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# Information causality as a tool for bounding the set of quantum correlations

Prabhav Jain  Department of Computer Science, Technical University of
Darmstadt, Darmstadt, 64289 Germany  Mariami Gachechiladze  Department of
Computer Science, Technical University of Darmstadt, Darmstadt, 64289 Germany
Nikolai Miklin  Heinrich Heine University Düsseldorf, Universitätsstraße 1,
40225 Düsseldorf, Germany  Institute for Quantum-Inspired and Quantum
Optimization, Hamburg University of Technology, Germany

###### Abstract

Information causality was initially proposed as a physical principle aimed at
deriving the predictions of quantum mechanics on the type of correlations
observed in the Bell experiment. In the same work, information causality was
famously shown to imply the Uffink inequality that approximates the set of
quantum correlations and rederives Tsirelson’s bound of the Clauser-Horne-
Shimony-Holt inequality. This result found limited generalizations due to the
difficulty of deducing implications of the information causality principle on
the set of nonlocal correlations. In this paper, we present a simple technique
for obtaining polynomial inequalities from information causality, bounding the
set of physical correlations in any Bell scenario. To demonstrate our method,
we derive a family of inequalities which non-trivially constrains the set of
nonlocal correlations in Bell scenarios with binary outcomes and equal number
of measurement settings. Finally, we propose an improved statement of the
information causality principle, obtain tighter constraints for the simplest
Bell scenario that goes beyond the Uffink inequality, and recovers a part of
the boundary of the quantum set.

##  I Introduction

Bell nonlocality is a celebrated phenomenon in modern physical sciences that
has given rise to an entire field of research called quantum information
processing Bell (1964). Many topics in this field, such as quantum
cryptography Acín  _et al._ (2007) and quantum self-testing Šupić and Bowles
(2020) rely on our understanding of Bell nonlocality directly, while others,
such as quantum computing Bravyi  _et al._ (2018) are influenced by it. Apart
from that, study of Bell nonlocality is motivated by a number of major open
problems Brunner  _et al._ (2014). In this work, we address two of such
problems.

The first is the question of what limitations on the types of correlations
observed in the Bell test are predicted by quantum mechanics Navascués  _et
al._ (2007). Tsirelson was the first to partially answer this question for the
simplest Bell scenario, proving his famous $2\sqrt{2}$ bound Cirel’son (1980)
on the Clauser-Horne-Shimony-Holt inequality Clauser  _et al._ (1969). In the
same paper, he coined the term _quantum Bell inequalities_ , which are
analytical expressions in terms of the observed statistics that approximate
the correlation set from the outside, or in other words, bound the set of
quantum correlations in the Bell test. Another notable example of a quantum
Bell inequality is the Uffink inequality, which is strictly stronger than the
Tsirelson inequality, and in a certain two-dimensional subspace of the
correlation space exactly describes the boundary of the quantum set Uffink
(2002).

The second open problem in Bell nonlocality that we address in this paper is
the ability of the so-called physical principles to predict the power of
quantum mechanics in the Bell test Pawłowski  _et al._ (2009); Navascués and
Wunderlich (2010); Fritz  _et al._ (2013); Brassard  _et al._ (2006). Physical
principles are intended to replace the formalism of quantum mechanics with a
set of postulates that appear natural in any reasonable physical theory.
Several physical principles have been proposed in the past, with the
information causality principle being the most promising and, at the same
time, the most difficult to apply Pawłowski  _et al._ (2009).

Information causality was proposed as a physical principle in 2009, and in the
same paper the authors showed that both the Tsirelson’s bound and the Uffink
inequality follow from it Pawłowski  _et al._ (2009). Only very recently,
these results found their generalizations beyond the simplest Bell scenario,
with the first infinite family of quantum Bell inequalities derived from a
physical principle Gachechiladze  _et al._ (2022). Still, the obtaining these
inequalities required extremely laborious calculations, leaving very little
room for deriving new quantum Bell inequalities using the same technique.
Around the same time, yet another simpler method to derive implications from
information causality was proposed, however, this approach only allowed to
numerically test if a given correlation satisfies the information causality
principle Miklin and Pawłowski (2021).

In this paper, we combine both, we present an easy to apply method for
deriving quantum Bell inequalities in any Bell scenario from the information
causality principle. This method makes information causality an efficient tool
for bounding the set of quantum correlations in the set of all nonsignaling
theories. As a demonstration of our method, we consider a family of symmetric
Bell scenarios in which both parties have an equal number of measurement
settings and binary outcomes. For each scenario, we provide a quadratic
quantum Bell inequalitiy, which gives a non-trivial constraints on the set of
nonlocal correlations. Additionally, with our new method, we rederive all the
previous findings in this research direction with significantly less effort.

Finally, we show that the proposed method for deriving quantum Bell
inequalities forges a new path in the research on physical principles. To this
end, we introduce a modification of the information causality principle and
apply our method to it. As a result, we obtain a quantum Bell inequality for
the simplest Bell scenario that is stronger than the Uffink inequality Uffink
(2002). Until now, it was open whether such a result was even possible. Yu and
Scarani (2022).

##  II Preliminaries

We start by describing the scenario of information causality (IC) Pawłowski
_et al._ (2009) shown in Fig. 1. In each round, Alice receives values of $n$
random variables $a_{0},\dots,a_{n-1}$, each taking values in
$[d]\coloneqq\\{0,1,\dots,d-1\\}$, while Bob receives a single dit $b$, taking
values in $[n]$. The goal of the game is for Bob to produce a guess $g$ of
Alice’s $b$-th dit. For example, if $b=0$, Bob tries to guess the value of
$a_{0}$. The parties have access to shared nonsignaling resources, which we
often refer to as a nonsignaling (NS) box. The inputs for an NS-box are
denoted as $\alpha$ and $\beta$, and the outputs as $A$ and $B$, and the box
is characterized by the collection of conditional probability distributions
$\mathrm{P}(A=k,B=l|\alpha=j,\beta=i)$, for all $k,l,j,i$ taken from the
corresponding sets. In each round, Alice sends a message $x$ to Bob through a
channel with the classical capacity $\mathcal{C}$. To encode her message $x$,
Alice can use any data available to her, that is $a_{0},\dots,a_{n-1}$, as
well as the output $A$ of the NS-box. Note, that since the communication
channel can be noisy, the random variables $x$ that Alice sends and
$x^{\prime}$ that Bob receives can take different values. Bob can use any
decoding strategy to produce his guess $g$, which can depend on $b$, $B$, and
the received message $x^{\prime}$.

The IC principle states that for mutually independent $a_{0},\dots,a_{n-1}$
the following bound should hold in any reasonable physical theory,

$\sum_{i=0}^{n-1}I(a_{i};g|b=i)\leq\mathcal{C},$ (1)

where $I(\cdot;\cdot)$ is the Shannon mutual information between $a_{i}$ and
$g$ Pawłowski  _et al._ (2009). In simple terms, the statement of the IC
principle says that the amount of information potentially accessible by Bob
cannot exceed the amount of information that Alice sends to him, which is
bounded by the capacity of the communication channel $\mathcal{C}$. In
Appendix A, we provide a derivation of the IC statement in Eq. (1) from the
axioms of the mutual information for noisy channels. Note that in the original
work of Ref. Pawłowski  _et al._ (2009), the IC statement was derived only for
the case of the noiseless channel.

Figure 1: The IC scenario: $a_{0},\dots,a_{n-1}$ are the input dits of Alice,
$b$ is the input dit of Bob; $\alpha,\beta,A,B$ are the inputs and outputs of
the NS-box; $x$ is the message sent by Alice and $x^{\prime}$ is the message
received by Bob; $g$ is the guess produced by Bob.

##  III Results

First, we describe our general and simple-to-apply method to obtain quantum
Bell inequalities from the information causality principle. The method
consists of a few steps in which some decisions must be made, while the
remaining calculations are straightforward and can be automated. As an
illustrative example, we consider the case of two independent and uniformly
distributed input bits $a_{0},a_{1}$ and an NS-box with binary inputs and
outputs, $\alpha,\beta,A,B\in\\{0,1\\}$, i.e., the simplest Bell scenario.
Naturally, in this case, we refaderive the famous Uffink inequality Uffink
(2002).

In the first step of our method, we choose an encoding-decoding protocol and a
communication channel type. For the former, we consider the van Dam protocol
Van Dam (2013),

$\displaystyle\alpha=a_{0}\oplus a_{1},\quad\beta=b,\quad x=a_{0}\oplus
A,\quad g=x^{\prime}\oplus B,$ (2)

where $\oplus$ denotes summation modulo $2$, and for the latter we choose the
binary symmetric channel. In the second step, we lower-bound each term on the
left-hand side in Eq. (1). In our illustrative example, we use the Fano’s
inequality Fano (1968) for this purpose, but other options are also available
Gachechiladze  _et al._ (2022). Denoting the probability of communicating a
correct bit in the communication channel as $\frac{1+e_{c}}{2}$, with
parameter $e_{c}\in[-1,1]$, we reduce the IC statement to

$\displaystyle 2-\sum_{i=0}^{1}h(\mathrm{P}(a_{i}=g|b=i))\leq
1-h\left(\frac{1+e_{c}}{2}\right),$ (3)

where $h(p)\coloneqq-p\log(p)-(1-p)\log(1-p)$ for $p\in[0,1]$ is the binary
Shannon entropy, and we chose the logarithm to be base-$2$. It is
straightforward to calculate the guessing probabilities
$\mathrm{P}(a_{i}=g|b=i)$ for both $i\in\\{0,1\\}$ having chosen the protocol,

$\displaystyle\mathrm{P}(g=a_{i}|b=i)=\frac{2+e_{c}\sum_{j=0}^{1}(-1)^{i\cdot
j}e_{j,i}}{4},$ (4)

