bell scenarios involving $n$ parties with
two binary measurements per party.

######  Corollary A.4 (Jordan’s lemma for extremal correlations in the CHSH
scenario).

Every extremal quantum correlations in the CHSH scenario can be realized by
measuring a single two-qubit state with real projective measurements.

###### Proof.

For an extremal point $\vec{P}\in\text{Ext}(\mathcal{Q})$, we can apply the
previous result to decompose $\vec{P}$ as a convex combination of correlations
$\vec{P}_{ij}$ obtained by measuring two qubit states with real unitary
operators. The extremality of $\vec{P}$ ensures that for all $i,j$,
$\vec{P}_{ij}=\vec{P}$ and as such any qubit realization of a given
$\vec{P}_{ij}$ provides a satisfying realization for $\vec{P}$. ∎

######  Proposition A.1 (Extremal points realization in the CHSH scenario).

Every extremal quantum correlations in the CHSH scenario can be realized by
measuring a real, pure, two-qubit state with real projective measurements.

###### Proof.

For $\vec{P}\in\text{Ext}(\mathcal{Q})$, one can apply A.4 to obtain a
realization of the form $(\rho_{AB},A_{x},B_{y})$ with $A_{x},B_{y}$ real
unitary operators and local Hilbert spaces of dimension 2. If $\rho_{AB}$ is
not equal to its complex conjugate $\bar{\rho}_{AB}$, then one can also
realize $\vec{P}$ by measuring the real state
$\tilde{\rho}_{AB}=(\rho_{AB}+\bar{\rho}_{AB})/2$, because all measurement
operators are real. Therefore, we can assume that $\rho_{AB}$ is real. A real
density matrix $\rho_{AB}$ can always be written as

$\rho_{AB}=\sum_{i}\lambda_{i}\outerproduct{\psi_{i}}{\psi_{i}}$ (23)

with $\lambda_{i}\in\mathbb{R}_{+}$, $\sum_{i}\lambda_{i}=1$, and where
$\ket{\psi_{i}}$ are real two-qubit states. Therefore, one has
$\vec{P}=\sum_{i}\lambda_{i}\vec{P}_{i}$, where $\vec{P}_{i}$ is obtained by
measuring $\outerproduct{\psi_{i}}{\psi_{i}}$ on $(A_{x},B_{y})$. The
extremality of $\vec{P}$ ensures that for all $i$, $\vec{P}_{i}=\vec{P}$, and
therefore the realization for any $i$ is a valid realization for $\vec{P}$. ∎

######  Remark.

An alternative proof of A.1 was recently given in [35, Appendix A].

######  Remark.

For the study of extremal point only, one does not necessarily need Naimark’s
dilation to impose that the measurement operators are unitary (see [51, Lemma
1] for an alternative argument).

######  Remark.

Corollaries A.3, A.4 and A.1 can be directly extended to Bell scenarios in
which $n$ parties each have two binary outcome measurements, proving that in
those scenarios all extremal quantum correlations can be achieved by measuring
real $n$-qubit states with real unitary measurements.

######  Lemma A.3 (Extremality under relabelings).

Consider a convex set $\mathcal{K}\subset\mathbb{R}^{n}$ stabilized by a
linear involution $S:\mathbb{R}^{n}\to\mathbb{R}^{n}$, i.e. such that
$S^{2}=1$ and $S(\mathcal{K})=\mathcal{K}$. Then, any point
$\vec{P}\in\mathcal{K}$ is extremal iff $S(\vec{P})$ is extremal.

###### Proof.

Suppose that $\vec{P}$ is extremal. For any convex decomposition
$S(\vec{P})=\lambda\vec{P}_{1}+(1-\lambda)\vec{P}_{2}$ of $S(\vec{P})$, we
have $\vec{P}=\lambda S(\vec{P}_{1})+(1-\lambda)S(\vec{P}_{2})$, because $S$
is linear and is an involution. By extremality of $\vec{P}$, we have
$\vec{P}=S(\vec{P}_{1})=S(\vec{P}_{2})$, and as such
$S(\vec{P})=\vec{P}_{1}=\vec{P}_{2}$ so $S(\vec{P})$ is extremal. Conversely,
one can use that $\vec{P}=S(S(\vec{P}))$. ∎

######  Remark.

The quantum set is invariant under the following involutions:

relabeling of parties: $\displaystyle S_{part}:A_{x}\longleftrightarrow
B_{x},$ (24a) relabeling of inputs: $\displaystyle
S_{in}:A_{0}\longleftrightarrow A_{1},$ (24b) relabeling of outputs:
$\displaystyle S_{out}:A_{0}\longrightarrow-A_{0}.$ (24c)

As such, the study of whether a quantum correlation $\vec{P}$ is extremal can
be reduced to the study of the extremality of any correlations of the form
$g\cdot\vec{P}$ where $g$ is an arbitrary element of the Bell group $G$
generated by $S_{part},S_{in},S_{out}$ (a group of order 32).

######  Proposition A.2 (Sufficient parametrization of extremal points in the
CHSH scenario).

The study of the extremal points of $\mathcal{Q}$ can be reduced to the study
of the extremality of correlations admitting a realization of the form:

state:
$\displaystyle\ket{\phi_{\theta}}=\cos(\theta)\ket{00}+\sin(\theta)\ket{11},$
(25a) measurements: $\displaystyle
A_{x}=\cos(a_{x})\sigma_{z}+\sin(a_{x})\sigma_{x},$ (25b) $\displaystyle
B_{y}=\cos(b_{y})\sigma_{z}+\sin(b_{y})\sigma_{x}.$ (25c)

where the real parameters $\theta,a_{x},b_{y}$ can be reduced, upon relabeling
symmetries, to the range

$\theta,a_{x},b_{y}\in[0,\pi),\ a_{0}\leq a_{1},\ b_{0}\leq b_{1},\ a_{0}\leq
b_{0}.$ (26)

###### Proof.

Thanks to A.1, all extremal points of $\mathcal{Q}$ admit a realization of the
form Eq. 25 for arbitrary real parameters $\theta,a_{x},b_{y}$. The
periodicity of the corresponding correlations allows reducing those to the
interval $\theta\in[0,\pi)$ and $a_{x},b_{y}\in[0,2\pi)$. Using A.3, one can
study the nature of such correlations by applying arbitrary symmetries of
$\mathcal{Q}$. In particular, one can use relabeling of outcomes to ensure
that $a_{x},b_{y}\in[0,\pi)$, relabeling of inputs to have $a_{0}\leq a_{1}$
and $b_{0}\leq b_{1}$ and relabeling of parties for $a_{0}\leq b_{0}$. ∎

##  Appendix B Steering of quantum realizations

In this appendix, we describe in more details the steering map introduced in
Fig. 2 of the main text. In the first section, we consider its action on
vectors of $\mathbb{C}^{2}$. In the second section, we discuss the action of
the map on qubit measurements and relate the statistics before and after the
transformation. In both cases, our analysis applies to arbitrary qubit states
and measurement settings (i.e. not necessarily real ones).

###  B.1 Steering of states

The non-linear steering map $T_{\theta}:\mathbb{C}^{2}\to\mathbb{C}^{2}$ is
defined by the action

$T_{\theta}:\ket{u}\mapsto\frac{\sqrt{2}(\innerproduct{\phi_{\theta}}{u})\ket{\phi^{+}}}{||\innerproduct{\phi_{\theta}}{u}||}$
(27)

on state $\ket{u}\in\mathcal{H}_{A}=\mathbb{C}^{2}$. In particular, in the
canonical parametrization

$\ket{u}=\cos(\frac{a}{2})\ket{0}+e^{i\psi}\sin(\frac{a}{2})\ket{1},$ (28)

the steering transformation returns

$T_{\theta}(\ket{u})=\frac{\cos(\theta)\cos(\frac{a}{2})\ket{0}+e^{i\psi}\sin(\theta)\sin(\frac{a}{2})\ket{1}}{\sqrt{\cos(\theta)^{2}\cos(\frac{a}{2})^{2}+\sin(\theta)^{2}\sin(\frac{a}{2})^{2}}}=\epsilon_{\theta}\left(\cos(\frac{\tilde{a}}{2})\ket{0}+e^{i\psi}\sin(\frac{\tilde{a}}{2})\ket{1}\right)$
(29)

where $\epsilon_{\theta}:=\text{sgn}(\cos(\theta))$ and $\tilde{\alpha}$ is
the transformed state angle. Note that the denominator can only cancel if
$\theta\in\\{0,\pi/2\\}$ and $a=2\theta\pm\pi$. Whenever this happens, we
define $T_{\theta}(\ket{u})$ to be the null vector (since this only happens
when $\ket{\phi_{\theta}}$ is a product state, we won’t be interested in this
case). The angle of the image state in the canonical basis is given by

$\tilde{\alpha}:=2\operatorname{atan}(\tan(\frac{a}{2})\tan(\theta))\in[-\pi,\pi].$
(30)

Note that in all cases the image vectors satisfy
$\langle\phi_{\theta}|u\rangle=\sqrt{2}||\innerproduct{\phi_{\theta}}{u}||\langle\phi^{+}|T_{\theta}(\ket{u})\rangle$,
and as such

$\begin{split}\forall B\in L(\mathcal{H}_{B}),\
\bra{\phi_{\theta}}\left(\outerproduct{u}{u}\otimes
B\right)\ket{\phi_{\theta}}={2||\innerproduct{\phi_{\theta}}{u}||^{2}}\bra{\phi^{+}}\left(\outerproduct{v}{v}\otimes
B\right)\ket{\phi^{+}},\end{split}$ (31)

where for clarity we denoted $\ket{v}=T_{\theta}(\ket{u})$. Finally, one can
notice that the inverse of the steering function is easy to compute. Indeed,
we have:

$T_{\pi/2-\theta}(T_{\theta}(\ket{u}))=\ket{u}.$ (32)

###  B.2 Steering of correlations and realizations

When considering binary non-degenerate projective measurements acting on a
pure state with local Hilbert spaces of dimension 2, the measured state can
always be written as $\ket{\phi_{\theta}}$ in the Schmidt basis. In this
basis, the projective measurements can be written as

$\begin{split}&A_{x}=\cos(a_{x})\sigma_{z}+\sin(a_{x})(\cos(\alpha_{x})\sigma_{x}+\sin(\alpha_{x})\sigma_{y}),\\\
&B_{y}=\cos(b_{y})\sigma_{z}+\sin(b_{y})(\cos(\beta_{y})\sigma_{x}+\sin(\beta_{y})\sigma_{y}),\end{split}$
(33)

and can be decomposed as the difference of two orthogonal projectors:

$\begin{split}&A_{x}=\outerproduct{u_{+,x}}{u_{+,x}}-\outerproduct{u_{-,x}}{u_{-,x}},\\\
&B_{y}=\outerproduct{w_{+,y}}{w_{+,y}}-\outerproduct{w_{-,y}}{w_{-,y}},\end{split}$
(34)

where
$\innerproduct{u_{\mathsf{a},x}}{u_{\mathsf{a}^{\prime},x}}=\delta_{\mathsf{a}\mathsf{a}^{\prime}}$,
$\innerproduct{w_{\mathsf{b},x}}{w_{\mathsf{b}^{\prime},x}}=\delta_{\mathsf{b}\mathsf{b}^{\prime}}$
and $\ket{u_{\mathsf{a},x}},\ket{w_{\mathsf{b},y}}\in\mathbb{C}^{2}$ are in
general complex vectors of the form Eq. 28 with $a\in(-\pi,\pi]$.

