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# Quantum Bell inequalities from Information Causality – tight for Macroscopic
Locality

Mariami Gachechiladze  Institute for Theoretical Physics, University of
Cologne, 50937 Cologne, Germany  Bartłomiej Bąk  Department of Mathematical
Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5,
02-093 Warsaw, Poland  Marcin Pawłowski  International Centre for Theory of
Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdańsk, Poland
Nikolai Miklin  International Centre for Theory of Quantum Technologies
(ICTQT), University of Gdansk, 80-308 Gdańsk, Poland

(August 27, 2024)

###### Abstract

In a Bell test, the set of observed probability distributions complying with
the principle of local realism is fully characterized by Bell inequalities.
Quantum theory allows for a violation of these inequalities, which is famously
regarded as Bell nonlocality. However, finding the maximal degree of this
violation is, in general, an undecidable problem. Consequently, no algorithm
can be used to derive quantum analogs of Bell inequalities, which would
characterize the set of probability distributions allowed by quantum theory.
Here we present a family of inequalities, which approximate the set of quantum
correlations in Bell scenarios where the number of settings or outcomes can be
arbitrary. We derive these inequalities from the principle of Information
Causality, and thus, we do not assume the formalism of quantum mechanics.
Moreover, we identify a subspace in the correlation space for which the
derived inequalities give the necessary and sufficient conditions for the
principle of Macroscopic Locality. As a result, we show that in this subspace,
the principle of Information Causality is strictly stronger than the principle
of Macroscopic Locality.

##  1 Introduction

At the most general level, experimenters’ interaction with physical
apparatuses is described by the probability distribution of their responses to
the operations performed. To predict which types of distributions occur in
nature is an important problem from both a conceptual and applied standpoints.
A Bell test is the most central and relevant scenario from both of these
perspectives [1]. A phenomenon known as nonlocality, that is demonstrated in a
Bell test, implies that the principle of local realism [2], which appears to
be inherent to all physical systems, must be rejected [1]. At the same time,
nonlocality promises unconditional communication security [3], randomness
amplification [4] and may even be the source of quantum computational
advantage [5].

To demonstrate nonlocality one needs to show that the observed correlations
cannot be reproduced by local hidden variable models, which is the name for
probabilistic theories satisfying the principle of local realism [1].
Theoretically, this task can be easily achieved since the set of distributions
corresponding to local models is fully characterized by Bell inequalities, and
quite a few examples of them are present in the literature [6, 7, 8, 9, 10,
11, 12]. Violation of a Bell inequality implies that the observed distribution
is nonlocal, which has also been experimentally demonstrated [13, 14, 15].
Although classification of all Bell inequalities remains a challenge, there is
a systematic way in which new relevant Bell inequalities can be identified
[16].

A much harder problem is characterizing the set of probability distributions
attainable in quantum theory. No algorithm or systematic method is known to
derive quantum analogs of Bell inequalities, especially non-linear ones, and
consequently a very few examples of relevant quantum Bell inequalities are
known [17, 18, 19, 20, 21]. Even the most straightforward approach of
translating “classical” Bell inequalities by substituting their local hidden
variable bounds by the maximum attainable in quantum theory is only possible
in very limited cases [22] due to the difficulty of the problem [23].
Important results in this direction come from a fast-growing field of quantum
self-testing [24]. The goal of self-testing protocol is to uniquely (up to
local isometries) identify a quantum state and (sometimes) corresponding
quantum measurements, solely from the observed correlations in a Bell test.
Often, the self-testing argument is formulated in the form of the maximal
violation of a Bell inequality attainable in quantum mechanics. The latter
value precisely singles out the self-tested state and measurements [25]. On
the other hand, such a quantum bound on the Bell inequality leads to a linear
quantum Bell inequality (see Re. [26] for a comprehensive review).

In this work, we present an infinite family of quantum Bell inequalities, that
generalize and outperform some previously known results [22, 19]. These
inequalities are derived from Information Causality [27], a physical principle
based on a set of information-theoretic assumptions that are satisfied by
quantum theory but do not rely on its full formalism. We show that these
inequalities are also implied by the principle of Macroscopic Locality [28],
and, moreover, we identify a subspace in the set of probability distributions
in which the derived inequalities also constitute the necessary condition for
this principle.

Bounding the set of quantum correlations from physical principles offers
several advantages. First, a physical principle might lead to a much more
efficient description of the set of correlations compatible with it, at the
expense of giving less tight approximation of the quantum set. We give a few
examples below in which this trade-off is apparent. Second, since the axioms
of quantum theory are not physically motivated, the problem of identifying
appropriate principles that lead to the same set of correlations as quantum
theory is of foremost importance. Information Causality is possibly the most
intriguing of the known principles, since it has not been established yet
whether it can be satisfied by any correlations stronger than quantum.
Finally, if quantum theory is eventually replaced with a more general
probabilistic theory [29], those inequalities that are based on physical
principles may still be valid.

Few examples of quantum Bell inequalities resulting from different physical
principles already appear in the literature. One of the most fundamental
principles, the principle of no-signaling [30], allows for an efficient
description of the set of correlations compatible with it. However, this
approximation tuns out to be too coarse since it allows for correlations
saturating the algebraic maximum of $4$ of Clauser-Horne-Shimony-Holt (CHSH)
inequality [6], while the maximal value permitted by quantum theory is
$2\sqrt{2}$ [22]. The principle of Information Causality (IC) [27], on the
other hand, recovers the so-called Tsirelson bound of $2\sqrt{2}$. However,
the set of correlations compatible with this principle is very hard to
describe, due to its formulation. The IC principle constraints the amount of
mutual information between a remote data and its guess in a communication game
[31] (See Section 3 for details). Despite this indirect formulation,
surprisingly, a simple quadratic quantum Bell inequality, known as Uffink
inequality [19] was shown to follow from the IC principle [27]. The principle
of Macroscopic Locality (ML) [28] offers a direct and relatively simple
formulation in terms of a semidefinite program, which also leads to the
correct bound on CHSH inequality. However, this formulation has a major
drawback that it requires solving an optimization problem. Since the number of
optimization variables grows with the size of the probability vector, this
means that an analytical description of the set of macroscopic local
correlations e.g., in terms of inequalities, is possible only in simple
scenarios [32, 21].

In this paper, we generalize the aforementioned results by presenting an
infinite family of quantum Bell inequalities in which the number of
measurement settings or outcomes can be taken arbitrary high. We show that
these inequalities, which take a simple form of quadratic expressions in terms
of the observed probabilities, follow from both the IC and ML principles. This
is especially intriguing since the two principles have very district
formulations. The interplay between the IC and ML principle was investigated
earlier by giving examples of correlations that satisfy the ML principle but
violate the principle of IC [33]. In this work we identify an entire subspace
of the correlation space in which the derived inequalities give the exact
boundary of the set of macroscopically local correlations, or in other words
provide the necessary and sufficient condition for the ML principle. It means
that everywhere in this subspace, which is defined by the symmetries of the
derived inequalities, the principle of IC is at least as strong as the
principle of ML.

Finally, in this work, we develop systematic methods that can be used to
derive quantum Bell inequalities from the both considered physical principles.
Concerning derivations from the IC principle, the main challenge is in
calculating the guessing probabilities resulting from using multiple copies of
the same probability distribution, the process called concatenation [27]. We
show that for a given communication protocol, this process can be formulated
as a Markov chain. Using a tool known as z-Transform [34, 35] we derive the
solution of a special type of Markov chains (originating from the
communication protocol) of growing dimension. We believe that the derivations
related to the Markov chains are elegant and can be of independent interest.
For obtaining the characterization of the set corresponding to the ML
principle, the main problem that one faces is the growing number of
optimization variables. In this work we show that if one considers a subspace
of the correlation space defined by some symmetry group, the condition of ML
can be broken into a set of conditions of lower dimensions, each of which can
be solved analytically. We provide an explicit construction for a specific
symmetry group in this paper, but the overall idea should be easily
generalizable to other groups.

The main part of the paper is divided into two parts. In the first part, we
give the formal statement of the IC principle and present our quantum Bell
inequalities with a high-level proof. In this proof, an explicit solution of a
particular Markov chain is given. In the second part, we give a formulation of
the principle of ML as a semidefinite program and show that the derived
inequalities constitute necessary and sufficient conditions for the considered
subspace of correlations. Technical details are left to Appendix.

Figure 1: Information Causality scenario. Alice receives a data set of random
dits $\\{a_{0},a_{1},\dots,a_{2^{n}-1}\\}$, while Bob is given $b$, which
takes values between $0$ and $2^{n}-1$. Alice and Bob have access to devices
which produce nonsignaling correlations with inputs denoted by $\alpha$ and
$\beta$ and outputs by $A$ and $B$. Bob’s guess of $a_{b}$ is denoted by $g$.
To facilitate with the guessing, Alice sends a classical message $x$ usually
taken to be of the same size as the data variables $a_{i}$.

##  2 Notations

Here we list some of the notations used throughout the paper. We use upper-
case and lower-case letters as well as Greek letters to denote random
variables. To avoid possible confusions, we always write explicitly when we
introduce a random variable. For the same reason, for events such as a random
variable $A$ taking a value $k$, i.e., $A=k$, or two random variables $A$ and
$B$ taking equal values, i.e., $A=B$, the probability is always spelled out as
$\mathrm{P}(A=i)$ or $\mathrm{P}(A=B)$. All other notations resembling
probabilities, e.g., $p(e|j)$ where $e$ and $j$ are indices (not random
variables), should be treated formally as functions. For correlations observed
in a bipartite Bell scenario, with inputs $\alpha$ and $\beta$ and outputs $A$
and $B$, the observed correlations are denoted as
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$ in accordance with our general notation
of probabilities.