In the above Eq. (4), we have introduced the notation $e_{j,i}$ for the
_biases_ of the NS-box, defined as $e_{j,i}\coloneqq 2\mathrm{P}(A\oplus
B=0|\alpha=j,\beta=i)-1$, for $i,j\in\\{0,1\\}$. Plugging in the probabilities
from Eq. (4) into Eq. (3) and dividing everything by the right-hand side,
which is manifestly positive, we obtain,

$\displaystyle\frac{2-\sum_{i=0}^{1}h\left(\frac{2+e_{c}\sum_{j=0}^{1}(-1)^{i\cdot
j}e_{j,i}}{4}\right)}{1-h\left(\frac{1+e_{c}}{2}\right)}\leq 1.$ (5)

This is where the main component of our method comes into play. The relation
in Eq. (5) has to hold for any $e_{c}\in[-1,1]$, which means that also the
limit of the ratio on the left-hand side of Eq. (5) for $e_{c}\to 0$ has to be
less than or equal to $1$. We notice that both the nominator and denominator
of the ratio in Eq. (5) as well as their derivatives with respect to $e_{c}$
vanish in the limit of $e_{c}\to 0$, which means that we can employ the
L’Hôpital’s rule twice to find the limit. This step allows for a significant
simplification of the condition in Eq. (5) and gives the Uffink inequality
Uffink (2002),

$\left(e_{0,0}+e_{1,0}\right)^{2}+(e_{0,1}-e_{1,1})^{2}\leq 4.$ (6)

The presented derivations are concise and simple to follow. Previously, in
order to derive a quantum Bell inequality from the IC principle one had to use
the cumbersome machinery of concatenation Pawłowski  _et al._ (2009);
Gachechiladze  _et al._ (2022), which also most likely would not be successful
for any protocol beyond the van Dam’s protocol in Eq. (2).

On the contrary, the presented simple method can be applied to any Bell
scenario and any encoding-decoding protocol, making information causality an
efficient tool for bounding the set of quantum correlations. As a
demonstration, we present a family of quantum Bell inequalities for the Bell
scenarios in which Alice and Bob have the same number of measurement settings
and binary outcomes, the so-called $nn22$ scenarios.

######  Result 1.

In the $nn22$ Bell scenario, the following inequality follows from the
information causality principle,

$\sum_{i=0}^{n-1}\left(e_{0,i}+\sum_{j=1}^{n-i}(-1)^{\delta_{j,n-i}}2^{j-1}e_{j,i}\right)^{2}\leq
4^{n-1},$ (7)

where $e_{j,i}\coloneqq 2\mathrm{P}(A=B|\alpha=j,\beta=i)-1$, for all
$i,j\in\\{0,1,\dots,n-1\\}$, and $e_{n,0}\coloneqq 0$.

In the statement of Result 1, $e_{n,0}$ is introduced just to present the
inequality in Eq. (7) is a concise form. Details on the derivation can be
found in Appendix B. It is easy to see that Result 1 recovers the Uffink
inequality for the case of $n=2$. At the same time, for each $n>2$, the
inequality in Eq. (7) is not implied by the Uffink inequality or any of the
inequalities in Eq. (7) for smaller $n$ (see Appendix B for the proof).
Additionally, an NS-box defined by $e_{0,i}=1$, $\forall i$, and
$e_{j,i}=(-1)^{\delta_{j,n-i}}$, $j>0,\forall i$ violates the inequality in
Result 1 by $\frac{4^{n}-4}{3}$.

At this point, we comment on the importance of deriving quadratic constrains
of the form in Eq. (7) from the statement of the information causality
principle. First, in order to apply the principle of IC as in Eq. (1) to a
given NS-box, one needs to find an optimal encoding-decoding protocol and an
optimal communication channel. Doing this analytically can be problematic if
one operates with the mutual information function. Quadratic constraints in
Eq. (7), on the other hand, can be applied directly to a given NS-box or a
family of NS-boxes to check whether they violate the IC principle, and
therefore are not physical. Moreover, for some classes of NS-boxes, the
approximation of the IC statement by the quadratic inequalities can be tight,
as it is the case for the Uffink inequality in the subspace
$(e_{0,0}+e_{1,0},e_{0,1}-e_{1,1})$. This is also the case for the class of
protocols that we considered in derivation of Result 1 (see Appendix B for
details). In other words, for some classes of protocols, our method provides
an exact characterization of the set of correlations satisfying the IC
principle.

Finally, once we have the implications of the IC principle in terms of
polynomial inequalities, such as in Eq. (7), we can make comparisons between
the IC and other physical principles. In particular, the Uffink inequality in
Eq. (6) also describes the boundary of the macroscopic local correlations
Navascués and Wunderlich (2010) in the subspace of
$(e_{0,0}+e_{1,0},e_{0,1}-e_{1,1})$ in the $2222$ Bell scenario. Recently, in
the $d2dd$ scenarios, the same equivalence results were obtained between the
quadratic inequalities implied by the IC statement and the macroscopic local
correlation set Gachechiladze  _et al._ (2022). Proving this result was
possible because the number of settings of Bob is exactly two. However, in the
current case of $nn22$ scenario, the list of quadratic inequalities obtained
with our method is less restrictive than the bounds that follow from the
macroscopic locality principle (see Appendix F for the details). A possible
reason, is that the boundary of the latter set in $nn22$ scenario for $n>2$ is
described by polynomial inequalities of order higher than two. It is an open
question whether such inequalities can be derived from the IC principle.

The presented method is not restricted in its application to Bell scenarios
with binary outcomes. In particular, the family of quantum Bell inequalities
for $d2dd$ scenario, which was the main result of Ref. Gachechiladze  _et al._
(2022), can be rederived by our method with much less steps (see Appendix C).
We also invite the interested reader to look at Appendix D for inequalities
that can be derived with our method for $nndd$ scenario. For Bell scenarios
with $d$-outcome measurements, a natural choice of the IC scenario is the one
in which Alice’s data is given by $d$-outcome random variables and the parties
communicate over a channel with $d$-dimensional input and output. This is also
the choice that we made when rederiving the inequalities for the $d2dd$
scenario. At the same time, from the results of Ref. Miklin and Pawłowski
(2021), we know that for particular NS-boxes the optimal channel capacity for
which the IC principle is violated by these boxes is nonzero. It is an
interesting open question whether tighter inequalities than the ones derived
in Ref. Gachechiladze  _et al._ (2022) can be obtained with our method with a
suitable choice of communication protocol.

Figure 2: Comparison between the Uffink inequality and the inequality in Eq.
(73) for a subspace of the correlation space given by a mixture of three NS-
boxes with the convex coefficients $q_{1},q_{2}$, and $1-q_{1}-q_{2}$, where
$q_{1}\in[0,1],q_{2}\in[0,0.3]$. The shaded regions correspond to the ranges
of values of $(q_{1},q_{2})$, for which the corresponding inequality, either
the Uffink or the one in Eq. (73), is satisfied. In this subspace, the
inequality in Eq. (73) describes the boundary of the quantum set.

Finally, we demonstrate that our method can be successfully applied to
modifications of the IC statement in Eq. (1). Here, we consider one such
modification, namely the case when Alice’s input data is correlated. In the
simplest case of two input bits, $a_{0}$,$a_{1}$, it is easy to see that the
respective IC statement takes the form,

$\displaystyle I(a_{0};g|b=0)+I(a_{1};g|b=1,a_{0})\leq\mathcal{C}.$ (8)

See Appendix A for details. We apply our method to the modified IC statement
to get a tighter bound than the Uffink inequality.

######  Result 2.

In the $2222$ Bell scenario, the following inequality follows from the
modified information causality principle,

$\left((1+\epsilon)e_{0,0}+(1-\epsilon)e_{1,0}\right)^{2}+(1-\epsilon^{2})(e_{0,1}-e_{1,1})^{2}\leq
4,$ (9)

for any parameter $\epsilon\in[-1,1]$.

See Appendix E for the derivation of this inequality. For the case
$\epsilon=0$ we recover the Uffink inequality, whereas some other values of
$\epsilon$ leads to a tighter bound, as we demonstrate in Fig. 2.

In Fig. 2, we consider a convex mixture of three NS-boxes with
$A,B,\alpha,\beta\in\\{0,1\\}$, an NS-box Popescu and Rohrlich (1994)
satisfying $A\oplus B=\alpha\cdot\beta$, a local NS-box where $A=\alpha$,
$B=0$, and a local NS-box where $A=\alpha$, $B=\beta$, with the coefficients
$1-q_{1}-q_{2}$, $q_{1}$, and $q_{2}$ respectively. The region of
$q_{1}+q_{2}>1$, in the top-right area of Fig. 2, is excluded as it does not
correspond to a valid convex mixture. As Fig. 2 shows, Result 2 provides a
tighter bound than the Uffink inequality on the set of quantum correlations.
In fact, in the considered subspace, the limiting case of $\epsilon\to 1$ of
Eq. (73) coincides with the Landau inequality Landau (1988), i.e., describes
the boundary of the quantum set.

##  IV Discussions and outlook

The topic of the information causality principle has been recently regaining
attention Miklin and Pawłowski (2021); Gachechiladze  _et al._ (2022); Yu and
Scarani (2022); Patra  _et al._ (2023). However, until now, working with the
information principle was associated with a lot of computational efforts. In
this paper, we show that the IC principle is also an efficient tool for
bounding the set of quantum correlations. This result is due to the proposed
general method of deriving quantum Bell inequalities from the IC statement. We
demonstrate the effectiveness of our method by deriving a family of quantum
Bell inequalities for a range of Bell scenarios. In particular, for every Bell
scenario with binary outcomes and an arbitrary high number of settings for
both parties, we give an example of a relevant nontrivial inequality that
bounds the set of quantum correlations. We also argue that the proposed method
allows to rederive all the known quantum Bell inequalities that were shown to
follow from the information causality principle but, this time, with much
fewer derivation steps.

The current work opens new possibilities of working with physical principles
that are based on communication scenarios. In this work, we demonstrate it for
a modification of the information causality scenario in which the inputs of
Alice are correlated. In particular, applying our method to this modification,
we were able to obtain a quantum Bell inequality that is tighter than the
Uffink inequality in the simplest Bell scenario. The other open possibility is
to consider communication complexity scenarios.