The expected probabilities
$p(\mathsf{a}\mathsf{b}|xy)=\frac{1}{4}(1+\mathsf{a}\langle
A_{x}\rangle+\mathsf{b}\langle B_{y}\rangle+\mathsf{a}\mathsf{b}\langle
A_{x}B_{y}\rangle)$ corresponding to this realization are given by

$p(\mathsf{a}\mathsf{b}|xy)=\bra{\phi_{\theta}}\left(\outerproduct{u_{\mathsf{a},x}}{u_{\mathsf{a},x}}\otimes\outerproduct{w_{\mathsf{b},y}}{w_{\mathsf{b},y}}\right)\ket{\phi_{\theta}}.$
(35)

Applying the steering map on the eigenstates $\ket{u_{\mathsf{a},x}}$ of
Alice’s measurements and using Eq. 31, we obtain:

$p(\mathsf{a}\mathsf{b}|xy)={2p(\mathsf{a}|x)}\bra{\phi^{+}}\left(\outerproduct{v_{\mathsf{a},x}}{v_{\mathsf{a},x}}\otimes\outerproduct{w_{\mathsf{b},y}}{w_{\mathsf{b},y}}\right)\ket{\phi^{+}},$
(36)

where $\ket{v_{\mathsf{a},x}}=T_{\theta}(\ket{u_{\mathsf{a},x}})$ for all
$\mathsf{a},x$.

Note that the vectors $\ket{v_{\mathsf{a},x}}$ are units or null but in full
generality $\ket{v_{1,x}}$ and $\ket{v_{-1,x}}$ are not orthogonal anymore. We
thus define a new unitary operator on Alice’s Hilbert space for each vector
$\ket{v_{\mathsf{a},x}}$:

$\tilde{A}^{\mathsf{a}}_{x}=\mathsf{a}\left(2\ket{v_{\mathsf{a},x}}\bra{v_{\mathsf{a},x}}-\mathds{1}\right).$
(37)

These new operators verify that for all $\mathsf{a},x,y$:

$\frac{\langle\phi_{\theta}|A_{x}B_{y}|\phi_{\theta}\rangle+\mathsf{a}\langle\phi_{\theta}|B_{y}|\phi_{\theta}\rangle}{1+\mathsf{a}\langle\phi_{\theta}|A_{x}|\phi_{\theta}\rangle}=\bra{\phi^{+}}\tilde{A}^{\mathsf{a}}_{x}B_{y}\ket{\phi^{+}}=:[\tilde{A}^{\mathsf{a}}_{x}B_{y}],$
(38)

which is well-defined as long as $\langle A_{x}\rangle\neq\pm 1$, i.e. at
least whenever $\theta\notin\\{0,\pi/2\\}$.

Upon non-linear transformation $T_{\theta}$, the original correlation vector
$\vec{P}$, obtained with a pure and projective two qubit realization, can thus
be interpreted as correlations of a quantum point $\vec{Q}$ realized by the
state $\ket{\phi^{+}}$, four measurements $\tilde{A}^{\mathsf{a}}_{x}$ for
Alice and two measurements $B_{y}$ for Bob:

$\vec{P}=\begin{array}[]{c|c|c}&\langle B_{0}\rangle&\langle B_{1}\rangle\\\
\hline\cr\langle A_{0}\rangle&\langle A_{0}B_{0}\rangle&\langle
A_{0}B_{1}\rangle\\\ \hline\cr\langle A_{1}\rangle&\langle
A_{1}B_{0}\rangle&\langle A_{1}B_{1}\rangle\end{array}\xrightarrow{\quad
T_{\theta}\quad}\vec{Q}=\begin{array}[]{c|c|c}&0&0\\\ \hline\cr
0&[\tilde{A}^{+}_{0}B_{0}]&[\tilde{A}^{+}_{0}B_{1}]\\\ \hline\cr
0&[\tilde{A}^{-}_{0}B_{0}]&[\tilde{A}^{-}_{0}B_{1}]\\\ \hline\cr
0&[\tilde{A}^{+}_{1}B_{0}]&[\tilde{A}^{+}_{1}B_{1}]\\\ \hline\cr
0&[\tilde{A}^{-}_{1}B_{0}]&[\tilde{A}^{-}_{1}B_{1}]\end{array}$ (39)

Note that all marginal terms of this new quantum point are obtained using the
property that marginals correlations are $0$ when measuring $\ket{\phi^{+}}$:
$\bra{\phi^{+}}\tilde{A}^{\mathsf{a}}_{x}\ket{\phi^{+}}=\bra{\phi^{+}}B_{y}\ket{\phi^{+}}=0$,
therefore both behaviors have the same number of degrees of freedom.

The point $\vec{Q}$ admits a realization that can be parameterized by

$\begin{split}\text{state:
}&\ket{\phi^{+}}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{11}),\\\ \text{measurements:
}&\tilde{A}^{\mathsf{a}}_{x}=\cos(\tilde{a}^{\mathsf{a}}_{x})\sigma_{z}+\sin(\tilde{a}^{\mathsf{a}}_{x})(\cos(\alpha_{x})\sigma_{x}+\sin(\alpha_{x})\sigma_{y}),\
\tilde{a}^{\mathsf{a}}_{x}\in[0,2\pi),\\\
&B_{y}=\cos(b_{y})\sigma_{z}+\sin(b_{y})(\cos(\beta_{y})\sigma_{x}+\sin(\beta_{y})\sigma_{y}),\
b_{y}\in[0,2\pi),\\\ \end{split}$ (40)

where if one assumes that $a_{x}\in[0,\pi)$ the new measurement angles are
given by

$\tilde{a}^{\mathsf{a}}_{x}=2\operatorname{atan}\left(\tan(\frac{a_{x}}{2})\tan(\theta)^{\mathsf{a}}\right).$
(41)

Note that when the initial state is the singlet state ($\theta=\pi/4$), then
the modified measurements are the same, i.e.
$\tilde{a}^{\mathsf{a}}_{x}=a_{x}$ for all $\mathsf{a},x$. The limit
$\theta\rightarrow 0$ is well-defined and gives modified angles
$\tilde{a}_{x}^{-1}=0$ when $a_{x}=0$ and $\tilde{a}_{x}^{-1}=\pi$ otherwise.
Furthermore, when $\theta\in[0,\pi/4]$, we have

$\displaystyle 0\leq\tilde{a}_{x}^{+}\leq a_{x}\leq\tilde{a}_{x}^{-}\leq\pi,$
(42a) $\displaystyle a_{x}\longrightarrow\tilde{a}_{x}^{+}\text{ is a
increasing function of $\theta$},$ (42b) $\displaystyle
a_{x}\longrightarrow\tilde{a}_{x}^{-}\text{ is an decreasing function of
$\theta$}.$ (42c)

and when $\theta\in(\pi/4,\pi/2)$:

$\displaystyle 0\leq\tilde{a}_{x}^{-}\leq a_{x}\leq\tilde{a}_{x}^{+}\leq\pi,$
(43a) $\displaystyle a_{x}\longrightarrow\tilde{a}_{x}^{+}\text{ is a
decreasing function of $\theta$},$ (43b) $\displaystyle
a_{x}\longrightarrow\tilde{a}_{x}^{-}\text{ is an increasing function of
$\theta$}.$ (43c)

###  B.3 Steering on Alice vs Bob

Everything in this appendix about the steering transformation could be done on
Bob’s side as well. As such, it is possible to associate to each measurement
$B_{y}$ two modified measurements $\tilde{B}_{y}^{\mathsf{b}}$ on Bob’s
Hilbert space such that one can map the initial realization on a partially
entangled state to a realization on a maximally entangled state. Those
modified measurements would also have modified angles with respect to
$\sigma_{z}$, following the same transformation

$\tilde{b}^{\mathsf{b}}_{y}=2\operatorname{atan}\left(\tan(\frac{b_{y}}{2})\tan(\theta)^{\mathsf{b}}\right),$
(44)

for all $\mathsf{b},y$. The realization associated to these measurements
$(A_{x},\tilde{B}_{y}^{\mathsf{b}})$ with angles $a_{x}$ and
$\tilde{b}_{y}^{\mathsf{b}}$ is in general different from the one involving
$(\tilde{A}_{x}^{\mathsf{a}},B_{y})$ where the steering is applied on Alice’s
side: the value of the correlators $[A_{x}\tilde{B}_{y}^{\mathsf{b}}]$ take
different values than the $[\tilde{A}_{x}^{\mathsf{a}}B_{y}]$.

However, the fact that the modified measurements angles are fully alternating
is unchanged by whether the transformation is applied on Alice or Bob’s side.
Indeed, for a fixed entanglement parameter $\theta$, the steering
transformation of the measurements angles verify the following:

  1. 1.

It is monotonous, i.e. for all $\mathsf{s}\in\\{\pm 1\\}$, $a\leq
b\Longrightarrow[\tilde{a}^{\mathsf{s}}]_{\pi}\leq[\tilde{b}^{\mathsf{s}}]_{\pi}$.

  2. 2.

Modifying an angle twice in a row with opposite outcome sign gives back the
original angle, as
$2\operatorname{atan}(\tan([2\operatorname{atan}(\tan(a/2)\tan(\theta))]/2)\tan(\theta)^{-1})=a$.