The addition modulo $d$ is denoted by $\oplus$ and we use $\overline{x}$ to
denote the inverse (or inverse random variable) of $x$, i.e.,
$\overline{x}{\oplus}x=0$. We also use $x\ominus y$ to denote
$x\oplus\overline{y}$. We use the convenient Dirac notation $|i\rangle$,
$i\in\\{0,1,\dots d\\}$ for the basis vectors in $\mathds{R}^{d}$. We also use
$|\psi\rangle$ for a general, not necessary normalized column vector $\psi$ in
the same space $\mathds{R}^{d}$. We use the common notations
$[n]\equiv\\{0,1,\dots,n-1\\}$, $\mathrm{i}=\sqrt{-1}$, $x^{*}$ for the
complex conjugate of $x\in\mathds{C}$, $\Re(x)$ and $\Im(x)$ for the real and
imaginary parts of $x$, and $\omega=\exp{(2\pi\mathrm{i}/d)}$ for the $d$-th
root of unity. Finally, we abbreviate a $d$-dimensional digit as “dit”.

##  3 Information causality principle

We start by summarizing the statement of Information Causality (IC) principle
[27]. The IC is defined in communication scenario, known as an entanglement-
assisted random access code [31], see Fig. 1. Two parties, Alice and Bob, face
a communication challenge. Alice is given a data set of $2^{n}$ $d$-outcome
random variables, which we denote as $\\{a_{0},a_{1},\dots,a_{2^{n}-1}\\}$
($a_{i}$ takes values in $[d]$ for all $i\in[2^{n}]$, $n$ is a positive
integer). Bob is asked to guess one of the dits from the data set given to
Alice, with the choice determined by another random variable $b$ which takes
values in $[2^{n}]$. All the random variables $a_{0},a_{1},\dots,a_{2^{n}-1}$,
and $b$ are considered independent and uniformly distributed. Their values are
drawn at random before each run of the experiment. In each run, Alice
communicates a message $x$ to Bob with possible information about some of her
data. At the end of each run, Bob outputs a value of his guess $g$, which also
takes values in $[d]$. Both parties succeed if $g=a_{b}$, by which we mean
(with a slight abuse of notation) that $g=a_{y}$, while $b=y$, $\forall
y\in[2^{n}]$.

Clearly, when the size of the message $x$ is smaller than the size of the data
set, $\\{a_{0},a_{1},\dots,a_{2^{n}-1}\\}$ no perfect guessing is possible.
This intuition, however, fails if the parties are allowed to facilitate their
communication with extreme nonsignaling correlations, also known as PR-boxes
[30]. In fact, Ref. [27] (and earlier works by van Dam [36, 37]) demonstrated
that, if parties have access to $(2^{n}-1)$ PR-boxes, a single bit of
communication (for the case of $d=2$) is enough to win the game with the unit
probability. The main idea there is that Alice uses the output(s) $A$ of the
PR-box(es) for the message encoding. By using output(s) $B$, Bob can
effectively choose which bit of Alice to decode.

The IC principle aims at restricting such non-physical behaviors based on the
set of natural assumptions also satisfied by quantum theory. More precisely,
it states that for nonsignaling theories equipped with a measure of
correlations, $I(\cdot;\cdot)$, that satisfies the chain rule, data processing
inequality, and which reduces to the Shannon mutual information for classical
variables, the following inequality holds [27],

$\sum_{y=0}^{2^{n}-1}I(a_{y};g|b=y)\leq|x|,$ (1)

where $|x|\equiv I(x;x)$ is the size of the message that Alice sends to Bob.
Equivalently, the bound on the right-hand side of Eq. (1) can be seen as the
capacity of the identity channel used for communication [38]. The statement in
Eq. (1) can be understood in the following way. Even if utilizing nonsignaling
correlations, in the discussed communication game, provides an advantage in
terms of guessing probability, there should be no benefit in terms of
information gain.

Considering the formulation of Eq. (1), it is not clear how to use it to bound
the set of correlations in a Bell test. The reason for this is that one needs
to consider all possible ways Alice and Bob can utilize the correlations
produced by, generally, an infinite number of copies of their devices in the
IC game. As a result, no characterization of the set of correlations complying
with the IC principle is known. Moreover, it is still possible that the IC
principle can single out the set of quantum correlations, and the description
of the latter is known to be an undecidable problem [23]. Despite these
difficulties, in the original work of Ref. [27] it was shown that in the case
of binary inputs and outputs, i.e., for $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$
with $k,l,i,j\in[2]$, a simple quantum Bell inequality, know as Uffink
inequality [19] can be derived from the IC principle. This approximation in
terms of Uffink inequality is tight in the sense that it predicts the
Tsirelson’s bound of $2\sqrt{2}$ as the maximum value of CHSH expression
attained in quantum mechanics [17, 6].

In this work, we consider a more general Bell scenario, in which the number of
settings or outcomes can be arbitrary high. More precisely we look at the
distributions $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, with $k,l,i\in[d]$, and
$j\in[2]$, for any possible $d$. In order to state our main result, let us
define the following linear expressions of the observed correlations
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, with $k,l,i\in[d]$, and $j\in[2]$,

$p(e|j)=\frac{1}{d}\sum_{i=0}^{d-1}\mathrm{P}(A\oplus B=i\cdot j\oplus
e|\alpha=i,\beta=j),$ (2)

for all $e\in[d]$ and $j\in[2]$. We use the lower-case letter $p$, which shall
not be confused for the probability $\mathrm{P}$ (see Notations). These
functions play the role of guessing probabilities for each individual pair of
the devices, however, for the moment they should be understood simply as a
notation convenient for presenting the following result.

######  Result 1.

Any nonsignaling theory satisfying the Information Causality principle
complies with the following inequalities in $\\{p(e|j)\\}_{e,j}$ for all
$i\in\left\\{1,\dots,\left\lfloor{d}/{2}\right\rfloor\right\\}$ and
$\omega=e^{{2\pi\mathrm{i}}/{d}}$,

$\left|\;\sum_{e=0}^{d-1}p(e|0)\omega^{e\cdot
i}\;\right|^{2}+\left|\;\sum_{e=0}^{d-1}p(e|1)\omega^{e\cdot
i}\;\right|^{2}\leq 1.$ (3)

Notably, for $d=2$, the inequality in Eq. (3) reduces to the famous quadratic
quantum Bell inequality by Uffink [19].

###### Proof.

The proof consists of several parts: (i) Communication protocol – Here a
detailed description of IC scenario and concatenation protocol are given; (ii)
Guessing probability – In this part, we express the success and error
probabilities of Bob’s guesses in terms of the nonsignaling correlations
shared between the parties; (iii) New inequalities from IC – In this final
part of the proof we connect the IC statement with the guessing probabilities
from part (ii), which leads us to our main result.

(i) Communication protocol. – Alice and Bob are in the IC scenario (see Fig.
1). They have access to a pair of devices producing nonlocal correlations, or
as we call it, nonsignaling box (NS-box). We denote the inputs of the shared
NS-box by $\alpha$ and $\beta$ and outputs by $A$ and $B$, respectively (see
Notations). In this work we focus on $d2dd$ NS-boxes, i.e., $\beta$ is binary
and $\alpha,A$, and $B$ take values in $[d]$. Then, as the communication
protocol for Alice and Bob, we can choose the direct generalization of the
case $2222$ [36], where Alice communicates to Bob a single dit message $x$,
and

$\displaystyle\begin{split}\alpha&=\overline{a}_{0}{\oplus}a_{1},\;x=a_{0}{\oplus}A,\\\
\beta&=b,\;g=x{\oplus}B.\end{split}$ (4)

If Alice and Bob share an NS-box, which exhibits the correlations $A\oplus
B=\alpha\cdot\beta$, then Bob’s guesses are perfect, i.e., $g=a_{0}$, if $b=0$
and $g=a_{1}$ if $b=1$. Consequently, such NS-boxes violate the IC principle,
as the left-hand side of Eq. (1) is $2$ while $|x|=1$. More generally, Eq. (1)
puts an upper bound on the probability $\mathrm{P}(A\oplus
B=\alpha\cdot\beta)$. However, for the above protocol for $n=1$, i.e., two
dits of Alice’s data, such bounds turn out to be loose, and they become
tighter when $n$ increases [27]. For example, in $d=2$ case, the bound one
obtains from IC for $n=1$ is approximately $0.8889$, while the true quantum
value is $1/2+1/2\sqrt{2}\simeq 0.8536$ [22], which one gets in the limit of
$n\rightarrow\infty$ [27].

The protocol in Eq. (4) can be generalized to an arbitrary $n$ through the
process called concatenation [27]. In the concatenation protocol, outputs of
some boxes are fed as inputs of subsequent ones, forming a layered structure
in this way. We use the superscript indices for inputs and outputs of a box to
specify the layer to which this NS-box belongs to. There are $n$ layers of NS-
boxes in total with $2^{m}$ boxes on each $m$-th layer, making it $2^{n}-1$
boxes for the whole protocol. For convenience, we enumerate the layers, $m$ in
descending order from $(n-1)$ to $0$.

We start with generalizing Alice’s encoding strategy. First, we group the data
set $\\{a_{0},a_{1},\dots,a_{2^{n}-1}\\}$ in consecutive pairs, i.e.,
$(a_{0},a_{1}),(a_{2},a_{3}),\dots,(a_{2^{n}-2},a_{2^{n}-1})$. We then assign
to each pair an NS-box with Alice’s output $A^{n-1}_{i}$. Here, the
superscript $(n-1)$ stands for the layer and the subscript $i\in[2^{n-1}]$
distinguishes NS-boxes within that layer. The input $\alpha^{n-1}_{i}$ to the
$i$-th box is $(\overline{a}_{2i}\oplus a_{2i+1})$, which resembles the
protocol in Eq. (4). At the end of the first step of encoding, we form
$2^{n-1}$ new dits $a_{2i}\oplus A_{i}^{n-1},i\in[2^{n-1}]$, which serve as
the data set for the $(n-2)$-th layer of NS-boxes. In other words, what used
to be the message $x$ in Eq. (4) plays the role of the data set on the
$(n-2)$-th layer of concatenation. The encoding strategy then proceeds
iteratively, the size of the data set halving on each lower layer of
concatenation, until there are only two dits of data on the $0$-th layer. The
output of the $0$-th layer of the encoding strategy is the message $x$ that
Alice sends to Bob.

