# Witnessing the non-objectivity of an unknown quantum dynamics

Davide Poderini  International Institute of Physics, Federal University of Rio
Grande do Norte, 59078-970, Natal, Brazil  Giovanni Rodari  Dipartimento di
Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy
George Moreno  International Institute of Physics, Federal University of Rio
Grande do Norte, 59078-970, Natal, Brazil  Departamento de Computação,
Universidade Federal Rural de Pernambuco, 52171-900, Recife, Pernambuco,
Brazil  Emanuele Polino  Dipartimento di Fisica - Sapienza Università di Roma,
P.le Aldo Moro 5, I-00185 Roma, Italy  Ranieri Nery  International Institute
of Physics, Federal University of Rio Grande do Norte, 59078-970, Natal,
Brazil  Alessia Suprano  Dipartimento di Fisica - Sapienza Università di Roma,
P.le Aldo Moro 5, I-00185 Roma, Italy  Cristhiano Duarte  International
Institute of Physics, Federal University of Rio Grande do Norte, 59078-970,
Natal, Brazil  School of Physics and Astronomy, University of Leeds, Leeds LS2
9JT, United Kingdom  Fabio Sciarrino<EMAIL_ADDRESS>Dipartimento
di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy
Rafael Chaves<EMAIL_ADDRESS>International Institute of Physics,
Federal University of Rio Grande do Norte, 59078-970, Natal, Brazil  School of
Science and Technology, Federal University of Rio Grande do Norte, Natal,
Brazil

###### Abstract

Quantum Darwinism offers an explanation for the emergence of classical
objective features – those we are used to at macroscopic scales – from quantum
properties at the microscopic level. The interaction of a quantum system with
its surroundings redundantly proliferates information to many parts of the
environment, turning it accessible and objective to different observers. But
given that one cannot probe the quantum system directly, only its environment,
how to witness whether an unknown quantum property can be deemed objective or
not? Here we propose a probabilistic framework to analyze this question and
show that objectivity implies a Bell-like inequality. Among several other
results, we show quantum violations of this inequality, a device-independent
proof of the non-objectivity of quantum correlations that give rise to the
phenomenon we name ”collective hallucination”: observers probing distinct
parts of the environment can agree upon their measurement outcome of a given
observable but such outcome can be totally uncorrelated from the property of
the quantum system that fixed observable should be probing. We also implement
an appealing photonic experiment where the temporal degree of freedom of
photons is the quantum system of interest, while their polarization acts as
the environment. Employing a fully black-box approach, we achieve the
violation of a Bell inequality, thus certifying the non-objectivity of the
underlying quantum dynamics in a fully device-independent framework.

##  I Introduction

Understanding how the quantum information encoded into a microscopic system
leads to classical features, those observed at the macroscopic scales, remains
a central question in the quantum foundations. In the early days of quantum
theory, the comprehension of the quantum-classical boundary relied on arguably
vague notions such as wave-particle duality [1], complementarity [2, 3] or
even that of a human observer [4]. Nowadays, the tools and concepts of quantum
information offer a more well-grounded framework to address those questions.

The study of decoherence [5, 6], for instance, shows that quantum properties,
such as coherence and entanglement, are degraded due to the interaction of a
quantum system with its surrounding environment, a process that becomes more
noticeable the larger the quantum system is [7], beautifully explaining some
crucial aspects of the quantum to classical transition [8, 9, 10]. Simply put,
decoherence selects the so-called pointer states [11]—natural candidates for
the macroscopically observed classical states obtained after a
measurement—while coherent superpositions of those are suppressed.
Decoherence, however, does not solve by itself the problem of how information
contained in the pointer states becomes available to different measurement
apparatuses, nor how this is turned into objective information, that is,
independent of observers.

That spreading of objective information is the central topic that gave rise to
the idea of quantum Darwinism [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26]. In quantum Darwinism, the environment—the same entity
responsible for decoherence— is also seen as a special carrier of information
about the quantum system, insofar as redundantly propagating the information
of the naturally selected pointer states to many external observers.
Crucially, the emergence of a classical notion of objectivity is a generic
feature of quantum dynamics [22]. Irrespective of the specific modelling for
the interaction with the environment, whenever the information about the
pointer states is accessible to sufficiently many observers, the evolution
will gradually resemble one where a specific observable is measured by all of
them.

But what if other measurements, not necessarily those related to a pointer
observable, are performed? In particular, if the system-environment dynamics
is not known, how can one test for objectivity or, rather, the absence of it?
Those are precisely the questions we address in this work.

Building on the results of [22], we propose a probabilistic framework to
address the question of an emergent notion of objectivity. In this
probabilistic setting, we associate an observer with each part of the
environment (see Fig. 1)), and we show that the ability of each observer to
encode and retrieve classical information about a quantum system translates
into the emergence of an objective value for a measurement outcome.
Objectivity here ought to be understood in the sense that it reflects a sort
of common knowledge among the observers— _a property of a quantum system is
objective when it is simultaneously agreed upon by all agents_. From that,
considering a particular case of two observers, we show that the Clauser-
Horne-Shimony-Holt (CHSH) inequality [27], a paradigmatic Bell inequality in
the study of quantum non-locality [28], can be also be turned into a witness
of non-objectivity.

More precisely, in our probabilistic setup, the violation of a CHSH inequality
implies the phenomenon we name “collective hallucination”. This collective
hallucination means that several observers can mutually agree upon their
outcome for the measurement of a given observable; still, that outcome can be
completely uncorrelated from the property of the quantum system it should be
related to. We also prove that if objectivity is demanded for all measurements
performed by the observers in the CHSH scenario, then it implies true
objectivity, reflecting not only agreement between observers but also to
properties of the quantum system under scrutiny. Finally, we provide a proof-
of-principle experimental realization of our framework. Employing birefringent
plates placed inside a Sagnac interferometer, the temporal degree of freedom
of photons gets entangled with their polarization, the first being the quantum
system of interest while the latter acts as its environment within the quantum
Darwinism scenario.

##  II Emergence of Objectivity in Quantum Darwinism

In the following, we review the basic notions of quantum Darwinism. We give
special emphasis to the standpoint of [22], where the authors prove that a
well-defined notion of objectivity is a generic property of any quantum
dynamics. We then move forward and prove our first result, a generalization of
the findings of [22] in a general probabilistic setting, that is, not
necessarily relying on (but certainly including) quantum theory.

We are interested in a general scenario where $n+1$ quantum systems interact
arbitrarily, being described at a certain instant of time by a density
operator $\rho_{AB_{1},\dots,B_{n}}$—at this level of generality, it is
irrelevant whether we refer to a closed system or a part of a larger system.
The sub-system $A$ describes the quantum system of interest, and $B_{i}$
stands for the different fractions of the environment. Each fraction $B_{i}$
interacts with $A$ and also possibly among themselves in such a way that the
quantum information originally contained in $A$ can be redundantly spread over
the joint system. In a quantum description of the process, this information
spreading is represented by a completely positive and trace preserving (CPTP)
map $\Lambda:\mathcal{D}(A)\rightarrow\mathcal{D}(B_{1}\otimes\dots\otimes
B_{n})$ where $\mathcal{D}(A)$ is the set of density operators on the Hilbert
space associated to system $A$ (similarly to the $B_{i}$’s). The scenario is
illustrated in Fig. 1

Figure 1: Quantum Darwinism scenario. The figure depicts the general scenario
considered in quantum Darwinism: one central system $A$ interacts with the
environment described by $n$ systems $B_{1},\ldots,B_{n}$. As a result of this
interaction, part of the information contained in $A$ is transferred to the
environment and replicated in each system $B_{i}$.

Within this context, Ref. [22] makes a distinction between two notions of
objectivity, that of observables and that of outcomes. The former states that
the observers should extract information about the same observable of the
system by probing parts of the environment, which would be associated with the
"pointer basis" selected by the system-environment interactions. The latter
considers that not only should the observable be the same, but also the value
of the measurement outcome should be agreed upon by the observers.

Regarding the objectivity of observables, it follows that, quite generally,
the map $\Lambda$ can be well approximated by a measure-and-prepare map, such
that the reduced map for most subsets of observers is given by

$\mathcal{E}_{\mathcal{B}}(\rho_{A})=\sum_{k}\operatorname{tr}[\rho_{A}\,F_{k}]\,\sigma_{\mathcal{B}}^{k},$
(1)

where $\mathcal{B}$ is the subset of observers (or degrees of freedom of the
environment being observed), $\\{F_{k}\\}_{k}$ is a POVM which _should be the
same_ for all subsets $\mathcal{B}$ of the same size,
$\sigma_{\mathcal{B}}^{k}$ is the (joint) quantum state for the observers in
$\mathcal{B}$, prepared according to the outcome $k$ of $F_{k}$ and
$\rho_{A}=\mathrm{Tr}_{B_{1}\dots,B_{n}}(\rho_{A,B_{1},\dots,B_{n}})$. More
precisely, the results in [22] $(i)$ provide an upper bound for how close a
family of measure-and-prepare maps sharing the same POVM are to the true
reduced evolution $\mathcal{E}_{\mathcal{B}}$ of smaller portions
$\mathcal{B}$’s and $(ii)$ show that for a suitable fraction of the observers
and for a large enough number of total observers, the bound gets closer to
zero, meaning that all observers would agree they are obtaining information
about the same property of $\rho_{A}$, determined by the observable described
via the POVM $\\{F_{k}\\}_{k}$.

Regarding the objectivity of outcomes, Ref. [22] introduces the guessing
probability of the outcome $k$ obtained with $F_{k}$ for all observers in the
subset $\mathcal{B}$—tacitly assuming that the dynamics for each environment’s
fraction of interest has exactly the form of Eq. (1). Consider a distribution
$\\{p_{i}\\}$ and a set of states $\\{\sigma_{i}\\}$ for
$i\in\\{1,\ldots,m\\}$ and let $p_{\mathrm{guess}}(p_{i},\sigma_{i})$ be the
guessing probability defined as:

$p_{\mathrm{guess}}(p_{i},\sigma_{i})=\max_{\\{E_{i}\\}}\sum_{i}p_{i}\operatorname{tr}(E_{i}\sigma_{i}),$
(2)

representing the capability of the ensemble of states
$\\{\sigma_{i}\\}_{i\in[m]}$ to properly encode $m$ classical states
distributed according to $\\{p_{i}\\}_{i\in[m]}$. It follows that if there
exists a positive $0<\delta<1$ such that for every observer $B_{k}$, with
$k\in\\{1,\ldots,n\\}$,

$\min_{\rho\in\mathcal{D}(A)}\left\\{p_{\mathrm{guess}}\left(\operatorname{tr}(F_{i}\rho),\sigma_{B_{k}}^{i}\right)\right\\}\geq
1-\delta,$ (3)

then there exists some POVM $\\{E^{k}_{i}\\}$ for each $B_{i}$ such that

$\min_{\rho\in\mathcal{D}(A)}\sum_{i}\operatorname{tr}(F_{i}\rho)\operatorname{tr}\left[\left(\bigotimes_{k=1}^{n}E^{k}_{i}\right)\sigma^{i}_{B_{1},\ldots,B_{n}}\right]\geq
1-6n\delta^{1/4},$ (4)

where $\\{F_{i}\\}$ is an appropriate POVM and $\sigma^{i}_{B_{k}}$ the
density matrix relative to the party $B_{k}$ only. Qualitatively, Eq. (3)
combined with Eq. (4) show that if each $B_{i}$ is capable of properly
encoding the outcomes of a measurement on the $A$ system, then one can assign
an objective value to it, shared by all the $B_{i}$, in the sense that
experimenters probing each single $B_{i}$ will all get the same value, with
high probability.

Our first goal is to extend this notion of objectivity beyond quantum
mechanical algebra machinery and instead rely on a purely probabilistic
approach 111A possible route to generalization of quantum Darwinism to GPTs
was proposed in [57]. Unfortunately, the authors are concerned with defining
what an idealized quantum Darwinism process would look like in GPTs, more
precisely, a general version of a fan-out gate, and do not consider the noisy
version of such a process.. There are two main reasons for that. The first is
that properties often seen as inherently quantum mechanical are, in fact, also
features of generalized probability theories, including monogamy of
correlations [30] and the impossibility of broadcasting information [31], just
to cite a few. Understanding informational principles in such generalized
settings often lead to deeper insights about quantum theory itself [32, 33].
The other main reason for our approach is of practical relevance and relates
to what is often called the device-independent approach to quantum information
[34], the paradigmatic examples of which are Bell inequalities violations,
non-contextuality inequality violations and their use in cryptographic
protocols [35, 36, 37]. In the device-independent setting, one can reach non-
trivial conclusions about the quantum states being prepared or the
measurements being performed by simply relying on the classical information
obtained by measurement outcomes—without resorting to a detailed description
of the experimental apparatus. In the particular case of quantum Darwinism, as
we will see, it will allow us to not only define the concept of objectivity
irrespective of any underlying dynamics or measurement setups, but also derive
testable constraints on whether the statistics observed in the experiment can
be deemed objective or not.

In our proposed setting, each agent $i$ has access to a portion $B_{i}$ of the
environment surrounding $A$. Additionally, each agent $i$ is free to
independently choose to measure one out of many possible observables
$x_{i}\in\\{x_{i}^{1},x_{i}^{2},...,x_{i}^{m_{i}}\\}$, obtaining the
corresponding outcome $b_{i}\in\\{b_{i}^{1},b_{i}^{2},...,b_{i}^{o_{i}}\\}$.
If we focus only on the aggregated statistics involved in this process, the
scenario is thus described by a joint probability distribution

$p(b_{1},\dots,b_{n}|x_{1},\dots,x_{n})=\sum_{a}p(a,b_{1},\dots,b_{n}|x_{1},\dots,x_{n}),$
(5)

where $a$ is the outcome one would observe if a direct measurement of the
system $A$, that measurement corresponding to the pointer-state observable
(assuming it exists) defined by a given dynamics, had been performed. Each
$x_{i}$ represents the random variable parametrizing the choice of which
observable the $i-$th agent having access to the portion $B_{i}$ of the
environment measures in a given run of the experiment.

According to the Born rule, a quantum description of the same scenario is
given by

$\displaystyle p(a,b_{1},\dots,b_{n}|x_{1},\dots,x_{n})=$ (6)
$\displaystyle\mathrm{Tr}[(F_{a}\otimes E^{1,x_{1}}_{b_{1}}\dots\otimes
E^{n,x_{n}}_{b_{n}})\,\rho_{A,B_{1},\dots,B_{n}}],$

where $\rho_{A,B_{1},\dots,B_{n}}$ is the density operator representing the
quantum state shared by all the environments $B_{i}$’s plus the central system
$A$, and where $\\{E_{b_{i}}^{i,x_{i}}\\}_{b_{i}}$ is the POVM representing a
possible choice of measurement that the $i-$th agent can realize on her
fraction of the environment. It is exactly Eq. (6) that motivates a general
probabilistic description where the joint distribution
$p(a,b_{1},\dots,b_{n}|x_{1},\dots,x_{n})$ has to fulfil three natural
assumptions.