Among open questions that follow from this work is tightness of our method for
approximating the set of correlations compatible with the information
causality principle. We have shown that this is the case for the inequalities
in Result 1 for the corresponding set of protocols. However, for scenarios
with non-bit communication, it is likely not to be the case. Moreover, with
our method, we are able to obtain inequalities that are quadratic in terms of
the observed correlations. It is interesting to find a generalization of our
approach that can produce polynomial inequalities of higher degree.

###### Acknowledgements.

We thank Marcin Pawłowski, Miguel Navascués, David Gross, and Robin Krebs for
fruitful discussions. This research was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), project number
441423094, 236615297 - SFB 1119.

## Appendix

In this Appendix, we provide technical details that support the statements
from the main text. In section (A), we show the derivation of the information
causality (IC) statement in Eq. (1), which was previously proven only for the
case of ideal channel. In the same section of the Appendix, we demonstrate how
to obtain a modification of the IC statement for the case of correlated input
data of Alice. In section B, we provide details on the derivation of Result 1
and the statements associated with it, specifically we prove the tightness of
our inequalities for a large class of protocols. In section C, using our
method we derive inequalities for the $d2dd$ case, which we further extend to
the $nndd$ case in section D. Section E gives the proof to Result 2, the
correlated input case. Section F gives comparison between all the inequalities
in the $3322$ case derived using our method and the bounds from the
macroscopic locality principle.

##  Appendix A Derivation of the information causality statement with noisy
channel and correlated inputs

Here we give a proof of the statement of information causality in Eq. (1). We
consider two systems $\mathrm{A}$ and $\mathrm{B}$, and we do not assume any
specific underlying probabilistic theory (Note that here we use a different
format of letters $\mathrm{A}$ and $\mathrm{B}$ for the systems than the
italic format $A$ and $B$, that denote outcomes of the measurements of a
nonsignaling (NS) box). The joint system AB is described by its state
$\rho_{\mathrm{AB}}$, which at this moment does not have to be a quantum
state, but any state in some abstract space $\mathcal{S}$ of a general
probabilistic theory. We require, however, that there exists a definition of a
composite system and the corresponding marginalization map, such that for any
state $\rho_{\mathrm{AB}}$ of a composite system $\mathrm{AB}$, there exist a
unique pair of marginal states $\rho_{\mathrm{A}}$ of subsystem $\mathrm{A}$
and $\rho_{\mathrm{B}}$ of $\mathrm{B}$. We also require existence of a notion
of classical system, classical state, and a map from a general state in
$\mathcal{S}$ to a classical state, which plays a role of a measurement map.

We require that there exists a measure $I(\cdot\;;\cdot|\cdot)$ that satisfies
the following properties for any systems $\mathrm{A},\mathrm{B},\mathrm{C}$,
and classical systems $X,Y$, and $Z$:

  1. (i)

(Non-negativity) $I(\mathrm{A};\mathrm{B}|\mathrm{C})\geq 0$,

  2. (ii)

(Chain rule)
$I(\mathrm{A},\mathrm{B};\mathrm{C})=I(\mathrm{B};\mathrm{C}|\mathrm{A})+I(\mathrm{A};\mathrm{C})$,

  3. (iii)

(Markov property) If the variables $\mathrm{A}$, $\mathrm{B}$, and
$\mathrm{C}$ satisfy the causal relation
$\mathrm{A}\rightarrow\mathrm{B}\rightarrow\mathrm{C}$, then
$I(\mathrm{A};\mathrm{C}|\mathrm{B})=0$,

  4. (iv)

(Data processing) $I(\mathrm{A};\mathrm{B})\geq I(\mathrm{A};X)$, if $X$ is a
random variable resulting from a measurement on B,

  5. (v)

(Consistency) $I(X;Y|Z)$ reduces to the conditional Shannon mutual information
for classical random variables $X,Y$, and $Z$.

Note that the above set of properties is not a minimal set since one can prove
the data processing inequality using the chain rule and the Markov property.
Importantly, all of the above constraints are satisfied for quantum mechanics
with $I(\cdot\;;\cdot|\cdot)$ being the von Neumann mutual information.

Using the above properties of $I(\cdot\;;\cdot|\cdot)$, we now derive the IC
statement. We recall that in the IC protocol, Alice receives dits
$a_{0},\dots,a_{n-1}$ and can send a message $x$ to Bob whose goal is to guess
an input dit of Alice, which corresponds to the value of $b$. The derivation
of Eq. (1) consists of finding an upper and a lower bound on the quantity
$I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})$. We start with a lower bound,

$\displaystyle I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})$
$\displaystyle=I(a_{0};x^{\prime},\mathrm{B})+I(a_{1},\dots,a_{n-1};x^{\prime},\mathrm{B}|a_{0})$
(10a)
$\displaystyle=I(a_{0};x^{\prime},\mathrm{B})+I(a_{1},\dots,a_{n-1};x^{\prime},\mathrm{B},a_{0})-I(a_{1},\dots,a_{n-1};a_{0})$
(10b) $\displaystyle\geq
I(a_{0};x^{\prime},\mathrm{B})+I(a_{1},\dots,a_{n-1};x^{\prime},\mathrm{B}),$
(10c)

where we used the chain rule in Eq. (10a). In Eq. (10b) since
$\\{a_{i}\\}_{i}$ are mutually independent, the third term vanishes. Finally,
going from Eq. (10b) to Eq. (10c) we use the chain rule and non-negativity of
$I(a_{0};a_{1},\dots,a_{n-1},x^{\prime},\mathrm{B})$.

Iterating this procedure $n$ times we get,

$\displaystyle
I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})\geq\sum^{n-1}_{i=0}I(a_{i};x^{\prime},\mathrm{B}).$
(11)

Now we add conditioning on particular value of $b$ to each term on the right-
hand side. Since $\mathrm{B}$ is a system and not result of a measurement of
Bob, it does not depend on $b$. The same is true for each $a_{i}$ and
$x^{\prime}$, in the latter case, due to the nonsignaling condition. Hence, we
can equate $I(a_{i};x^{\prime},\mathrm{B})$ to
$I(a_{i};x^{\prime},\mathrm{B}|b=i)$, for each $i\in[n]$. Lastly, due to data
processing inequality, if Bob makes a measurement and obtains an outcome $B$,
and subsequently produces a guess $g$, which is a function of $x^{\prime}$ and
$B$, we have that $I(a_{i};x^{\prime},\mathrm{B}|b=i)\geq I(a_{i};g|b=i)$, for
every $i\in[n]$. As a result, we have,

$I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})\geq\sum^{n-1}_{i=0}I(a_{i};g|b=i).$
(12)

Now, we find an upper bound on $I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})$.
Consider the quantity,

$\displaystyle
I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})-I(a_{0},\dots,a_{n-1},\mathrm{B};x^{\prime}|x)$
$\displaystyle=I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})-I(x,a_{0},\dots,a_{n-1},\mathrm{B};x^{\prime})+I(x;x^{\prime})$
(13a)
$\displaystyle=I(a_{0},\dots,a_{n-1};\mathrm{B})+I(a_{0},\dots,a_{n-1};x^{\prime}|\mathrm{B})$
(13b)
$\displaystyle-I(x,a_{0},\dots,a_{n-1};\mathrm{B})-I(x,a_{0},\dots,a_{n-1};x^{\prime}|\mathrm{B})+I(x;x^{\prime})$
(13c) $\displaystyle\leq
I(a_{0},\dots,a_{n-1};x^{\prime}|\mathrm{B})-I(x,a_{0},\dots,a_{n-1};x^{\prime}|\mathrm{B})+I(x;x^{\prime})\leq
I(x;x^{\prime}).$

Here we used the chain rule in Eq. (13a) and Eq. (13b). Since
$a_{0},\dots,a_{n-1},x,x^{\prime}$ are independent of $\mathrm{B}$, the first
term in Eq. (13b) vanishes. Finally, in the last step we again we use the
chain rule and the positivity of
$I(x;x^{\prime}|a_{0},\dots,a_{n-1},\mathrm{B})$. At the same time, since
$x^{\prime}$ is correlated with $a_{0},\dots,a_{n-1},\mathrm{B}$ only through
$x$, the conditional mutual information
$I(a_{0},\dots,a_{n-1},\mathrm{B};x^{\prime}|x)=0$ because of the Markov
property.

We can now combine the lower bound in Eq. (12) and the upper bound in Eq.
(13),

$\displaystyle\sum^{n-1}_{i=0}I(a_{i};g|b=i)\leq
I(a_{0},\dots,a_{n-1};x^{\prime},\mathrm{B})\leq
I(x;x^{\prime})\leq\sup_{p(x)}I(x;x^{\prime})=\mathcal{C},$ (14)

where we used the definition of the channel capacity
$\mathcal{C}=\sup_{p(x)}I(x^{\prime},x)$ in which the supremum is to be taken
over all the possible probability distributions $p(x)$ of $x$. This completes
the proof of the IC statement in Eq. (1) for non-ideal channels.

### Modified IC statement for correlated inputs

We can easily modify the above derivation to accommodate for the case of
correlated input dits $\\{a_{i}\\}_{i}$. If we again consider Eq. (10) for the
case of two dits, we have,

$\displaystyle
I(a_{0},a_{1};x^{\prime},\mathrm{B})=I(a_{0};x^{\prime},\mathrm{B})+I(a_{1};x^{\prime},\mathrm{B}|a_{0})\geq
I(a_{0};g|b=0)+I(a_{1};g|b=0,a_{0}),$ (15)

where we essentially stopped at the first derivation step in Eq. (10) and
applied data processing to $\mathrm{B}$. Since we can use the same upper bound
on $I(a_{0},a_{1};x^{\prime},\mathrm{B})$ we obtain the IC statement for tow
correlated dits,

$\displaystyle I(a_{0};g|b=0)+I(a_{1};g|b=1,a_{0})\leq\mathcal{C}.$ (16)

One can notice that Eq. (16) is not symmetric with respect to the input dits
$a_{0},a_{1}$. However, it is evident that one can also obtain a similar
condition for the reverse order of conditioning. It is also clear how to
obtain a generalization of Eq. (16) for higher number $n$ of input dits of
Alice, where we have freedom of choosing a sequence in which the conditioning
on $\\{a_{i}\\}_{i}$ occurs.