The decoding strategy of Bob is best to be understood in terms of walks.
First, let us introduce $\vec{b}=(b_{0},b_{1},\dots,b_{n-1})$, the binary form
of $b$, i.e., $b=\sum^{n-1}_{m=0}b_{m}2^{m}$. Each bit $b_{m}$ dictates the
“left" $(0)$ or “right" $(1)$ direction of the $m$-th step that Bob needs to
take in unscrambling the $m$-th layer of Alice’s encoding. In the $0$-th step
of his decoding, Bob inputs $\beta^{0}_{0}=b_{0}$ to the only layer-$0$ NS box
and adds his output dit $B^{0}_{0}$ to the message $x$. If now $b_{1}=0$, he
“moves left" (by which we mean that he uses $0$-th box of layer $1$ with an
input $\beta^{1}_{0}=0$) and he adds $B^{1}_{0}$ to $x\oplus B^{0}_{0}$.
Alternatively, if $b_{1}=1$, Bob “moves right" (i.e., he inputs
$\beta^{1}_{1}=1$ to the $1$-st box of layer $1$) and he adds $B^{1}_{1}$ to
$x\oplus B^{0}_{0}$. He proceeds by moving up the layers till the $(n-1)$-th
layer, using only one box $i_{m}$ per each layer $m$. This way, on the $m$-th
layer with totally $2^{m}$ boxes, Bob chooses $i_{m}$-th box, where
$i_{m}=2^{m}b_{m}+i_{m-1}$ can be calculated recursively for $1\leq
m\leq(n-1)$. Finally, the sum of the outputs, $g=x\oplus B^{0}_{0}\oplus\dots$
is the guess $g$ that Bob makes.

For reader’s convenience, we give an explicit form of encoding-decoding
strategy for $n=2$ below:

$\displaystyle\begin{split}\alpha^{1}_{0}&=\overline{a}_{0}{\oplus}a_{1},\;\alpha^{1}_{1}=\overline{a}_{2}{\oplus}a_{3},\\\
\alpha^{0}_{0}&=\overline{(a_{0}{\oplus}A^{1}_{0})}\oplus(a_{2}{\oplus}A^{1}_{1}),\\\
x&=(a_{0}{\oplus}A^{1}_{0})\oplus A^{0}_{0},\\\
\beta^{0}_{0}&=b_{0},\;\beta^{1}_{0}=b_{1},\;\beta^{1}_{1}=b_{1},\\\
g&=x{\oplus}B^{0}_{0}\oplus B^{1}_{b_{0}},\end{split}$ (5)

where by $B^{1}_{b_{0}}$ we mean $B^{1}_{0}$, if $b_{0}=0$, and $B^{1}_{1}$,
otherwise.

(ii) Guessing probability. – Here we work out the guessing probabilities
$\\{\mathrm{P}(g=a_{y}|b=y)\\}_{y}$ from the concatenation protocol described
in (i), and also the probabilities of errors. We define the probability of
making an error $E$ for the fixed value of $b$,

$P(E|y)\equiv\mathrm{P}(g=a_{y}\oplus E|b=y),$ (6)

$\forall E\in[d],\forall y\in[2^{n}]$ where $P(0|y)$ is the probability of Bob
guessing $a_{y}$ correctly. For $d=2$ there is only one probability of error,
which does not need to be calculated as $P(1|y)=1-P(0|y)$. However, for the
further derivations, for $d>2$, we need to calculate error probabilities in
Eq. (6) for all $E$.

In our derivations, we assume that all $(2^{n}-1)$ NS-boxes are identical,
i.e., they all exhibit the same correlations. For the simplest case, $n=1$, by
employing the protocol in Eq. (4), we can explicitly calculate the probability
$P(E|y)$,

$\displaystyle P(E|y)$ $\displaystyle=\mathrm{P}(x\oplus B=a_{y}\oplus E|b=y)$
$\displaystyle=\mathrm{P}(a_{0}\oplus A\oplus B=a_{y}\oplus E|b=y)$
$\displaystyle=\mathrm{P}(A\oplus
B=a_{y}\oplus\overline{a}_{1}\oplus\alpha\oplus E|b=y)$
$\displaystyle=p(E|y).$ (7)

In the last equality, on the right-hand side we have the correlation of the
NS-box defined in Eq. (2) and the equality holds since
$a_{y}\oplus\overline{a}_{1}\oplus\alpha=\alpha\cdot y$ for all
$y\leavevmode\nobreak\ \in\leavevmode\nobreak\ [2]$.

Our goal is to compute $P(E|y)$ for an arbitrary $n$, however, this proves to
be an intricate task. The largest effort is to count the resulting error $E$
as a sum of the individual errors $e^{m}$ corresponding to the NS-box used on
every $m$-th layer. To give an example, if we use the protocol in Eq. (5) for
$n=2$, we get two errors: $e^{0}$ and $e^{1}$ that sum up to $E=e^{0}\oplus
e^{1}$. Furthermore, there are $d$ possible pairs of $e^{0}$ and $e^{1}$ that
sum to the same resulting error $E$. More generally, for $n$ boxes, we have
$d^{n-1}$ combinations of errors, and we have to account for all of them in
the final guessing probability.

Finally, apart from listing all combinations of errors
$\\{e^{m}\\}_{m=0}^{n-1}$ which contribute to $E$, we are also interested in
the type of boxes used, i.e., the value of $\beta$ inserted in the box that
produces $e^{m}$. For such an error-handling, it is again beneficial to think
of Bob’s decoding strategy as a walk.

By looking at his input $b=y$, Bob finds that the guess of the value of
$a_{y}$ involves aiming $(n-k)$ times at the left (inserting $\beta=0$) and
$k$ times at the right (inserting $\beta=1$), where $k$ is the weight of $y$,
i.e., the number of $1$s in the binary form of $y$. Since the addition
${\oplus}$ is symmetric and associative, every $P(E|y)$ with input of the same
weight is going to be the same. Hence, in full generality, we can redefine
$P(E|y)$ in Eq. (6) as $P(E|k)$, with $k\in[n]$.

In order to calculate $P(E|k)$, we introduce one more notation

$|Q^{(k)}(j)\rangle=\sum_{e=0}^{d-1}Q^{(k)}_{e}(j)|e\rangle,\quad j\in[2],$
(8)

where $Q^{(k)}_{e}(j)$ is the probability of making an error $e$ when aiming
$k$ times in $j$-th direction, and $\\{|e\rangle\\}_{e=0}^{d-1}$ is the
computational basis in $\mathds{R}^{d}$. If we also consider the errors
accumulating when aiming $(n-k)$ times in $\bar{j}$ (opposite to $j$)
direction, we get the total probability for each error $E$,

$P(E|k)=\langle Q^{(k)}(0)|S_{E}|Q^{(n-k)}(1)\rangle,$ (9)

where $S_{E}=\sum_{i=0}^{d-1}|i\rangle\langle\overline{i}\oplus E|$ is a
permutation matrix.

Let us consider some examples of $Q_{e}^{(k)}(j)$ for $k=1$ and $k=2$ and
arbitrary $d$. Starting from the simplest case $k=1$, the probability of
making $e$ errors, when aiming one time in $j$-th direction is
$Q_{e}^{(1)}(j)=p(e|j)$, $\forall e\in[d]$. For $k=2$, we have

$Q_{e}^{(2)}(j)=\sum_{{e^{\prime}}=0}^{d-1}p(e\oplus
e^{\prime}|j)p(\bar{e^{\prime}}|j).$ (10)

More generally, the error probability vector, $|Q^{(k)}(j)\rangle$ can be
expressed as the $k$-th state vector in the Markov process with the transition
matrix depending on $j$,

$M_{j}=\sum_{e=0}^{d-1}\sum_{e^{\prime}=0}^{d-1}p(e^{\prime}\oplus\bar{e}|j)|e^{\prime}\rangle\langle
e|.$ (11)

In other words, we can obtain the vector for $k=2$ from $k=1$ as
$|Q^{(2)}(j)\rangle=M_{j}|Q^{(1)}(j)\rangle$, and more generally,

$|Q^{(k)}(j)\rangle=M^{k}_{j}|0\rangle.$ (12)

Hence, we have reduced the problem to finding $M^{k}_{j}$, the $k$-th power of
the doubly stochastic matrix $M_{j}$, which can be managed using the formalism
of the z-Transforms [34, 35]. With the help of z-Transform formalism, we
calculated arbitrary $k$-th power of $M_{j}$ for small dimensions. Luckily,
already from working out the cases of $d=3$ and $d=4$ we were able to
conjecture the general form of $|Q^{({k})}(j)\rangle$ for any $d$, which we
give below.

For $d$ odd, $e\in[d]$, the entries of $|Q^{({k})}(j)\rangle$, amount to

$Q^{(k)}_{e}(j)=\frac{1}{d}+\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\Re{\left(l^{k}_{i}(j)\omega^{e\cdot
i}\right)},$ (13)

where $l_{i}(j)=\sum_{e=0}^{d-1}p(e|j)\omega^{-e\cdot i}$ is a complex-valued
function of the probabilities $p(e|j)$. The formula has a concise proof by
induction (See Appendix A.1). A slight variation of the above formula for $d$
even is also given in Appendix A.1.