The first, called _no-superdeterminism_ states that

$p(a|x_{1},\dots,x_{n})=p(a),$ (7)

for every $i\in[n],$ and for every
$x_{i}\in\\{x_{i}^{1},x_{i}^{2},...,x_{i}^{m_{i}}\\}$. In other words, the
choices of which observable to measure can be made by each agent independently
of how the $A$ system has been prepared or which are the pointer observables
defined by a given dynamics. This is reminiscent of the measurement
independence (also called ”free-will” assumption) in Bell’s theorem [38, 39].

The second assumption, named _no-signalling_ , states that

$p(b_{i}|a,x_{1},\dots,x_{n})=p(b_{i}|a,x_{i}),$ (8)

for all $i\in[n]$ and for all
$b_{i}\in\\{b_{i}^{1},b_{i}^{2},...,b_{i}^{o_{i}}\\}$. This is analogous to
the no-signalling condition in Bell’s theorem and implies that the choice of
observables made by a given observer should not have any direct causal effect
on the statistics of all other observers.

Our final assumption, which we name _$\delta$ -objectivity_, is structured as
follows. Let $\delta>0$ represent an error parameter. For each agent $i$,
denote $x^{*}_{i}$ their choice of measurement corresponding to the case where
their outcome should be correlated with the outcome $a$. In a quantum
description, that would precisely correspond to the pointer-state observable
on system $A$, that is, corresponding to a POVM $\\{E_{k}\\}_{k}$ reproducing
as reliably as possible the observable $\\{F_{k}\\}_{k}$ emerging in the
effective measure-and-prepare dynamics in Eq. (1). The outcome $b_{i}$ is
$\delta$-objective, if for each observer, we have that

$\sum_{a}p(a)\,p(b_{i}=a|a,x_{i}^{*})\geq 1-\delta.$ (9)

The fact that this assumption brings a clearer notion of objectivity will
become justified after our first result below. For now, notice that as there
is always a POVM attaining the optimal value for the guessing probability, we
can create a parallel involving Eq.(6), the equation defining
$p_{\mathrm{guess}}$, and the quantity in Eq.(4) as shown below:

$\displaystyle
p_{\mathrm{guess}}(\operatorname{tr}(F_{i}\rho_{A}),\sigma_{i})=\max_{\\{E_{i}\\}}\sum_{i}\operatorname{tr}(F_{i}\rho_{A})\operatorname{tr}(E_{i}\sigma_{i})$
$\displaystyle\longleftrightarrow\sum_{a}p(a)p(b_{i}=a|a,x^{*})$ (10)
$\displaystyle\min_{\rho_{A}}\sum_{i}\operatorname{tr}(F_{i}\rho_{A})\operatorname{tr}\left(\bigotimes_{k}E^{k}_{i}\sigma_{i}^{1,\ldots,n}\right)$
$\displaystyle\longleftrightarrow\sum_{a}p(a)p(b_{1}=b_{2}=\cdots=b_{n}=a|a,x_{1}^{*},\ldots,x_{n}^{*})$
(11)

With that, we can state our first result, proven in Appendix VIII.1,
justifying our $\delta$-objectivity assumption.

######  Result 1.

If there exists a positive $\delta\leq 1$ such that for every
$k\in\\{1,\ldots,n\\}$:

$\sum_{a}p(a)p(b_{k}=a|a,x_{k}^{*})\geq 1-\delta.$ (12)

Then we have

$\displaystyle\sum_{a}p(a)p(b_{1}=b_{2}=\cdots=b_{n}=a|a,x_{1}^{*}$
$\displaystyle,\ldots,x_{n}^{*})$ $\displaystyle\geq 1-n\delta$

Remark: Result 1 says that a result analogous to Eq. (4) continues to hold,
even in a fully probabilistic setting. Put another way, the inequality
$\sum_{a}p(a)p(b_{1}=b_{2}=...=b_{n}=a|x_{1}^{*},...,x_{n}^{*})\geq 1-n\delta$
expresses the possibility of assigning an objective nature to the outcome
obtained by each observer. Recall that objectivity here means that regardless
of the outcome obtained by each agent, that outcome is agreed upon among all
the $B_{i}$’s, that is,
$p(b_{1}=b_{2}=\cdots=b_{n}|x_{1}^{*},\ldots,x_{n}^{*})=1$. What is more, it
also reflects a property related to an observable described by a POVM
$\\{F_{k}\\}_{k}$ acting on the subsystem $A$. In particular, when there is a
perfect local agreement (i.e., when $\delta=0$, implying
$\sum_{a}p(b_{k}=a|x_{k}^{*})=1$ for every agent), Result 1 guarantees that
$\sum_{a}p(b_{1}=b_{2}=\cdots=b_{n}=a|x_{1}^{*},\ldots,x_{n}^{*})=1$. One can
read this implication as saying that perfect local agreement implies perfect
global agreement.

##  III Bell-like inequalities witnessing non-objectivity

The conditions of no-superdetermism, no-signalling and $\delta$-objectivity,
eqs. (7), (8) and (9) respectively, clearly define a notion for objectivity of
outcomes in the probabilistic setting. Notwithstanding, note that those
conditions involve the outcome $a$ that by assumption is not directly
observable, as any information about it can only be obtained indirectly, by
correlations of it with the outcomes $b_{i}$’s. Thus, similarly to Bell’s
theorem, $a$ plays the role of a latent or hidden variable. However, the
conjunction of assumptions (7), (8) and (9) do imply testable constraints for
the observed correlations among the outcomes $b_{i}$. We approach those
testable constraints in this section.

For doing so, we consider the particular case of only two observers ($n=2$).
Each observer has two possible measurements available to them. After each
measurement, a single value, out of a list of two, is flashed out. Put another
way, $x_{1},x_{2},b_{1},b_{2}\in\\{0,1\\}$. Moreover, we specify $x_{1}^{*}$
as $x_{1}=0$ and $x_{2}^{*}$ as $x_{2}=0$—recall that each $x_{i}^{\ast}$
corresponds to the special case where the outcome should be correlated with
the outcome $a$. We can then state our second result.

######  Result 2.

Any observed correlation $p(b_{1},b_{2}|x_{1},x_{2})$ compatible with the
conditions (7), (8) and (9), fulfills the inequality

$\mathrm{CHSH}_{\delta,\epsilon}=\left<B^{0}_{1}B^{0}_{2}\right>+\left<B^{0}_{1}B^{1}_{2}\right>\\\
-\left<B^{1}_{1}B^{0}_{2}\right>+\left<B^{1}_{1}B^{1}_{2}\right>\leq
2+4\delta-2\epsilon,$ (13)

with $\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$ where
$\left<B^{x_{1}}_{1}B^{x_{2}}_{2}\right>=\sum(-1)^{b_{1}+b_{2}}p(b_{1},b_{2}|x_{1},x_{2})$
is the expectation value of the observables corresponding to inputs $x_{1}$
and $x_{2}$.

Notice that Eq.(13) is a relaxed version of the CHSH inequality [27] with one
additional constraint. In Eq. (13) we impose that
$\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$ to mean that both observers are
in agreement (up to a discordance factor of $2\epsilon$) whenever they decide
to measure the special inputs $x_{1}^{*}=0$ and $x_{2}^{*}=0$ respectively.

Notice that our Result 1 implies that $\delta\geq\epsilon/2$ while
$\delta=\epsilon/2$ corresponds to the "darwinistic" case where the
disagreement $\delta$ between the observers and the latent observable follows
directly from the observable disagreement $\epsilon$ between the observers
themselves. Thus, any observed value $\operatorname{CHSH}_{\delta,\epsilon}>2$
implies that $\delta>\epsilon/2$, witnessing non-objectivity even in the case
of non-perfect agreement between the observers ($\epsilon>0$).

Considering the case where $\delta=0$ and $\epsilon=0$, our next result shows
that quantum theory can violate the $\operatorname{CHSH}_{0,0}$ inequality
while respecting $\left<B^{0}_{1}B^{0}_{2}\right>=1$. That is, the observers
agree among themselves but their outcomes do not reflect a property of the
system $A$ to which they assume to be fully correlated with—a phenomenon we
call ”collective hallucination”.

######  Result 3.

Quantum theory allows a violation of $\operatorname{CHSH}_{0,0}$ up to the
value $5/2$ while respecting $\left<B^{0}_{1}B^{0}_{2}\right>=1$. In
particular, the maximal violation, allows to self-test a maximally two-qubit
entangled state, which at the same time certifies one bit of randomness and
also implies a monogamy relation. That is, even though the observers agree
among themselves, the outcome of each one of them is completely uncorrelated
from system A.

In the following, we will discuss in more depth the consequences of these
results while a detailed proof is presented in the Appendix VIII.3. Notice
that the violation $\operatorname{CHSH}_{0,0}=5/2$ is achieved considering
state

$\left|\psi\right\rangle_{B_{1}B_{2}}=\frac{1}{\sqrt{2}}(\left|00\right\rangle+\left|11\right\rangle),$
(14)

and choosing $B_{j}^{0}=\sigma_{z}$ for $j=1,2$ and

$\displaystyle B_{1}^{1}$
$\displaystyle=-\frac{\sigma_{z}}{2}-\frac{\sqrt{3}}{2}\sigma_{x}$ (15)
$\displaystyle B_{2}^{1}$
$\displaystyle=\frac{\sigma_{z}}{2}-\frac{\sqrt{3}}{2}\sigma_{x}\;,$ (16)

As detailed in the Appendix VIII.3, the proof that
$\operatorname{CHSH}_{0,0}=5/2$ is the maximum quantum violation relies on the
idea that together with the agreement condition
$\left<B^{0}_{1}B^{0}_{2}\right>=1$, it allows to self-test the maximally
entangled state. Recall that the possibility of performing a self-test is a
sufficient condition to ensure that the quantum probability distribution
achieving $\operatorname{CHSH}_{0,0}=5/2$ is unique, as discussed in ref.
[40]. Combining that uniqueness with the convex nature of the set of quantum
correlations, it thus follows that $\operatorname{CHSH}_{0,0}=5/2$ is the
maximal possible violation. Otherwise, if there was a distribution leading to
a higher violation, there would be different manners to mix it with other
probability distributions (say the ones leading to maximal violation of other
symmetries of this inequality) in order to obtain two different correlations
reaching $\operatorname{CHSH}_{0,0}=5/2$, a situation that would forbid the
possibility of self-testing.

Furthermore, following the arguments of Ref. [41], we can state that being an
extreme point of the set of quantum behaviors assures that any third part
event is uncorrelated with the outcomes of the observers, i.e. it holds that
any realization $a$ of some third variable $A$ is such that
$p(a,b_{1},b_{2}|x_{1},x_{2})=p(b_{1},b_{2}|x_{1},x_{2})p(a)$. Finally,
because the $\operatorname{CHSH}_{0,0}$ inequality is invariant under the
transformation $b_{1}^{\prime}=(b_{1}+1)\bmod 2$ (the same holds for a similar
transformation of $b_{2}$), and the behavior leading to its maximal violation
is unique, we can certify a bit of randomness [42], either $b_{1}$ or $b_{2}$.
In particular, the certification of a random bit and the fact that any third
party is uncorrelated implies that the probability of guessing the outcome of
one of the participants is always $1/2$.

Figure 2: a, b) Minimal values possible for $\delta$ as a function of
$\operatorname{CHSH}_{\delta,\epsilon}$ value corresponding to the observable
distribution $p(b_{1},b_{2}|x_{1},x_{2})$. Results were obtained using the
third level of NPA hierarchy [43]. Different curves correspond to different
values for outcome agreement between observers, given by the constraint
$\sum_{b_{1}}p(b_{1}=b_{2}|x_{1}=x^{*},\,x_{2}=x^{*})=1-\epsilon$, or
equivalently $\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$. A change in the
behavior for the maximal violation of CHSH can be seen between values
$\left<B^{0}_{1}B^{0}_{2}\right>\leq 1/\sqrt{2}$ (shown in panel a), and
values $\left<B^{0}_{1}B^{0}_{2}\right>\geq 1/\sqrt{2}$ (panel b), where the
former increases with increasing $\epsilon$, while the latter decreases with
increasing $\epsilon$. At the same time, the restriction
$\delta\geq\epsilon/2$ from Eq. (1) can be observed throughout the graphics,
with saturation occurring in all cases for
$\operatorname{CHSH}_{\delta,\epsilon}=2$. It also can be seen that a sharp
rise in $\delta$ occurs near maximal violation for each $\epsilon$, which
leads to numerical instabilities in these regions. For this reason, no curve
reaches $\delta=0.5$ and terminal points are different for each curve. c)
Optimal values for the violation of CHSH inequality with the constraint
$\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$, with explicit quantum
realizations found numerically. The points achieve the self-testing criterion
of [44], satisfying
$\mathrm{asin}(\left<B^{0}_{1}B^{0}_{2}\right>)+\mathrm{asin}(\left<B^{0}_{1}B^{1}_{2}\right>)+\mathrm{asin}(\left<B^{1}_{1}B^{0}_{2}\right>)-\mathrm{asin}(\left<B^{1}_{1}B^{1}_{2}\right>)=\pi$,
together with the constraints
$\left<B^{0}_{1}B^{1}_{2}\right>=\left<B^{1}_{1}B^{0}_{2}\right>=-\left<B^{1}_{1}B^{1}_{2}\right>$.
The analytical curve is obtained by combining the equations, resulting in
$\operatorname{CHSH}_{\delta,\epsilon}=1-2\epsilon+3\sin(\frac{\pi}{3}-\frac{1}{3}\mathrm{asin}(1-2\epsilon))$.

It is worth noting that seen from Bell’s theorem perspective, Result 3 also
has interesting consequences for randomness certification. Differently from
the usual setup that requires a violation of
$\operatorname{CHSH}_{0,0}=2\sqrt{2}$ to certify one bit of randomness, the
agreement condition $\left<B^{0}_{1}B^{0}_{2}\right>=1$ permits the same to
reach with a smaller CHSH inequality violation. Furthermore, the standard
scenario with $\operatorname{CHSH}_{0,0}=2\sqrt{2}$ requires that a third
input is measured by either of the parties if one wishes to directly establish
a common secret bit between them. This follows from the fact that in this
case,
$\left<B^{0}_{1}B^{0}_{2}\right>=\left<B^{0}_{1}B^{1}_{2}\right>=-\left<B^{1}_{1}B^{0}_{2}\right>=\left<B^{1}_{1}B^{1}_{2}\right>=1/\sqrt{2}$,
that is, the measurement outcomes are not completely correlated. In turn, in
our case, the condition $\left<B^{0}_{1}B^{0}_{2}\right>=1$ already guarantees
the perfectly correlated secret bit.