With those two properties, one can verify that

$\forall a,b\in[0,\pi),\ [a^{\mathsf{s}}]_{\pi}\leq b\Longleftrightarrow
a\leq[b^{-\mathsf{s}}]_{\pi}$ (45)

by application of the steering transformation. As such, the fully alternating
condition when the steering is applied on Alice’s measurement angles is
equivalent to the full alternating condition after application of the steering
condition on Bob’s side:

$\left(\forall\mathsf{s},\mathsf{t},\ 0\leq[a_{0}^{\mathsf{s}}]_{\pi}\leq
b_{0}\leq[a_{1}^{\mathsf{t}}]_{\pi}\leq
b_{1}<\pi\right)\Longleftrightarrow\left(\forall\mathsf{s},\mathsf{t},\ 0\leq
a_{0}\leq[b_{0}^{\mathsf{s}}]_{\pi}\leq
a_{1}\leq[b_{1}^{\mathsf{t}}]_{\pi}<\pi\right).$ (46)

###  B.4 Proof of Proposition 1

A subset of $\mathcal{Q}$ that is very well characterized is the subspace of
zero marginals distributions. In this case, the following statement holds:

######  Proposition B.1 (Masanes Theorem, [21]).

In the CHSH scenario, the set of quantum points with zero marginals is exactly
the set of points verifying the following eight inequalities:

$\begin{array}[]{r}-\pi\leq-\operatorname{asin}\langle
A_{0}B_{0}\rangle+\operatorname{asin}\langle
A_{1}B_{0}\rangle+\operatorname{asin}\langle
A_{0}B_{1}\rangle+\operatorname{asin}\langle A_{1}B_{1}\rangle\leq\pi\\\
-\pi\leq\operatorname{asin}\langle
A_{0}B_{0}\rangle-\operatorname{asin}\langle
A_{1}B_{0}\rangle+\operatorname{asin}\langle
A_{0}B_{1}\rangle+\operatorname{asin}\langle A_{1}B_{1}\rangle\leq\pi\\\
-\pi\leq\operatorname{asin}\langle
A_{0}B_{0}\rangle+\operatorname{asin}\langle
A_{1}B_{0}\rangle-\operatorname{asin}\langle
A_{0}B_{1}\rangle+\operatorname{asin}\langle A_{1}B_{1}\rangle\leq\pi\\\
-\pi\leq\operatorname{asin}\langle
A_{0}B_{0}\rangle+\operatorname{asin}\langle
A_{1}B_{0}\rangle+\operatorname{asin}\langle
A_{0}B_{1}\rangle-\operatorname{asin}\langle
A_{1}B_{1}\rangle\leq\pi.\end{array}$ (47)

Since any point obtained by measuring a maximally entangled two qubits state
$\ket{\phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$ has zero marginals, it must
verify the above inequalities.

###### Proof of Proposition 1.

For any point $\vec{P}\in\mathcal{Q}$ which admits a pure projective two qubit
realization, the steering transformation allows one to consider a non-linear
transformation of $\vec{P}$ granting a vector of correlation
$\vec{Q}=\\{[A_{x}^{\mathsf{a}}B_{y}]\\}_{\mathsf{a},x,y}$ with zero
marginals, in a scenario with four binary measurements for Alice and two
binary measurement for Bob. Thus, for each pair
$(A_{0}^{\mathsf{s}},A_{1}^{\mathsf{t}})$ of Alice’s new measurements, B.1
holds and grants:

$\begin{array}[]{r}-\pi\leq-\operatorname{asin}[A_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[A_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[A_{0}^{\mathsf{s}}B_{0}]-\operatorname{asin}[A_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[A_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[A_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{0}]-\operatorname{asin}[A_{0}^{\mathsf{s}}B_{1}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{1}]\leq\pi\\\
-\pi\leq\operatorname{asin}[A_{0}^{\mathsf{s}}B_{0}]+\operatorname{asin}[A_{1}^{\mathsf{t}}B_{0}]+\operatorname{asin}[A_{0}^{\mathsf{s}}B_{1}]-\operatorname{asin}[A_{1}^{\mathsf{t}}B_{1}]\leq\pi\end{array}$
(48)

for all $\mathsf{s},\mathsf{t}\in\\{-1,1\\}$, where:

$[A_{x}^{\mathsf{a}}B_{y}]=\frac{\langle A_{x}B_{y}\rangle+\mathsf{a}\langle
B_{y}\rangle}{1+\mathsf{a}\langle A_{x}\rangle}$ (49)

∎

##  Appendix C Proof of 1

This section is organized in the following way: in the first two subsections
we prove some preliminary results. In particular, we prove in C.6 that the
conditions of 1 along with the assumption that the underlying realization is
pure, projective and and nondegenerate in $\mathcal{Q}_{2}$ is sufficient to
fully identify a unique realization. The third subsection is dedicated to the
proof of 1. To prove extremality, we consider convex decompositions into sub-
correlations. We make use of the steering transformation, introduced in
Appendix B, to identify four underlying realizations on four different states.
We then use geometrical considerations to understand how the conditions Eq. 7
relate these different realizations to each other. Finally, we transpose these
conclusions onto the original quantum realization and infer its properties.

###  C.1 Preliminary Geometrical Considerations

Let us recall some elements of geometry. Here, we consider normalized states
$\ket{\psi_{1}},\ket{\psi_{2}},\ket{\psi_{3}}$ in a real Hilbert space
$\mathcal{H}=\mathbb{R}^{d}$.

######  Definition C.1.

The angular distance between $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ is defined
as the angle $\theta\in[0,\pi]$ s.t.

$\cos\theta=\innerproduct{\psi_{1}}{\psi_{2}}.$ (50)

The angular distance $\theta$ is 0 iff $\ket{\psi_{1}}=\ket{\psi_{2}}$, $\pi$
iff $\ket{\psi_{1}}=-\ket{\psi_{2}}$, and $\pi/2$ iff $\ket{\phi_{1}}$ and
$\ket{\psi_{2}}$ are orthogonal.

######  Lemma C.1.

The pairwise angular distance between three vectors
$\ket{\psi_{1}},\ket{\psi_{2}},\ket{\psi_{3}}$ satisfies the triangle
inequality

$|\theta_{12}-\theta_{23}|\leq\theta_{13}\leq\theta_{12}+\theta_{23},$ (51)

with equality only if the vectors are coplanar.

###### Proof.

Clearly, two vectors define a plane, and therefore if either pair of states
are colinear ($\theta=0,\pi$), then all vectors are coplanar. Furthermore,
whenever $\theta_{ij}=0$ for some $i\neq j$, then $\theta_{ik}=\theta_{jk}$
($k\neq i,j$) and all versions of the inequalities Eq. 51 are verified.
Similarly, whenever $\theta_{ij}=\pi$ for some $i\neq j$, then
$\theta_{ik}=\pi-\theta_{jk}$ ($k\neq i,j$) and all versions of the
inequalities Eq. 51 are verified. We can thus assume that the angular distance
between every pair of states is strictly contained in the interval
$0<\theta_{ij}<\pi$.

In this case, $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ define a plane. In
particular, there exist two orthonormal vectors $\ket{\phi_{1}}$,
$\ket{\phi_{2}}$ such that

$\displaystyle\ket{\psi_{2}}$ $\displaystyle=\ket{\phi_{1}}$ (52)
$\displaystyle\ket{\psi_{1}}$
$\displaystyle=\cos\theta_{12}\ket{\phi_{1}}+\sin\theta_{12}\ket{\phi_{2}}.$
(53)

Without loss of generality, the third vector can then be written

$\ket{\psi_{3}}=\cos\theta_{23}\ket{\phi_{1}}+\sin\theta_{23}\cos\alpha\ket{\phi_{2}}+\sin\theta_{23}\sin\alpha\ket{\phi_{3}}$
(54)

for $\ket{\phi_{3}}$ orthogonal to $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$ and
an angle $\alpha\in[0,\pi]$.

The angular distance $\theta_{13}$ between $\ket{\psi_{1}}$ and
$\ket{\psi_{3}}$ then satisfies

$\displaystyle\cos\theta_{13}$
$\displaystyle=\cos\theta_{12}\cos\theta_{23}+\sin\theta_{12}\sin\theta_{23}\cos\alpha.$
(55)

In order to find the extremal values of $\theta_{13}$, notice that setting the
derivative of the rhs with respect to $\alpha$ to zero yields the values
$\cos(\theta_{12}-\theta_{23})=\cos(|\theta_{12}-\theta_{23}|)$ for $\alpha=0$
and $\cos(\theta_{12}+\theta_{23})$ for $\alpha=\pi$. Using
$|\theta_{12}-\theta_{23}|<\theta_{12}+\theta_{23}$ and the fact that the
cosine function is strictly decreasing on $[0,\pi]$, one obtains the lower and
upper bounds of Eq. 51. Moreover, since these extremal values are only reached
for $\alpha=0,\pi$, saturation of Eq. 51 is only possible when all vectors
belong to the plane spanned by $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$. ∎

######  Lemma C.2.

Consider normalized vectors
$\vec{m}_{\mathsf{a},x},\vec{n}^{\mathsf{a}}_{x,y}\in\mathbb{R}^{12}$ for all
$\mathsf{a}\in\\{-1,1\\}$, $x,y\in\\{0,1\\}$. Suppose that:

  1. 1.

$\forall(\mathsf{s},\mathsf{t})\in\\{-1,1\\}^{2},\
\operatorname{asin}\langle\vec{m}_{\mathsf{s},0}|\vec{n}^{\mathsf{s}}_{0,0}\rangle+\operatorname{asin}\langle\vec{m}_{\mathsf{t},1}|\vec{n}^{\mathsf{t}}_{1,0}\rangle-\operatorname{asin}\langle\vec{m}_{\mathsf{s},0}|\vec{n}^{\mathsf{s}}_{0,1}\rangle+\operatorname{asin}\langle\vec{m}_{\mathsf{t},1}|\vec{n}^{\mathsf{t}}_{1,1}\rangle=\pi$

  2. 2.

$\exists\lambda_{x}\in(0,1),l\in\mathbb{R}$ s.t.
$\lambda_{x}\langle\vec{n}^{+}_{x,0}|\vec{n}^{+}_{x,1}\rangle+(1-\lambda_{x})\langle\vec{n}^{-}_{x,0}|\vec{n}^{-}_{x,1}\rangle=l$
(doesn’t depend on $x$).

Then all triples of vectors $\vec{m}_{\mathsf{a},x}$,
$\vec{n}^{\mathsf{a}}_{x,0}$, $\vec{n}^{\mathsf{a}}_{x,1}$ lie in the same
plane and
$\langle\vec{n}^{\mathsf{a}}_{x,0}|\vec{n}^{\mathsf{a}}_{x,1}\rangle$ doesn’t
depend on $\mathsf{a},x$.