Knowing $|Q^{(k)}(j)\rangle$, we can calculate the probabilities $P(E|k)$ in
Eq. (9),

$\displaystyle P($ $\displaystyle
E|k)-\frac{1}{d}=\sum_{e=0}^{d-1}Q^{k}_{e}(0)Q^{n-k}_{\bar{e}\oplus
E}(1)-\frac{1}{d}$ (14) $\displaystyle=$
$\displaystyle\frac{4}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-1}{2}}\sum_{e=0}^{d-1}\Re{\left(\omega^{i_{1}e}l_{i_{1}}^{k}(0)\right)}\Re{\left(\omega^{i_{2}(\bar{e}\oplus
E)}l_{i_{2}}^{n-k}(1)\right)}$ (15) $\displaystyle=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left(\cos\left(\frac{2\pi
Ei}{d}\right)A_{i}-\sin\left(\frac{2\pi Ei}{d}\right)B_{i}\right).$ (16)

Eq. (15) looks cumbersome, but it simplifies greatly once we use Euler’s
formula for the root of unity and work with real and imaginary parts of
$l_{i_{1}}^{k}(0)$ and $l_{i_{2}}^{n-k}(1)$. An important observation is that
if one first takes the sum over $e$ in Eq. (15), the sums of cosine and sine
functions are nonzero if and only if $i_{1}=i_{2}$, which leads to Eq. (16)
(See Appendix A.2 for more detailed calculation). In Eq. (16) we introduced
the following useful notations,
$A_{i}=\Re{\left(l_{i}^{k}(0)\right)}\Re{\left(l_{i}^{n-k}(1)\right)}-\Im{\left(l_{i}^{k}(0)\right)}\Im{\left(l_{i}^{n-k}(1)\right)}$
and
$B_{i}=\Re{\left(l_{i}^{k}(0)\right)}\Im{\left(l_{i}^{n-k}(1)\right)}+\Im{\left(l_{i}^{k}(0)\right)}\Re{\left(l_{i}^{n-k}(1)\right)}.$
Again, we do not give here the formula for $P(E|k)$ for $d$ even due to its
similarity with the odd case. For the formula and its proof see Appendix A.2.

(iii) New inequalities from IC. – In this work, we study the consequences of
the IC principle for nonsignaling boxes (NS-boxes) with $d\geq 2$ outcomes. To
this end, we assume the statement of IC as in Eq. (1), where
$I(a;b)=H(a,b)-H(a)-H(b)$ for two random variables $a$ and $b$, $|x|=H(x)$,
and $H(\cdot)$ is the Shannon entropy, which for convenience we define with
natural logarithm, i.e., $H(x)=\sum_{i}\mathrm{P}(x=i)\ln\mathrm{P}(x=i)$.

Having computed general $|Q^{({k})}(j)\rangle$, the only missing part in our
derivation is to connect the guessing probabilities with the IC statement.
Ref. [27] used Fano’s inequality [39] to this end. Here, we use a similar
inequality for two $d$-outcome random variables $a$ and $b$, where $a$ is
uniformly distributed, $b=a\oplus e$, and
$H(e)=-\sum_{i=0}^{d-1}\mathrm{P}(e=i)\ln\mathrm{P}(e=i)$,

$I(a;b)\geq\ln d-H(e).$ (17)

See a short proof in Appendix A.3. If we take $|x|=\ln d$,
$H(E|b=y)=-\sum_{E=0}^{d-1}P(E|y)\ln P(E|y)$ and use the new inequality, we
can approximate the terms, which appear in the formulation of IC in Eq. (1).

$\sum_{y=0}^{2^{n}-1}H(E|b=y)\geq(2^{n}-1)\ln d.$ (18)

On the other hand, we can also upper bound the entropy with a polynomial
function. In particular, the following inequality holds

$H(E|b=y)\leq\ln d-\kappa\sum_{E=0}^{d-1}\left(P(E|y)-\frac{1}{d}\right)^{2}$
(19)

for $\kappa=\frac{d^{2}(\ln d-1)+d}{(d-1)^{2}}>0$. See Appendix A.4 for the
proof. In the part (ii), we redefined $P(E|y)$ as $P(E|k)$ (see Eq. (9)),
where $k$ is the weight of $y$. Then, using Eqs. (18,19) and the fact that
$\binom{n}{k}$ terms corresponding to $b=y$ have the same weight, we arrive at
the following approximation of the statement of the IC principle:

$\sum_{k=0}^{n}\binom{n}{k}\sum_{E=0}^{d-1}\left(P(E|k)-\frac{1}{d}\right)^{2}\leq\frac{(d-1)^{2}\ln
d}{d^{2}(\ln d-1)+d}.$ (20)

In our proof, we are not concerned with the exact bound on the right-hand side
of Eq. (20), but rather with the fact that it does not grow with $n$.
Interestingly, the right-hand side of Eq. (20) is maximal for $d=12$ and
approximately equals $1.331$, while the limit of $d\rightarrow\infty$ is equal
to $1$.

We can now insert probabilities from Eq. (16) into Eq. (20). Using
trigonometric identities for the sums over $E$ we obtain (See details in
Appendix A.5),

$\displaystyle\sum_{E=0}^{d-1}\left(P(E|k)-\frac{1}{d}\right)^{2}=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left(A_{i}^{2}+B_{i}^{2}\right)$
(21) $\displaystyle=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left|l_{i}(0)\right|^{2k}\left|l_{i}(1)\right|^{2(n-k)}.$

The binomial sum in Eq. (20) then amounts to

$\sum_{k=0}^{n}\scalebox{0.9}{$\displaystyle\binom{n}{k}$}\left|l_{i}(0)\right|^{2k}\left|l_{i}(1)\right|^{2(n-k)}=\left(\left|l_{i}(0)\right|^{2}+\left|l_{i}(1)\right|^{2}\right)^{n}$
(22)

Since this is true for any
$i\in\\{1,2,\dots,\left\lfloor{d}/{2}\right\rfloor\\}$ and arbitrarily large
$n$, our main results, the statement of Eq. (3) follows. For even $d$ the
derivations are very similar, and we include them in Appendix A.5. ∎

In the next section, we first describe the set of correlations compatible with
the Macroscopic Locality principle, and then compare the two results.

##  4 Macroscopic Locality principle

The principle of Macroscopic Locality (ML) postulates that correlations
resulting from measurements on microscopic systems should obey the laws of
classical mechanics, in particular local realism [1], if a large number of
copies of these systems are considered, making them macroscopic [28]. We do
not give the formal statement of the ML principle in this paper. Instead, for
its quantitative formulation we use the fact that the set of macroscopic local
correlations coincides with the first level, often denoted as Q1, of the
hierarchy of Ref. [32]. Note that there exit other notions of “macroscopicity”
in quantum mechanics [40, 41], which are used in other scenarios than Bell’s.

The set Q1 can be described as follows. Let the nonsignaling correlations
shared between parties be described by the conditional probability
distribution $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$. At the moment we do not
need to specify the finite sets in which the inputs or outputs take their
values. Let $\vec{p}_{A}$ be a (column) vector of marginal probabilities
$\mathrm{P}(A=k|\alpha=i)$ of Alice, and let $\vec{p}_{B}$ be the similar
vector for Bob. Let the matrix $P$ be the matrix arrangement of probabilities
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, such that for every row of $P$ the
outcome and input of Alice, i.e., $k$ and $i$ are fixed. Similarly, let the
input and output of Bob, $l,j$ be the same for every element on the same
column of $P$. The correlations are then said to belong to the set Q1, if the
following matrix,

$\Gamma=\left(\begin{array}[]{c c c}1&\vec{p_{A}}^{T}&\vec{p_{B}}^{T}\\\
\vec{p_{A}}&V&P\\\ \vec{p_{B}}&P^{T}&W\end{array}\right),$ (23)

can be made positive semidefinite ($\Gamma\geq 0$) by an appropriate choice of
square matrices $V$ and $W$, which themselves are subject to additional
constraints:

$\mathrm{diag}(V)=\vec{p_{A}},\;\mathrm{diag}(W)=\vec{p_{B}}.$ (24)

With $\mathrm{diag}(\cdot)$ we denoted a map from the space of $d\times d$
matrices to $\mathds{R}^{d}$ that takes the diagonal entries of the input
matrix. Additionally, if an off-diagonal entry of $V$ corresponds to the same
input of Alice, but two different outputs, then this entry has to be $0$. The
same constraints are applied to the off-diagonal entries of $W$ and the inputs
and outputs of Bob. A more detailed description of $\Gamma$, in application to
our problem, can be found in Appendix B.

We are now ready to present our second main result, which is also stated in
terms of the probabilities in Eq. (2).

######  Result 2.

The set of constraints from Result 1 is necessary and sufficient for
probabilities $\\{p(e|j)\\}_{e,j}$ to satisfy the Macroscopic Locality
principle.

###### Proof.

As introduced earlier in this section, the set of the correlations complying
with the ML principle coincides with the set $Q^{1}$ [32], which is defined by
the condition $\Gamma\geq 0$ in Eq. (23). Positivity of $\Gamma$ matrix is in
general difficult to resolve for probabilities
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$ that are not specified. More precisely,
the condition $\Gamma\geq 0$ is equivalent to a system of polynomial
inequalities in $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$ as well as $V$ and $W$.
In order to resolve this system analytically one needs to eliminate the
variables $V$ and $W$ from this system. This problem is an instance of the so-
called quantifier elimination problems for which the best-known algorithm
requires double exponential time [42]. However, in this work we are interested
in deriving the set of constraints that the principle of ML sets on the
probabilities $p(e|j)$ defined in Eq. (2). Since the latter forms only a
subspace in the space of correlations $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$,
one might hope for simplification of the considered problem.