Moving beyond the case of the maximal violation
$\operatorname{CHSH}_{0,0}=5/2$, we can also probe via a semi-definite program
detailed in Appendix VIII.4, the minimum value of $\delta$ required to explain
a given value of $\operatorname{CHSH}_{\delta,\epsilon}$, with the results
shown in Fig. 2. There we also consider the effect of imperfections on the
agreement between the observers, that is, we allow
$\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$, a condition of relevance for
experimental tests of our witness. Interestingly, we observe that for any
value of $\epsilon$ there is always a quantum violation of the
$\operatorname{CHSH}_{\delta,\epsilon}$ inequality leading to $\delta=1/2$,
that is, the outcomes of the observers are completely uncorrelated from the
outcome $a$ of the central system they are supposedly probing. Between the
range $0\leq\epsilon\leq\frac{1}{2}(1-1/\sqrt{2})$, as we increase the
$\epsilon$ we also increase the maximum quantum violation of
$\operatorname{CHSH}_{0,0}$. From this point on, that corresponds to
$\operatorname{CHSH}_{\delta,\epsilon}=2\sqrt{2}$ and
$\left<B^{0}_{1}B^{0}_{2}\right>=1/\sqrt{2}$ we see the opposite behavior,
since the maximum possible violation of
$\operatorname{CHSH}_{\delta,\epsilon}$ decreases as we increase $\epsilon$ in
the range $\frac{1}{2}(1-1/\sqrt{2})\leq\epsilon\leq 1/2$.

Generalizing our scenario, we can now consider the case where system $A$ has
not only one but actually, two properties, corresponding to the outcomes
$a_{0}$ and $a_{1}$, that we assume to be correlated with the outcomes of
measurements performed by the two observers. In this case, the conditions (7),
(8) and (9) now assume a joint probability distribution
$p(a_{0},a_{1},b_{1},b_{2}|x_{1},x_{2})$ and in particular, assuming
$\delta=0$ for simplicity, the objectivity condition implies that

$\displaystyle\left\\{\begin{array}[]{ll}p(a_{i}=b_{1}|x_{1}=i,x_{2})&=1\\\
p(a_{i}=b_{2}|x_{1},x_{2}=i)&=1\end{array}\right.$ (19)

Put differently, if $x_{i}=0$ then the outcome $b_{i}$ should be correlated
with $a_{0}$; if $x_{i}=1$ then the outcome $b_{i}$ should be correlated with
$a_{1}$. Similarly to the previous case, it follows that
$\operatorname{CHSH}_{\delta,0}$ constraints the correlations compatible with
this scenario. However, as stated in our next result, proven in the Appendix
VIII.5, differently from the previous case, if we impose a stronger notion of
agreement of the observers for both possible measurements, that is,
$p(b_{1}=b_{2}|x_{1}=x_{2})$, then there are no quantum violations of
$\operatorname{CHSH}_{\delta,0}$.

######  Result 4.

If we impose that $p(b_{1}=b_{2}|x_{1}=x_{2})=1$ for all $x_{1}$ and $x_{2}$,
then all quantum correlations are compatible with the assumptions (7), (8) and
(19).

This shows that if the observers agree on the outcomes of all measurements
being performed, necessarily then, the observed correlations are compatible
with an underlying statistics where the observed outcomes are correlated with
the property of the system they are probing, represented by the probability
distribution $p(a)$.

It is worth remarking here the fact that such non-objectivity bounds can also
serve as witnesses of post-quantum correlations. Differently from Result 4 for
quantum correlations, no-signalling correlations, those respecting eqs. (7)
and (8), a set of correlations that includes the quantum set, do allow for
violations even in the case where the observers agree on the outcomes of all
measurements being performed. To illustrate this, it is enough to consider the
paradigmatic PR-box [32], given by

$p(b_{1},b_{2}|x_{1},x_{2})=\frac{1}{2}\delta_{b_{1}\oplus
b_{2},x_{1}\bar{x_{2}}}\;.$ (20)

The PR-box is such that the observers agree on the outcomes of both possible
measurement inputs but at the same time can violate the CHSH inequality up to
its algebraic maximum of $\operatorname{CHSH}_{0,0}=4$. In general, via our
Result 4, the violation of the CHSH inequality under the constraint of
concordance between the observers implies directly the post-quantum nature of
the correlations.

##  IV Proof-of-principle experiment: setup

In the same spirit of Bell’s theorem, the constraints we want to test do not
depend on any specific dynamics and do not need to assume any precise physical
theory. In the following we describe a proof-of-principle photonic experiment
realizing a physical interaction dynamics whose output state can be naturally
mapped into the quantum Darwinism scenario.

The quantum mechanical description of the scheme and the interaction between
the involved photonic degrees of freedom provide only an intuitive picture,
and we stress that the conclusions drawn from the experimental data are fully
device-independent and the dynamics could in principle be unknown.

In our scheme, we consider the temporal degree of freedom of photons as the
observed system $A$, while the polarization of two photons represents a pair
of observer systems $B_{1}$ and $B_{2}$, as the environment (see Fig. 3a). To
simplify the presentation we consider a model where the temporal modes of the
photons are treated separately, for a more in-depth analysis the reader can
refer to Appendix VIII.6. More specifically, two photons interact with a
birefringent crystal, thus coupling the polarization and the temporal delay.
Both the generation and the interaction occur within a Sagnac interferometer,
after which the photons are spatially separated and their (possibly) entangled
polarization carries information on the temporal delay, as shown in Fig. 3b.
The action $\hat{T}$ of the birefringent plate on a single photon with state
$\left|\Psi\right\rangle=\alpha\left|H\right\rangle+\beta\left|V\right\rangle$
is to introduce a delay between the $\left|H\right\rangle$ and
$\left|V\right\rangle$ polarization states, coupling them with a temporal
wavefunction $\Phi(t)$, such that

$\begin{split}\hat{T}\left|\Psi\right\rangle=\hat{T}[(\alpha\left|H\right\rangle+\beta\left|V\right\rangle)\,\left|\Phi(t)\right\rangle]=\\\
=\alpha\left|H\right\rangle\left|\Phi(t+\Delta
t_{H})\right\rangle+\beta\left|V\right\rangle\left|\Phi(t)\right\rangle\;,\end{split}$
(21)

where $\Delta t_{H}$ is the temporal delay, induced by the birefringent plate,
on the horizontal polarization with respect to the vertical polarization.
Notice that this operation realizes a dephasing channel, a paradigmatic model
to formalize decoherence in realistic physical situations, among which
measurements are particular cases [45]. Indeed, a complex interaction
involving many degrees of freedom with discrete or continuous spectrum (all
realistic measurements involve several degrees of freedom of different nature)
can be modeled as a unitary operation involving the considered qubit states,
in our case the polarizations, and an external state corresponding to the
system $A$, that in our case corresponds to the joint temporal degree of
freedom.

More specifically, the state of the qubit interacting with the birefringent
plate, modeled as the dephasing channel involving time as the system $A$, can
be described by the following evolution:

$\begin{split}&\left|H\right\rangle\otimes\left|0\right\rangle_{A}\xrightarrow{\hat{T}}\Delta\left|H\right\rangle\otimes\left|0\right\rangle_{A}+\sqrt{1-|\Delta|^{2}}\left|H\right\rangle\otimes\left|1\right\rangle_{A}\;,\\\
&\left|V\right\rangle\otimes\left|0\right\rangle_{A}\xrightarrow{\hat{T}}\left|V\right\rangle\otimes\left|0\right\rangle_{A}\;,\end{split}$
(22)

where $\Delta$ is the dephasing parameter related to the overlap between,
respectively, the two-photon joint temporal wavefunction where the horizontal
polarization is retarded with respect to the vertical polarization and the
wavefunction without retardation. The retarded states
$\left|0\right\rangle_{A}$ and $\left|1\right\rangle_{A}$ are orthonormal
states of the observed system ${}_{A}\left\langle i\mid
j\right\rangle_{A}=\delta_{ij}$, with $i,j=0,1$ (here, $\delta$ is the
Kronecker delta) corresponding to non-overlapping terms of the photonic
temporal wavefunction (see Appendix VIII.6).

Consider the scheme in Fig. 3. A nonlinear crystal is placed within the
interferometer and a laser pump beam enters the interferometer along an input
of a dual-wavelength polarizing beam splitter (DPBS) and its interaction with
the crystal generates pairs of photons in the state $\left|HV\right\rangle$.
The pump passes through the interferometer along a clockwise and a counter-
clockwise direction, and the relative amplitude of these contributions depends
on the polarization state of the pump beam at the input of the DPBS. Note that
this represents a common scheme for the generation of polarization-entangled
photon pairs [46]. Then, a birefringent plate is inserted in a Sagnac
interferometer after (seen from the anti-clockwise path) the crystal. The pump
is unaffected by the plate, which induces only a negligible phase shift of the
pump field, since it has a coherence time much larger than the introduced
delay. On the other hand, the anti-clockwise generated $\left|HV\right\rangle$
state is affected by the dephasing channel as in (LABEL:eq:dephasing), while
the clockwise term is unaffected since it does not pass through the plate.
Hence, for the counter-clockwise generation, the state is given by

$\begin{split}&\left|H\right\rangle\left|V\right\rangle\otimes\left|0\right\rangle_{A}\left|0\right\rangle_{A}\xrightarrow{\hat{T}}\Delta\left|H\right\rangle\left|V\right\rangle\otimes\left|0\right\rangle_{A}\left|0\right\rangle_{A}+\\\
&+\sqrt{1-|\Delta|^{2}}\left|H\right\rangle\left|V\right\rangle\otimes\left|1\right\rangle_{A}\left|0\right\rangle_{A}\;.\end{split}$
(23)

(a) Conceptual scheme.

(b) Experimental setup

Figure 3:  Proof-of-principle experimental setup. a) Conceptual scheme of the
experiment, where the polarization of photons represents the environments
while the temporal delay in the joint wavefunction represents the quantum
system of interest. b) Experimental setup. A Sagnac-based polarization
entangled photon source generates pairs of degenerate photons at $808$nm.
Inside the Sagnac interferometer, the interaction of the photons generated in
the anti-clockwise direction interacts with the birefringent plates which
couple the polarizations with the temporal degree of freedom. The strength of
the interaction is parametrized by $\Delta$ indicating the overlap between the
temporal wavefunctions of the horizontal and vertical polarizations and is
varied by changing the thickness of the birefringent plates. Finally, the
photons are collected and detected by single photon detectors. In detail, M =
mirror; PBS = polarizing beam splitter; HWP= half-waveplate; QWP = quarter-
waveplate; DM = dichroic mirror; DHWP = dual-wavelength half-waveplate; APDs =
avalanche photo-diode detectors.

Considering a pump beam in a diagonal polarization state, the (non-normalized)
final state after the DPBS of the Sagnac interferometer will be:

$\begin{split}&\frac{1}{\sqrt{2}}[\Delta\left|H\right\rangle\left|V\right\rangle\left|0\right\rangle_{A}\left|0\right\rangle_{A}+\\\
&+\sqrt{1-|\Delta|^{2}}\left|H\right\rangle\left|V\right\rangle\otimes\left|1\right\rangle_{A}\left|0\right\rangle_{A}-\left|V\right\rangle\left|H\right\rangle\otimes\left|0\right\rangle_{A}\left|0\right\rangle_{A}]\;.\end{split}$
(24)

Tracing out the time degree of freedom, that nonetheless encodes its
information in the polarization state of the photons, the final state of the
two observer systems is given by

$\displaystyle\rho_{\mathrm{f}}=|\Delta|^{2}\left|\Psi^{-}\right\rangle\left\langle\Psi^{-}\right|+(1-|\Delta|^{2})\rho_{\mathrm{mix}}\;,$
(25)

where $\left|\Psi^{-}\right\rangle$ is the singlet polarization state, and
$\rho_{\mathrm{mix}}$ is the mixed state
$\rho_{\mathrm{mix}}=(\left|HV\right\rangle\left\langle
HV\right|+\left|VH\right\rangle\left\langle VH\right|)/2$.

In this way, an interaction occurs between the polarization of the two photons
and their time degree of freedom by means of the same birefringent plate.
After such an interaction, the polarization of the photons, i.e. the
environment systems $B_{1}$ and $B_{2}$, carry information about the time
degree of freedom defined as the observed system $A$ (see Appendix VIII.6 for
a detailed description of the mapping between the quantum Darwinism scenario
and the experimental realization). The strength of this interaction can be
tuned by changing the thickness of the birefringent plates. To illustrate
this, we describe two extremal conditions. When no birefringent plate is
present no interaction occurs, thus the temporal state of the photons is
uncorrelated with respect to the polarization. In this case, $\Delta=1$ and,
from Eq.(25), the final polarization state of the photons, is a maximally
entangled state
$\frac{1}{\sqrt{2}}(\left|HV\right\rangle-\left|VH\right\rangle)$.

Conversely, when the thickness of the birefringent plate introduces a temporal
delay greater than the coherence time of the photons, $\Delta=0$ and then the
global state, from Eq. (LABEL:eq:unitaryt2), will be
$(\left|H\right\rangle\left|V\right\rangle\otimes\left|1\right\rangle_{A}\left|0\right\rangle_{A}-\left|V\right\rangle\left|H\right\rangle\otimes\left|0\right\rangle_{A}\left|0\right\rangle_{A})/\sqrt{2}$.
From Eq. (25), tracing out the time degree of freedom, the polarization state
will be the mixed state $(\left|HV\right\rangle\left\langle
HV\right|+\left|VH\right\rangle\left\langle VH\right|)/2$.

To resume, considering the state of the photonic polarizations in Eq.(25), we
have that, when there is maximum coupling ($\Delta=0$) the polarization values
of the two photons effectively correspond to the presence
($\left|H\right\rangle$ for the first photon and $\left|V\right\rangle$ for
the second one) or absence ($\left|V\right\rangle$ for the first photon and
$\left|H\right\rangle$ for the second one) of a temporal delay of the
wavefunction. Conversely, when no coupling is present, no information on the
presence of temporal delay is stored in the polarization of the photons. From
a Quantum Darwinism perspective, polarization plays the role of an
environment, mediating the interactions between the (indirectly) observed
system, here the temporal delay, and the measurement apparatus.

Figure 4:  Experimental data. Optimal values of the experimental CHSH
parameter as a function of the constraint $\left\langle B^{0}_{1}\mid
B^{0}_{2}\right\rangle=1-2\epsilon$ for different values of temporal overlap
$\Delta$ between the wavefunctions of the two polarizations. The dashed lines
represent the optimal values calculated from the theoretical model of the
experimental state. Moreover, for $\Delta=1$ and $\Delta=0.91$, there are also
reported results obtained using the ab-initio approach and are indicated with
the black and blue points, respectively. The error bars are calculated by
assuming Poissonian statistics.

##  V Experimental results

We performed the measurements using four different delays between the two
polarizations due to the birefringent plates. For each delay, there is a
correspondent strength of the interaction, parameterized by the overlap
$\Delta_{i}$, with $i=1,2,3,4$, from Eq. (LABEL:eq:unitaryt).

Once the overlap $\Delta_{i}$ is fixed, the measurements are performed by
varying the agreement $\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$ between
the measurements of the two observers. For each agreement, the violation of
the CHSH inequality is optimized, using the information from a quantum state
tomography [47]. More specifically, from the tomography, we extract the values
of the rotation angles of the measurement waveplates able to reach, within
experimental errors, the desired agreement and the corresponding maximum CHSH
parameter achievable by the generated state.