##  Appendix B Details on the derivation of Result 1 for the $nn22$ scenario

In this section, we provide details on the derivations of Result 1. For
convenience, we restate this result below.

######  Result 1.

In the $nn22$ Bell scenario, the following inequality follows from the
information causality principle,

$\sum_{i=0}^{n-1}\left(e_{0,i}+\sum_{j=1}^{n-i}(-1)^{\delta_{j,n-i}}2^{j-1}e_{j,i}\right)^{2}\leq
4^{n-1},$ (17)

where $e_{j,i}\coloneqq 2\mathrm{P}(A=B|\alpha=j,\beta=i)-1$, for all
$i,j\in\\{0,1,\dots,n-1\\}$, and $e_{n,0}\coloneqq 0$.

To derive the inequalities in Eq. (17), we prove a more general result.

######  Result 3.

In the $nn22$ Bell scenario, the following family of inequalities follows from
the information causality principle,

$\sum_{i=0}^{n-1}\left(\sum_{j=0}^{n-1}\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\delta_{f(\mathbf{k}),j}(-1)^{h(\mathbf{k})\oplus
k_{i}}e_{j,i}\right)^{2}\leq 4^{n},$ (18)

where $e_{j,i}\coloneqq 2\mathrm{P}(A=B|\alpha=j,\beta=i)-1$, for all
$i,j\in[n]$, and discrete functions $f:\\{0,1\\}^{n}\to[n]$, and
$h:\\{0,1\\}^{n}\to\\{0,1\\}$, can be chosen arbitrarily. In Eq. (18), $k_{i}$
denotes the $i$-th element of $n$-dimensional binary vector $\mathbf{k}$.

###### Proof.

The derivation of Result 3 follows the same steps as the derivation of the
Uffink inequality in the main text. However, this time we consider a more
general class of encoding-decoding protocols for the scenario with $n$ input
bits for Alice. In particular, we take the following,

$\displaystyle\alpha=f(\mathbf{a}),\quad\beta=b,\quad x=h(\mathbf{a})\oplus
A,\quad g=x^{\prime}\oplus B,$ (19)

where $f:\\{0,1\\}^{n}\to[n]$ and $h:\\{0,1\\}^{n}\to\\{0,1\\}$ are discrete-
valued functions of the inputs bits $\\{a_{i}\\}_{i}$, which we arranged as a
binary vector $\mathbf{a}\in\\{0,1\\}^{n}$. We choose the communication of
Alice to Bob to be over a symmetric binary channel with the probability of
transmitting a correct message being $\frac{1+e_{c}}{2}$, i.e.,
$\mathrm{P}(x^{\prime}=0|x=0)=\mathrm{P}(x^{\prime}=1|x=1)=\frac{1+e_{c}}{2}$.
Note, that one can consider an even more general class of protocols for the
$nn22$ scenario. However, we leave analysis of such protocols for future
investigations, because their number is infinite.

For the second step of our protocol, we take the Fano’s inequality for lower-
bounding the mutual information terms in the IC statement, as we did in the
main text. We also take the explicit formula for the capacity of a binary
symmetric channel. As a result, we arrive at the following implication of the
IC principle for the considered scenario,

$n-\sum_{i=0}^{n-1}h\left(\mathrm{P}(g=a_{i}|b=i)\right)\leq
1-h\left(\frac{1+e_{c}}{2}\right),$ (20)

which is a straightforward generalization of Eq. (3).

As a next step, we calculate the guessing probability
$\mathrm{P}(g=a_{i}|b=i)$ by inserting into it the protocol from Eq. (19).
First, we write that,

$\displaystyle\begin{split}&\mathrm{P}(g=a_{i}|b=i)=\mathrm{P}(x^{\prime}\oplus
B=a_{i}|b=i)=\mathrm{P}(x^{\prime}=x)\mathrm{P}(x\oplus
B=a_{i}|b=i)+\mathrm{P}(x^{\prime}\neq x)\mathrm{P}(x\oplus B\neq
a_{i}|b=i)\\\ &=\frac{1+e_{c}}{2}\mathrm{P}(x\oplus
B=a_{i}|b=i)+\frac{1-e_{c}}{2}\mathrm{P}(x\oplus B\neq
a_{i}|b=i)=\frac{1}{2}+\frac{e_{c}}{2}\left(2\mathrm{P}(x\oplus
B=a_{i}|b=i)-1\right).\end{split}$ (21)

Next, we calculate the probability $\mathrm{P}(x\oplus B=a_{i}|b=i)$,

$\displaystyle\mathrm{P}(x\oplus B=a_{i}|b=i)$
$\displaystyle=\mathrm{P}(A\oplus B=h(\mathbf{a})\oplus
a_{i}|b=i)=\frac{1}{2^{n}}\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\mathrm{P}(A\oplus
B=h(\mathbf{k})\oplus k_{i}|b=i,\mathbf{a}=\mathbf{k}),$ (22a)
$\displaystyle=\frac{1}{2^{n}}\sum_{j=0}^{n-1}\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\mathrm{P}(\alpha=j|\mathbf{a}=\mathbf{k})\mathrm{P}(A\oplus
B=h(\mathbf{k})\oplus k_{i}|\alpha=j,\beta=i),$ (22b)
$\displaystyle=\frac{1}{2^{n}}\sum_{j=0}^{n-1}\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\delta_{f(\mathbf{k}),j}\mathrm{P}(A\oplus
B=h(\mathbf{k})\oplus k_{i}|\alpha=j,\beta=i),$ (22c)

where in Eq. (22a) we expanded the probability as a sum, in which each term is
conditioned on specific values $\mathbf{k}$ of the input bits of Alice, and in
Eq. (22b) we introduced the sum over values of $\alpha$. In Eq. (22b), we
removed the conditioning of the probability of $A\oplus B=h(\mathbf{k})\oplus
k_{i}$ on the values of $\mathbf{a}$, because the probability of the outcomes
$A$ and $B$ for an NS-box depends only on the choices of settings $\alpha$ and
$\beta$.

Finally, we use the notation $e_{j,i}=2\mathrm{P}(A=B|\alpha=j,\beta=i)-1$,
which we introduced in the statement of Result 1. In this notation, we can
express the probability $\mathrm{P}(A\oplus B=h(\mathbf{k})\oplus
k_{i}|\alpha=j,\beta=i)$ as

$\mathrm{P}(A\oplus B=h(\mathbf{k})\oplus
k_{i}|\alpha=j,\beta=i)=\frac{1+e_{j,i}(-1)^{h(\mathbf{k})\oplus k_{i}}}{2}.$
(23)

Plugging Eq. (23) back into Eq. (22), and subsequently to Eq.
(LABEL:eq:app_nn22_guess_prob_1), we obtain the final expression for the
guessing probability

$\mathrm{P}(g=a_{i}|b=i)=\frac{1}{2}+\frac{e_{c}}{2}\left(\frac{1}{2^{n}}\sum_{j=0}^{n-1}\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\delta_{f(\mathbf{k}),j}(-1)^{h(\mathbf{k})\oplus
k_{i}}e_{j,i}\right).$ (24)

In the last step of the derivation of Result 3, we take the limit of $e_{c}\to
0$, just as we did in the main text when deriving the Uffink inequality. ∎

Now, we can derive Result 1 from Result 3 by making a particular choice of $f$
and $h$. Namely, we take,

$h(\mathbf{a})=a_{0},\quad
f(\mathbf{a})=n-1-\sum_{i=1}^{n-1}\prod_{l=1}^{i}(a_{0}\oplus a_{l}).$ (25)

For convenience, let us denote the coefficients in front of $e_{j,i}$ in Eq.
(18) as $c_{j,i}$, i.e.,

$c_{j,i}\coloneqq\sum_{\mathbf{k}\in\\{0,1\\}^{n}}\delta_{f(\mathbf{k}),j}(-1)^{h(\mathbf{k})\oplus
k_{i}},\quad i,j\in[n].$ (26)

Looking at the function $f$ in Eq. (25), it is clear that the case $j=0$ is
special, because $f(\mathbf{a})=0$ only occurs when $(a_{0}\oplus a_{i})=1$
for all $i\in\\{1,2,\dots,n-1\\}$. It means that for $j=0$, in the sum over
$\mathbf{k}$ in Eq. (26), each term enters with a minus sign, i.e.,
$(-1)^{h(\mathbf{k})\oplus k_{i}}=-1$, since $h(\mathbf{k})=k_{0}$, and there
are only two such terms, namely $k_{0}=0$, $k_{i}=1$,
$i\in\\{1,2,\dots,n-1\\}$, and $k_{0}=1$, $k_{i}=0$,
$i\in\\{1,2,\dots,n-1\\}$. Therefore, we can conclude that $c_{0,i}=-2$,
$\forall i\in\\{1,2,\dots,n-1\\}$. The case of $j=0$, and $i=0$ is even easier
to analyze, and one concludes that $c_{0,0}=2$.

Continuing derivation of coefficients in Eq. (26), we consider another special
case of $i=0$. In that case, all the terms in the sum over $\mathbf{k}$ in Eq.
(26) are positive for all $j\in[n]$. Doing a bit of counting, one can realize
that the case $f(\mathbf{k})=j$ occurs exactly for $2^{j}$ values of
$\mathbf{k}$. Therefore, $c_{j,0}=2^{j}$, for all $j\in[n]$. Notice, that in
the formulation of Result 1, in Eq. (17) the sum over $j$ goes to the limit
$n-i$, which in case of $i=0$ exceeds the number of values that $j$ can take.
However, to make the formula in Eq. (17) more concise, we artificially
included the case of $i=0$, $j=n$, by setting $e_{n,0}\coloneqq 0$.