###### Proof.

Let’s consider the angular distances
$\gamma_{x,y}^{\mathsf{a}},\theta_{\mathsf{a},x}\in[0,\pi]$ defined as:

$\cos(\gamma_{x,y}^{\mathsf{a}})=\langle\vec{m}_{\mathsf{a},x}|\vec{n}^{\mathsf{a}}_{x,y}\rangle,\quad\cos(\theta_{\mathsf{a},x})=\langle\vec{n}^{\mathsf{a}}_{x,0}|\vec{n}^{\mathsf{a}}_{x,1}\rangle.$
(56)

Using $\operatorname{asin}=\pi/2-\operatorname{acos}$, the first assumption of
the lemma then becomes:

$\forall(\mathsf{s},\mathsf{t})\in\\{-1,1\\}^{2},\
\gamma^{\mathsf{s}}_{0,0}+\gamma^{\mathsf{t}}_{1,0}-\gamma^{\mathsf{s}}_{0,1}+\gamma^{\mathsf{t}}_{1,1}=0$
(57)

Moreover, C.1 applied to $\vec{n}^{\mathsf{a}}_{x,0}$,
$\vec{n}^{\mathsf{a}}_{x,1}$ and $\vec{m}_{\mathsf{a},x}$ grants:

$|\gamma_{x,0}^{\mathsf{a}}-\gamma_{x,1}^{\mathsf{a}}|\leq\theta_{\mathsf{a},x}\leq\gamma_{x,0}^{\mathsf{a}}+\gamma_{x,1}^{\mathsf{a}}.$
(58)

Let us now consider the orderings of $\theta_{\mathsf{a},x}$ which are
compatible with the second requirement of the lemma. Note that this implies
that there exist convex combinations of $\cos(\theta_{\mathsf{a},x})$ for
$\mathsf{a}=\pm$ that doesn’t depend on $x$. In particular, this implies that
it is impossible to have $\theta_{\mathsf{t},1}<\theta_{\mathsf{s},0}$ for all
$\mathsf{s},\mathsf{t}$. Without loss of generality, let’s consider that
$\theta_{+,0}\leq\theta_{+,1}$ (the other cases are treated similarly). We
obtain from Eqs. 57 and 58:

$0\leq\gamma_{1,0}^{+}+\gamma_{1,1}^{+}=\gamma_{0,1}^{+}-\gamma_{0,0}^{+}\leq\theta_{+,0}\leq\theta_{+,1}\leq\gamma_{1,0}^{+}+\gamma_{1,1}^{+},$
(59)

which implies that $\theta_{+,0}=\theta_{+,1}$. The second hypothesis of the
lemma then implies that either
$\theta_{-,0},\theta_{-,1}\leq\theta_{+,0}=\theta_{+,1}$ or
$\theta_{-,0},\theta_{-,1}\geq\theta_{+,0}=\theta_{+,1}$. Let’s consider the
first case, where we have $\theta_{-,0}\leq\theta_{+,1}$ (again, the other one
is similar). We obtain from Eqs. 57 and 58:

$0\leq\gamma_{1,0}^{+}+\gamma_{1,1}^{+}=\gamma_{0,1}^{-}-\gamma_{0,0}^{-}\leq\theta_{-,0}\leq\theta_{+,1}\leq\gamma_{1,0}^{+}+\gamma_{1,1}^{+},$
(60)

and thus $\theta_{-,0}=\theta_{+,1}$. Since three of the thetas match,
hypothesis 2 implies that all four are equal, and that

$0\leq\gamma_{1,0}^{-}+\gamma_{1,1}^{-}=\gamma_{0,1}^{-}-\gamma_{0,0}^{-}\leq\theta_{-,0}=\theta_{-,1}\leq\gamma_{1,0}^{-}+\gamma_{1,1}^{-},$
(61)

The saturation of the triangle inequalities in Eqs. 59, 60 and 61 implies that
$\vec{m}_{\mathsf{a},x}$, $\vec{n}_{0,\mathsf{a},x}$ and
$\vec{n}_{1,\mathsf{a},x}$ must be co-planar for all $\mathsf{a},x$.
Furthermore, we obtained
$\theta_{\mathsf{a},x}=\theta_{a^{\prime},x^{\prime}}$ for all
$\mathsf{a},\mathsf{a}^{\prime},x,x^{\prime}$, i.e. that
$\langle\vec{n}_{0,\mathsf{a},x}|\vec{n}_{1,\mathsf{a},x}\rangle$ doesn’t
depend on $\mathsf{a},x$. ∎

###  C.2 Preliminary Quantum Considerations

######  Lemma C.3.

[30, Theorem 2.5] In the CHSH scenario, suppose that a correlation vector
verifies the following conditions:

  1. 1.

It is extremal in $\mathcal{Q}$,

  2. 2.

Up to local unitaries, there exists a unique pure, projective and non-
degenerate realisation $(\ket{\phi_{\theta}},A_{x},B_{y})$ of local dimension
two giving those correlations.

Then this correlation vector self-tests the realisation
$(\ket{\phi_{\theta}},A_{x},B_{y})$.

######  Remark.

The fact that the correlation vector is supposed extremal is key to the
previous Lemma, which of course does not hold if it’s not the case.

######  Lemma C.4.

[47, Theorem 1] Any correlation verifying $\langle A_{x}\rangle=\langle
B_{y}\rangle=0$ for all $x,y$,

$\displaystyle\operatorname{asin}\langle
A_{0}B_{0}\rangle+\operatorname{asin}\langle
A_{1}B_{0}\rangle-\operatorname{asin}\langle
A_{0}B_{1}\rangle+\operatorname{asin}\langle A_{1}B_{1}\rangle=\pi$ (62)

and such that at most one $\langle A_{x}B_{y}\rangle$ is equal to $\pm 1$,
self-tests the maximally entangled two qubit state $\ket{\phi^{+}}$ along with
alternating measurements on the $(\sigma_{z},\sigma_{x})$-plane.

######  Remark.

If one obtains correlations verifying the conditions of the lemma with a
realization of local dimension 2, self-testing ensures the existence of
unitaries $U_{A},U_{B}\in L(\mathbb{C}^{2})$ s.t:

$\begin{split}(U_{A}\otimes U_{B})\ket{\psi}&=\ket{\phi^{+}},\\\
U_{A}A_{x}U_{A}^{\dagger}&=\cos(a_{x})\sigma_{z}+\sin(a_{x})\sigma_{x},\\\
U_{B}B_{1}U_{B}^{\dagger}&=\cos(b)\sigma_{z}+\sin(b)\sigma_{x},\\\ \quad
U_{B}B_{0}U_{B}^{\dagger}&=\sigma_{z}.\end{split}$ (63)

where $0\leq a_{0}\leq b\leq a_{1}\leq\pi$ are fully determined by $\langle
A_{x}B_{y}\rangle$.

######  Lemma C.5.

Consider a nonlocal correlation of the set $\mathcal{Q}$, then:

  1. 1.

$\langle A_{x}\rangle\neq\pm 1$ and as such the correlators
$[\tilde{A}^{\mathsf{a}}_{x}B_{y}]=\frac{\langle
A_{x}B_{y}\rangle+\mathsf{a}\langle B_{y}\rangle}{1+\mathsf{a}\langle
A_{x}\rangle}$ are well-defined

  2. 2.

There is always at least two equations in the four equations of 1 in which the
number of correlators $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]$ equal to $\pm 1$ is
at most 1. Moreover, the number of correlators equal to $\pm 1$ in the other
two equations is at most 2.

###### Proof.

The proof mainly relies on a result from Ref. [30] where the authors prove
that if a quantum correlation has two zero of probabilities for a fixed input
choice $x$ of Alice, then it is local. In terms of correlations, it means that
if two equations of the form $1+\mathsf{a}\langle
A_{x}\rangle+\mathsf{b}\langle B_{y}\rangle+\mathsf{a}\mathsf{b}\langle
A_{x}B_{y}\rangle=0$ are satisfied for the same $(\mathsf{a},x)$ or the same
$(\mathsf{b},y)$, then the correlation is local. They also prove that a non-
local point can have at most three zeros of positivities.

Therefore, one can verify that if $\langle A_{x}\rangle=\pm 1$, non-signaling
condition imply that $\mathsf{b}(\langle B_{y}\rangle\mp\langle
A_{x}B_{y}\rangle)\geq 0$ for all $\mathsf{b}$ and thus that $\langle
B_{y}\rangle\mp\langle A_{x}B_{y}\rangle=0$. Then both equations $1\mp\langle
A_{x}\rangle+\mathsf{b}\langle B_{y}\rangle\mp\mathsf{b}\langle
A_{x}B_{y}\rangle=0$ for $\mathsf{b}=\pm 1$ are satisfied, and the point is
local. Conversely, if the point is non-local, then $\langle
A_{x}\rangle\neq\pm 1$.

For the second point, we have that $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]=\pm 1$
iff $1+\mathsf{a}\langle A_{x}\rangle\mp\mathsf{a}\langle
B_{y}\rangle\mp\langle A_{x}B_{y}\rangle=0$, and as such verifying
$[\tilde{A}^{\mathsf{a}}_{x}B_{y}]=\pm 1$ is equivalent to having a zero of
probability for tuples $(\mathsf{a},x)$ and $(\mathsf{b}=\mp\mathsf{a},y)$.
Let’s consider a non-local point. It can have at most three zeros of
probabilities. We suppose that this is the case (other cases with fewer zeros
can be treated similarly). Since a non-local point cannot have two zeros of
probability for a fix tuple $(\mathsf{a},x)$, two zeros occur for a given $x$
and both values of $\mathsf{a}$ and one for the other input value $x^{\prime}$
and a single output value $\mathsf{a}^{\prime}$. Without loss of generality,
let’s consider that two zeros happen for $x=0$ and both $\mathsf{a}\in\\{\pm
1\\}$ and the last one for $x^{\prime}=1$ and $\mathsf{a}^{\prime}=+1$. Now,
we obtain that $[\tilde{A}^{+}_{0}B_{y_{+,0}}]$,
$[\tilde{A}^{-}_{0}B_{y_{-,0}}]$, $[\tilde{A}^{+}_{1}B_{y_{+,1}}]$ are equal
to $\pm 1$ for some given inputs $y_{+,0}$, $y_{-,0}$, $y_{+,1}$. Considering
the four equations of 1, both equations for tuple $(\mathsf{s},+)$ have two
terms s.t. $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]=\pm 1$
($[\tilde{A}^{\mathsf{s}}_{0}B_{y_{\mathsf{s},0}}]$ and
$[\tilde{A}^{+}_{1}B_{y_{+,1}}]$), and both equations for tuple
$(\mathsf{s},-)$ only have one
($[\tilde{A}^{\mathsf{s}}_{0}B_{y_{\mathsf{s},0}}]$). Since no other zeros of
probabilities can happen, no other terms $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]$
can be equal to $\pm 1$. ∎

######  Lemma C.6.

For any pure, projective and non-degenerate realization of local dimension two
$(\ket{\psi},A_{x},B_{y})$ verifying the assumptions of 1, there exists local
unitaries $U_{A},U_{B}\in L(\mathbb{C}^{2})$ such that:

$\begin{split}(U_{A}\otimes U_{B})\ket{\psi}&=\ket{\phi_{\theta}},\\\
U_{A}A_{x}U_{A}^{\dagger}&=\cos(a_{x})\sigma_{z}+\sin(a_{x})\sigma_{x},\\\
U_{B}B_{y}U_{B}^{\dagger}&=\cos(b_{y})\sigma_{z}+\sin(b_{y})\sigma_{x},\end{split}$
(64)

and the parameters $\theta,a_{x},b_{y}$ are fully determined by the
correlators $\langle A_{x}\rangle$, $\langle B_{y}\rangle$ and $\langle
A_{x}B_{y}\rangle$.