The results are shown in Fig. 4. For the case with maximum interaction,
$\Delta=0$, the polarization of the two photons became maximally entangled
with the time degree of freedom and so, from the monogamy of entanglement
[48], no entanglement is possible between the polarizations. Thus, no
violation of the CHSH inequality is observed (see green points in Fig. 4). In
this case, one can argue the emergence of objectivity, since the observers
measuring the polarization of the two photons, not only agree among themselves
but their measurement outcomes can indeed correspond to an objective property,
in our proof-of-principle experiment represented by the time degree of freedom
of the photons.

Conversely, when the interaction is absent ($\Delta=1$), the experimental
entangled state is able to violate the CHSH inequality up to a value
$S^{exp}=2.475\pm 0.008$ with an observed agreement $\left\langle
B^{0}_{1}\mid B^{0}_{2}\right\rangle=1-2\epsilon=0.956\pm 0.002$, that is
$\epsilon=0.022\pm 0.001$. Using the $\operatorname{CHSH}_{\delta.\epsilon}$
inequality (13), valid for general no-signalling correlations, we see that
this corresponds to $\delta\geq 0.124\pm 0.002$. The effect becomes more
pronounced when assuming the validity of quantum theory for all systems
involved. A numerical computation approximating the quantum set by a superset
of allowed distributions [43] returns $\delta=0.49\pm 0.01$, revealing that
the observed variables should be almost completely uncorrelated to any
candidate for the system $A$ property. We thus obtain solid experimental
evidence of the (noisy) "collective hallucination", where the two observers,
even if they have an agreement on the supposedly measured value, the latter
cannot correspond to an objective property of the quantum system of interest.

In order to bring our experimental evidence for non-objectivity closer to the
spirit of the device-independent paradigm, we also perform measurements using
the ab-initio approach introduced in [49], where experimental violations of
classical constraints are found and optimized in a fully black-box scenario,
without any knowledge the generated state and the measurement apparatuses.

More specifically, while usually in an experiment one tries to violate some
inequality using precise knowledge of the employed experimental apparatus, in
an ab-initio approach one does not assume any prior information and, based
only on the (noisy) output statistics, adaptively learns the optimal values of
some controllable parameters, in order to optimize a given cost function, such
as the violation of a Bell-like inequality.

In our experiment there are $8$ parameters to be optimized by the algorithm,
corresponding to the values of the angles of pairs of waveplates (one pair for
each measurements station) for each of the $4$ measurements needed to evaluate
the CHSH parameter. In particular, the optimization firstly reaches the target
value of the agreement $\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$ tuning
the $4$ involved waveplate parameters, and then it reaches a global optimum
for the CHSH value tuning the other $4$ parameters associated with the
remaining CHSH measurements $\\{B^{1}_{1},B^{1}_{2}\\}$. Details on the ab-
initio optimization protocol can be found in Appendix VIII.7. For each run of
the protocol, the algorithm performs $350$ iterations, i.e. number of points
sampled in a pair of $4$-dimensional parameters space associated to the CHSH
measurements $\\{\\{B^{0}_{1},B^{0}_{2}\\},\\{B^{1}_{1},B^{1}_{2}\\}\\}$. The
first four parameters are related to the measurements
$\\{B^{0}_{1},B^{0}_{2}\\}$, fixing a value for the agreement between the
observers; the other parameters are related to the remaining operators
$\\{B^{1}_{1},B^{1}_{2}\\}$. The results on the values of the CHSH
experimentally achieved with the ab-initio approach are shown in Fig.4.

The experimental points collected with the ab-initio approach can reach values
higher than the ones achieved using quantum state tomography information. This
is possible because within an ab-initio framework errors in the
characterization of the optical setup, such as the optical axes of waveplates,
can be compensated automatically by the optimization process. For all the
curves with $\Delta\neq 0$, for each value of the observed agreement, the CHSH
is violated, consequently witnessing a degree of non-objectivity in a device-
independent way.

##  VI Discussion

Comprehending how microscopic quantum features give rise to the observed
macroscopic properties is a central goal of decoherence theory and in
particular of quantum Darwinism. Importantly, the emergence of objectivity,
that is, the fact that different observers agree on the properties of a
quantum system under observation, can be seen as a generic aspect as long as
the information of the quantum system is successfully outspread to the
environment it is interacting with. It is unclear, however, how to witness the
presence or rather the absence of such objectivity in practice. Can we witness
non-objectivity by simply probing the environment, without any knowledge of
the underlying dynamics?

To answer in positive to this question, we establish a probabilistic framework
to cast objectivity through operational lens, building up on the results of
[22]. Within this setting, we propose three properties defining what is to be
expected from a generic objective behaviour: no-superdeterminism, no-
signalling and the $\delta$-objectivity, the latter stating that
$p(b_{i}=a|x_{i}^{*})\geq 1-\delta$, where $x_{i}^{*}$ denotes the measurement
for which the observer should try to recover as best as possible the
information about the system $A$ as encoded in the probability $p(a)$. Those
conditions play a role similar to what the concept of local realism implies
for Bell’s theorem [28]. In particular, the notion of $\delta$-objectivity is
justified by our first result stating that the local agreement between a given
observer and the quantum system of interest, translates into a global notion
of agreement between all observers having access to some part of the
environment.

We then showed that a generalization of the seminal CHSH inequality [27]
constraints the set of possible correlations compatible with the three
aforementioned assumptions. Furthermore, the violation of this inequality
offers a device-independent witness of the non-objectivity of the underlying
process, at the same time that it naturally quantifies how much one should
give up objectivity in order to explain the observed correlations. Further, we
proved that quantum mechanics allow for violations of this inequality and, in
particular, leads to a monogamy relation implying that even though the
observers agree among themselves, their outcomes are completely uncorrelated
from the system they supposedly should be correlated with. A phenomenon we
name "collective hallucination" and that we have experimentally probed using a
photonic setup where the quantum property of interest is encoded in the
temporal degree of freedom of photons, the polarization of which plays the
role of the environment to which the information should redundantly
proliferate.

For scenarios where the probed system has more than one property of interest,
we demonstrated that if the observers have to agree on measurement outcomes of
all performed measurements, then quantum correlations are compatible with the
assumptions no-superdeterminism, no-signalling and objectivity, that is, they
cannot violate any Bell-like inequality. A result that can be violated by
correlations beyond those allowed by quantum theory and that thus can be
employed as a test for post-quantumness.

To our knowledge, this is the first connection between quantum Darwinism, and
the notion of objectivity it entails, with Bell inequalities and device-
independent quantum information processing, a bridge that deserves further
investigation. For instance, it would be interesting to generalize our results
to a larger number of observers and consider measurements with more outcomes.
At the same time, one should understand paradigmatic dynamics considered in
the literature of quantum Darwinism [13, 14, 15, 16, 17, 18, 19, 20] under
this new perspective, and explore the connections with other objectivity
measures [50] valid in the quantum framework. It is worth remarking also that
our approach could lead to substantial refinements in recent tests for the
emergence of objectivity [51, 52, 53]. That said, we should also note that
another bridge connecting quantum Darwinism, Spekkens contextuality and
quantum information has also been recently erected. Adapting the prepare-and-
measure scenario into the usual environment as a witness framework, in Ref.
[24], the authors managed to prove that Spekkens non-contextuality for each
observer follows through whenever the environment proliferates the information
about the central system appropriately. Their notion of classicality differs
from ours, insofar as here we consider mutual agreement rather than non-
contextuality as a signature of classicality—and our connection with
foundations of quantum mechanics is via Bell scenarios. Additionally, our work
goes a step further, as we investigate our theoretical findings with a proof-
of-principle experiment.

Finally, we notice that the $\delta$-objectivity constraint we consider here
is mathematically very similar to the notion of absoluteness of events
employed to analyze a generalization of the Wigner’s friend experiment [54,
55, 56] in the foundations of quantum theory. Apprehending further the
connections between quantum Darwinism/objectivity and Wigner’s friend
experiment/absoluteness of events is another relevant research direction that
we hope our results might trigger.

##  VII Acknowledgements

This work was supported by the Serrapilheira Institute (Grant No.
Serra-1708-15763), by the Simons Foundation (Grant Number 884966, AF), the
Brazilian National Council for Scientific and Technological Development (CNPq)
via the National Institute for Science and Technology on Quantum Information
(INCT-IQ) and Grants No. 406574/2018-9 and 307295/2020-6 This research was
also supported by the Fetzer Franklin Fund of the John E. Fetzer Memorial
Trust and by grant number FQXi-RFP-IPW-1905 from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation. We acknowledge support from the Templeton Foundation,
The Quantum Information Structure of Spacetime (QISS) Project (qiss.fr) (the
opinions expressed in this publication are those of the author(s) and do not
necessarily reflect the views of the John Templeton Foundation) Grant
Agreement No. 61466.

## References

  * Broglie [1924] L. d. Broglie, A tentative theory of light quanta, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47, 446 (1924).
  * Bohr  _et al._ [1928] N. Bohr _et al._ ,  _The quantum postulate and the recent development of atomic theory_ , Vol. 3 (Printed in Great Britain by R. & R. Clarke, Limited, 1928).
  * Wootters and Zurek [1979] W. K. Wootters and W. H. Zurek, Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of bohr’s principle, Physical Review D 19, 473 (1979).
  * Wigner [1963] E. P. Wigner, The problem of measurement, American Journal of Physics 31, 6 (1963).
  * Schlosshauer [2007] M. A. Schlosshauer,  _Decoherence: and the quantum-to-classical transition_ (Springer Science & Business Media, 2007).
  * Joos  _et al._ [2013] E. Joos, H. D. Zeh, C. Kiefer, D. J. Giulini, J. Kupsch, and I.-O. Stamatescu,  _Decoherence and the appearance of a classical world in quantum theory_ (Springer Science & Business Media, 2013).
  * Aolita  _et al._ [2008] L. Aolita, R. Chaves, D. Cavalcanti, A. Acín, and L. Davidovich, Scaling laws for the decay of multiqubit entanglement, Physical Review Letters 100, 080501 (2008).
  * Brune  _et al._ [1996] M. Brune, E. Hagley, J. Dreyer, X. Maitre, A. Maali, C. Wunderlich, J. M. Raimond, and S. Haroche, Observing the progressive decoherence of the “meter” in a quantum measurement, Physical Review Letters 77, 4887 (1996).
  * Arndt  _et al._ [1999] M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. Van der Zouw, and A. Zeilinger, Wave–particle duality of c60 molecules, nature 401, 680 (1999).
  * Sonnentag and Hasselbach [2007] P. Sonnentag and F. Hasselbach, Measurement of decoherence of electron waves and visualization of the quantum-classical transition, Physical review letters 98, 200402 (2007).
  * Zurek [2003] W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical, Reviews of modern physics 75, 715 (2003).
  * Zurek [2009] W. H. Zurek, Quantum darwinism, Nature physics 5, 181 (2009).
  * Ollivier  _et al._ [2004] H. Ollivier, D. Poulin, and W. H. Zurek, Objective properties from subjective quantum states: Environment as a witness, Physical review letters 93, 220401 (2004).
  * Ollivier  _et al._ [2005] H. Ollivier, D. Poulin, and W. H. Zurek, Environment as a witness: Selective proliferation of information and emergence of objectivity in a quantum universe, Physical review A 72, 042113 (2005).
  * Blume-Kohout and Zurek [2006] R. Blume-Kohout and W. H. Zurek, Quantum darwinism: Entanglement, branches, and the emergent classicality of redundantly stored quantum information, Physical review A 73, 062310 (2006).
  * Korbicz  _et al._ [2014] J. K. Korbicz, P. Horodecki, and R. Horodecki, Objectivity in a noisy photonic environment through quantum state information broadcasting, Physical review letters 112, 120402 (2014).
  * Riedel and Zurek [2010] C. J. Riedel and W. H. Zurek, Quantum darwinism in an everyday environment: Huge redundancy in scattered photons, Phys. Rev. Lett. 105, 020404 (2010).
  * Blume-Kohout and Zurek [2008] R. Blume-Kohout and W. H. Zurek, Quantum darwinism in quantum brownian motion, Phys. Rev. Lett. 101, 240405 (2008).
  * Zwolak  _et al._ [2009] M. Zwolak, H. T. Quan, and W. H. Zurek, Quantum darwinism in a mixed environment, Phys. Rev. Lett. 103, 110402 (2009).
  * Riedel and Zurek [2011] C. J. Riedel and W. H. Zurek, Redundant information from thermal illumination: quantum darwinism in scattered photons, New Journal of Physics 13, 073038 (2011).
  * Horodecki  _et al._ [2015] R. Horodecki, J. K. Korbicz, and P. Horodecki, Quantum origins of objectivity, Physical review A 91, 032122 (2015).
  * Brandao  _et al._ [2015] F. G. Brandao, M. Piani, and P. Horodecki, Generic emergence of classical features in quantum darwinism, Nature communications 6, 1 (2015).
  * Qi and Ranard [2021] X.-L. Qi and D. Ranard, Emergent classicality in general multipartite states and channels, Quantum 5, 555 (2021).
  * Baldijão  _et al._ [2021] R. Baldijão, R. Wagner, C. Duarte, B. Amaral, and M. T. Cunha, Emergence of noncontextuality under quantum darwinism, PRX Quantum 2 (2021).
  * Çakmak  _et al._ [2021] B. Çakmak, Ö. E. Müstecaplıoğlu, M. Paternostro, B. Vacchini, and S. Campbell, Quantum darwinism in a composite system: Objectivity versus classicality, Entropy 23, 995 (2021).
  * Touil  _et al._ [2022] A. Touil, B. Yan, D. Girolami, S. Deffner, and W. H. Zurek, Eavesdropping on the decohering environment: quantum darwinism, amplification, and the origin of objective classical reality, Physical Review Letters 128, 010401 (2022).
  * Clauser  _et al._ [1969] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Physical review letters 23, 880 (1969).
  * Brunner  _et al._ [2014] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics 86, 419 (2014).
  * Note [1] A possible route to generalization of quantum Darwinism to GPTs was proposed in [57]. Unfortunately, the authors are concerned with defining what an idealized quantum Darwinism process would look like in GPTs, more precisely, a general version of a fan-out gate, and do not consider the noisy version of such a process.
  * Barrett  _et al._ [2006] J. Barrett, A. Kent, and S. Pironio, Maximally nonlocal and monogamous quantum correlations, Physical review letters 97, 170409 (2006).
  * Barnum  _et al._ [2007] H. Barnum, J. Barrett, M. Leifer, and A. Wilce, Generalized no-broadcasting theorem, Physical review letters 99, 240501 (2007).
  * Popescu and Rohrlich [1994] S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Foundations of Physics 24, 379 (1994).
  * Müller and Masanes [2016] M. P. Müller and L. Masanes, Information-theoretic postulates for quantum theory, in  _Quantum Theory: Informational Foundations and Foils_, edited by G. Chiribella and R. W. Spekkens (Springer Netherlands, Dordrecht, 2016) pp. 139–170.
  * Pironio  _et al._ [2016] S. Pironio, V. Scarani, and T. Vidick, Focus on device independent quantum information, New Journal of Physics 18, 100202 (2016).
  * Ekert [1991] A. K. Ekert, Quantum cryptography based on bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
  * 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, _et al._ , Random numbers certified by bell’s theorem, Nature 464, 1021 (2010).
  * Cabello  _et al._ [2011] A. Cabello, V. D’Ambrosio, E. Nagali, and F. Sciarrino, Hybrid ququart-encoded quantum cryptography protected by kochen-specker contextuality, Phys. Rev. A 84, 030302 (2011).
  * Hall [2016] M. J. Hall, The significance of measurement independence for bell inequalities and locality, in  _At the frontier of spacetime_ (Springer, 2016) pp. 189–204.
  * Chaves  _et al._ [2021] R. Chaves, G. Moreno, E. Polino, D. Poderini, I. Agresti, A. Suprano, M. R. Barros, G. Carvacho, E. Wolfe, A. Canabarro, _et al._ , Causal networks and freedom of choice in bell’s theorem, PRX Quantum 2, 040323 (2021).
  * Šupić and Bowles [2020] I. Šupić and J. Bowles, Self-testing of quantum systems: a review, Quantum 4, 337 (2020).
  * Masanes  _et al._ [2006] L. Masanes, A. Acin, and N. Gisin, General properties of nonsignaling theories, Phys. Rev. A 73, 012112 (2006).
  * Dhara  _et al._ [2013] C. Dhara, G. Prettico, and A. Acín, Maximal quantum randomness in bell tests, Phys. Rev. A 88, 052116 (2013).
  * Navascués  _et al._ [2007] M. Navascués, S. Pironio, and A. Acín, Bounding the set of quantum correlations, Phys. Rev. Lett. 98, 010401 (2007).
  * 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).
  * Preskill [1998] J. Preskill, Lecture notes for physics 229: Quantum information and computation, California Institute of Technology 16, 1 (1998).
  * Fedrizzi  _et al._ [2007] A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, A wavelength-tunable fiber-coupled source of narrowband entangled photons, Optics Express 15, 15377 (2007).
  * James  _et al._ [2005] D. F. James, P. G. Kwiat, W. J. Munro, and A. G. White, On the measurement of qubits, in  _Asymptotic Theory of Quantum Statistical Inference: Selected Papers_ (World Scientific, 2005) pp. 509–538.
  * Osborne and Verstraete [2006] T. J. Osborne and F. Verstraete, General monogamy inequality for bipartite qubit entanglement, Physical review letters 96, 220503 (2006).
  * Poderini  _et al._ [2022] D. Poderini, E. Polino, G. Rodari, A. Suprano, R. Chaves, and F. Sciarrino, Ab initio experimental violation of bell inequalities, Physical Review Research 4, 013159 (2022).
  * Girolami  _et al._ [2022] D. Girolami, A. Touil, B. Yan, S. Deffner, and W. H. Zurek, Redundantly amplified information suppresses quantum correlations in many-body systems, arXiv preprint arXiv:2202.09328 (2022).
  * Ciampini  _et al._ [2018] M. A. Ciampini, G. Pinna, P. Mataloni, and M. Paternostro, Experimental signature of quantum darwinism in photonic cluster states, Physical Review A 98, 020101 (2018).
  * Chen  _et al._ [2019] M.-C. Chen, H.-S. Zhong, Y. Li, D. Wu, X.-L. Wang, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, Emergence of classical objectivity of quantum darwinism in a photonic quantum simulator, Science Bulletin 64, 580 (2019).
  * Unden  _et al._ [2019] T. K. Unden, D. Louzon, M. Zwolak, W. H. Zurek, and F. Jelezko, Revealing the emergence of classicality using nitrogen-vacancy centers, Phys. Rev. Lett. 123, 140402 (2019).
  * Bong  _et al._ [2020] K.-W. Bong, A. Utreras-Alarcón, F. Ghafari, Y.-C. Liang, N. Tischler, E. G. Cavalcanti, G. J. Pryde, and H. M. Wiseman, A strong no-go theorem on the wigner’s friend paradox, Nature Physics 16, 1199 (2020).
  * Moreno  _et al._ [2021] G. Moreno, R. Nery, C. Duarte, and R. Chaves, Events in quantum mechanics are maximally non-absolute (2021).
  * Wiseman  _et al._ [2022] H. M. Wiseman, E. G. Cavalcanti, and E. G. Rieffel, A" thoughtful" local friendliness no-go theorem: a prospective experiment with new assumptions to suit, arXiv preprint arXiv:2209.08491 (2022).
  * Baldijão  _et al._ [2022] R. D. Baldijão, M. Krumm, A. J. Garner, and M. P. Müller, Quantum darwinism and the spreading of classical information in non-classical theories, Quantum 6, 636 (2022).
  * 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).
  * Boyd [2020] R. W. Boyd,  _Nonlinear optics_ (Academic press, 2020).
  * Chang [2012] K.-H. Chang, Stochastic nelder–mead simplex method–a new globally convergent direct search method for simulation optimization, European journal of operational research 220, 684 (2012).