Finally, let us consider the general case $i>0,j>0$. Setting
$f(\mathbf{a})=j$, we get that the satisfying values of $\mathbf{a}$ are such
that $\prod_{l=1}^{n-j-1}(a_{0}\oplus a_{l})=1$, and
$\prod_{l=1}^{n-j}(a_{0}\oplus a_{l})=0$, which is the case when
$a_{l}=a_{0}\oplus 1$ for $l\in\\{1,2,\dots,n-j-1\\}$, and $a_{n-j}=a_{0}$. It
means that for $c_{j,i}$ in Eq. (26) with $i\in\\{1,2,\dots,n-j-1\\}$, we have
that all the terms in the sum over $\mathbf{k}$ enter with a minus sign, and,
therefore, in that case $c_{j,i}=-2^{j}$. For $i=n-j$, on the contrary, all
the terms in the sum over $\mathbf{k}$ in Eq. (26) are with the plus sign, and
hence $c_{j,n-j}=2^{j}$. Lastly, for $i>n-j-1$, half of the terms in the sum
over $\mathbf{k}$ Eq. (26) come with a plus sign and half with a minus, i.e.,
in that case $c_{j,i}=0$. The calculated coefficients $c_{j,i}$ recover the
inequality in Result 1.

### Protocols for which the method provides an exact characterization of the
IC set

Here, we prove the tightness of the obtained quadratic inequalities in Result
1 in approximating the set of correlations compatible with the IC principle
for the considered protocol in Eq. (19) and with functions $f$ and $h$
specified by Eq. (25). In fact, we show this result for an entire class of
protocols in Eq. (19) with $h(\mathbf{a})$ being a balanced function, i.e.,
$h(\mathbf{a})=0$ for exactly half of the values of $\mathbf{a}$. We also
assume that the input bits of Alice are uniformly distributed. We start by
showing that in that case, the guess $g$ of Bob is uniformly distributed,

$\displaystyle\begin{split}\mathrm{P}(g=0|b=i)&=\mathrm{P}(x^{\prime}\oplus
B=0|b=i)=\frac{1+e_{c}}{2}\mathrm{P}(x\oplus
B=0|b=i)+\frac{1-e_{c}}{2}\mathrm{P}(x\oplus B=1|b=i)\\\
&=\frac{1}{2}+\frac{e_{c}}{2}\Big{(}2\mathrm{P}(A\oplus
B=h(\mathbf{a})|b=i)-1\Big{)}=\frac{1}{2}.\end{split}$ (27)

Since each $a_{i}$ is also uniformly distributed, the Fano’s inequality is
tight for each $I(a_{i};g|b=i)$, and the statement of the IC principle is
exactly captured by the formula in Eq. (20). Let us reexpress the formula in
Eq. (24) as,

$\mathrm{P}(g=a_{i}|b=i)=\frac{1}{2}+\frac{e_{c}}{2}e_{i},$ (28)

where $e_{i}$ is the expression in the brackets that is multiplied by
$\frac{e_{c}}{2}$ in Eq. (24). In terms of $e_{i}$, the inequality resulting
from our method is simply

$\sum_{i=0}^{n-1}e_{i}^{2}\leq 1.$ (29)

Now we show that this is the tightest constraint that one can obtain from Eq.
(20). Let us consider the expression in Eq. (20) as a function of $e_{c}$, and
call it $F(e_{c})$,

$F(e_{c})\coloneqq
n-\sum_{i=0}^{n-1}h\left(\mathrm{P}(g=a_{i}|b=i)\right)-1+h\left(\frac{1+e_{c}}{2}\right),$
(30)

with the IC statement being $F(e_{c})\leq 0$. One can easily show that the
second derivative of $F(e_{c})$ with respect to $e_{c}$ is non-positive for
all $e_{c}\in[-1,1]$, whenever Eq. 29 holds,

$\displaystyle F^{\prime\prime}(e_{c})$
$\displaystyle=\sum_{i=0}^{n-1}\frac{e_{i}^{2}}{1-e_{c}^{2}e_{i}^{2}}-\frac{1}{1-e_{c}^{2}}=\sum_{i=0}^{n-1}\left(\frac{e_{i}^{2}}{1-e_{c}^{2}e_{i}^{2}}-\frac{e_{i}^{2}}{1-e_{c}^{2}}\right)+\sum_{i=0}^{n-1}\frac{e_{i}^{2}}{1-e_{c}^{2}}-\frac{1}{1-e_{c}^{2}}$
(31)
$\displaystyle\leq\sum_{i=0}^{n-1}e_{i}^{2}\left(\frac{-e_{c}^{2}(1-e_{i}^{2})}{(1-e_{c}^{2})(1-e_{c}^{2}e_{i}^{2})}\right)\leq
0.$ (32)

Let us now assume that one can obtain a tighter constraint than the one in Eq.
(29) for some value $e_{c}^{*}$. It means, that there is at least one point in
the space of $\\{e_{i}\\}_{i}$ that satisfies Eq.(29), but violates the IC
inequality for $e_{c}=e_{c}^{*}$, i.e., $F(e_{c}^{*})>0$. Since
$F^{\prime\prime}(e_{c})\leq 0$, for all $e_{c}\in[-1,1]$ and
$\\{e_{i}\\}_{i}$ satisfying Eq.(29), $F(e_{c})$ is a concave function in the
entire interval of $e_{c}\in[-1,1]$, and, therefore, has at most two roots. We
also know that $e_{c}=0$ is one of the roots, and hence if there is a point
$e_{c}^{*}$, such that $F(e_{c}^{*})>0$, then $F(e_{c})>0$ in the neighborhood
of $e_{c}=0$ (either for $e_{c}\in(0,\epsilon)$ or $e_{c}\in(-\epsilon,0)$).
But in that case, the inequality in Eq.(29) is violated for the considered
point in the space of $\\{e_{i}\\}_{i}$, i.e., we reach a contradiction.

### Each inequality in Result 1 provides a relevant constraint on the set of
nonlocal correlations

Here we show that for each $n$, the inequality in Result 1 provides a
constraint on the set of nonlocal correlations that is not implied by
inequalities in Result 1 for smaller $n$, in particular, the Uffink
inequality. We show this by providing a particular NS-box for each $n$, which
violates the inequality in Result 1 for that $n$, but cannot violate any of
the inequalities in Result 1 for smaller $n$.

Consider an NS-box that is obtained by mixing white noise with the NS-box
given by $e^{\mathrm{NS}}_{0,i}=1$, $\forall i$, and
$e^{\mathrm{NS}}_{j,i}=(-1)^{\delta_{j,n-i}}$, $j>0,\forall i$, which gives
the maximal violation of the inequality in Result 1. If the parameter of
mixing is $q\in[0,1]$, then the corresponding values of $e_{j,i}$ the mixture
is just $e_{j,i}=qe^{\mathrm{NS}}_{j,i}$, for all $i,j\in[n]$. Direct
calculation shows that the inequality in Result 1 provides the following upper
bound on $q$,

$q\leq\frac{3}{7-4^{2-n}}.$ (33)

An important observation, is that this bound decreases with $n$. It means,
that if we applied any of the inequalities in Result 1 for smaller $n$, we
would get a looser bound on $q$, regardless of the strategy in which we would
apply it, because we already took the algebraic maximum of the left-hand side
when calculating the bound in Eq. (33). Therefore, taking any value of
$q\in\left(\frac{3}{7-4^{2-n}},\frac{3}{7-4^{3-n}}\right)$ provides an NS-box
that is detected by the inequality in Result 1 for that $n$ and not any
inequality for smaller $n$.

##  Appendix C Details of derivation in the $d2dd$ scenario

######  Result 4 (Ref. Gachechiladze  _et al._ (2022)).

In $d2dd$ scenario, the following family of inequalities follows from the
information causality principle,

$\displaystyle\sum_{i=0}^{1}\left|\sum_{j,k=0}^{d-1}e^{i\cdot j\oplus
k}_{j,i}\omega^{k\cdot l}\right|^{2}\leq d^{4},\quad\forall
l\in\\{1,\dots,\lfloor d/2\rfloor\\}.$ (34)

where $\omega=e^{2\pi\mathbbm{i}/d}$ is the $d$-th root of unity.

Here we consider the scenario in which Alice has $d$ measurement settings and
outcomes while Bob has 2 measurement settings and $d$ outcomes. The input dits
$a_{0},a_{1}$ are assumed to be independent and uniformly distributed. We
consider the protocol,

$\displaystyle\alpha=\overline{a_{0}}\oplus a_{1},\quad\beta=b,\quad
x=a_{0}\oplus A,\quad g=x^{\prime}\oplus B,$ (35)

where $\overline{a}$ denotes the additive inverse of $a$ modulo $d$. For this
scenario we use a strongly symmetric channel which we will refer to as the
clock channel. For this channel the probability for confusing any two symbols
depends on the distance between them. More precisely we define the clock
distance between any two dits to be

$D(i,j)=\min\\{i\ominus j,j\ominus i\\},$ (36)

where all arithmetic is assumed to be modulo $d$. Thus the transition matrix
is given by $\mathcal{T}_{ij}=p_{D(i,j)}$, where we see that $p_{m}=p_{d-m}$
for $0\leq m\leq d-1$.