###### Proof.

Up to local unitaries $U_{A},U_{B}\in L(\mathbb{C}^{2})$, one can map any
realization $(\ket{\psi},A_{x},B_{y})$ to:

$\begin{split}\text{state: }&\ket{\phi_{\theta}},\quad\theta\in[0,\pi/4],\\\
\text{measurements:
}&A_{x}=\cos(a_{x})\sigma_{z}+\sin(a_{x})(\cos(\alpha_{x})\sigma_{x}+\sin(\alpha_{x})\sigma_{y}),\\\
&B_{0}=\cos(b_{0})\sigma_{z}+\sin(b_{0})(\cos(\beta)\sigma_{x}+\sin(\beta)\sigma_{y}),\\\
&B_{1}=\cos(b_{1})\sigma_{z}+\sin(b_{1})\sigma_{x}.\end{split}$ (65)

The case where the marginals $\langle A_{x}\rangle=\langle B_{y}\rangle=0$
vanish is due to the self-testing result [47] (see C.4) so we can exclude the
zero marginals case. Since the correlations are supposed non-local, we have
$\theta\neq 0,\pi/4$. One can apply the steering transformation introduced in
Appendix B to ensure that all $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]$ are
correlators measured on a singlet state $\ket{\phi^{+}}$, where:

$\begin{split}&\tilde{A}^{\mathsf{a}}_{x}=\cos(\tilde{a}^{\mathsf{a}}_{x})\sigma_{z}+\sin(\tilde{a}^{\mathsf{a}}_{x})(\cos(\alpha_{x})\sigma_{x}+\sin(\alpha_{x})\sigma_{y}),\\\
&B_{0}=\cos(b_{0})\sigma_{z}+\sin(b_{0})(\cos(\beta)\sigma_{x}+\sin(\beta)\sigma_{y}),\\\
&B_{1}=\cos(b_{1})\sigma_{z}+\sin(b_{1})\sigma_{x}.\end{split}$ (66)

Note that since the correlations are non-local, we have $a_{0}\ncong
a_{1}[\pi]$ and the fact that the state is not maximally entangled ensures
that either $\tilde{a}^{+}_{0}\neq\tilde{a}^{-}_{0}$ or
$\tilde{a}^{+}_{1}\neq\tilde{a}^{-}_{1}$.

The fact that correlations are taken non-local ensures by C.5 that one can
apply the self-testing result C.4 for at least two pairs of
$(\mathsf{s},\mathsf{t})$, certifying the angle of measurements $B_{0},B_{1}$
with respect to $(\tilde{A}_{\mathsf{s},0},\tilde{A}_{\mathsf{s},0})$ in those
cases. Note that if C.4 doesn’t apply for the other pairs, it means that some
correlators are equal to $\pm 1$, i.e. $[\tilde{A}^{\mathsf{a}}_{x}B_{y}]=\pm
1$. Since the vector $\tilde{A}^{\mathsf{a}}_{x}\ket{\phi^{+}}$ and
$B_{y}\ket{\phi^{+}}$ are unit and $\ket{\phi^{+}}$ is full rank, this would
mean that $\tilde{A}^{\mathsf{a}}_{x}=B_{y}$, fully identifying the missing
measurements. Finally, since the initial state is already $\ket{\phi^{+}}$,
there exists $U\in L(\mathbb{C}^{2})$ s.t:

$\begin{split}&\bar{U}\tilde{A}^{\mathsf{a}}_{x}\bar{U}^{\dagger}=\cos(w_{\mathsf{a},x})\sigma_{z}+\sin(w_{\mathsf{a},x})\sigma_{x},\\\
&UB_{0}U^{\dagger}=\cos(b)\sigma_{z}+\sin(b)\sigma_{x},\quad
UB_{1}U^{\dagger}=\sigma_{z}\end{split}$ (67)

where $0\leq w_{\mathsf{s},0}\leq b\leq w_{\mathsf{t},1}\leq\pi$ for all
$\mathsf{s},\mathsf{t}$ are determined by the correlators
$[\tilde{A}^{\mathsf{a}}_{x}B_{y}]$.

In particular, we obtain that all vectors
$\tilde{A}^{\mathsf{a}}_{x}\ket{\phi^{+}}$, $\bar{B_{y}\ket{\phi^{+}}}$ lie in
the same plane. If one consider $x$ such that
$\tilde{A}^{+}_{x}\neq\tilde{A}^{-}_{x}$, then
$\tilde{A}^{+}_{x}\ket{\phi^{+}}$ and $\tilde{A}^{-}_{x}\ket{\phi^{+}}$ form a
plane of angle $\alpha_{x}$, with respect to $\sigma_{x}-\sigma_{y}$. This
implies that it is always possible to perform unitaries
$V_{A}=\operatorname{diag}(1,e^{-i\alpha_{x}})$ and $V_{B}=V_{A}^{\dagger}$
such that measurements $A_{x},B_{y}$ are real and the state is unchanged
($(V_{A}\otimes V_{B})\ket{\phi_{\theta}}=\ket{\phi_{\theta}}$).

We are now left with only real matrices and thus we can parametrize $U$ as a
real unitary matrix of the form:

$U=\begin{pmatrix}\cos(\gamma)&-\epsilon_{u}\sin(\gamma)\\\
\sin(\gamma)&\epsilon_{u}\cos(\gamma)\\\ \end{pmatrix}$ (68)

where $\epsilon_{u}\in\\{-1,1\\}$ and $\gamma\in[0,2\pi)$. Since a global sign
flip doesn’t change the measurement, one can assume $\gamma\in[-\pi/2,\pi/2]$.
Likewise, the sign $\epsilon_{u}$ can be absorbed by applying a $\sigma_{z}$
on both parties (which preserves the state $\ket{\phi_{\theta}}$). Therefore,
for the rest of the proof, we assume that $U$ is a rotation of angle $\gamma$.

Let us write the initial measurements
$A_{x}=\outerproduct{u_{+,x}}{u_{+,x}}-\outerproduct{u_{-,x}}{u_{-,x}}$ and
denote $\ket{v_{\mathsf{a},x}}:=T_{\theta}(\ket{u_{\mathsf{a},x}})$. Since the
vectors $\ket{v_{\mathsf{a},x}}$ correspond to the eigenvectors of
$\tilde{A}^{\mathsf{a}}_{x}$ of eigenvalue $\mathsf{a}$, they are fixed by Eq.
67:

$\begin{split}&\ket{v_{+,x}}=U^{\dagger}(\cos(\frac{w_{+,x}}{2})\ket{0}+\sin(\frac{w_{+,x}}{2})\ket{1}):=U^{\dagger}\ket{W_{+,x}},\\\
&\ket{v_{-,x}}=U^{\dagger}(\sin(\frac{w_{-,x}}{2})\ket{0}-\cos(\frac{w_{-,x}}{2})\ket{1}):=U^{\dagger}\ket{W_{-,x}}.\end{split}$
(69)

Note that in Appendix B, we showed that
$\ket{u_{\mathsf{a},x}}=T_{\pi/2-\theta}(\ket{v_{\mathsf{a},x}})$. As such the
measurements $A_{x}$ are fully determined by the knowledge of $\theta$ and
$\gamma$. More precisely:

$\ket{u_{\mathsf{a},x}}\propto(\sin(\theta)\langle
0|U^{\dagger}|W_{\mathsf{a},x}\rangle\ket{0}+\cos(\theta)\langle
1|U^{\dagger}|W_{\mathsf{a},x}\rangle\ket{1}).$ (70)

where the normalization constants are given by
$\innerproduct{u_{\mathsf{a},x}}{u_{\mathsf{a},x}}=1$. Since the original
measurements $A_{x}$ are unitary, we have $\langle u_{+,x}|u_{-,x}\rangle=0$
which implies that for all $x$:

$\sin^{2}(\theta)\langle 0|U^{\dagger}|W_{+,x}\rangle\langle
0|U^{\dagger}|W_{-,x}\rangle+\cos^{2}(\theta)\langle
1|U^{\dagger}|W_{+,x}\rangle\langle 1|U^{\dagger}|W_{-,x}\rangle=0.$ (71)

The lhs can be simplified by introducing
$\gamma_{x}=\gamma-(w_{+,x}+w_{-,x})/4$, leading to

$\frac{1}{2}\left[\cos(2\theta)\cos(2\gamma_{x})-\sin(\frac{w_{+,x}+w_{-,x}}{2})\right]=0.$
(72)

By linear combination of those equations, one obtains

$\tan(\gamma)=\frac{\sin(\frac{w_{+,0}-w_{-,0}}{2})\cos(\frac{w_{+,1}+w_{-,1}}{2})-\sin(\frac{w_{+,1}-w_{-,1}}{2})\cos(\frac{w_{+,0}+w_{-,0}}{2})}{\sin(\frac{w_{+,0}-w_{-,0}}{2})\sin(\frac{w_{+,1}+w_{-,1}}{2})-\sin(\frac{w_{+,1}-w_{-,1}}{2})\sin(\frac{w_{+,0}+w_{-,0}}{2})},$
(73)

where we allow both sides to be infinite. This equation fixes a unique
$\gamma\in[-\pi/2,\pi/2]$ and Eq. 72 fixes $\theta$ to be either
$\theta^{\star}$ or $\pi-\theta^{\star}$, and only one of those two solutions
can belong to $(0,\pi/4)$.

∎

###  C.3  Proof of 1

Let us consider non-local correlations in the CHSH scenario satisfying Eq. 7.
A.1 reduces the search for extremal points to the ones admitting a pure real
realization of local dimension 2 and since every correlation is a convex
combination of extremal correlations, one can write:

$\forall x,y\in\\{-1,0,1\\},\ \langle
A_{x}B_{y}\rangle=\sum_{i=1}^{N}p_{i}\langle A_{x}^{(i)}B_{y}^{(i)}\rangle$
(74)

where $p_{i}\geq 0$, $\sum_{i}p_{i}=1$, $A_{-1}=B_{-1}=\mathds{1}$ and on each
block $i$ the correlation vector $\\{\langle
A_{x}^{(i)}B_{y}^{(i)}\rangle\\}_{x,y}$ is in $\mathcal{Q}_{2}$. To prove that
the initial correlations are extremal in $\mathcal{Q}$, we need to prove that
the sub-correlations $\\{\langle A_{x}^{(i)}B_{y}^{(i)}\rangle\\}_{x,y}$ don’t
depend on the register $i$.