##  VIII Appendix

###  VIII.1 Proof of Result 1

###### Proof.

We start by summing all the conditions $p(b_{k}=a,a|x_{k}^{*})\geq 1-\delta$
and expanding them in terms of the joint distribution
$p(b_{1}=a,b_{2}=a,\ldots,b_{n}=a,a|x_{1}^{*},\ldots,x_{n}^{*}),p(b_{1}\neq
a,b_{2}=a,\ldots,b_{n}=a,a|x_{1}^{*},\ldots,x_{n}^{*}),\ldots$. To simplify
the notation we use $p_{s}=p_{\beta_{1},\beta_{2},\ldots,\beta_{n}}$, where
$s$ is a string of bits $\beta_{i}\in\\{0,1\\}$, denoting the term in which
$b_{i}=a$ or $b_{i}\neq a$, respectively with $\beta_{i}=0$ and $\beta_{i}=1$.
So for example $p_{0}=p_{00\ldots 0}$ represents the term where all the
$b_{i}$ agree with $a$, while $p_{1}=p_{10\ldots 0}$ represents the one where
only $b_{1}$ does not. Calling $H(s)$ the hamming distance between $00\ldots
0$ and $s$, i.e. the number of $1$s in the bitstring $s$, we obtain:

$\displaystyle n(1-\delta)$
$\displaystyle\leq\sum_{k}p(b_{k}=a|x_{k}^{*})=\sum_{s\in\\{0,1\\}^{n}}(n-H(s))p_{s}\leq$
(26) $\displaystyle\leq p_{0}+(n-1)\left(p_{0}+\sum_{s:H(s)\geq
1}p_{s}\right)\leq p_{0}+(n-1)$ (27)

where we used the normalization condition for the $p_{s}$. From this we
directly obtain (1). ∎

#### Thightness of bound

###### Proof.

To prove the tightness of the bound consider the distribution
$p(a,b_{1},\ldots,b_{n}|x_{1},\ldots,x_{n})$ defined as follows by
highlighting the settings and the variables associated with special settings
$x^{*}_{i}$:

$p(a,b_{1},\ldots,b_{n}|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}},x_{i_{k+1}},\ldots,x_{i_{n}})=p_{\delta}(a,b_{i_{1}},\ldots,b_{i_{k}})p_{U}(b_{i_{k+1}},\ldots,b_{i_{n}})$
(28)

where

$p_{\delta}(a,b_{i_{1}},\ldots,b_{i_{k}})=\left\\{\begin{aligned}
&(1-k\delta)/2&\text{for}\quad a=b_{i_{l}}\forall l\leq k\\\
&\delta/2&\text{for}\quad a=b_{i_{l}}\forall l\leq k,l\neq m,b_{i_{m}}\neq
a\\\ &0&\text{elsewhere}\end{aligned}\right.$ (29)

while $p_{U}(b_{i_{k+1}},\ldots,b_{i_{n}})=2^{k-n}$ is the uniform
distribution over the remaining variables. First, notice that this
distribution for $n+1$ variables reduces to the one for $n$ variables by
marginalizing on any $b_{i}$. This is obvious when $x_{n+1}\neq x^{*}_{n+1}$,
because of the definition (28), while for $x_{n+1}=x^{*}_{n+1}$ we have

$\displaystyle
p(a,b_{1},\ldots,b_{n}|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}})=p(a,b_{1},\ldots,b_{n},b_{n+1}=a|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}},x^{*}_{n+1})+$
(30) $\displaystyle+p(a,b_{1},\ldots,b_{n},b_{n+1}\neq
a|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}},x^{*}_{n+1})=(1-k\delta)/2\quad\text{when}\quad
a=b_{i_{l}}\forall l\leq k$ (31) $\displaystyle
p(a,b_{1},\ldots,b_{n}|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}})=p(a,b_{1},\ldots,b_{n},b_{n+1}=a|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}},x^{*}_{n+1})+$
(32) $\displaystyle+p(a,b_{1},\ldots,b_{n},b_{n+1}\neq
a|x^{*}_{i_{1}},\ldots,x^{*}_{i_{k}},x^{*}_{n+1})=\delta/2\quad\text{when}\quad
a=b_{i_{l}}\forall l\leq k,l\neq m,b_{i_{m}}\neq a$ (33)

It is now easy to show that these distributions satisfy (9). Indeed the
condition is trivially valid for the case $n=1$, while for a general one, (9)
applies to the marginal on all but one of the $b_{i}$, which corresponds to
the distribution (28) for $n=1$ as we just proved.

Similarly to show that (29) satisfies the no-signaling condition
$\sum_{a,b_{j}}p(a,b_{i},b_{j}|x_{i},x_{j})=\sum_{a,b_{j}}p(a,b_{i},b_{j}|x_{i},x_{j}^{\prime})\quad\forall\;x_{j},x_{j}^{\prime}$,
we again use the fact that marginalizing on $b_{j}$ gives (28) for $n=1$,
which is independent of the choice of $x_{j}$. ∎

###  VIII.2 Proof of Result 2

###### Proof.

The proof follows closely Ref. [55]. Given a distribution
$p(b_{1},b_{2}|x_{1},x_{2},a)$, let us define the following quantities:

$\displaystyle\gamma^{(a)}_{x_{1},x_{2}}\equiv
p(b_{1}=0,b_{2}=0|x_{1},x_{2},a),$ (34)
$\displaystyle\beta^{(a)}_{x_{1}}\equiv p(b_{1}=0|x_{1},a),$ (35)
$\displaystyle\eta^{(a)}_{x_{2}}\equiv p(b_{2}=0|x_{2},a).$ (36)

Now, using the definition $\langle
B^{x_{1}}_{1}B^{x_{2}}_{2}\rangle_{a}=\sum(-1)^{b_{1}+b_{2}}p(b_{1},b_{2}|x_{1},x_{2},a)$,
the NS condition, and the fact that
$\max\\{0,\beta^{(a)}_{x_{1}}+\eta^{(a)}_{x_{2}}-1\\}\leq\gamma^{(a)}_{x_{1},x_{2}}\leq\min\\{\beta^{(a)}_{x_{1}},\eta^{(a)}_{x_{2}}\\}$,
one can show that

$\displaystyle
2\left|\beta^{(a)}_{x_{1}}+\eta^{(a)}_{x_{2}}-1\right|-1\leq\langle
B^{x_{1}}_{1}B^{x_{2}}_{2}\rangle_{a}\leq
1-2\left|\beta^{(a)}_{x_{1}}-\eta^{(a)}_{x_{2}}\right|.$ (37)

which can be used to bound the quantity
$\operatorname{CHSH}_{\delta,\epsilon}^{(a)}=\langle
B^{0}_{1}B^{0}_{2}\rangle_{a}+\langle B^{0}_{1}B^{1}_{2}\rangle_{a}-\langle
B^{1}_{1}B^{0}_{2}\rangle_{a}+\langle B^{1}_{1}B^{1}_{2}\rangle_{a}$, as

$\displaystyle\operatorname{CHSH}_{\delta,\epsilon}^{(a)}\leq\langle
B^{0}_{1}B^{0}_{2}\rangle_{a}+3-2J^{(a)}_{\delta},$ (38)

where

$\displaystyle J^{(a)}_{\delta}$
$\displaystyle\equiv\left|\beta^{(a)}_{0}-\eta^{(a)}_{1}\right|+\left|-\beta^{(a)}_{1}+\eta^{(a)}_{1}\right|+\left|\beta^{(a)}_{1}+\eta^{(a)}_{0}-1\right|$
$\displaystyle\geq\left|\beta_{0}^{(a)}-\beta_{1}^{(a)}\right|+\left|\beta^{(a)}_{1}+\eta^{(a)}_{0}-1\right|$
$\displaystyle\geq\left|\beta_{0}^{(a)}+\eta_{0}^{(a)}-1\right|=\left|p(b_{1}=0|x_{1}=0,a)+p(b_{2}=0|x_{2}=0,a)-1\right|.$

All the inequalities above are obtained via a direct application of the
triangle inequality. In the second line above, equality can be obtained by
setting $\eta_{1}^{(a)}=\frac{\beta_{0}^{(a)}-\beta_{1}^{(a)}}{2}$, which is
always allowed under the hypothesis we are working with. The inequality of the
third line can always be saturated if we set
$\beta_{1}^{(a)}=\frac{\beta_{0}^{(a)}-\eta_{0}^{(a)}}{2}$, also an available
choice.

Now, looking at the quantity $\operatorname{CHSH}_{\delta,\epsilon}$ and
recalling that $\sum_{a}p(a)\langle
B^{0}_{1}B^{0}_{2}\rangle_{a}=1-2\epsilon$, it follows that

$\displaystyle\operatorname{CHSH}_{\delta,\epsilon}$
$\displaystyle=\sum_{a}p(a)\operatorname{CHSH}_{\delta}^{(a)}$
$\displaystyle\leq 4-2\,\epsilon-2\sum_{a}p(a)J^{(a)}_{\delta}$
$\displaystyle\leq
4-2\,\epsilon-2\sum_{a}p(a)\left|\beta_{0}^{(a)}+\eta_{0}^{(a)}-1\right|.$
(39)

Let $\tilde{\beta}_{0}^{(a)}=p(b_{1}=1|x_{1}=0,a)$ and
$\tilde{\eta}_{0}^{(a)}=p(b_{2}=1|x_{2}=0,a)$, due to normalization, it holds
that
$\left|\beta_{0}^{(a)}+\eta_{0}^{(a)}-1\right|=\left|\tilde{\beta}_{0}^{(a)}+\tilde{\eta}_{0}^{(a)}-1\right|$.
This allows rewriting the last inequality of Eq. (39) as

$\displaystyle\operatorname{CHSH}_{\delta,\epsilon}\leq
4-2\,\epsilon-\sum_{a}p(a)\left(\,\left|\beta_{0}^{(a)}+\eta_{0}^{(a)}-1\right|+\left|\tilde{\beta}_{0}^{(a)}+\tilde{\eta}_{0}^{(a)}-1\right|\,\right).$
(40)

Using the triangle inequality to combine the $a=b_{i}$ terms and to combine
the $a\neq b_{i}$ terms leads to

$\displaystyle\operatorname{CHSH}_{\delta,\epsilon}\leq
4-2\,\epsilon-\left|p(b_{1}=a|x_{1}=0)+p(b_{2}=a|x_{2}=0)-1\right|-\left|p(b_{1}=1-a|x_{1}=0)+p(b_{2}=1-a|x_{2}=0)-1\right|,$
(41)

which relates the terms in the moduli to the $\delta$-objectivity inequality
(9) and its complement. The expression in the first modulus satisfies
$\sum_{i}\,\sum_{a}p(b_{i}=a,a|x_{i})-1\geq 1-2\delta$, while the expression
in the second modulus gives $\sum_{i}\,\sum_{a}p(b_{i}=1-a,a|x_{i})-1\leq
2\delta-1$. Since $\delta\leq 1/2$, we arrive at

$\displaystyle\operatorname{CHSH}_{\delta,\epsilon}\leq
2-2\,\epsilon+4\,\delta,$ (42)

which concludes the proof.