This channel is manifestly symmetric and has $\lfloor d/2\rfloor$ independent
parameters and its capacity amount to,

$\displaystyle\mathcal{C}=\log d+\sum_{m=0}^{d-1}p_{m}\log p_{m}.$ (37)

Next, we define the global error $E$ to be a random variable such that
$g=a_{i}\oplus E$. Using an improved version of the Fano’s inequality from
Ref. Gachechiladze  _et al._ (2022) and the channel capacity from Eq. (37) the
IC statement looks as follows,

$\displaystyle\sum_{i=0}^{1}\left(\log
d-H(E|b=i)\right)\leq\sum_{i=0}^{1}I(a_{i},g|b=i)\leq\log
d+\sum_{m=0}^{d-1}p_{m}\log p_{m}.$ (38)

We parameterize our probabilities by introducing biases $\\{e_{m}\\}$ such
that

$\displaystyle
p_{0}=\frac{1-\sum_{m=1}^{d-1}e_{m}}{d},\;p_{m}=\frac{1+e_{m}}{d},\quad
p_{m}=p_{d-m}\quad\forall m\in\\{1,\dots,d-1\\}.$ (39)

We also parameterize the box distribution as

$\mathrm{P}(A\oplus B=k|\alpha=j,\beta=i)=\frac{1+e^{k}_{j,i}}{d}.$ (40)

Using this parametrization the RHS of Eq. 38 simplifies as follows:

$\displaystyle\begin{split}\log d+\sum_{m=0}^{d-1}p_{m}\log p_{m}&=\log
d+\left(\frac{1-\sum_{m=1}^{d-1}e_{m}}{d}\right)\log\left(\frac{1-\sum_{m=1}^{d-1}e_{m}}{d}\right)+\sum_{m=1}^{d-1}\frac{1+e_{m}}{d}\log\frac{1+e_{m}}{d}\\\
&=\frac{1}{d}\left[\left(1-\sum_{m=1}^{d-1}e_{m}\right)\log\left(1-\sum_{m=1}^{d-1}e_{m}\right)+\sum_{m=1}^{d-1}(1+e_{m})\log(1+e_{m})\right].\end{split}$
(41)

Now we can work out the error probability $\mathrm{P}(E=k|b=i)$ in LHS of Eq.
(38) using the protocol in Eq. (35),

$\displaystyle\begin{split}\mathrm{P}(E=k|b=i)&=\mathrm{P}(g=a_{i}\oplus
k|b=i)=\mathrm{P}(x^{\prime}\oplus B=a_{i}\oplus
k|b=i)\cr&=\sum_{m=0}^{d-1}\sum_{j=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus
m)\mathrm{P}(\alpha=j)\mathrm{P}(A\oplus B=\overline{a_{0}}\oplus a_{i}\oplus
k\oplus\overline{m}|\alpha=j,\beta=i)\cr&=\sum_{m=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus
m)\left[\frac{1}{d}\sum_{j=0}^{d-1}\frac{1+e^{i\cdot j\oplus
k\oplus\overline{m}}_{j,i}}{d}\right]\cr&=\frac{1-\sum_{m=1}^{d-1}e_{m}}{d}\left[\frac{1}{d}\sum_{j=0}^{d-1}\frac{1+e^{i\cdot
j\oplus
k}_{j,i}}{d}\right]+\sum_{m=1}^{d-1}\frac{1+e_{m}}{d}\left[\frac{1}{d}\sum_{j=0}^{d-1}\frac{1+e^{i\cdot
j\oplus k\oplus\overline{m}}_{j,i}}{d}\right],\end{split}$ (42)

where we used the fact that $\alpha$ is uniformly distributed. We also define
the Fourier transform of the collection of probabilities in the square
brackets above as follows

$q_{s}(i):=\sum_{k=0}^{d-1}\left[\frac{1}{d}\sum_{j=0}^{d-1}\frac{1+e^{i\cdot
j\oplus k}_{j,i}}{d}\right]\omega^{k\cdot s},$ (43)

and then its inverse transformation is given by,

$\frac{1}{d}\sum_{j=0}^{d-1}\frac{1+e^{i\cdot j\oplus
k}_{j,i}}{d}=\frac{1}{d}\sum_{s=0}^{d-1}q_{s}(i)\omega^{-k\cdot s}.$ (44)

We substitute this inverse transformation back into the derivation of
$\mathrm{P}(E=k|b=i)$ to obtain,

$\displaystyle\mathrm{P}(E=k|b=i)$
$\displaystyle=\frac{1}{d^{2}}\left[\left(1-\sum_{m=1}^{d-1}e_{m}\right)\sum_{s=0}^{d-1}q_{s}(i)\omega^{-k\cdot
j}+\sum_{m=1}^{d-1}(1+e_{m})\sum_{s=0}^{d-1}q_{s}(i)\omega^{-(k+d-m)\cdot
j}\right]$ (45)
$\displaystyle=\frac{1}{d^{2}}\sum_{s=0}^{d-1}q_{s}(i)\omega^{-k\cdot
j}\left[\left(1-\sum_{m=1}^{d-1}e_{m}\right)+\sum_{m=1}^{d-1}(1+e_{m})\omega^{m\cdot
j}\right]$ (46)
$\displaystyle=\frac{1}{d^{2}}\sum_{s=0}^{d-1}q_{s}(i)\omega^{-k\cdot
j}\left[d\delta_{j,0}+\sum_{m=1}^{d-1}e_{m}(\omega^{m\cdot j}-1)\right]$ (47)
$\displaystyle=\frac{1}{d}\left(1+\frac{1}{d}\sum_{s=1}^{d-1}q_{s}(i)\omega^{-k\cdot
j}\sum_{m=1}^{d-1}e_{m}(\omega^{m\cdot
j}-1)\right)=:\frac{1}{d}\left(1+F_{k}^{i}\right),$ (48)

where we defined a new notation for the later use. We are now ready to plug in
this expression back in the LHS of Eq. (38) to obtain,

$\displaystyle 2\log
d+\sum_{i=0}^{1}\sum_{k=0}^{d-1}\frac{1+F_{k}^{i}}{d}\log\frac{1+F_{k}^{i}}{d}=\frac{1}{d}\sum_{i=0}^{1}\sum_{k=0}^{d-1}\left(1+F_{k}^{i}\right)\log\left(1+F_{k}^{i}\right),$
(49)

where we used the fact that $\sum_{k=0}^{d-1}F_{k}^{i}=0$. Thus our simplified
IC statement looks as follows,

$\sum_{i=0}^{1}\sum_{k=0}^{d-1}\left(1+F_{k}^{i}\right)\log\left(1+F_{k}^{i}\right)\leq\left(1-\sum_{m=1}^{d-1}e_{m}\right)\log\left(1-\sum_{m=1}^{d-1}e_{m}\right)+\sum_{m=1}^{d-1}(1+e_{m})\log(1+e_{m}).$
(50)

We notice that in the limit $e_{m}\rightarrow 0$, both LHS and RHS of Eq. (38)
vanish. Further since the RHS is a channel capacity it is always non-negative
and hence we can divide both sides of Eq. (38) by it. We make the substitution
$e_{m}\equiv\epsilon g_{m}$ and study this inequality in the limit
$\epsilon\rightarrow 0$ using L’Hôpital’s rule. For simplicity of the
derivations, we set the logarithm to be natural logarithm.

$\lim_{\epsilon\rightarrow
0}\frac{\text{LHS}}{\text{RHS}}=\lim_{\epsilon\rightarrow
0}\frac{\text{LHS'}}{\text{RHS'}}=\lim_{\epsilon\rightarrow
0}\frac{\sum_{i=0}^{1}\sum_{k=0}^{d-1}F_{k}^{i}\left(1+\log\left(1+\epsilon
F_{k}^{i}\right)\right)}{\sum_{m=1}^{d-1}g_{m}\left(\log(1+\epsilon
g_{m})-\log\left(1-\epsilon\sum_{m=1}^{d-1}g_{m}\right)\right)}\leq 1.$ (51)

We see that after the single derivation, the numerator and denominator both
still vanish when $\epsilon\rightarrow 0$ and we can again use L’Hôpital’s
rule,

$\lim_{\epsilon\rightarrow
0}\frac{\text{LHS'}}{\text{RHS'}}=\lim_{\epsilon\rightarrow
0}\frac{\text{LHS''}}{\text{RHS''}}=\frac{\sum_{i=0}^{1}\sum_{k=0}^{d-1}(F_{k}^{i})^{2}}{\sum_{m=1}^{d-1}g_{m}^{2}+\left(\sum_{m=1}^{d-1}g_{m}\right)^{2}}\leq
1.$ (52)

This form of the inequality already looks close to the final result, however,
we need to do a bit more work to simplify the numerator.

$\displaystyle\begin{split}\sum_{i=0}^{1}\sum_{k=0}^{d-1}(F_{k}^{i})^{2}&=\sum_{i=0}^{1}\sum_{k=0}^{d-1}\left(\frac{1}{d}\sum_{s=1}^{d-1}q_{s}(i)\omega^{-k\cdot
s}\sum_{m=1}^{d-1}g_{m}(\omega^{m\cdot
s}-1)\right)\left(\frac{1}{d}\sum_{s^{\prime}=1}^{d-1}q_{s^{\prime}}(i)\omega^{-k\cdot
s_{2}}\sum_{m^{\prime}=1}^{d-1}g_{m^{\prime}}(\omega^{m^{\prime}\cdot
s^{\prime}}-1)\right)\cr&=\frac{1}{d^{2}}\sum_{i=0}^{1}\sum_{k=0}^{d-1}\sum_{m,m^{\prime}=1}^{d-1}\sum_{s,s^{\prime}=1}^{d-1}q_{s}(i)q_{s^{\prime}}(i)\omega^{-k\cdot(s+s^{\prime})}g_{m}g_{m^{\prime}}(\omega^{m\cdot
s}-1)(\omega^{m^{\prime}\cdot
s^{\prime}}-1)\cr&=\frac{1}{d}\sum_{i=0}^{1}\sum_{s=1}^{d-1}|q_{s}(i)|^{2}\left(\sum_{m,m^{\prime}=1}^{d-1}g_{m}g_{m^{\prime}}(\omega^{m\cdot
s}-1)(\omega^{-m^{\prime}\cdot s}-1)\right),\end{split}$ (53)

where we used the fact that the whole expression vanishes, unless
$s+s^{\prime}=0$, and in such cases $q_{s}(i)q_{s^{\prime}}(i)=|q_{s}(i)|^{2}$
for each $i$. If we now choose $g_{m}=(\omega^{l\cdot m}+\omega^{-l\cdot
m})/2$, where $l\in\\{1,\dots,\lfloor{d/2}\rfloor\\}$, the summation in the
inner brackets above simplifies to,