Such a decomposition allows considering an overall underlying realization with
local Hilbert spaces encoding a qubit space and a classical register:
$\mathcal{H}_{A(B)}=\mathcal{R}^{2}\otimes\mathbf{N}$ (where $\mathbf{N}$ is
the set of natural numbers from $1$ to $N$). The overall state and
measurements are:

$\begin{split}\text{state:
}&\ket{\psi}=\sum_{i=1}^{N}p_{i}\ket{\psi^{i}}\otimes\ket{ii},\\\
\text{measurements:
}&A_{x}=\sum_{i=1}^{N}A_{x}^{(i)}\otimes\outerproduct{i}{i},\quad
B_{y}=\sum_{i=1}^{N}B_{y}^{(i)}\otimes\outerproduct{i}{i}.\end{split}$ (75)

For each $i$ one can find unitaries $U_{A}^{i}$, $U_{B}^{i}$ such that the
underlying state is $\ket{\phi_{\theta^{i}}}$, with $\theta^{i}\in[0,\pi)$.
Considering the local unitaries
$U_{A(B)}=\sum_{i}U_{A(B)}^{i}\otimes\outerproduct{i}{i}$, the overall state
can then be written

$(U_{A}\otimes
U_{B})\ket{\psi}=\sum_{i}p_{i}\ket{\phi_{\theta^{i}}}\otimes\ket{ii}.$ (76)

Let us now consider the steering transformation $T_{\theta^{i}}$ introduced in
Appendix B on each block. We know that it allows to consider new operators
$\tilde{A}^{\mathsf{a}\,(i)}_{x}\in L(\mathbb{R}^{2})$ (see Eq. 38) such that

$\langle\phi_{\theta^{i}}|A_{x}^{(i)}B_{y}^{(i)}|\phi_{\theta^{i}}\rangle+\mathsf{a}\langle\phi_{\theta^{i}}|B_{y}^{(i)}|\phi_{\theta^{i}}\rangle=({1+\mathsf{a}\langle\phi_{\theta^{i}}|A_{x}^{(i)}|\phi_{\theta^{i}}\rangle})\bra{\phi^{+}}\tilde{A}^{\mathsf{a}\,(i)}_{x}B_{y}^{(i)}\ket{\phi^{+}},$
(77)

Let us now consider that

$1+\mathsf{a}\langle
A_{x}\rangle=1+\mathsf{a}\sum_{i=1}^{N}p_{i}\langle\phi_{\theta^{i}}|A_{x}^{(i)}|\phi_{\theta^{i}}\rangle=\sum_{i=1}^{N}p_{i}({1+\mathsf{a}\langle\phi_{\theta^{i}}|A_{x}^{(i)}|\phi_{\theta^{i}}\rangle}).$
(78)

Since the overall correlation is non-local, C.5 ensures that
$1+\mathsf{a}\langle A_{x}\rangle\neq 0$. One can therefore define

$\alpha_{\mathsf{a},x}^{i}=\frac{p_{i}({1+\mathsf{a}\langle\phi_{\theta^{i}}|A_{x}^{(i)}|\phi_{\theta^{i}}\rangle})}{1+\mathsf{a}\langle
A_{x}\rangle}$ (79)

which verifies $\alpha_{\mathsf{a},x}^{i}\geq 0$ and
$\sum_{i}\alpha_{\mathsf{a},x}^{i}=1$. Now using Eq. 77 we have:

$\begin{split}\frac{\langle A_{x}B_{y}\rangle+\mathsf{a}\langle
B_{y}\rangle}{1+\mathsf{a}\langle
A_{x}\rangle}&=\sum_{i=1}^{N}\alpha_{\mathsf{a},x}^{i}\bra{\phi^{+}}\tilde{A}^{\mathsf{a}\,(i)}_{x}B_{y}^{(i)}\ket{\phi^{+}}.\end{split}$
(80)

We now define a four new overall states and measurements operators as

$\begin{split}\ket{\Phi^{+}_{\mathsf{a},x}}&=\sum_{i}\sqrt{\alpha_{\mathsf{a},x}^{i}}\ket{\phi^{+}}\otimes\ket{ii},\quad\tilde{A}^{\mathsf{a}}_{x}=\sum_{i}\tilde{A}^{\mathsf{a}}_{x^{i}}\otimes\outerproduct{i}{i},\quad
B_{y}=\sum_{i}B_{y}^{(i)}\otimes\outerproduct{i}{i}.\\\ \end{split}$ (81)

Denoting expectations values on $\ket{\Phi^{+}_{\mathsf{a},x}}$ as
$\langle\cdot\rangle_{\mathsf{a},x}$, we obtain:

$\begin{split}&\langle\tilde{A}^{\mathsf{a}}_{x}\rangle_{\mathsf{a},x}=0,\langle
B_{y}\rangle_{\mathsf{a},x}=0,\\\ &\frac{\langle
A_{x}B_{y}\rangle+\mathsf{a}\langle B_{y}\rangle}{1+\mathsf{a}\langle
A_{x}\rangle}=\langle\tilde{A}^{\mathsf{a}}_{x}B_{y}\rangle_{\mathsf{a},x}.\end{split}$
(82)

Note that the marginals are always $0$ because for every classical index $i$
the measured state $\ket{\phi^{+}}$ is maximally entangled.

Since the measurement operators in Eq. 81 are real and unitary, and the state
$\ket{\Phi_{\mathsf{a},x}^{+}}$ is real and normalized, all vectors
$\tilde{A}^{\mathsf{a}}_{x}\ket{\Phi_{\mathsf{a},x}^{+}}$,
$B_{y}\ket{\Phi_{\mathsf{a},x}^{+}}$ are real and normalized. Notice that we
have:

$\displaystyle\langle
B_{0}B_{1}\rangle_{\mathsf{a},x}=\sum_{i}\alpha_{\mathsf{a},x}^{i}\bra{\phi^{+}}B_{0}^{i}B_{1}^{i}\ket{\phi^{+}},$
(83a) $\displaystyle(1+\langle A_{x}\rangle)\alpha_{+,x}^{i}+(1-\langle
A_{x}\rangle)\alpha_{-,x}^{i}=2p_{i}$ (83b)

Therefore, with $\lambda_{x}=(1+\langle A_{x}\rangle)/2\in(0,1)$ and utilizing
the assumption of 1, we obtain:

  1. 1.

$\forall(\mathsf{s},\mathsf{t})\in\\{-1,1\\}^{2},\
\operatorname{asin}\langle\tilde{A}_{0}^{\mathsf{s}}B_{0}\rangle_{\mathsf{s},0}+\operatorname{asin}\langle\tilde{A}_{1}^{\mathsf{t}}B_{0}\rangle_{\mathsf{t},1}-\operatorname{asin}\langle\tilde{A}_{0}^{\mathsf{s}}B_{1}\rangle_{\mathsf{s},0}+\operatorname{asin}\langle\tilde{A}_{1}^{\mathsf{t}}B_{1}\rangle_{\mathsf{t},1}=\pi$

  2. 2.

$\lambda_{x}\langle B_{0}B_{1}\rangle_{+,x}+(1-\lambda_{x})\langle
B_{0}B_{1}\rangle_{-,x}=\sum_{i}p_{i}\bra{\phi^{+}}B_{0}^{i}B_{1}^{i}\ket{\phi^{+}}$
doesn’t depend on $x$.

We can thus use C.2 with
$\vec{m}_{\mathsf{a},x}=\tilde{A}_{x}^{\mathsf{a}}\ket{\Phi_{\mathsf{a},x}^{+}}$
and $\vec{n}^{\mathsf{a}}_{x,y}=B_{y}\ket{\Phi_{\mathsf{a},x}^{+}}$ to obtain
that the following triples of vectors must be coplanar for all $\mathsf{a},x$:

$\displaystyle
B_{0}\ket{\Phi_{\mathsf{a},x}^{+}},B_{1}\ket{\Phi_{\mathsf{a},x}^{+}},\tilde{A}_{x}^{\mathsf{a}}\ket{\Phi_{\mathsf{a},x}^{+}}.$
(84)

and that $\langle B_{0}B_{1}\rangle_{\mathsf{a},x}:=C$ doesn’t depend on
$\mathsf{a},x$.

This allows us to write for all $\mathsf{a},x$:

$\tilde{A}^{\mathsf{a}}_{x}\ket{\Phi^{+}_{\mathsf{a},x}}=r_{\mathsf{a},x}^{0}B_{0}\ket{\Phi^{+}_{\mathsf{a},x}}+r_{\mathsf{a},x}^{1}B_{1}\ket{\Phi^{+}_{\mathsf{a},x}}$
(85)

where $r_{\mathsf{a},x}^{y}\in\mathbb{R}$. By projecting on each classical
register $\outerproduct{i}{i}$, we get

$(\tilde{A}^{\mathsf{a}}_{x})^{i}\ket{\phi^{+}}=r_{\mathsf{a},x}^{0}B_{0}^{i}\ket{\phi^{+}}+r_{\mathsf{a},x}^{1}B_{1}^{i}\ket{\phi^{+}}$
(86)

Now we can sum over all classical registers but with weights
$\sqrt{\alpha_{\mathsf{a}^{\prime},x^{\prime}}^{i}}$ to obtain

$\tilde{A}^{\mathsf{a}}_{x}\ket{\Phi^{+}_{\mathsf{a}^{\prime},x^{\prime}}}=r_{\mathsf{a},x}^{0}B_{0}\ket{\Phi^{+}_{\mathsf{a}^{\prime},x^{\prime}}}+r_{\mathsf{a},x}^{1}B_{1}\ket{\Phi^{+}_{\mathsf{a}^{\prime},x^{\prime}}}$
(87)

And finally we can compute

$\langle\tilde{A}^{\mathsf{a}}_{x}B_{y}\rangle_{\mathsf{a}^{\prime},x^{\prime}}=r_{\mathsf{a},x}^{y}\langle
B_{y}^{2}\rangle_{\mathsf{a}^{\prime},x^{\prime}}+r_{\mathsf{a},x}^{0}r_{\mathsf{a},x}^{1}\langle
B_{0}B_{1}\rangle_{\mathsf{a}^{\prime},x^{\prime}}=r_{\mathsf{a},x}^{y}+r_{\mathsf{a},x}^{0}r_{\mathsf{a},x}^{1}C$
(88)