∎

###  VIII.3 Proof of Result 3 and the self-test of the state maximally
violating $\mathrm{CHSH}_{0,0}$

The idea of performing a self-test of a target state $|\psi\rangle$ from the
knowledge about the existence of a given state $|\Psi\rangle$ fulfilling some
special property, consists of exploring this property to build a local
isometry which, when applied to $|\Psi\rangle$ returns the tensor product of
$|\psi\rangle$ with some other ancillary and non-relevant state. Here we will
follow similar steps to that shown in section 4 of Ref. [40], which is an up-
to-date review of self-testing protocols.

Let $|\Psi^{*}\rangle$ be a state leading to the maximal quantum violation of
the $\operatorname{CHSH}_{0,0}$ inequality when one of the parts measures the
observables (and hence Hermitian operators) $B^{0}_{1}$ and $B^{1}_{1}$ and
the other part measures $B^{0}_{2}$ and $B^{1}_{2}$, i.e.,

$\displaystyle\langle\Psi^{*}|B^{0}_{1}B^{0}_{2}+B^{0}_{1}B^{1}_{2}-B^{1}_{1}B^{0}_{2}+B^{1}_{1}B^{1}_{2}|\Psi^{*}\rangle=\frac{5}{2}.$
(43)

in which, by hypothesis, it holds that
$\langle\Psi^{*}|B^{0}_{1}B^{0}_{2}|\Psi^{*}\rangle=1$, a condition which
entails that

$\displaystyle B^{0}_{1}|\Psi^{*}\rangle=B^{0}_{2}|\Psi^{*}\rangle.$ (44)

Our first step is to prove the following relation,

$\displaystyle\langle\Psi^{*}|\left[(B^{0}_{1}-B^{0}_{2})^{2}+\frac{1}{2}\left[(B^{1}_{2}-B^{0}_{2})-B^{1}_{1}\right]^{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[\frac{5}{2}\mathbbm{1}-\operatorname{CHSH}_{0,0}\right]|\Psi^{*}\rangle$
(45)

This relation can be achieved via a straightforward calculation using equation
44 and the condition $\langle\Psi^{*}|B^{0}_{1}B^{0}_{2}|\Psi^{*}\rangle=1$:

$\displaystyle\langle\Psi^{*}|\left[(B^{0}_{1}-B^{0}_{2})^{2}+\frac{1}{2}\left[(B^{1}_{2}-B^{0}_{2})-B^{1}_{1}\right]^{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[(B^{0}_{1})^{2}-2B^{0}_{1}B^{0}_{2}+(B^{0}_{2})^{2}+\frac{1}{2}(B^{1}_{2}-B^{0}_{2})^{2}\right.$
$\displaystyle-$
$\displaystyle\left.B^{1}_{1}(B^{1}_{2}-B^{0}_{2})+\frac{1}{2}(B^{1}_{1})^{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[2\mathbbm{1}-2B^{0}_{1}B^{0}_{2}+\frac{1}{2}((B^{1}_{2})^{2}-B^{1}_{2}B^{0}_{2}-B^{0}_{2}B^{1}_{2}+(B^{0}_{2})^{2})\right.$
$\displaystyle-$
$\displaystyle\left.B^{1}_{1}B^{1}_{2}+B^{1}_{1}B^{0}_{2}+\frac{1}{2}\mathbbm{1}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[\frac{7}{2}\mathbbm{1}-2B^{0}_{1}B^{0}_{2}-\frac{1}{2}(B^{1}_{2}B^{0}_{2}+B^{0}_{2}B^{1}_{2})-B^{1}_{1}B^{1}_{2}+B^{1}_{1}B^{0}_{2}\right]|\Psi^{*}\rangle,$

now we use that $\langle\Psi^{*}|B^{0}_{1}B^{0}_{2}|\Psi^{*}\rangle=1$ to
obtain,

$\displaystyle\langle\Psi^{*}|\left[(B^{0}_{1}-B^{0}_{2})^{2}+\frac{1}{2}\left[(B^{1}_{2}-B^{0}_{2})-B^{1}_{1}\right]^{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[\frac{5}{2}\mathbbm{1}-B^{0}_{1}B^{0}_{2}-\frac{1}{2}(B^{1}_{2}B^{0}_{2}+B^{0}_{2}B^{1}_{2})-B^{1}_{1}B^{1}_{2}+B^{1}_{1}B^{0}_{2}\right]|\Psi^{*}\rangle,$

and then we use Eq. 44, from which we get

$\displaystyle\langle\Psi^{*}|\left[(B^{0}_{1}-B^{0}_{2})^{2}+\frac{1}{2}\left[(B^{1}_{2}-B^{0}_{2})-B^{1}_{1}\right]^{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[\frac{5}{2}\mathbbm{1}-B^{0}_{1}B^{0}_{2}-B^{0}_{1}B^{1}_{2}+B^{1}_{1}B^{0}_{2}-B^{1}_{1}B^{1}_{2}\right]|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle\langle\Psi^{*}|\left[\frac{5}{2}\mathbbm{1}-\operatorname{CHSH}_{0,0}\right]|\Psi^{*}\rangle,$

which concludes the proof of Eq. 45.

By combining 45 and eq 43, we conclude that:

$\displaystyle(B^{1}_{2}-B^{0}_{2})|\Psi^{*}\rangle=B^{1}_{1}|\Psi^{*}\rangle.$
(46)

and using again relation 44,

$\displaystyle(B^{0}_{1}+B^{1}_{1})|\Psi^{*}\rangle=B^{1}_{2}|\Psi^{*}\rangle.$
(47)

Now, using 44, 46, and 47,

$\displaystyle\langle\Psi^{*}|\\{B^{0}_{1},B^{1}_{1}\\}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{0}_{1}B^{1}_{1}+B^{1}_{1}B^{0}_{1}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{0}_{1}(B^{1}_{2}-B^{0}_{2})+B^{1}_{1}B^{0}_{2}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|(B^{1}_{2}-B^{0}_{2})B^{0}_{1}+B^{0}_{2}B^{1}_{1}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{1}_{2}B^{0}_{1}-B^{0}_{2}B^{0}_{1}+B^{0}_{2}(B^{1}_{2}-B^{0}_{2})|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{1}_{2}B^{0}_{2}-B^{0}_{2}B^{0}_{2}+B^{0}_{2}B^{1}_{2}-B^{0}_{2}B^{0}_{2}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{1}_{2}B^{0}_{2}+B^{0}_{2}B^{1}_{2}-2\mathbbm{1}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|\\{B^{0}_{2},B^{1}_{2}\\}-2\mathbbm{1}|\Psi^{*}\rangle.$
(48)

in which, because $\langle\Psi^{*}|[B^{0}_{1},B^{1}_{1}]|\Psi^{*}\rangle=0$
and $\langle\Psi^{*}|[B^{0}_{2},B^{1}_{2}]|\Psi^{*}\rangle=0$, it holds that

$\displaystyle\left\\{\begin{array}[]{l}\langle\Psi^{*}|\\{B^{0}_{1},B^{1}_{1}\\}|\Psi^{*}\rangle=2\langle\Psi^{*}|B^{1}_{1}B^{0}_{1}|\Psi^{*}\rangle=2\langle\Psi^{*}|B^{1}_{1}B^{0}_{2}|\Psi^{*}\rangle\\\
\langle\Psi^{*}|\\{B^{0}_{2},B^{1}_{2}\\}|\Psi^{*}\rangle=2\langle\Psi^{*}|B^{0}_{2}B^{1}_{2}|\Psi^{*}\rangle=2\langle\Psi^{*}|B^{0}_{1}B^{1}_{2}|\Psi^{*}\rangle\end{array}\right.,$
(51)

and so

$\displaystyle 2\langle\Psi^{*}|B^{1}_{1}B^{0}_{2}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|2B^{0}_{1}B^{1}_{2}-2\mathbbm{1}|\Psi^{*}\rangle$
$\displaystyle\implies\langle\Psi^{*}|B^{0}_{1}B^{1}_{2}-B^{1}_{1}B^{0}_{2}|\Psi^{*}\rangle=1.$

Because of the constraint
$\langle\Psi^{*}|B^{0}_{1}B^{0}_{2}|\Psi^{*}\rangle=1$ and the Eq. 43, this
means that

$\displaystyle\frac{1}{2}$
$\displaystyle=\langle\Psi^{*}|B^{1}_{1}B^{1}_{2}|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{1}_{1}(B^{0}_{1}+B^{1}_{1})|\Psi^{*}\rangle$
$\displaystyle=\langle\Psi^{*}|B^{1}_{1}B^{0}_{1}+\mathbbm{1}|\Psi^{*}\rangle$
$\displaystyle\implies\langle\Psi^{*}|B^{1}_{1}B^{0}_{1}|\Psi^{*}\rangle=-\frac{1}{2}.$

Going back to Eq. 51,

$\displaystyle\langle\Psi^{*}|\\{B^{0}_{1},B^{1}_{1}\\}|\Psi^{*}\rangle=-1\;\;\;\implies\;\;\;\\{B^{0}_{1},B^{1}_{1}\\}|\Psi^{*}\rangle=-|\Psi^{*}\rangle,$
(52)

which can be replaced in Eq. VIII.3 to obtain

$\displaystyle\langle\Psi^{*}|\\{B^{0}_{2},B^{1}_{2}\\}|\Psi^{*}\rangle=1\;\;\;\implies\;\;\;\\{B^{0}_{2},B^{1}_{2}\\}|\Psi^{*}\rangle=|\Psi^{*}\rangle$
(53)

Now let us define the following operators:

$\displaystyle Z_{1}\equiv B^{0}_{1}$ $\displaystyle
X_{1}\equiv-\frac{\sqrt{3}}{3}\left(2B^{1}_{1}+B^{0}_{1}\right)$
$\displaystyle Z_{2}\equiv B^{0}_{2}$ $\displaystyle
X_{2}\equiv\frac{\sqrt{3}}{3}\left(B^{0}_{2}-2B^{1}_{2}\right).$ (54)

These operators are such that,

$\displaystyle Z_{1}|\Psi^{*}\rangle=Z_{2}|\Psi^{*}\rangle,$ (55)

from Eq. 44. Additionally, from eqs.(46), and (47):

$\displaystyle X_{1}|\Psi^{*}\rangle$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{3}\left(2B^{1}_{1}+B^{0}_{1}\right)|\Psi^{*}\rangle$
(56) $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{3}\left(2(B^{1}_{2}-B^{0}_{2})+B^{0}_{2}\right)|\Psi^{*}\rangle$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{3}\left(2B^{1}_{2}-B^{0}_{2}\right)|\Psi^{*}\rangle$
$\displaystyle=$ $\displaystyle X_{2}|\Psi^{*}\rangle,$

Also, the effect of the anticommutator of $Z_{1}$ and $X_{1}$,
$\\{Z_{1},X_{1}\\}$, and the anticommutator of $Z_{2}$ and $X_{2}$ are related
when acting on $|\Psi^{*}\rangle$:

$\displaystyle\\{Z_{1},X_{1}\\}|\Psi^{*}\rangle$
$\displaystyle=-\frac{\sqrt{3}}{3}\left(2B^{0}_{1}B^{1}_{1}+B^{0}_{1}B^{0}_{1}+2B^{1}_{1}B^{0}_{1}+B^{0}_{1}B^{0}_{1}\right)|\Psi^{*}\rangle$
$\displaystyle=-\frac{\sqrt{3}}{3}\left(2\\{B^{0}_{1},B^{1}_{1}\\}+2\mathbbm{1}\right)|\Psi^{*}\rangle$
$\displaystyle=0$ (57)

and similarly,

$\displaystyle\\{Z_{2},X_{2}\\}|\Psi^{*}\rangle$
$\displaystyle=\frac{\sqrt{3}}{3}\left(B^{0}_{2}B^{0}_{2}-2B^{0}_{2}B^{1}_{2}+B^{0}_{2}B^{0}_{2}-2B^{1}_{2}B^{0}_{2}\right)|\Psi^{*}\rangle$
$\displaystyle=\frac{\sqrt{3}}{3}\left(-2\\{B^{0}_{2},B^{1}_{2}\\}+2\mathbbm{1}\right)|\Psi^{*}\rangle$
$\displaystyle=0$ (58)

in which we used relations 52 and 53.

Furthermore, $Z_{1}$ and $Z_{2}$ are unitary by construction, and $X_{1}$ and
$X_{2}$ act in $|\Psi^{*}\rangle$ as a unitary operator:

$\displaystyle X_{1}^{\dagger}X_{1}|\Psi^{*}\rangle$
$\displaystyle=\frac{1}{3}\left(2B^{1}_{1}+B^{0}_{1}\right)\left(2B^{1}_{1}+B^{0}_{1}\right)|\Psi^{*}\rangle$
$\displaystyle=\frac{1}{3}\left(4B^{1}_{1}B^{1}_{1}+2B^{1}_{1}B^{0}_{1}+2B^{0}_{1}B^{1}_{1}+B^{0}_{1}B^{0}_{1}\right)|\Psi^{*}\rangle$
$\displaystyle=\frac{1}{3}\left(5\mathbbm{1}+2(B^{1}_{1}B^{0}_{1}+B^{0}_{1}B^{1}_{1})\right)|\Psi^{*}\rangle$
$\displaystyle=\frac{1}{3}\left(5\mathbbm{1}-2\mathbbm{1}\right)|\Psi^{*}\rangle$
$\displaystyle=|\Psi^{*}\rangle.$ (59)

The proof for $X_{2}$ follows from equation 56. Finally, as the sub-indices on
the B’s label different parties, it also holds true that $[Z_{1},Z_{2}]=0$ as
well as $[X_{1},X_{2}]=0$.

Now consider the local isotropic transformation depicted in Fig. 5. The final
state is dubbed $|\Psi_{final}\rangle$ and given by:

Figure 5: The partial swap circuit, a local isotropic transformation usually
employed in self-testing protocols.