Such simplification is indeed possible, if one is able to identify symmetries
of the problem, which is a known technique in semidefinite programming [43,
44]. According to Ref. [44], if the optimization problem formulated in the
hierarchy of Ref. [32] exhibits symmetries defined by a group $G$, then the
following holds,

$\Gamma\geq
0\quad\Leftrightarrow\quad\tilde{\Gamma}\equiv\frac{1}{|G|}\sum_{g\in
G}g.\Gamma\geq 0,$ (25)

where an action of each group element $g$ on $\Gamma$ corresponds to a
permutation of its entries. The implications $\Leftrightarrow$ in Eq. 25
should be understood in the sense of constraints that the positivity of
$\tilde{\Gamma}$ and $\Gamma$ imply on the symmetric subspace, defined by the
group $G$, of compatible correlations $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$.
As a result of the symmetrization, the number of optimization parameters in
$\tilde{\Gamma}$ is reduced compared to the ones in $\Gamma$. Below we
identify the symmetry group $G$ in our problem.

First, permutation of Alice’s outcome can be counteracted by permutation of
Bob’s outcome, in a way that the sum of both outcomes remains constant.

$\displaystyle\mathrm{P}((A\oplus r)\oplus(B\oplus\bar{r})=i\cdot j\oplus
e|\alpha=i,\beta=j)$ $\displaystyle=$ $\displaystyle\mathrm{P}(A\oplus
B=i\cdot j\oplus e|\alpha=i,\beta=j),$ (26)

$\forall r\in[d]$. Second, permutation of Alice’s setting can be counteracted
by permutation of Bob’s outcome, conditioned on Bob’s input $j$ in the
following way,

$\displaystyle\sum_{i=0}^{d-1}\scalebox{0.95}{$\mathrm{P}(A\oplus B\oplus
s\cdot j=(i\oplus s)j\oplus e|\alpha=i\oplus s,\beta=j)$}$
$\displaystyle=\sum_{i=0}^{d-1}\mathrm{P}(A\oplus B=i\cdot j\oplus
e|\alpha=i,\beta=j),$ (27)

$\forall s\in[d]$. These two types of permutations generate a group $G$, the
action of which leaves $p(e|j)$ invariant for all $e$ and $j$. Importantly,
the resulting symmetrized matrix $\tilde{\Gamma}$ in Eq. (25) contains only
the probabilities $p(e|j)$, constant terms, and optimization variables, number
of which is reduced (See Eq. (90) in Appendix B for the exact form of
$\tilde{\Gamma}$). Note also that we normalize $\tilde{\Gamma}$ by the factor
of $d$ in our derivations instead of $|G|=d^{2}$.

As a next step, we use unitary transformation, $U$ to block-diagonalize the
matrix $\tilde{\Gamma}$. In general, diagonalizing such a large matrix is a
difficult task, however, $\tilde{\Gamma}$ itself has an easy-to-handle
$d\times d$ block structure (if we do not consider the first row and column),
which dictates the form of the unitary $U$, given explicitly in Eq. (93) in
Appendix B.

$\displaystyle\tilde{\Gamma}\stackrel{{\scriptstyle
U}}{{\longmapsto}}\bigoplus_{m=0}^{d+1}\Gamma_{m},$ (28)

where $\Gamma_{0}$ is a $(d+1)\times(d+1)$ constant positive matrix,
independent of $p(e|j)$ (see Eq. (98) in Appendix B) and $\Gamma_{m}$, for
$1\leq m\leq d+1$, are $d\times d$ Hermitian matrices. To be concise, we only
give the upper-triangular part of $\Gamma_{m}$ here,

$\Gamma_{m}=\begin{pmatrix}1&\nu_{m}^{1}&\dots&\nu_{m}^{d-1}&p_{m}^{0}&p_{m}^{1}\\\
&1&\ddots&\vdots&p_{m}^{0}&\omega^{m}p_{m}^{1}\\\
&&\ddots&\nu_{m}^{1}&\vdots&\vdots\\\ &&&1&p_{m}^{0}&\omega^{(d-1)\cdot
m}p_{m}^{1}\\\ &&&&1&0\\\ &&&&&1\end{pmatrix},$ (29)

where $p^{j}_{m}=\sum_{e=0}^{d-1}\omega^{m\cdot(1+e)}p(e|j)$ and $v_{m}^{i}$
are optimization variables.

We have, thus, reduced checking compatibility of $p(e|j)$ with the ML
principle to positivity of block-diagonal matrices:

$\Gamma\geq 0\quad\Leftrightarrow\quad\Gamma_{m}\geq 0,\quad 1\leq m\leq d+1.$
(30)

Now we can finally show that the inequalities in Eq. (3) derived from IC
principle, are necessary and sufficient for $\Gamma\geq 0$. First, we prove
that if $\Gamma_{m}\geq 0$, $\forall m$, then the condition in Eq. (3) must
hold. If we use the notation, where
$V_{m}=\sum_{i,i^{\prime}=0}^{d-1}\nu^{i\ominus
i^{\prime}}_{m}|i\rangle\langle i^{\prime}|$, with the convention that
$\nu^{0}_{m}=1$ and $\nu^{i-i^{\prime}}_{m}=(\nu^{i^{\prime}-i}_{m})^{*}$, and
$P_{m}$ is a $d\times 2$ matrix with the first column given by $p^{0}_{m}$ and
the second by the vector $\sum_{i=0}^{d-1}\omega^{m\cdot
i}p^{1}_{m}|i\rangle$, we can formulate the positivity of $\Gamma_{m}$ in
terms of the Schur complement:

$\Gamma_{m}=\begin{pmatrix}V_{m}&P_{m}\\\
P_{m}^{\dagger}&\mathbbm{1}\end{pmatrix}\geq
0\;\Leftrightarrow\;V_{m}-P_{m}P_{m}^{\dagger}\geq 0.$ (31)

The latter condition implies $1-|p_{m}^{0}|^{2}-|p_{m}^{1}|^{2}\geq 0$ from
the positivity of the diagonal elements. Inserting the values of $p_{m}^{0}$
and $p_{m}^{1}$ for $1\leq m\leq d-1$, we get the familiar expression,

$\displaystyle\left|\sum_{e=0}^{d-1}p(e|0)\omega^{e\cdot
m}\right|^{2}+\left|\sum_{e=0}^{d-1}p(e|1)\omega^{e\cdot m}\right|^{2}\leq 1.$
(32)

However, since these inequalities are symmetric under conjugation, we can
remove redundant ones by just considering
$m\in\\{1,\dots\left\lfloor\frac{d}{2}\right\rfloor\\}$. This finishes one
direction of the proof.

For the reverse direction, we need to show that if inequalities in Eq.(3)
hold, then we can always find $v^{i}_{m}$ in Eq. (29), such that
$\Gamma_{m}\geq 0$, or alternatively:

$\displaystyle V_{m}-P_{m}P_{m}^{\dagger}\geq 0.$ (33)

If we choose the variables of $V_{m}$ to be equal to the off-diagonal entries
of $P_{m}P_{m}^{\dagger}$,

$\nu_{m}^{i}=|p_{m}^{0}|^{2}+|p_{m}^{1}|^{2}\omega^{-i\cdot m},$ (34)

then the condition $\Gamma_{m}\geq 0$ indeed follows from
$|p_{m}^{0}|^{2}+|p_{m}^{1}|^{2}\leq 1$. ∎

Combining the two main results of this work, Results 1 and 2, we can directly
conclude that in the entire subspace of probabilities $p(e|j)$, the principle
of IC provides at least as good approximation of the set of quantum
correlations as the ML principle. On the other hand, results of Refs. [33, 45]
showed that there exist macroscopically local correlations
$\\{p(e|j)\\}_{e,j}$ that violate the IC principle. Hence, we can conclude the
following:

######  Result 3.

In the subspace of correlations, defined by the probabilities
$\\{p(e|j)\\}_{e,j}$, the principle of Information Causality is strictly
stronger than that of Macroscopic Locality.

Figure 2: The regions constrained by the IC and ML principles for
$p(0|0)=\mu,p(1|0)=p(2|0)=\frac{\eta}{2}$, and $p(e|0)=p(e|1)$ for all
$e\in[d]$, where $d=4$. ML $\&$ IC1 (dark blue) is the set constrained by both
the IC and ML principles as follows from Result 1. IC2 (light green) is an
alternative approximation of the set constrained by IC taken from Ref. [45].

In Fig. 2 we give a simple example of an inclusion of the region of
correlations constrained by IC in that of ML. We consider the assignment of
probabilities $p(0|0)=\mu,p(1|0)=p(2|0)=\frac{\eta}{2}$, and $p(e|0)=p(e|1)$
for all $e\in[d]$, where $d=4$. The set of correlations constrained by IC is
included in the intersection of the sets depicted by ML $\&$ IC1 and IC2 in
Fig. 2, while the correlations obeying the ML principle are exactly those
denoted by ML $\&$ IC1 in Fig. 2. One can notice two small regions that are
included in ML $\&$ IC1 but excluded from IC2. Examples of the parameter
values belonging to these regions are $\mu=\frac{2+3\sqrt{2}}{8}$,
$\eta=\frac{2-\sqrt{2}}{4}$ and $\mu=\frac{2-\sqrt{2}}{8}$,
$\eta=\frac{2-\sqrt{2}}{4}$.

Here IC2 is an alternative approximation of the set constrained by the IC
principle taken from Ref. [45]. In Ref. [45] one is only concerned with
optimally bounding the “guessing” probabilities $p(0|0)$ and $p(0|1)$, rather
than aiming at a tight approximation of the whole set of probabilities. To
that end, Ref. [45] uses Fano’s inequality [39] instead of Eq. (17). For
further information on this approach, see Discussions.