Since the right-hand term doesn’t depend on $\mathsf{a}^{\prime},x^{\prime}$
neither can the left-hand one. Thus, we have proven that $\langle
A_{x}^{\mathsf{a}}B_{y}\rangle_{\mathsf{a}^{\prime},x^{\prime}}$ doesn’t
depend on the indexes $\mathsf{a}^{\prime},x^{\prime}$. Notably, we can
rewrite the assumption of 1 for a single arbitrary state
$\ket{\Phi^{+}}:=\ket{\Phi^{+}_{+,0}}$:

$\forall(\mathsf{s},\mathsf{t})\in\\{-1,1\\}^{2},\
\operatorname{asin}\langle\tilde{A}_{0}^{\mathsf{s}}B_{0}\rangle_{\Phi^{+}}+\operatorname{asin}\langle\tilde{A}_{1}^{\mathsf{t}}B_{0}\rangle_{\Phi^{+}}-\operatorname{asin}\langle\tilde{A}_{0}^{\mathsf{s}}B_{1}\rangle_{\Phi^{+}}+\operatorname{asin}\langle\tilde{A}_{1}^{\mathsf{t}}B_{1}\rangle_{\Phi^{+}}=\pi$
(89)

For all $(\mathsf{s},\mathsf{t})$, we are now dealing with a realization on
the state $\ket{\Phi^{+}}$, with two measurements
$\tilde{A}_{0}^{\mathsf{s}}$, $\tilde{A}_{1}^{\mathsf{t}}$ and $B_{0}$,
$B_{1}$ for each party. The corresponding quantum correlations can be
decomposed as a convex mixture of $N$ sub-correlations due to the classical
register encoded in $\ket{\Phi^{+}}$. Since it has zero marginals and verifies
Eq. 89, the work of [21] ensures that the overall correlation is extremal for
all $\mathsf{s},\mathsf{t}$ and thus all sub-correlations are equals.
Therefore:

$\\{\bra{\phi^{+}}\tilde{A}^{\mathsf{a}\,(i)}_{x}B_{y}^{(i)}\ket{\phi^{+}}\\}_{\mathsf{a},x,y}\
\text{doesn't depend on the register {\hbox{i}}.}$ (90)

Moreover, it means that every initial sub-correlations, obtained by measuring
state $\ket{\phi_{\theta^{i}}}$ with real measurements
$(A_{x}^{(i)},B_{y}^{(i)})$, satisfy the condition of 1. Since C.6 ensures
that such a realization is determined by the values of
$\bra{\phi^{+}}\tilde{A}^{\mathsf{a}\,(i)}_{x}B_{y}^{(i)}\ket{\phi^{+}}$, we
obtain that $\theta^{i}$ and $(A_{x}^{(i)},B_{y}^{(i)})$ don’t depend on $i$.
This implies that the sub-correlations $\\{\langle
A_{x}^{(i)}B_{y}^{(i)}\rangle\\}_{x,y}$ don’t depend on $i$ and thus the
extremality of the initial correlations. The self-testing part is then
obtained combining extremality and C.6 together with C.3.

##  Appendix D Proof of 2

The correlation distribution corresponding to a realization Eq. 25 in
$\mathcal{Q}_{2}$ is:

$\vec{P}_{\theta,a_{x},b_{y}}=\begin{array}[]{c|c|c}&\cos({2\theta})\cos({b_{0}})&\cos({2\theta})\cos({b_{1}})\\\
\hline\cr\cos({2\theta})\cos({a_{0}})&\lx@intercol\hfil\hbox{\multirowsetup$\cos({a_{x}})\cos({b_{y}})+\sin({2\theta})\sin({a_{x}})\sin({b_{y}})$}\hfil\lx@intercol\\\
\cline{1-1}\cr\cos({2\theta})\cos({a_{1}})&\lx@intercol\hfil\hfil\lx@intercol\end{array},\quad\theta,a_{x},b_{y}\in\mathbb{R}$
(91)

While A.2 guarantees that every extremal point in $\mathcal{Q}$ is of this
form, it is not granted that every such point is extremal in $\mathcal{Q}$. In
the following, we express a condition on the parameters $\theta,a_{x},b_{y}$
for such a point to be non-exposed, i.e. not to be the unique maximizer of any
Bell expression.

In all generality, a Bell expression in the CHSH scenario can be denoted by a
real vector $\vec{\beta}\in\mathbb{R}^{8}$. The value of the Bell expression
on $\vec{P}$ is given by the scalar product $\vec{\beta}\cdot\vec{P}$ and thus
$\vec{P}$ is non-exposed in $\mathcal{Q}$ iff

$\forall\vec{\beta}\in\mathbb{R}^{8},\ \exists\vec{P^{\prime}}\in\mathcal{Q},\
\text{s.t}\ \vec{\beta}\cdot\vec{P^{\prime}}\geq\vec{\beta}\cdot\vec{P},\
\vec{P}^{\prime}\neq\vec{P}$ (92)

If we denote by $\mathcal{C}_{P}$ the set of all Bell expressions for which
$\vec{P}$ reaches their maximal quantum value, i.e.
$\mathcal{C}_{P}\leavevmode\nobreak\ =\leavevmode\nobreak\
\\{\beta\in\mathbb{R}^{8}|\vec{\beta}\cdot\vec{P}=\max_{P^{\prime}\in\mathcal{Q}}\vec{\beta}\cdot\vec{P^{\prime}}\\}$.
Then the condition for $\vec{P}$ to be non-exposed reduces to

$\forall\vec{\beta}\in\mathcal{C}_{P},\
\exists\vec{P^{\prime}}\in\mathcal{Q},\ \text{s.t}\
\vec{\beta}\cdot\vec{P^{\prime}}=\vec{\beta}\cdot\vec{P},\
\vec{P}^{\prime}\neq\vec{P}.$ (93)

The proof of Theorem 3 is divided in two main parts, which are developped in
the sections below. In the first one, we find necessary conditions on Bell
expressions to be in the subspace $\mathcal{C}_{P}$. In the second one, we
identify a point verifying Eq. 93 for every Bell expression in this subset. In
what follows, we fix a choice of parameters $\theta,a_{x},b_{y}$ in the region
Eq. 5 and denote by $\vec{P}$ the distribution $\vec{P}_{\theta,a_{x},b_{y}}$.
Furthermore, since all realizations with $\theta=0$ only give local
correlations, we assume $\theta>0$.

###  D.1 Identification of Bell expressions maximized by $\vec{P}$

For a given choice of measurements we introduce the measurement vector

$\vec{M}=\\{A_{0},A_{1},B_{0},B_{1},A_{0}B_{0},A_{1}B_{0},A_{0}B_{1},A_{1}B_{1}\\}\in
L(\mathcal{H}_{A}\otimes\mathcal{H}_{B})^{8}.$ (94)

For every Bell expression $\beta$ with $\\{\beta\\}_{-1,-1}=0$, one can
construct the Bell operator associated with this measurement choice as
$S=\vec{\beta}\cdot\vec{M}$, which is an hermitian operator. For any state
$\ket{\psi}$, the correlation distribution is given by
$\vec{P^{\prime}}=\bra{\psi}\vec{M}\ket{\psi}$ and the value of the Bell
expression is

$\vec{\beta}\cdot\vec{P^{\prime}}=\bra{\psi}S\ket{\psi}.$ (95)

As such, if the point $\vec{P}$ is to give the maximal quantum value of
$\beta$, then the state $\ket{\phi_{\theta}}$ must be an eigenstate of $S$ (of
maximal eigenvalue). This implies that for all vector $\ket{\psi^{\bot}}$
orthogonal to $\ket{\phi_{\theta}}$, $\ket{\psi^{\bot}}$ is also orthogonal to
$S\ket{\phi_{\theta}}$, and as such

$0=\bra{\psi^{\bot}}S\ket{\phi_{\theta}}=\vec{\beta}\cdot\bra{\psi^{\bot}}\vec{M}\ket{\phi_{\theta}}=\vec{\beta}\cdot\vec{T}_{\ket{\psi^{\bot}}}$
(96)

where we denoted
$\vec{T}_{\ket{\psi^{\bot}}}=\bra{\psi^{\bot}}\vec{M}\ket{\phi_{\theta}}$.
Note that this condition was recently used for solving optimization problems
and prove that specific quantum points are non-exposed, see [29, 30].

Another condition to get a maximal violation is that for any small variation
of the parameters $\theta,a_{x},b_{y}$, the value of the Bell expression
should be non-increasing. At first order, this gives

$\begin{split}0=\vec{\beta}\cdot\frac{\partial\vec{P}}{\partial\theta}=\vec{\beta}\cdot\frac{\partial\vec{P}}{\partial
a_{x}}=\vec{\beta}\cdot\frac{\partial\vec{P}}{\partial b_{y}}\end{split}$ (97)

We now consider a (possible) subset of the necessary conditions Eq. 96 and Eq.
97 by considering only the three orthogonal states
$\ket{\psi_{\theta}}=\sin(\theta)\ket{00}-\cos(\theta)\ket{11}$, $\ket{01}$,
$\ket{10}$, and variations along measurement angles $a_{0}$ and $b_{0}$. This
allows us to say that if $\vec{P}$ maximizes $\vec{\beta}$, then

$\vec{\beta}\in V^{\bot},\quad\text{where}\
V=\text{Vect}\left\langle\vec{T}_{\ket{\psi_{\theta}}},\vec{T}_{\ket{01}},\vec{T}_{\ket{10}},\frac{\partial\vec{P}}{\partial
a_{0}},\frac{\partial\vec{P}}{\partial b_{0}}\right\rangle$ (98)

This allows us to conclude that $\mathcal{C}_{P}\subset V^{\bot}$ and to
reduce the range of Bell expressions to this linear subspace for the rest of
the argument.