$\displaystyle|\Psi_{final}\rangle=\frac{1}{4}\sum_{j,k=0}^{1}|jk\rangle\otimes\left[Z_{1}^{j}(\mathbbm{1}+(-1)^{j}X_{1})Z_{2}^{k}(\mathbbm{1}+(-1)^{k}X_{2})|\Psi^{*}\rangle\right].$
(60)

Because of relation 56 and also because $X_{1}$ commutes with $X_{2}$, all
terms for which $j+k=1$ vanish, leading to

$\displaystyle|\Psi_{final}\rangle=\frac{1}{4}|00\rangle\otimes\left[(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})|\Psi^{*}\rangle\right]+\frac{1}{4}|11\rangle\otimes\left[Z_{1}(\mathbbm{1}-X_{1})Z_{2}(\mathbbm{1}-X_{2})|\Psi^{*}\rangle\right],$
(61)

in which, we can use the fact that $Z_{1}$ and $X_{1}$, as well as $Z_{2}$ and
$X_{2}$, anticommute to perform the following calculation,

$\displaystyle
Z_{1}(\mathbbm{1}-X_{1})Z_{2}(\mathbbm{1}-X_{2})|\Psi^{*}\rangle$
$\displaystyle=(Z_{1}-Z_{1}X_{1})(Z_{2}-Z_{2}X_{2})|\Psi^{*}\rangle$
$\displaystyle=(Z_{1}+X_{1}Z_{1})(Z_{2}+X_{2}Z_{2})|\Psi^{*}\rangle$
$\displaystyle=(\mathbbm{1}+X_{1})Z_{1}(\mathbbm{1}+X_{2})Z_{2}|\Psi^{*}\rangle$
$\displaystyle=(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})Z_{1}Z_{2}|\Psi^{*}\rangle,$
$\displaystyle=(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})Z_{1}Z_{1}|\Psi^{*}\rangle,$
(62)
$\displaystyle=(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})\mathbbm{1}|\Psi^{*}\rangle,$
(63)

which leads to

$\displaystyle|\Psi_{final}\rangle$
$\displaystyle=\frac{1}{4}|00\rangle\otimes\left[(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})|\Psi^{*}\rangle\right]+\frac{1}{4}|11\rangle\otimes\left[(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})|\Psi^{*}\rangle\right]$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)\otimes\left[\frac{1}{2\sqrt{2}}(\mathbbm{1}+X_{1})(\mathbbm{1}+X_{2})|\Psi^{*}\rangle\right]$
$\displaystyle=|\phi^{+}\rangle\otimes|\zeta_{junk}\rangle,$

thus completing our self-test.

###  VIII.4 A semi-definite program formulation of the problem

Given the distribution $p(b_{1},b_{2}|x_{1},x_{2})$ one may search for the
least $\delta$ necessary to describe it with a $\delta$-objective distribution
$p(a,b_{1},b_{2}|x_{1},x_{2})$. Using a general distribution, this is an
optimization that can be realized with a linear program, the restriction to
quantum attainable distributions $p(a,b_{1},b_{2}|x_{1},x_{2})$, however, is a
hard task in general. Ideally, the objective would be finding the best
combination of density operator $\rho_{AB_{1}B_{2}}$ and POVMs $M_{A}^{(a)}$,
$M^{(b_{1})}_{B_{1},\,x_{1}}$ and $M^{(b_{2})}_{B_{2},\,x_{2}}$ such that

$p(a,b_{1},b_{2}|x_{1},x_{2})=\operatorname{tr}\left(\rho_{AB_{1}B_{2}}M_{A}^{(a)}\otimes
M^{(b_{1})}_{B_{1},\,x_{1}}\otimes
M^{(b_{2})}_{B_{2},\,x_{2}}\right),\,\forall a,b_{1},b_{2},x_{1},x_{2},$ (64)

satisfying constraint (9) with the lowest value possible for $\delta$. Since
no restriction is made on the dimensions of the involved Hilbert spaces, these
would also become parameters to be included in the optimization. Upper bounds
on the optimal $\delta$ can be done simply by providing a generic realization
for $p(a,b_{1},b_{2}|x_{1},x_{2})$ that is not necessarily the optimal one
(noticing that we are minimizing $\delta$). A way to lower bound the optimal
$\delta$ can be realized by relaxing the problem and imposing some but not all
necessary conditions that any quantum-attainable distribution should satisfy.

A standard method in the literature is to employ the NPA hierarchy [43] of
semidefinite program tests, which is proven to converge to the set of quantum
distributions [58]. The method proposes to further constrain the possible
distributions $p(a,b_{1},b_{2}|x_{1},x_{2})$ by requiring that it should also
be compatible with a matrix of statistical moments that can always be made
positive semidefinite whenever there exists a quantum realization for $p$. For
a given $p$ with unknown realization, the moment matrix is built up by
considering free variables and the constraints implied if a quantum
realization exists: a sequence of observables $O_{1},\ldots,O_{n}$ and a
quantum state $|\psi\rangle_{AB_{1}B_{2}}$ are assumed, which can be chosen
pure by using a high enough dimension for the Hilbert space, and each entry in
the moment matrix $\Gamma$ is given by
$\Gamma_{ij}=\langle\psi|O_{i}^{\dagger}O_{j}|\psi\rangle$. In general, the
observables are chosen to be the measurement operators $M_{A}^{(a)}$,
$M^{(b_{1})}_{B_{1},\,x_{1}}$ and $M^{(b_{2})}_{B_{2},\,x_{2}}$ (which can
also be assumed projective by using a dimension big enough) and strings formed
by them, with length up to a given $k\geq 1$. In some cases, the entries can
be associated with components of the distribution
$p(a,b_{1},b_{2}|x_{1},x_{2})$, or they can be identified with other entries.
If a quantum realization for $p$ exists, $k$ can be taken to an arbitrarily
high value; failure to obtain a positive semidefinite moment matrix $\Gamma$
for a given $k$ then certificates that the behavior is post-quantum.

Employing this method to bound a distribution $p(a,b_{1},b_{2}|x_{1},x_{2})$
which reduces to an observable distribution $p(b_{1},b_{2}|x_{1},x_{2})$ with
specific values for the CHSH expression, allows us to bound the objectivity
that can be associated to the observed values when $x_{1}=0$ or $x_{2}=0$. In
Fig. 2 we show different curves for the minimum $\delta$ as a function of the
CHSH value, considering also different agreements between the observers, given
by the extra (linear) constraint
$\sum_{b_{1}}p(b_{1}=b_{2}|x_{1}=0,\,x_{2}=0)=1-\epsilon$, with
$0\leq\epsilon\leq 0.5$.

###  VIII.5 Proof of result 4

To prove Result 4, we are going to show that it is always possible to define a
classical joint distribution $p(a_{1},a_{2})$ in a way that $(a_{1},a_{2})$
plays the rule of a hidden-variable in a classical model that can explain any
distribution $p(b_{1},b_{2}|x_{1},x_{2})$ obtained in an experiment with
quantum systems.

First of all, we take advantage of the fact that we are not making any
constrain in the dimension of the systems under consideration, to work only
with pure states. So consider a pure state
$|\Psi\rangle\in(\mathcal{H}_{1}\otimes\mathcal{H}_{2})$. Let
$\\{M^{(1)}_{0|0},M^{(1)}_{1|0},M^{(1)}_{0|1},M^{(1)}_{1|1},M^{(2)}_{0|0},M^{(2)}_{1|0},M^{(2)}_{0|1},M^{(2)}_{1|1}\\}$
be a set of projectors, such that $\sum_{b=0}^{1}M^{(i)}_{b,x}=\mathbbm{1}$
for $i,x\in\\{0,1\\}$, $M^{(i)}_{b,x}\in\mathcal{L}(\mathcal{H}_{i})$, and

$\displaystyle
M^{(i)}_{b,x}=\sum_{k}|\phi^{(i)}_{k,b,x}\rangle\langle\phi^{(i)}_{k,b,x}|$
(65)

in which

$\displaystyle\langle\phi^{(i)}_{k,b,x}|\phi^{(i)}_{k^{\prime},b^{\prime},x}\rangle=\delta_{b,b^{\prime}}\delta_{k,k^{\prime}}.$
(66)

The condition $p(b_{1}=b_{2}|x_{1}=x_{2})$ implies that $E_{00}=E_{11}=1$ and
hence, no symmetry of the CHSH having these two terms with different signs can
be violated. It also implies that,

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{k,l}\alpha_{k,l}|\phi^{(0)}_{k,0,0}\rangle\otimes|\phi^{(1)}_{l,0,0}\rangle+\sum_{r,s}\beta_{r,s}|\phi^{(0)}_{r,1,0}\rangle\otimes|\phi^{(1)}_{s,1,0}\rangle$
(67)

and, at the same time,

$\displaystyle|\Psi\rangle$ $\displaystyle=$
$\displaystyle\sum_{p,q}\alpha^{\prime}_{p,q}|\phi^{(0)}_{p,0,1}\rangle\otimes|\phi^{(1)}_{q,0,1}\rangle+\sum_{u,v}\beta^{\prime}_{u,v}|\phi^{(0)}_{u,1,1}\rangle\otimes|\phi^{(1)}_{v,1,1}\rangle.$
(68)

Now, define $\tilde{p}(a_{0},a_{1})$ as follows,

$\displaystyle\tilde{p}(a_{0},a_{1})$ $\displaystyle\equiv$ $\displaystyle
p(b_{1}=a_{0},b_{2}=a_{1}|x_{1}=0,x_{2}=1)$ $\displaystyle=$
$\displaystyle\langle\Psi|M_{a_{0}|0}^{(0)}\otimes
M_{a_{1}|1}^{(1)}|\Psi\rangle$ $\displaystyle=$
$\displaystyle\langle\Psi|\left[\left(\sum_{j}|\phi^{(0)}_{j,a_{0},0}\rangle\langle\phi^{(0)}_{j,a_{0},0}|\right)\otimes\left(\sum_{k}|\phi^{(1)}_{k,a_{1},1}\rangle\langle\phi^{(1)}_{k,a_{1},1}|\right)\right]\left(\sum_{p,q}\alpha^{\prime}_{p,q}|\phi^{(0)}_{p,0,1}\rangle\otimes|\phi^{(1)}_{q,0,1}\rangle+\sum_{u,v}\beta^{\prime}_{u,v}|\phi^{(0)}_{u,1,1}\rangle\otimes|\phi^{(1)}_{v,1,1}\rangle\right)$
$\displaystyle=$
$\displaystyle\langle\Psi|\left[\left(\sum_{j}|\phi^{(0)}_{j,a_{0},0}\rangle\langle\phi^{(0)}_{j,a_{0},0}|\right)\otimes\mathbbm{1}^{(1)}\right]\left(\sum_{k,p,q}\left(\alpha^{\prime}_{pq}\delta_{a_{1},0}+\beta^{\prime}_{p,q}\delta_{a_{1},1}\right)\delta_{k,q}|\phi_{p,a_{1},1}^{(0)}\rangle\otimes|\phi^{(1)}_{k,a_{1},1}\rangle\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{u,v}\alpha_{u,v}^{*}\langle\phi^{(0)}_{u,0,0}|\otimes\langle\phi^{(1)}_{v,0,0}|+\sum_{r,s}\beta_{r,s}^{*}\langle\phi^{(0)}_{r,1,0}|\otimes\langle\phi^{(1)}_{s,1,0}|\right)\left[\left(\sum_{j}|\phi^{(0)}_{j,a_{0},0}\rangle\langle\phi^{(0)}_{j,a_{0},0}|\right)\otimes\mathbbm{1}^{(1)}\right]$
$\displaystyle\times$
$\displaystyle\left(\sum_{k,p,q}\left(\alpha^{\prime}_{pq}\delta_{a_{1},0}+\beta^{\prime}_{p,q}\delta_{a_{1},1}\right)\delta_{k,q}|\phi_{p,a_{1},1}^{(0)}\rangle\otimes|\phi^{(1)}_{k,a_{1},1}\rangle\right)$
$\displaystyle=$
$\displaystyle\left(\sum_{j,u,v}\left(\alpha_{u,v}^{*}\delta_{a_{0},0}+\beta_{u,v}^{*}\delta_{a_{0}},1\right)\delta_{v,j}\langle\phi^{(0)}_{j,a_{0},0}|\otimes\langle\phi^{(1)}_{j,a_{0},0}|\right)\left(\sum_{k,p}\left(\alpha^{\prime}_{pq}\delta_{a_{1},0}+\beta^{\prime}_{p,q}\delta_{a_{1},1}\right)|\phi_{p,a_{1},1}^{(0)}\rangle\otimes|\phi^{(1)}_{k,a_{1},1}\rangle\right)$
$\displaystyle=$
$\displaystyle\sum_{j,u,k,p}\left(\alpha_{u,v}^{*}\delta_{a_{0},0}+\beta_{u,v}^{*}\delta_{a_{0}},1\right)\left(\alpha^{\prime}_{pq}\delta_{a_{1},0}+\beta^{\prime}_{p,q}\delta_{a_{1},1}\right)\langle\phi^{(0)}_{j,a_{0},0}|\phi_{p,a_{1},1}^{(0)}\rangle\langle\phi^{(1)}_{j,a_{0},0}|\phi^{(1)}_{k,a_{1},1}\rangle$
$\displaystyle=$ $\displaystyle\left(\langle\Psi|M_{a_{1}|1}^{(0)}\otimes
M_{a_{0}|0}^{(1)}|\Psi\rangle\right)^{*}$ $\displaystyle=$
$\displaystyle\langle\Psi|M_{a_{1}|1}^{(0)}\otimes
M_{a_{0}|0}^{(1)}|\Psi\rangle$ $\displaystyle=$ $\displaystyle
p(b_{1}=a_{1},b_{2}=a_{0}|x_{1}=1,x_{2}=0)$

Hence, given a quantum distribution (which always satisfies the NS condition)
obeying $p(b_{1}=b_{2}|x_{1}=x_{2})=1$ one can always define a distribution
$\tilde{p}(a_{0},a_{1})$ such that, by construction
$\tilde{p}(a_{0},a_{1}|x_{1},x_{2})=\tilde{p}(a_{0},a_{1})$, i.e., it
satisfies the NSD condition. More precisely, for any quantum distribution
obeying $p(b_{1}=b_{2}|x_{1}=x_{2})=1$, it holds that,

$\displaystyle
p(b_{1},b_{2}|x_{1},x_{2})=\left\\{\begin{array}[]{ll}\delta_{b_{1},b_{2}}\tilde{p}(a_{0}=b_{1})&\mbox{,
if }x_{1}=0,\mbox{ and }x_{2}=0\\\ \tilde{p}(a_{0}=b_{1},a_{1}=b_{2})&\mbox{,
if }x_{1}=0,\mbox{ and }x_{2}=1\\\ \tilde{p}(a_{1}=b_{1},a_{0}=b_{2})&\mbox{,
if }x_{1}=1,\mbox{ and }x_{2}=0\\\
\delta_{b_{1},b_{2}}\tilde{p}(a_{1}=b_{1})&\mbox{, if }x_{1}=1,\mbox{ and
}x_{2}=1\end{array}\right.,$ (73)

in which we have just proven the second and the third equations, and one can
obtain the first and the fourth by applying the NS condition combined with the
imposition $p(b_{1}=b_{2}|x_{1}=x_{2})=1$ to the second and third equations.