##  5 Discussions

In this work we consider the problem of bounding the set of quantum
correlations, observed in a Bell test, in the space of all nonsignaling
correlations. To this end, we provide a family of quantum Bell inequalities.
We derive our inequalities from the principle of Information Causality and
show that in the subspace of probabilities, defined by symmetries of these
inequalities, they provide necessary and sufficient conditions for the
principle of Macroscopic Locality. From these two results we conclude that in
the aforementioned subspace the principle of Information Causality is strictly
stronger than that of Macroscopic Locality.

Although our main results are tailored to the scenario in which Bob has binary
settings, the derived inequalities can be applied to arbitrary Bell scenarios.
For instance, one can adjust our inequalities to the scenarios with lower
number of outcomes by simply setting the probabilities of some outcomes to
$0$. On the other hand, by using the derived inequalities for each pair of
Bob’s inputs, one can bound correlations in the scenarios with an arbitrary
number of Bob’s settings.

The derivation of our main result uses concatenation protocol, while in the
recent paper of Ref. [45] it was claimed that all known results can be re-
derived and improved by replacing this protocol with communication over a
noisy classical channel. A natural question to ask is whether concatenation is
really necessary here, or could we have gone without it. It happens to be the
former since the results obtained in the current work differ qualitatively
from all the bounds published before and referenced in Ref. [45]. The idea of
Ref. [45] is to look at the concatenation procedure as a communication channel
taking the outputs of the topmost ($(n-1)$-th) layer on Alice’s side and
producing the message that Bob uses with his final box to obtain his guess.
The transition matrix of this channel is given by the probability distribution
of Bob’s guesses given Alice’s data. If a protocol involving concatenation
procedure leads to a violation of Information Causality, then there exists a
protocol, which also gives a violation, but involves only a single box and a
communication over a channel described by this transition matrix. However,
Ref. [45] uses an implicit assumption that the transition matrix is the same
for all Alice’s dits. This is the case for Refs. [27, 31, 33] but not for the
general protocol studied in the current work.

Our results can be generalized in many directions. One could use the
techniques of this paper to derive inequalities for the scenario in which Bob
has ternary inputs by extending the concatenation protocol from Ref. [31]. The
techniques used in the proof regarding ML principle can be employed to derive
quantum Bell inequalities displaying other symmetries. Moreover, a tighter
approximation of the entropy function will give tighter inequalities, which
might shed light on the long-standing problem of whether almost quantum
correlations can violate the IC principle [46].

Resolving this problem is hard for a few reasons. First, there are numerous
ways to employ nonsignaling correlations in the communication scenario of the
IC, which makes it difficult to decide whether a given nonsignaling box
violates the IC. Second, the set of almost quantum correlations is believed to
give a tight approximation to the quantum set, allowing for no room for
relaxations of the IC statement. In this case, one might need to consider
combinations of various nonsignaling boxes in the same communication protocol.
We believe that the approach of deriving quantum Bell inequalities is
advantageous over testing individual candidates of nonsignaling boxes as the
former is more systematic for high-dimensional correlations spaces.

## Acknowledgments

We thank Costantino Budroni and David Gross for fruitful discussions. We
acknowledge partial support by the Foundation for Polish Science (IRAP
project, ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth
Operational Programme). This research was made possible by funding from
QuantERA, an ERA-Net cofund in Quantum Technologies (www.quantera.eu), under
the project eDICT. M.P. acknowledges the support of NCN through grant SHENG
(2018/30/Q/ST2/00625). Funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy – Cluster of
Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 –
390534769

## APPENDIX

In this Appendix, we provide technical details regarding proofs and
derivations of the results stated in the main text.

One of the most useful identities, which we use in the proofs, are the sums
related to Dirichlet kernel [47]. See Ref. [48] (section 1.341) for more
exhaustive list.

$\displaystyle\sum_{k=1}^{n-1}\sin(x+ky)=$
$\displaystyle\sin{\left(x+\frac{n-1}{2}y\right)}\sin\left({\frac{ny}{2}}\right)\csc\left({\frac{y}{2}}\right).$
(35) $\displaystyle\sum_{k=1}^{n-1}\cos(x+ky)=$
$\displaystyle\cos{\left(x+\frac{n-1}{2}y\right)}\sin\left({\frac{ny}{2}}\right)\csc\left({\frac{y}{2}}\right).$
(36) $\displaystyle\sum_{k=1}^{n-1}(-1)^{k}\sin(x+ky)=$
$\displaystyle\sin{\left(x+\frac{n-1}{2}(y+\pi)\right)}\sin\left({\frac{n(y+\pi)}{2}}\right)\sec\left({\frac{y}{2}}\right).$
(37) $\displaystyle\sum_{k=1}^{2n-1}(-1)^{k}\cos(x+ky)=$
$\displaystyle\sin{\left(x+\frac{2n-1}{2}y\right)}\sin({ny})\sec\left({\frac{y}{2}}\right).$
(38)

We refer to these identities in several parts of this Appendix.

We also use the following notation in this Appendix: $\mathbbm{J}$ is $d\times
d$ matrix of all $1$’s, i.e., $\mathbbm{J}=\sum_{i,j=0}^{d-1}|i\rangle\langle
j|$. The matrix $S_{E}=\sum_{i=0}^{d-1}|i\rangle\langle\bar{i}\oplus E|$.
Generalized $d$-dimensional Pauli matrices are defined as follows,

$X^{m}=\leavevmode\nobreak\ \sum_{i=0}^{d-1}|i\rangle\langle i\ominus
m|\quad\mbox{ and }\quad Z^{m}=\leavevmode\nobreak\
\sum_{i=0}^{d-1}\omega^{i\cdot m}|i\rangle\langle i|.$ (39)

Generalized $d$-dimensional Hadamard is given by

$H=\frac{1}{\sqrt{d}}\sum_{i,j=0}^{d-1}\omega^{-i\cdot j}|i\rangle\langle j|.$
(40)

Finally, the following identities will prove useful,

$H^{\dagger}X^{m}H=Z^{m},\qquad S_{i}S_{d-1}=X^{i+1}\quad\mbox{ and }\quad
S_{d-1}^{\dagger}X^{m}S_{d-1}=X^{-m}.$ (41)

##  Appendix A Proof of Result 1

###  A.1 Proof of Eq. (13) and the even case

Here we prove the formula for $|Q^{(k)}(j)\rangle$ given in Eq. (13) for $d$
odd and its even counterpart:

$\displaystyle Q^{(k)}_{e}(j)$
$\displaystyle=\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-1}{2}}{\left(l^{k}_{i}(j)\omega^{e\cdot
i}+{l^{*}_{i}}^{k}(j)\omega^{-e\cdot i}\right)},\quad\forall e\in[d],\quad
d\;\text{odd},$ (42) $\displaystyle Q^{(k)}_{e}(j)$
$\displaystyle=\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-2}{2}}{\left(l^{k}_{i}(j)\omega^{e\cdot
i}+{l^{*}_{i}}^{k}(j)\omega^{-e\cdot
i}\right)}+\frac{1}{d}l^{k}_{\frac{d}{2}}(j)(-1)^{e},\quad\forall
e\in[d],\quad d\;\text{even},$ (43)

where

$l_{i}(j)=\sum_{e=0}^{d-1}p(e|j)\omega^{-e\cdot
i},\;i\in\left\\{1,\dots,\left\lfloor\frac{d}{2}\right\rfloor\right\\}.$ (44)

###### Proof.

We split the proof in two parts, first we prove the above expressions for $d$
odd, and then for $d$ even. It is clear that the proof is independent of the
choice of Bob’s setting $j$.

For odd $d$, we start by proving the base case of $k=1$.

$\displaystyle\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-1}{2}}\sum_{e^{\prime}=0}^{d-1}p(e^{\prime}|j)\left(\omega^{(e-e^{\prime})\cdot
i}+\omega^{(e^{\prime}-e)\cdot
i}\right)=\frac{1}{d}+\frac{1}{d}\sum_{e^{\prime}=0}^{d-1}p(e^{\prime}|j)\cdot
2\sum_{i=1}^{\frac{d-1}{2}}\cos\left(\frac{2\pi(e-e^{\prime})\cdot
i}{d}\right).$ (45)

We separately write out the last sum and use Eq. (36) to evaluate it,

$2\sum_{i=1}^{\frac{d-1}{2}}\cos\left(\frac{2\pi(e-e^{\prime})\cdot
i}{d}\right)=\frac{\sin\left(\frac{2\pi(e-e^{\prime})d}{2d}\right)}{\sin\left(\frac{2\pi(e-e^{\prime})}{2d}\right)}-1.$
(46)

In this expression, if $e=e^{\prime}$, the sum amounts to $d-1$, and if $e\neq
e^{\prime}$, to $-1$. Inserting these two cases in Eq. (45), we prove the base
case $Q^{(1)}_{e}(j)=p(e|j)$,

$\displaystyle\frac{1}{d}-\frac{1}{d}(1-p(e|j))+\frac{1}{d}p(e|j)(d-1)=p(e|j).$
(47)

Now we can prove the induction step by assuming the form of $Q^{(k)}_{e}(j)$
in Eq. (42) for some $k$, and using the general formula
$|Q^{(k+1)}(j)\rangle=M_{j}|Q^{(k)}(j)\rangle$ to calculate the $(k+1)$ case.