###  D.2 Non-exposed sufficient condition

Let’s suppose there exists a vector $\vec{v}\in V$ and a local vector
$\vec{L}\in\mathcal{Q}$ such that $\vec{L}=\vec{P}+\vec{v}$, then

$\forall\beta\in V^{\bot},\vec{\beta}\cdot\vec{L}=\vec{\beta}\cdot\vec{P}$
(99)

Therefore if such a decomposition exists, and $\vec{L}\neq\vec{P}$, condition
Eq. 92 is satisfied and the point $\vec{P}$ is non-exposed in $\mathcal{Q}$. A
sufficient condition to our problem is therefore to find a decomposition of
the form

$\vec{L}=\vec{P}+\vec{v},\ \vec{L}\in\mathcal{Q},\ \vec{v}\in V-\\{0\\}.$
(100)

We further restrict the choice of $\vec{L}$ to be included in one of the
following four subspaces:

$\forall\mathsf{s},\mathsf{t}\in\\{-1,+1\\},\
\mathcal{L}_{\mathsf{s}\mathsf{t}}=\left\\{\begin{array}[]{c|c|c}&\alpha_{0}&\alpha_{1}\\\
\hline\cr\mathsf{s}&\mathsf{s}\,\alpha_{0}&\mathsf{s}\,\alpha_{1}\\\
\hline\cr\mathsf{t}&\mathsf{t}\,\alpha_{0}&\mathsf{t}\,\alpha_{1}\\\
\end{array},\alpha_{0},\alpha_{1}\in\mathbb{R}\right\\}$ (101)

These spaces have nice properties. First, one can verify that the positivity
constraints imply that $\alpha_{0},\alpha_{1}\in[-1,1]$, and that under these
conditions, the value of all variants of the CHSH expression is upper bounded
by 2. Therefore, the local, quantum and non-signaling sets coincide in these
subspaces, and we can express the condition for a point
$\vec{L}\in\mathcal{L}_{\mathsf{s}\mathsf{t}}$ to be local (or quantum) simply
as $\alpha_{0},\alpha_{1}\in[-1,1]$, or equivalently as $1-\alpha_{x}^{2}\geq
0$ for all $x$. Second, due to the fact that $\theta>0$, we have $|\langle
A_{x}\rangle|<1$ for all $x$ and thus
$\vec{P}\notin\mathcal{L}_{\mathsf{s}\mathsf{t}}$ for all
$\mathsf{s},\mathsf{t}$. This means that finding a decomposition of the form
Eq. 100 for $\vec{L}\in\mathcal{L}_{\mathsf{s}\mathsf{t}}$ always ensures that
$\vec{v}\neq 0$.

As any point in $V$ can be written as the linear combination

$\vec{v}_{x,y,z,a,b}=x\vec{T}_{\ket{\psi_{\theta}}}+y\vec{T}_{\ket{01}}+z\vec{T}_{\ket{10}}+a\frac{\partial\vec{P}}{\partial
a_{0}}+b\frac{\partial\vec{P}}{\partial b_{0}}$ (102)

where $x,y,z,a,b$ are five real parameters, finding a decomposition
$\vec{L}=\vec{P}+\vec{v}$ where $\vec{L}\in\mathcal{L}_{\mathsf{s}\mathsf{t}}$
can be translated to the following linear system

$\left\\{\begin{split}&\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{0,-1}=\mathsf{s}\\\
&\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{1,-1}=\mathsf{t}\\\
&\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{-1,0}=\mathsf{s}\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{0,0}=\mathsf{t}\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{1,0}\\\
&\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{-1,1}=\mathsf{s}\\{\vec{P}+\vec{v}_{x,y,z,a,b})_{0,1}=\mathsf{t}\\{\vec{P}+\vec{v}_{x,y,z,a,b}\\}_{1,1}\\\
\end{split}\right.,$ (103)

where we recall that ${\vec{P}}_{x,y}=\langle A_{x}B_{y}\rangle$ for
$x,y\in\\{-1,0,1\\}$ with the convention that
$A_{-1}=\mathds{1}_{\mathcal{H}_{A}}$, $B_{-1}=\mathds{1}_{\mathcal{H}_{B}}$.

We first focus on the case where $\theta\leq\pi/4$. In this case, the linear
system admits a solution for any parameters verifying Eq. 5 and
$(\mathsf{s},\mathsf{t})\neq(-1,1)$, given by:

$\displaystyle
x=-\cos\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi+(a_{1}+\frac{1-\mathsf{t}}{2}\pi)}{2}\right)\sin({2\theta})/D_{\mathsf{s}\mathsf{t}},$
(104a) $\displaystyle
y=\sin\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi}{2}\right)\cos\left(\frac{a_{1}+\frac{1-\mathsf{t}}{2}\pi}{2}\right)\sin({\theta})/D_{\mathsf{s}\mathsf{t}},$
(104b) $\displaystyle
z=\cos\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi}{2}\right)\sin\left(\frac{a_{1}+\frac{1-\mathsf{t}}{2}\pi}{2}\right)\cos({\theta})/D_{\mathsf{s}\mathsf{t}},$
(104c) $\displaystyle
a=-\sin\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi-(a_{1}+\frac{1-\mathsf{t}}{2}\pi)}{2}\right)/D_{\mathsf{s}\mathsf{t}},$
(104d) $\displaystyle b=0,$ (104e)

where:

$D_{\mathsf{s}\mathsf{t}}=\cos\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi-(a_{1}+\frac{1-\mathsf{t}}{2}\pi)}{2}\right)+\cos\left(\frac{a_{0}+\frac{1-\mathsf{s}}{2}\pi+(a_{1}+\frac{1-\mathsf{t}}{2}\pi)}{2}\right)\cos({2\theta}).$
(105)

This solution gives a decomposition of the form $\vec{L}=\vec{P}+\vec{v}$
where $\vec{L}\in\mathcal{L}_{\mathsf{s}\mathsf{t}}$ is given by the two
parameters:

$\begin{split}\alpha_{y}^{\mathsf{s}\mathsf{t}}=\frac{\left(\left(\mathsf{s}\cos(a_{0})+\cos(2\theta)\right)\sin(b_{y})-\mathsf{s}\cos(b_{y})\sin(a_{0})\sin(2\theta)\right)\left(\left(\mathsf{t}\cos(a_{1})+\cos(2\theta)\right)\sin(b_{y})-\mathsf{t}\cos(b_{y})\sin(a_{1})\sin(2\theta)\right)}{D_{\mathsf{s}\mathsf{t}}}.\end{split}$
(106)

Note that for the parameters we chose here ($0<\theta\leq\pi/4$, $0\leq
a_{0}\leq a_{1}<\pi$) the denominator $D_{\mathsf{s}\mathsf{t}}$ is never $0$.

To verify Eq. 100, we now need to look at when the point $\vec{L}$ belongs to
the quantum set. As we said, this is equivalent to ask that their exist some
$(\mathsf{s},\mathsf{t})$ such that
$-1\leq\alpha_{y}^{\mathsf{s}\mathsf{t}}\leq 1$ for $y\in\\{0,1\\}$, or
equivalently that

$\forall y\in\\{0,1\\},\ 1-(\alpha_{y}^{\mathsf{s}\mathsf{t}})^{2}\geq 0.$
(107)

The validity of this inequality is unchanged by multiplication with a positive
scalar and as such one can look at the sign of

$\Delta_{\mathsf{s}\mathsf{t}}=\frac{D_{\mathsf{s}\mathsf{t}}^{2}}{(1+\mathsf{s}\langle
A_{0}\rangle)(1+\mathsf{t}\langle
A_{1}\rangle)}(1-(\alpha_{y}^{\mathsf{s}\mathsf{t}})^{2}).$ (108)

This quantity can be written simply as

$\Delta_{\mathsf{s}\mathsf{t}}=\mathsf{s}\,\mathsf{t}\,\sin(\tilde{a}^{\mathsf{s}}_{0}-b_{y})\sin(\tilde{a}^{\mathsf{t}}_{1}-b_{y}).$
(109)

by considering the following change of variables:

$\begin{split}a_{0}\longrightarrow\tilde{a}^{\mathsf{s}}_{0},\quad
a_{1}\longrightarrow\tilde{a}^{\mathsf{t}}_{1},\end{split}$ (110)

where $\tilde{a}^{\mathsf{a}}_{x}$ are defined in Eq. 41. Thus, the positivity
conditions for $\vec{L}$ to be local now become equivalent to

$\forall y\in\\{0,1\\},\
\Delta_{\mathsf{s}\mathsf{t}}=\mathsf{s}\,\mathsf{t}\,\sin(\tilde{a}^{\mathsf{s}}_{0}-b_{y})\sin(\tilde{a}^{\mathsf{t}}_{1}-b_{y})\geq
0.$ (111)

We now think by contradiction and look for conditions that ensure that there
is no choice of $(\mathsf{s},\mathsf{t})$ such that the above is verified:

  1. 1.

The choice of parameters that we made allows us to state that $b_{y}\leq
a_{0}\leq\tilde{a}^{+}_{0}$ for all $y$. Therefore,
$\sin(\tilde{a}^{+}_{0}-b_{y})\leq 0$ and the above condition for
$(\mathsf{s},\mathsf{t})=(1,1)$ is verified whenever $\tilde{a}^{+}_{1}\leq
b_{0}$. We thus impose $\tilde{a}^{+}_{1}\geq b_{0}$.

  2. 2.

Now, the condition for $(\mathsf{s},\mathsf{t})=(1,-1)$ is verified whenever
$\sin(\tilde{a}^{-}_{1}-b_{y})\geq 0$ for both $y$. As $\theta\leq\pi/4$,
$\tilde{a}^{-}_{1}\geq\tilde{a}^{+}_{1}\geq 0$ and thus it is always true for
$y=0$. Therefore, we need to have $\tilde{a}^{-}_{1}\leq b_{1}$.

  3. 3.

Last, the condition for $(\mathsf{s},\mathsf{t})=(-1,-1)$ and $y=1$ is now
verified as $\tilde{a}^{-}_{0}\leq\tilde{a}^{-}_{1}\leq b_{1}$. Then the
inequality for $y=0$ does not hold only when $\tilde{a}^{-}_{0}\leq b_{0}$.

Finally, we can conclude that none of the conditions Eq. 111 is verified only
when the parameters verify:

$0\leq\tilde{a}^{\mathsf{s}}_{0}\leq b_{0}\leq\tilde{a}^{\mathsf{t}}_{1}\leq
b_{1}<\pi$ (112)

for all $\mathsf{s},\mathsf{t}\in\\{-1,1\\}$, i.e. when then modified angles
on Alice’s side $\tilde{a}^{\mathsf{a}}_{x}$ and the angles on Bob’s side
$b_{y}$ alternate for all choices of $(\mathsf{s},\mathsf{t})$. Conversely,
one can conclude that when this full alternating property is not verified,
there exists a solution to the problem Eq. 100 and as such the point $\vec{P}$
is non-exposed in $\mathcal{Q}$.

The cases $\theta\in[\pi/4,\pi/2)$, $\theta\in(\pi/2,3\pi/4]$ and
$\theta\in[3\pi/4,\pi)$ were left aside, but the proof goes exactly as the
previous case, but considering the solutions of all three linear systems for
$(\mathsf{s},\mathsf{t})\neq(1,-1)$, $(\mathsf{s},\mathsf{t})\neq(1,-1)$ and
$(\mathsf{s},\mathsf{t})\neq(-1,1)$ respectively.