Another way of writing 73 is as follows

$\displaystyle
p(b_{1},b_{2}|x_{1},x_{2})=\sum_{a_{1},a_{2}}\delta_{b_{1},f_{a_{1},a_{2}}(x_{1})}\delta_{b_{2},f_{a_{1},a_{2}}(x_{2})}p(a_{1},a_{2}),$
(74)

in which

$\displaystyle f_{a_{1},a_{2}}(x)=a_{x}.$ (75)

By definition, eq 74 is a local hidden-variable model for
$p(b_{1},b_{2}|x_{1},x_{2})$, implying that this distribution cannot violate
the CHSH inequality, as announced in Result 4.

###  VIII.6 Mapping the experimental scheme to the Darwin scenario

In section IV we described the dynamics of the system treating independently
the temporal modes of the two photons to simplify the presentation. Here we
will describe more in detail the evolution of the system and environment in
terms of their joint temporal mode. In the experimental scheme, two photons,
generated by a collinear SPDC (Spontaneous Parametric Down-Conversion) process
of type II, in a ppKTP (Periodically-Poled Potassium Titanyl Phosphate)
crystal (see section IV), interact with a birefringent crystal in one of the
branches of the Sagnac interferometer. In this interaction, the polarization
and the temporal mode of the two photons, get entangled. The strength of the
interaction can be tuned by enlarging the thickness of the birefringent plate.
In order to map the scheme into the quantum Darwinism scenario described in
section II, we identify the polarization of the two photons after they leave
the interferometer as two fragments of the environment, identified by the
four-dimensional Hilbert space
$\mathcal{H}_{B_{1}}\otimes\mathcal{H}_{B_{2}}$. On the other hand, the joint
temporal mode of the two photons represents the quantum system probed by the
environment, corresponding to the Hilbert space $H_{A}$. A joint temporal
state of the two photons generated in our scheme can be written, in general,
in the following form:

$\left|f_{HV}\right\rangle=\int
d\omega_{H}d\omega_{V}\tilde{f}(\omega_{H},\omega_{V})a^{\dagger}_{H,\omega_{H}}a^{\dagger}_{V,\omega_{V}}\left|0\right\rangle=\int
dt_{H}dt_{V}f(t_{H},t_{V})A^{\dagger}_{H,t_{H}}A^{\dagger}_{V,t_{V}}\left|0\right\rangle\;,$
(76)

where $a^{\dagger}_{H,\omega_{H}},a^{\dagger}_{V,\omega_{V}}$ are the creation
operators for photons with horizontal and vertical polarization respectively,
while $A^{\dagger}_{P,t}$ is defined as

$A^{\dagger}_{P,t}=\int\frac{d\omega}{2\pi}e^{-i\omega
t}a^{\dagger}_{P,\omega}\;.$ (77)

If we consider a monochromatic pump with $\omega_{p}$, we have that the joint
spectral amplitude
$\tilde{f}(\omega_{H},\omega_{V})=\delta(\omega_{p}-\omega_{H}-\omega_{V})\tilde{f}^{\prime}(\omega_{H},\omega_{V})$,
($\tilde{f}^{\prime}(\omega_{H},\omega_{V})$ corresponds to the phase matching
function in the SPDC process [59]) and its Fourier transform, i.e. the joint
temporal function, will depend only on the difference $\Delta t=t_{H}-t_{V}$,
a part from a global phase shift
$f(t_{H},t_{V})=e^{i\omega_{p}(t_{H}+t_{V})/2}g(t_{H}-t_{V})$.

The birefringent crystal will change this temporal mode by introducing a shift
$t_{H}\rightarrow t_{H}+\tau$. The overlap with the original time mode is then
given by

$\Delta=\left\langle f_{HV}\mid
f_{HV}^{\tau}\right\rangle=e^{i\omega_{p}\tau/2}\int d\Delta t\;g^{*}(\Delta
t)g(\Delta t+\tau),$ (78)

where $\left|f_{HV}^{\tau}\right\rangle$ is the state with the translated
temporal mode $f(t_{H}+\tau,t_{V})$. Using the notation just introduced we can
describe the state in the interferometer, after the birefringent crystal, as

$\frac{1}{\sqrt{2}}\left(e^{-i\phi}\left|f_{HV},\mathrm{CW}\right\rangle+\left|f_{HV}^{\tau},\mathrm{CCW}\right\rangle\right)=\frac{1}{\sqrt{2}}\left(e^{-i\phi}\left|f_{HV},\mathrm{CW}\right\rangle+\Delta\left|f_{HV},\mathrm{CCW}\right\rangle+\gamma\left|f_{HV}^{\perp},\mathrm{CCW}\right\rangle\right)$
(79)

where $\mathrm{CCW},\mathrm{CW}$ represent the counter-clockwise and clockwise
orientations inside the interferometer, and
$\gamma\left|f_{HV}^{\perp}\right\rangle=(\mathds{I}-\left|f_{HV}\right\rangle\left\langle
f_{HV}\right|)\left|f_{HV}^{\tau}\right\rangle$ is the projection of
$\left|f_{HV}^{\tau}\right\rangle$ on the space orthogonal to
$\left|f_{HV}\right\rangle$, with $\gamma=\sqrt{1-|\Delta|^{2}}$. The phase
$\phi$ can be appropriately tuned using a liquid crystal (not shown in Fig 3).
If we now associate a qubit
$\left|0\right\rangle_{A},\left|1\right\rangle_{A}$ to
$\left|f_{HV}\right\rangle$ and its orthogonal state, respectively, we end up,
after the interferometer, with the state

$\frac{1}{\sqrt{2}}\left(e^{-i\phi}\left|V\right\rangle_{B_{1}}\left|H\right\rangle_{B_{2}}+\Delta\left|H\right\rangle_{B_{1}}\left|V\right\rangle_{B_{2}}\right)\left|0\right\rangle_{A}+\frac{\gamma}{\sqrt{2}}\left|H\right\rangle_{B_{1}}\left|V\right\rangle_{B_{2}}\left|1\right\rangle_{A}\;,$
(80)

which, after tracing out $A$, corresponds exactly to the state (25).

This shows that the internal degree of freedom probed by the environment is
precisely the _joint_ temporal mode of the two photons and, in particular, the
mode $\left|f_{HV}\right\rangle$ and its orthogonal complement.

###  VIII.7 Ab-initio optimization protocol

The gradient-free optimization algorithm employed in our proof-of-principle
experiment is based on the Stochastic Nelder-Mead (SNM) introduced in Ref.
[60], considered an improvement over the well-known Nelder-Mead algorithm when
dealing with noisy cost functions
$F(\vec{x})=f(\vec{x})+\mathcal{N}(\vec{x})$. In particular, it tends to avoid
getting stuck in a local optimum of the function provided to the algorithm,
which could be due to statistical fluctuations in the noise term
$\mathcal{N}(\vec{x})$. This algorithm has been recently used for the ab-
initio optimization of the violation of Bell inequalities and randomness
generation, extremizing the device-independent approach in experimental tasks
[49]. In this approach, the experimental apparatus is treated as a black box,
that is, the action of the controllable parameters (see Fig. 6 a) is unknown
to the algorithm and only the noisy output statistics is used to reach the
optimal value of a given cost function, that in our case is composed of the
agreement between the observers and subsequently the CHSH parameter.

Figure 6: $\bm{a)}$ Conceptual scheme of the ab-initio optimization process in
an experimental scenario. Both the photonic system being produced and the
measurement stations performing polarization measurement are unknown to the
algorithm, i.e. they are treated as black boxes. Nonetheless, the goal of the
algorithm will be to find the maximal value of a noisy function $S(\vec{x})$
\- e.g. the CHSH quantity - tuning progressively the parameters $\vec{x}$,
which have an apriori unknown relation with the measurements which are being
carried out. In this case, the parameters $\vec{x}$ have a one-to-one
correspondence with the rotation of two motorized waveplates. $\bm{b)}$
Pictorial representation of possible evolutions of the geometrical simplex
tracked by the SNM algorithm as described carried out by the SNM algorithm in
Appendix G. First, the barycenter $x_{bar}$ of the simplex is computed,
together with a reflection point $x_{ref}$. If such a point is promising, i.e.
$F(x_{ref})\leq F(x_{min})$, an expansion point $x_{exp}$ is computed. On the
contrary, if $F(x_{max})\leq F(x_{ref})$, a contraction point $x_{contr}$ is
tested. If such a step fails to find a point such that $F(x_{contr})\leq
F(x_{max})$, an Adaptive Random Search is performed. With probability $1-P$, a
local search occurs: a new point $x_{RNG}$ is uniformly sampled in a
hypersphere of radius $\epsilon$ centered in a randomly chosen simplex point.

The SNM algorithm performs the optimization of a given cost function over a
d-dimensional parameter space, finding

$\min_{\vec{x}}F(\vec{x})\quad\mathrm{where}\quad
F(\vec{x}):\mathbb{R}^{d}\rightarrow\mathbb{R},$ (81)

starting from an initial (d+1)-dimensional simplex of points
$\Sigma_{0}=\\{x_{0},\dots x_{d}\\}$, sampled in a given parameter range
through the Latin Hypercube Sampling algorithm [60].

The algorithm flow proceeds as follows (see also Fig. 6 b for a graphical
representation of the simplex evolutions):

  1. 0.

At first, the cost function $F(\cdot)$ is evaluated in each point of the
initial simplex $\Sigma_{0}$.

  2. 1.

If $\dim(\Sigma_{k})>d+1$, the point in which the cost function assumes the
highest value is discarded.

  3. 2.

The remaining points of the simplex $\Sigma_{k}$ are ranked with respect to
the value of the cost function $F(x_{i})$. $x_{max}$, $x_{2ndmax}$, $x_{min}$;
i.e. the points in which the cost function assumes the highest, the second
highest and the lowest values, are identified.

  4. 3.

The barycenter of the simplex points is computed as

$x_{bar}=\sum_{x\in\Sigma_{k}\setminus x_{max}}x/d,$

and the reflection point is computed as

$x_{ref}=(1+\alpha)x_{bar}-\alpha x_{max};\quad\alpha>0.$

The cost function $F(x_{ref})$ is evaluated. If $F(x_{min})\leq
F(x_{ref})<F(x_{2ndmax})$, then $\Sigma_{k+1}=\Sigma_{k}\cup x_{ref}$. Else:

    1. (a)

If $F(x_{ref})\leq F(x_{min})$, further expand the simplex in the direction of
$x_{ref}$, computing an expansion point:

$x_{exp}=(1-\gamma)x_{bar}+\gamma x_{ref};\quad\gamma>1.$

Compute the cost function in $x_{exp}$: if $F(x_{exp})<F(x_{ref})$ then
$\Sigma_{k+1}=\Sigma_{k}\cup x_{exp}$, else $\Sigma_{k+1}=\Sigma_{k}\cup
x_{ref}$. Go back to step 1.

    2. (b)

If $F(x_{2ndmax})\leq F(x_{ref})<F(x_{max})$, perform an external contraction:

$x_{contr}=(1-\beta)x_{bar}+\beta x_{ref};\quad 0\leq\beta\leq 1.$

Compute the cost function in the contracted point $x_{contr}$: if
$F(x_{contr})\leq F(x_{ref})$, impose $\Sigma_{k+1}=\Sigma_{k}\cup x_{contr}$
and go back to step 1.

    3. (c)

If instead $F(x_{max})\leq F(x_{ref})$, perform an internal contraction:

$x_{contr}=(1-\beta)x_{bar}+\beta x_{max};\quad 0\leq\beta\leq 1.$

Compute the cost function in the contracted point $x_{contr}$: if
$F(x_{contr})\leq F(x_{max})$, impose $\Sigma_{k+1}=\Sigma_{k}\cup x_{contr}$
and go back to step 1.

  5. 4.

If the contraction fails, perform an Adaptive Random Search (ARS),
stochastically sampling the parameter space:

    1. (a)

With probability $P$, a global search is performed, choosing a point $x_{RNG}$
uniformly sampling the parameter space within given bounds for each parameter.

    2. (b)

With probability $1-P$, a local search is performed: a point of the simplex
$x_{i}$ is randomly chosen and $x_{RNG}$ is uniformly sampled within an
hypersphere of radius $\epsilon$ centered in $x_{i}$.

The adaptive random search is performed until a point such that
$F(x_{RNG})<F(x_{max})$ is found. Impose $\Sigma_{k+1}=\Sigma_{k}\cup x_{RNG}$
and go back to step 1.

Within our experimental approach, performing polarization measurements of
single photon pairs, each observable $B^{i}_{j}$ is associated to two
parameters $(\delta^{i}_{j},\phi^{i}_{j})$ corresponding to the rotation
angles of the optical axes of the two waveplates implementing, together the
PBS and single-photon detectors, such measurements. Albeit the SNM algorithms
has no knowledge of this physical equivalence, the parameters on which it has
to optimize a given cost function will have a one-to-one correspondence to a
set of angles $(\delta^{i}_{j},\phi^{i}_{j})$ which are then provided to the
waveplates’ motorized rotation mounts, while expectation values
$\left<B^{k}_{1}B^{l}_{2}\right>$ are obtained by recording photon-coincidence
counts on single photon avalanche photodiodes within a window of $2.4$ ns.
Moreover, note that in our proof-of-principle experiment, we need to perform a
constrained optimization of the CHSH parameter:

$\mathrm{CHSH}=\left<B^{0}_{1}B^{0}_{2}\right>+\left<B^{0}_{1}B^{1}_{2}\right>\\\
-\left<B^{1}_{1}B^{0}_{2}\right>+\left<B^{1}_{1}B^{1}_{2}\right>$ (82)

under the constraint $\left<B^{0}_{1}B^{0}_{2}\right>=1-2\epsilon$, for a
given $\epsilon$. To address this problem, we perform first an optimization of
the agreement between the two observers and then the optimization of the CHSH
value, according to the following two-step process:

  1. 1.

First, fix a value of $\epsilon$ and perform the 4-parameter minimization of
the function:

$A=\left\lvert\left<B^{0}_{1}B^{0}_{2}\right>-(1-2\epsilon)\right\rvert$

The minimization is performed for 100 proposed points, with early stopping if
$A<0.03$.

  2. 2.

Keeping fixed the parameters related to $\\{B^{0}_{1},B^{0}_{2}\\}$, perform a
minimization of:

$\mathrm{S}=-\left\lvert\left<B^{0}_{1}B^{0}_{2}\right>+\left<B^{0}_{1}B^{1}_{2}\right>\\\
-\left<B^{1}_{1}B^{0}_{2}\right>+\left<B^{1}_{1}B^{1}_{2}\right>\right\rvert$

Extract $S$ and $\epsilon=\frac{1-\left<B^{0}_{1}B^{0}_{2}\right>}{2}$ from
the best point in the simplex after 250 explored points in the 4-dimensional
parameter space corresponding to $\\{B^{1}_{1},B^{1}_{2}\\}$.