$\displaystyle\begin{split}\frac{1}{4}\left(\sum_{m=1}^{d-1}\left(\omega^{l\cdot
m}+\omega^{-l\cdot m}\right)\left(\omega^{m\cdot
s}-1\right)\right)\left(\sum_{m^{\prime}=1}^{d-1}\left(\omega^{l\cdot
m^{\prime}}+\omega^{-l\cdot m^{\prime}}\right)\left(\omega^{-m^{\prime}\cdot
s}-1\right)\right)=\frac{d^{2}}{4}\left(\delta_{s,l}+\delta_{\bar{s},l}\right).\end{split}$
(54)

For the denominator we have for $l\in\\{1,\dots,\lfloor{d/2}\rfloor\\}$,

$\displaystyle\frac{1}{4}\sum_{m=1}^{d-1}\left(\omega^{l\cdot
m}+\omega^{-l\cdot
m}\right)^{2}+\frac{1}{4}\left(\sum_{m=1}^{d-1}\omega^{l\cdot
m}+\omega^{-l\cdot m}\right)^{2}$
$\displaystyle=\frac{1}{4}(2d+d^{2}\delta_{l,0})=d/2.$ (55)

Plugging the simplified forms of the nominator and the denominator in Eq. (52)
gives us,

$\displaystyle\frac{\frac{1}{d}\sum_{i=0}^{1}\sum_{s=1}^{d-1}|q_{s}(i)|^{2}d^{2}\left(\delta_{s,l}+\delta_{\bar{s},l}\right)}{2d}=\sum_{i=0}^{1}|q_{l}(i)|^{2}\leq
1.$ (56)

Putting back the definition of $q_{l}(i)$ we finally have the desired
inequalities

$\displaystyle\sum_{i=0}^{1}\left|\sum_{j,k=0}^{d-1}e^{i\cdot j\oplus
k}_{j,i}\omega^{k\cdot l}\right|^{2}\leq d^{4},\quad\quad\forall
l\in\\{1,\dots\lfloor d/2\rfloor\\}.$ (57)

##  Appendix D Details of derivation in the $nndd$ scenario

Next, we generalize Result 3 to the case of arbitrary number of measurement
outcomes, i.e., to the $nndd$ Bell scenarios.

######  Result 5.

In the $nndd$ Bell scenario, the following family of inequalities follows from
the information causality principle,

$\displaystyle\sum_{i=0}^{n-1}\left|\sum_{l,m=0}^{d-1}\sum_{j=0}^{n-1}c^{l}_{j,i}\omega^{(l\oplus
m)\cdot t}e^{m}_{j,i}\right|^{2}\leq d^{2(n+1)},$ (58)

for all $t\in\\{1,\dots,\lfloor d/2\rfloor\\}$, where $e^{k}_{j,i}\coloneqq
d\mathrm{P}(A\oplus B=k|\alpha=j,\beta=i)-1$, for all $i,j\in[n]$ and
$l\in[d]$, and the coefficients $c^{l}_{j,i}\in\mathbb{Z}$ are determined by
discrete functions $f:[d]^{n}\to[n]$, $h:[d]^{n}\to[d]$, and $r:[n]\to[d]$, as

$c^{l}_{j,i}:=\sum_{\mathbf{k}\in[d]^{n}}\delta_{f(\mathbf{k}),j}\delta_{h(\mathbf{k})\oplus
k_{i},l\oplus r(i)},$ (59)

$\omega\coloneqq e^{2\pi\mathbbm{i}/d}$ is the $d$-th root of unity, and
$\oplus$ denotes summation modulo $d$.

###### Proof.

The derivation for the case of $nndd$, that is for $n$ measurement settings
and $d$ outcomes for both parties will closely follow the previous two
examples. The input dits $a_{0},\dots,a_{n-1}$ are again assumed to be
independent and uniformly distributed. We consider a general protocol:

$\displaystyle\alpha=f(\mathbf{a}),\quad\beta=b,\quad
x=\overline{h(\mathbf{a})}\oplus A,\quad g=x^{\prime}\oplus B\oplus r(b),$
(60)

and we choose to use the clock channel again. Now for two random variables
$a,b$ such that $a$ is uniformly distributed and $d$ is the internal dimension
of the variables, their mutual information satisfies

$\displaystyle I(a;b)$ $\displaystyle\geq\log d-H(e),$ (61)

where $b=a\oplus e$. We begin by introducing the global error $E$ to be random
variable such that $g=a_{i}\oplus E$. Thus using the above inequality and the
channel capacity from (37), we have the IC statement expressed as,

$\displaystyle\sum_{i=0}^{n-1}\left(\log d-H(E|b=i)\right)\leq\log
d+\sum_{l=0}^{d-1}p_{l}\log p_{l}.$ (62)

We parameterize our probabilities by introducing biases $\\{e_{l}\\}$ such
that

$\displaystyle
p_{0}=\frac{1-\sum_{l=1}^{d-1}e_{l}}{d},\;p_{k}=\frac{1+e_{k}}{d},\quad\forall
k\in\\{1,\dots,d-1\\}.$ (63)

Now we calculate the error probability that appears in the entropy on the LHS
as follows,

$\displaystyle\begin{split}&\mathrm{P}(E=k|b=i)=\mathrm{P}(g=a_{i}\oplus
k|b=i)=\mathrm{P}(x^{\prime}\oplus B\oplus r(i)=a_{i}\oplus
k|b=i)\cr&=\sum_{l=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus l)\mathrm{P}(x\oplus
B\oplus r(i)\oplus l=a_{i}\oplus
k|b=i)\cr&=\sum_{l=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus
l)\sum_{j=0}^{n-1}\mathrm{P}(\alpha=j)\mathrm{P}(A\oplus B=h\oplus
a_{i}\oplus\overline{l}\oplus\overline{r(i)}\oplus
k|\alpha=j,\beta=i)\cr&=\sum_{l=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus
l)\sum_{j=0}^{n-1}\mathrm{P}(\alpha=j)\sum_{s=0}^{d-1}\mathrm{P}(h\oplus
a_{i}=s|\alpha=j)\mathrm{P}(A\oplus
B=s\oplus\overline{l}\oplus\overline{r(i)}\oplus
k|\alpha=j,\beta=i)\cr&=\sum_{l=0}^{d-1}\mathrm{P}(x^{\prime}=x\oplus
l)\left[\sum_{j=0}^{n-1}\sum_{s=0}^{d-1}\frac{c^{s}_{j,i}}{d^{n}}\frac{1+e^{s\oplus\overline{l}\oplus\overline{r(i)}\oplus
k}_{j,i}}{d}\right]\cr&=\frac{1-\sum_{l=1}^{d-1}e_{l}}{d}\left[\sum_{j=0}^{n-1}\sum_{s=0}^{d-1}\frac{c^{s}_{j,i}}{d^{n}}\frac{1+e^{s\oplus\overline{r(i)}\oplus
k}_{j,i}}{d}\right]+\sum_{l=1}^{d-1}\frac{1+e_{l}}{d}\left[\sum_{j=0}^{n-1}\sum_{s=0}^{d-1}\frac{c^{s}_{j,i}}{d^{n}}\frac{1+e^{s\oplus\overline{l}\oplus\overline{r(i)}\oplus
k}_{j,i}}{d}\right],\end{split}$ (64)

where we first used the fact that $\mathbf{a}$ is independent of $\beta$ and
introduced the notation,

$\displaystyle\mathrm{P}(\alpha=j)=\frac{c_{j}}{d^{n}},\quad\mathrm{P}(h\oplus
a_{i}=k|\alpha=j)=\frac{c^{k}_{j,i}}{c_{j}},\quad\mathrm{P}(A\oplus
B=k|\alpha=j,\beta=i)=\frac{1+e^{k}_{j,i}}{d}.$ (65)

We also define the Fourier transform of the collection of probabilities in the
square brackets above as follows

$u_{t}(i):=\sum_{k=0}^{d-1}\left[\sum_{j=0}^{n-1}\sum_{s=0}^{d-1}\frac{c^{s}_{j,i}}{d^{n}}\frac{1+e^{s\oplus\overline{r(i)}\oplus
k}_{j,i}}{d}\right]\omega^{k\cdot t}.$ (66)

Thus the inverse transformation is given by

$\left[\sum_{j=0}^{n-1}\sum_{s=0}^{d-1}\frac{c^{s}_{j,i}}{d^{n}}\frac{1+e^{s\oplus\overline{r(i)}\oplus
k}_{j,i}}{d}\right]=\frac{1}{d}\sum_{t=0}^{d-1}u_{t}(i)\omega^{-k\cdot t}.$
(67)

We substitute this inverse transformation back into the definition of
$\mathrm{P}(E=k|b=i)$, and similarly to the $d2dd$ case, we obtain,

$\displaystyle\mathrm{P}(E=k|b=i)=\frac{1}{d}\left(1+F_{k}^{i}\right),\quad\text{where
}F_{k}^{i}=\frac{1}{d}\sum_{t=1}^{d-1}u_{t}(i)\omega^{-k\cdot
t}\sum_{l=1}^{d-1}e_{l}(\omega^{l\cdot t}-1).$ (68)

Plugging this expression back in the LHS we have, and simplifying as in $d2dd$
case, gives us the simplified IC statement,

$\sum_{i=0}^{n-1}\sum_{k=0}^{d-1}\left(1+F_{k}^{i}\right)\log\left(1+F_{k}^{i}\right)\leq\left(1-\sum_{l=1}^{d-1}e_{l}\right)\log\left(1-\sum_{l=1}^{d-1}e_{l}\right)+\sum_{l=1}^{d-1}(1+e_{l})\log(1+e_{l}).$
(69)

We again use our method of differentiating twice in the limiting case when the
channel capacity goes to zero. To this end, we make a substitution
$e_{l}\equiv\epsilon g_{l}$ and take $\epsilon\rightarrow 0$ to get almost the
final result,

$\frac{\sum_{i=0}^{n-1}\sum_{k=0}^{d-1}(F_{k}^{i})^{2}}{\sum_{l=1}^{d-1}g_{l}^{2}+\left(\sum_{l=1}^{d-1}g_{l}\right)^{2}}\leq
1.$ (70)

At this stage we notice that the expression for the inequalities is exactly
analogous to (52) and thus after carrying out similar steps for the
simplification of the numerator and denominator we end up with the following
inequalities,

$\sum_{i=0}^{n-1}|u_{t}(i)|^{2}\leq 1,\quad\forall t\in\\{1,\dots,\lfloor
d/2\rfloor\\}.$ (71)

Plugging in the form of $u_{t}(i)$ in the above inequalities we get,

$\displaystyle\sum_{i=0}^{n-1}\left|\sum_{k,s=0}^{d-1}\sum_{j=0}^{n-1}c^{s}_{j,i}e^{s\oplus
r(i)\oplus k}_{j,i}\omega^{k\cdot l}\right|^{2}\leq d^{2}d^{2n},\quad\forall
l\in\\{1,\dots,\lfloor d/2\rfloor\\}.$ (72)

∎

##  Appendix E The 2222 Bell scenario with correlated inputs

For convenience, we restate the statement of Result 2.