$\displaystyle\begin{split}Q^{(k+1)}_{e}(j)&=\sum_{e^{\prime}=0}^{d-1}Q^{(k)}_{e^{\prime}}(j)p(e\oplus\bar{e^{\prime}}|j)=\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-1}{2}}l^{k}_{i}(j)\left(\sum_{e^{\prime}=0}^{d-1}\omega^{e^{\prime}\cdot
i}p(e\oplus\bar{e^{\prime}}|j)\right)\\\
&+\frac{1}{d}\sum_{i=1}^{\frac{d-1}{2}}{l^{*}_{i}}^{k}(j)\left(\sum_{e^{\prime}=0}^{d-1}\omega^{-e^{\prime}\cdot
i}p(e\oplus\bar{e^{\prime}}|j)\right).\end{split}$ (48)

It should be clear that if we take factors $\omega^{e\cdot i}$ and
$\omega^{-e\cdot i}$ out of the sums over $e^{\prime}$ above, these sums will
become $l_{i}(j)$ and $l^{*}_{i}(j)$ respectively, since in the definition of
$l_{i}(j)$ in Eq. (44) we can always shift the index $e$ of the sum by some
constant $e^{\prime}$.

For even $d$, we repeat the procedure above and start with the base case of
$k=1$. This time we need to show that

$\displaystyle\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-2}{2}}\sum_{e^{\prime}=0}^{d-1}p(e^{\prime}|j)\left(\omega^{(e-e^{\prime})\cdot
i}+\omega^{(e^{\prime}-e)\cdot
i}\right)+\frac{1}{d}\sum_{e^{\prime}=0}^{d-1}(-1)^{(e-e^{\prime})}p(e^{\prime}|j)=p(e|j).$
(49)

We again use the Dirichlet identity from Eq. (36) to calculate the sum of
$\omega^{(e-e^{\prime})\cdot i}+\omega^{(e^{\prime}-e)\cdot i}$ over $i$ and
$e^{\prime}$.

$2\sum_{i=1}^{\frac{d-2}{2}}\cos\left(\frac{2\pi(e-e^{\prime})\cdot
i}{d}\right)=\frac{\sin\left(\frac{2\pi(e-e^{\prime})(d-1)}{2d}\right)}{\sin\left(\frac{2\pi(e-e^{\prime})}{2d}\right)}-1=\frac{\sin\left(\pi(e-e^{\prime})-\frac{\pi(e-e^{\prime})}{d}\right)}{\sin\left(\frac{\pi(e-e^{\prime})}{d}\right)}-1.$
(50)

This time the conclusions are different. If $e=e^{\prime}$, the sum is equal
to $d-2$. If $(e-e^{\prime})$ is even (except for $0$), then the expression in
Eq. (50) is equal to $-2$. If, on the other hand, $(e-e^{\prime})$ is odd, the
expression is equal to $0$. We can express the last findings in the following
way:

$\sum_{i=1}^{\frac{d-2}{2}}\sum_{e^{\prime}=0}^{d-1}p(e^{\prime}|j)\left(\omega^{(e-e^{\prime})\cdot
i}+\omega^{(e^{\prime}-e)\cdot i}\right)=(d-2)p(e|j)-\sum_{e^{\prime}\neq
e}(1+(-1)^{(e-e^{\prime})})p(e^{\prime}|j).$ (51)

The last sum on the left-hand side of Eq. (49) separates into the term
$p(e|j)$, corresponding to $e=e^{\prime}$ and the rest, which is
$\sum_{e^{\prime}\neq e}(-1)^{(e-e^{\prime})}p(e^{\prime}|j)$. It is clear
then, that summing all the terms up, leads to the proof of Eq. (49). Now, the
missing part of the proof is the induction step for the case of even $d$. We
again assume the form of $Q^{(k)}_{e}(j)$ and explicitly write the $k+1$ case
as in Eq. (48).

$\displaystyle\begin{split}Q^{(k+1)}_{e}(j)&=\frac{1}{d}+\frac{1}{d}\sum_{i=1}^{\frac{d-2}{2}}l^{k}_{i}(j)\left(\sum_{e^{\prime}=0}^{d-1}\omega^{e^{\prime}\cdot
i}p(e\oplus\bar{e^{\prime}}|j)\right)+\frac{1}{d}\sum_{i=1}^{\frac{d-2}{2}}{l^{*}_{i}}^{k}(j)\left(\sum_{e^{\prime}=0}^{d-1}\omega^{-e^{\prime}\cdot
i}p(e\oplus\bar{e^{\prime}}|j)\right)\\\
&+\frac{1}{d}l^{k}_{\frac{d}{2}}\sum_{e^{\prime}=0}^{d-1}(-1)^{e^{\prime}}p(e\oplus\bar{e^{\prime}}|j).\end{split}$
(52)

As in the case of odd $d$, one can easily see by taking the factors of
$\omega^{e\cdot i}$, $\omega^{-e\cdot i}$, and $(-1)^{e}$ out of the sums over
$e^{\prime}$ that these sums are equal to $l_{i}(j)$, $l^{*}_{i}(j)$, and
$l_{\frac{d}{2}}(j)$ respectively. ∎

###  A.2 Proof of Eq. (16)

Here we rigorously derive Eq. (16) as well as give the similar expression for
even $d$. We start with odd $d$.

$\displaystyle\begin{split}P(E|k)-\frac{1}{d}=&\sum_{e=0}^{d-1}Q^{k}_{e}(0)Q^{n-k}_{\bar{e}\oplus
E}(1)-\frac{1}{d}\\\
=&\frac{1}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-1}{2}}\sum_{e=0}^{d-1}{\left(\omega^{i_{1}e}l_{i_{1}}^{k}(0)+\omega^{-i_{1}e}l_{i_{1}}^{*k}(0)\right)}{\left(\omega^{i_{2}(\bar{e}\oplus
E)}l_{i_{2}}^{n-k}(1)+\omega^{-i_{2}(\bar{e}\oplus
E)}l_{i_{2}}^{*(n-k)}(1)\right)}\\\
=&\frac{4}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-1}{2}}\sum_{e=0}^{d-1}\left(\cos\left(\frac{2\pi
ei_{1}}{d}\right)m^{(k)}_{i_{1}}-\sin\left(\frac{2\pi
ei_{1}}{d}\right)n^{(k)}_{i_{1}}\right)\\\
&\cdot\left(\cos\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)m^{(n-k)}_{i_{2}}-\sin\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)n^{(n-k)}_{i_{2}}\right).\end{split}$ (53)

In Eq. (53) the expressions $m_{i_{1}}^{(k)}$ and $n_{i_{1}}^{(k)}$ denote the
real and imaginary parts of $l_{i1}^{k}(0)$. In the same way, we denote the
real and imaginary parts of $l_{i2}^{n-k}(1)$ by $m_{i_{2}}^{(n-k)}$ and
$n_{i_{2}}^{(n-k)}$, respectively. First we take a sum over $e$. Note that,
conveniently, all $l_{i}(j)$ are constant in $e$.

$\displaystyle\sum_{e=0}^{d-1}\cos\left(\frac{2\pi
ei_{1}}{d}\right)\cos\left(\frac{2\pi(\bar{e}\oplus E)i_{2}}{d}\right)$
$\displaystyle=\frac{d}{2}\cos\left(\frac{2\pi Ei}{d}\right),$ (54)
$\displaystyle\sum_{e=0}^{d-1}\cos\left(\frac{2\pi
ei_{1}}{d}\right)\sin\left(\frac{2\pi(\bar{e}\oplus E)i_{2}}{d}\right)$
$\displaystyle=\frac{d}{2}\sin\left(\frac{2\pi Ei}{d}\right),$ (55)
$\displaystyle\sum_{e=0}^{d-1}\sin\left(\frac{2\pi
ei_{1}}{d}\right)\cos\left(\frac{2\pi(\bar{e}\oplus E)i_{2}}{d}\right)$
$\displaystyle=\frac{d}{2}\sin\left(\frac{2\pi Ei}{d}\right),$ (56)
$\displaystyle\sum_{e=0}^{d-1}\sin\left(\frac{2\pi
ei_{1}}{d}\right)\sin\left(\frac{2\pi(\bar{e}\oplus E)i_{2}}{d}\right)$
$\displaystyle=-\frac{d}{2}\cos\left(\frac{2\pi Ei}{d}\right),$ (57)

where $i=i_{1}=i_{2}$ (See identities in Eqs. (35,36)). Putting all the values
in Eq. (53) we get,

$\displaystyle\begin{split}P(E|k)-\frac{1}{d}=&\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left(\cos\left({\frac{2\pi
Ei}{d}}\right)(m_{i}^{(k)}m_{i}^{(n-k)}-n_{i}^{(k)}n_{i}^{(n-k)})-\sin\left({\frac{2\pi
Ei}{d}}\right)(m_{i}^{(k)}n_{i}^{(n-k)}+n_{i}^{(k)}m_{i}^{(n-k)})\right)\\\
=&\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left(\cos\left({\frac{2\pi
Ei}{d}}\right)A_{i}-\sin\left({\frac{2\pi
Ei}{d}}\right)B_{i}\right).\end{split}$ (58)

Even $d$:

$\displaystyle P(E|k)-\frac{1}{d}$
$\displaystyle=\sum_{e=0}^{d-1}Q^{k}_{e}(0)Q^{n-k}_{\bar{e}\oplus
E}(1)-\frac{1}{d}=\frac{(-1)^{E}}{d}l_{\frac{d}{2}}^{k}(0)l_{\frac{d}{2}}^{n-k}(1)$
$\displaystyle+\frac{4}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-2}{2}}\sum_{e=0}^{d-1}\left(\cos\left(\frac{2\pi
ei_{1}}{d}\right)m^{(k)}_{i_{1}}-\sin\left(\frac{2\pi
ei_{1}}{d}\right)n^{(k)}_{i_{1}}\right)$
$\displaystyle\cdot\left(\cos\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)m^{(n-k)}_{i_{2}}-\sin\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)n^{(n-k)}_{i_{2}}\right)$
$\displaystyle+\frac{2}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-2}{2}}\sum_{e=0}^{d-1}(-1)^{\bar{e}\oplus
E}\left(\cos\left(\frac{2\pi
ei_{1}}{d}\right)m^{(k)}_{i_{1}}-\sin\left(\frac{2\pi
ei_{1}}{d}\right)n^{(k)}_{i_{1}}\right)l_{\frac{d}{2}}^{n-k}(1)$ (59)
$\displaystyle+\frac{2}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-2}{2}}\sum_{e=0}^{d-1}(-1)^{e}l_{\frac{d}{2}}^{k}(0)\left(\cos\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)m^{(n-k)}_{i_{2}}-\sin\left(\frac{2\pi(\bar{e}\oplus
E)i_{2}}{d}\right)n^{(n-k)}_{i_{2}}\right).$ (60)