######  Result 2.

In the $2222$ Bell scenario, the following inequality follows from the
modified information causality principle,

$\left((1+\epsilon)e_{0,0}+(1-\epsilon)e_{1,0}\right)^{2}+(1-\epsilon^{2})(e_{0,1}-e_{1,1})^{2}\leq
4,$ (73)

for any parameter $\epsilon\in[-1,1]$.

###### Proof.

We resort again to the van Dam protocol Van Dam (2013) in Eq. (2). The only
difference is that now the input bits of Alice are correlated. To account for
such a correlation, we introduce the following notation for two input bits
distribution,

$\mathrm{P}(a_{0}=l,a_{1}=m)=\frac{1+(-1)^{l\oplus m}\epsilon}{4}.$ (74)

For such a distribution we have $\text{corr}(a_{0},a_{1})=\epsilon$. We can
now calculate the following familiar expression,

$\displaystyle\begin{split}&\mathrm{P}(g=k|a_{0}=l,a_{1}=m,b=i)=\mathrm{P}(x^{\prime}\oplus
B=k|a_{0}=l,a_{1}=m,b=i)\cr=&\mathrm{P}(x^{\prime}=x)\mathrm{P}(x\oplus
B=k|a_{0}=j,a_{1}=k,b=l)+\mathrm{P}(x^{\prime}\neq x)\mathrm{P}(x\oplus
B=k\oplus
1|a_{0}=j,a_{1}=k,b=l)\cr=&\mathrm{P}(x^{\prime}=x)\mathrm{P}(A\oplus
B=a_{0}\oplus k|a_{0}=l,a_{1}=m,b=i)+\mathrm{P}(x^{\prime}\neq
x)\mathrm{P}(A\oplus B=a_{0}\oplus{k\oplus
1}|a_{0}=l,a_{1}=m,b=i)\cr=&\mathrm{P}(x^{\prime}=x)\sum_{j=0}^{1}\mathrm{P}(\alpha=j|a_{0}=l,a_{1}=m,b=i)\mathrm{P}(A\oplus
B=l\oplus k|\alpha=j,\beta=i)\cr+&\mathrm{P}(x^{\prime}\neq
x)\sum_{j=0}^{1}\mathrm{P}(\alpha=j|a_{0}=l,a_{1}=m,b=i)\mathrm{P}(A\oplus
B=l\oplus k\oplus
1x|\alpha=j,\beta=i)\cr=&\frac{1}{2}\sum_{j=0}^{1}\delta_{j,l\oplus
m}[1+e_{c}(2\mathrm{P}(A\oplus B=l\oplus
k|\alpha=j,\beta=i)-1)]=\frac{1+(-1)^{l\oplus k}e_{c}e_{l\oplus
m,i}}{2}.\end{split}$ (75)

Here we used the independence of $a_{0},a_{1}$ and $b$, and the protocol
identity $\alpha=a_{0}\oplus a_{1}$. We now express the joint distribution,

$\displaystyle\mathrm{P}(g=k,a_{0}=l,a_{1}=m|b=i)=\frac{1+(-1)^{l\oplus
m}\epsilon}{4}\frac{1+(-1)^{l\oplus k}e_{c}e_{l\oplus m,i}}{2}.$ (76)

Using the final expression, we easily compute following relevant marginals:

$\displaystyle\begin{split}\mathrm{P}(g=k,a_{0}=l|b=0)&=\frac{1}{8}\left(2+(-1)^{k\oplus
l}e_{c}(e_{j,0}+e_{j\oplus 1,0}+(-1)^{j}\epsilon(e_{j,0}-e_{j\oplus
1,0}))\right).\end{split}$ (77)

$\displaystyle\begin{split}\mathrm{P}(g=a_{0}|b=0)&=\sum_{j=0}^{1}\mathrm{P}(g=j,a_{0}=j|b=0)=\frac{1}{2}+\frac{e_{c}}{2}\left(\frac{e_{00}(1+\epsilon)}{2}+\frac{e_{10}(1-\epsilon)}{2}\right).\end{split}$
(78)

$\displaystyle\sum_{m=0}^{1}\mathrm{P}(g=k,a_{1}=m|b=1,a_{0}=l)=\frac{1}{2}+\frac{e_{c}}{2}(-1)^{k\oplus
l}\left(\frac{e_{l,1}(1+(-1)^{l}\epsilon)}{2}+\frac{e_{l\oplus
1,1}(1-(-1)^{l}\epsilon)}{2}\right).$ (79)

Now we have calculated expression for each probability distribution entering
the correlated IC statement,

$\displaystyle I(a_{0};g|b=0)+I(a_{1};g|b=1,a_{0})\leq\mathcal{C}.$ (80)

Inserting the corresponding distributions from above we calculate the mutual
information explicitly and as a next step we again use our algorithm to
extract the inequality:

$\left((1+\epsilon)e_{00}+(1-\epsilon)e_{10}\right)^{2}+(1-\epsilon^{2})(e_{01}-e_{11})^{2}\leq
4.$ (81)

∎

##  Appendix F Comparison between the obtained inequalities and the
macroscopic locality principle in the $3322$ scenario

We now compare the derived inequalities with the bounding power of the
macroscopic locality principle in the $3322$ case. We use the following common
notation introduced in Ref. Collins and Gisin (2004) to represent an NS-box
as,

$p=\begin{tabular}[]{ c | ccc }&$\mathrm{P}(A_{0})$&$\mathrm{P}(A_{1})$&$\mathrm{P}(A_{2})$\\\ \hline\cr$\mathrm{P}(B_{0})$&$\mathrm{P}(A_{0}B_{0})$&$\mathrm{P}(A_{1}B_{0})$&$\mathrm{P}(A_{2}B_{0})$\\\ $\mathrm{P}(B_{1})$&$\mathrm{P}(A_{0}B_{1})$&$\mathrm{P}(A_{1}B_{1})$&$\mathrm{P}(A_{2}B_{1})$\\\ $\mathrm{P}(B_{2})$&$\mathrm{P}(A_{0}B_{2})$&$\mathrm{P}(A_{1}B_{2})$&$\mathrm{P}(A_{2}B_{2})$ ,\\\ \end{tabular}$ (82)

where $\mathrm{P}(A_{i}B_{j})$ is a short hand notation for a joint
probability of getting the outcome ${}^{\prime}0^{\prime}$ for both
measurements. We now choose the following three distributions:

$p_{1}=\frac{1}{2}\begin{tabular}[]{ c | ccc }&$1$&$1$&$1$\\\ \hline\cr$1$&$1$&$1$&$1$\\\ $1$&$1$&$0$&$1$\\\ $1$&$1$&$1$&$0$\\\ \end{tabular},\quad p_{2}=\frac{1}{2}\begin{tabular}[]{ c | ccc }&$1$&$1$&$1$\\\ \hline\cr$1$&$0$&$1$&$1$\\\ $1$&$1$&$0$&$1$\\\ $1$&$1$&$1$&$0$\\\ \end{tabular},\quad p_{N}=\frac{1}{4}\begin{tabular}[]{ c | ccc }&$2$&$2$&$2$\\\ \hline\cr$2$&$1$&$1$&$1$\\\ $2$&$1$&$1$&$1$\\\ $2$&$1$&$1$&$1$\\\ \end{tabular}.$ (83)

The first two are PR boxes which maximally violate the $I_{3322}$ inequality
while the last one is a local one. We consider the family of boxes with a
single parameter,

$p_{c}:=c\left(\frac{p_{1}+p_{2}}{2}\right)+(1-c)p_{N}.$ (84)

The $I_{3322}$ value $I=2c-1$ for such boxes. In Ref. Cavalcanti  _et al._
(2010), it was shown that all boxes satisfying macroscopic locality give
$I\leq 1/5$. By considering all the possible deterministic protocols, we can
calculate for which value of $c$ the IC inequalities are violated by the box
$p_{c}$. Choosing the smallest of them will give us the tightest bound on this
set.

We optimize over all protocols, and thus, over all the inequalities in Result
3 in the $3322$ case, to find the tightest one bounding the parameter $c$.
This leads to the following inequality,

$(e_{10}+e_{20})^{2}+(2e_{01}-e_{11}+e_{21})^{2}+(2e_{02}+e_{12}-e_{22})^{2}\leq
16.$ (85)

Plugging in the distribution $p_{c}$ gives us a bound of $c\leq 2/3$ which
implies $I\leq 1/3$. Thus, the bound we get using our method from the IC
principle is strictly weaker than the ML bound in this particular slice.

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