We first show that the summands in Eqs. (59,60) amount to $0$. Using Eqs.
(37,38), we show that

$\displaystyle\sum_{e=0}^{d-1}(-1)^{\bar{e}}\cos\left(\frac{2\pi
ei_{1}}{d}\right)=$ $\displaystyle\sin\left(\frac{(d-1)\pi
i_{1}}{d}\right)\sin(\pi i_{1})\sec\left({\frac{\pi i_{1}}{d}}\right)=0.$ (61)
$\displaystyle\sum_{e=0}^{d-1}(-1)^{\bar{e}}\sin\left(\frac{2\pi
ei_{1}}{d}\right)=$ $\displaystyle\sin\left(\frac{(d-1)(2\pi
i_{1}+d\pi)}{2d}\right)\sin\left(\pi(i_{1}+\frac{d}{2})\right)\sec\left({\frac{\pi
i_{1}}{d}}\right)=0.$ (62)

Similarly, the summand in Eq. (60) amount to $0$ too. Hence, the expression
for $P(E|k)$ becomes

$\displaystyle
P(E|k)-\frac{1}{d}=\frac{2}{d}\sum_{i=1}^{\frac{d-2}{2}}\left(\cos\left({\frac{2\pi
Ei}{d}}\right)A_{i}-\sin\left({\frac{2\pi
Ei}{d}}\right)B_{i}\right)+\frac{(-1)^{E}}{d}l_{\frac{d}{2}}^{k}(0)l_{\frac{d}{2}}^{n-k}(1).$
(63)

###  A.3 Proof of inequality in Eq. (17)

First, we repeat claim: For two random variables $a$ and $b$ taking values in
$[d]$, and $a$ being uniformly distributed, define variable $e$, such that
$\mathrm{P}(b=a\oplus e)=1$. Then, the following inequality holds:

$I(a;b)\geq\ln d-H(e).$ (64)

###### Proof.

Let $\\{p(a=i,b=j)\\}_{i,j}$ be a joint probability distribution of $a$ and
$b$ and $\\{p_{a}(a=i)\\}_{i}$ and $\\{p_{b}(b=j)\\}_{j}$ the corresponding
marginal distributions of $a$ and $b$ respectively. In the following we use
the shorthand notation $p(i,j)\equiv p(a=i,b=j)$, and $p_{a}(i)\equiv
p_{a}(a=i)$, $p_{b}(j)\equiv p_{b}(b=j)$. Since $a$ is uniformly distributed,
$p_{a}(i)=\frac{1}{d},\forall i\in[d]$. We start by writing explicitly the
form of $I(a;b)$:

$I(a;b)=\sum_{i=0}^{d-1}\sum_{j=0}^{d-1}p(i,j)\ln\frac{p(i,j)}{p_{a}(i)p_{b}(j)}=\sum_{k=0}^{d-1}\sum_{i=0}^{d-1}p(i,i\oplus
k)\ln\frac{p(i,i\oplus k)}{p_{a}(i)p_{b}(i\oplus k)},$ (65)

Now we use the log sum inequality (see e.g., Ref. [49]) for
$c_{i},d_{i}\in\mathds{R}$:

$\sum_{i}c_{i}\ln\frac{c_{i}}{d_{i}}\geq\left(\sum_{i}c_{i}\right)\ln\frac{\sum_{i}c_{i}}{\sum_{i}d_{i}},$
(66)

to obtain the following:

$I(a;b)\geq\sum_{k=0}^{d-1}\left(\sum_{i=0}^{d-1}p(i,i\oplus
k)\right)\ln\frac{\sum_{i=0}^{d-1}p(i,i\oplus
k)}{\sum_{i=0}^{d-1}p_{a}(i)p_{b}(i\oplus
k)}=\sum_{k=0}^{d-1}{p}(e=k)\ln\frac{{p}(e=k)}{\frac{1}{d}},$ (67)

where we used the fact that $a$ is uniformly distributed. The probabilities
${p}(e=k)\equiv\sum_{i=0}^{d-1}p(i,i\oplus k)$ obviously form a distribution
for $e$, corresponding to $\\{p(i,j)\\}_{i,j}$. Hence,
$H(e)=-\sum_{k=0}^{d-1}{p}(e=k)\ln{p}(e=k)$, which proves the inequality. ∎

###  A.4 Approximation of entropy with polynomial of degree 2

Given a random variable $a$ taking values in $[d]$, and $\\{p(a=i)\\}_{i}$ a
distribution of $a$, there exist $\kappa>0$, such that the following
inequality holds:

$H(a)=-\sum_{i=0}^{d-1}p(a=i)\ln p(a=i)\leq\ln
d-\kappa\sum_{i=0}^{d-1}\left(p(a=i)-\frac{1}{d}\right)^{2},$ (68)

where it is sufficient to take $\kappa=\frac{d^{2}(\ln d-1)+d}{(d-1)^{2}}$.

###### Proof.

Let $x_{i}=dp(a=i)-1,\forall i\in[d]$. Let us write the expression for
$p(a=i)\ln p(a=i)$ in terms of $x_{i}$:

$p(a=i)\ln
p(a=i)=\left(\frac{x_{i}}{d}+\frac{1}{d}\right)\ln\left(\frac{x_{i}}{d}+\frac{1}{d}\right)=\frac{1}{d}(x_{i}+1)(-\ln
d+\ln(x_{i}+1)).$ (69)

Since $\sum_{i=0}^{d-1}p(a=i)=1$, we have that $\sum_{i=0}^{d-1}x_{i}=0$.
Also, due to positivity of probability, we have that $-1\leq x_{i}\leq
d-1,\forall i\in[d]$. On the other hand, the function
$(x_{i}+1)\ln(x_{i}+1)-x_{i}$ behaves like $\frac{x_{i}^{2}}{2}$ around
$x_{i}=0$. Hence, we can find $\kappa^{\prime}$, such that
$(x_{i}+1)\ln(x_{i}+1)-x_{i}\geq\kappa^{\prime}x_{i}^{2}$ for the domain of
$x_{i}$. The sufficient coefficient $\kappa^{\prime}$ can be determined by
taking $x_{i}=(d-1)$, which gives:

$\kappa^{\prime}=\frac{d\ln d-d+1}{(d-1)^{2}}.$ (70)

We finish the proof by substituting this bound in Eq. (69) and summing over
$i$ (and remembering that $\sum_{i=0}^{d-1}x_{i}=0$):

$\displaystyle\begin{split}\sum_{i=0}^{d-1}p(a=i)\ln p(a=i)&\geq-\ln
d+\frac{1}{d}(1-\ln
d)\sum_{i=0}^{d-1}x_{i}+\frac{k^{\prime}}{d}\sum_{i=0}^{d-1}x_{i}^{2}\\\
&\geq-\ln d+\frac{d^{2}(\ln
d-1)+d}{(d-1)^{2}}\sum_{i=0}^{d-1}\left(p(a=i)-\frac{1}{d}\right)^{2}.\end{split}$
(71)

∎

###  A.5 Proof of Eq. (21)

The techniques employed in this proof are very similar to the ones used in Eq.
(16) and rely on trigonometric identities in Eq. (35,36). We consider the
simpler case first, the case of odd $d$.

$\displaystyle\sum_{E=0}^{d-1}\left(P(E|k)-\frac{1}{d}\right)^{2}=$
$\displaystyle\frac{4}{d^{2}}\sum_{i_{1},i_{2}=1}^{\frac{d-1}{2}}\sum_{E=0}^{d-1}\left(\cos\left({\frac{2\pi
Ei_{1}}{d}}\right)A_{i_{1}}-\sin\left({\frac{2\pi
Ei_{1}}{d}}\right)B_{i_{1}}\right)$
$\displaystyle\cdot\left(\cos\left({\frac{2\pi
Ei_{2}}{d}}\right)A_{i_{2}}-\sin\left({\frac{2\pi
Ei_{2}}{d}}\right)B_{i_{2}}\right)$ $\displaystyle=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left(A_{i}^{2}+B_{i}^{2}\right)$
(72) $\displaystyle=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left((m_{i}^{(k)}m_{i}^{(n-k)})^{2}+(n_{i}^{(k)}n_{i}^{(n-k)})^{2}+(m_{i}^{(k)}n_{i}^{(n-k)})^{2}+(n_{i}^{(k)}m_{i}^{(n-k)})^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left((m_{i}^{(k)})^{2}+(n_{i}^{(k)})^{2}\right)\left((m_{i}^{(n-k)})^{2}+(n_{i}^{(n-k)})^{2}\right)=\frac{2}{d}\sum_{i=1}^{\frac{d-1}{2}}\left|l_{i}(0)\right|^{2k}\left|l_{i}(1)\right|^{2(n-k)}.$

Next, we give the expression for even $d$. Similarly to the odd $d$ case, all
the cross terms sum to $0$, but here we have an additional term, due to
$\frac{(-1)^{E}}{d}l_{\frac{d}{2}}^{k}(0)l_{\frac{d}{2}}^{n-k}(1)$ in Eq.
(63).

$\displaystyle\sum_{E=0}^{d-1}\left(P(E|k)-\frac{1}{d}\right)^{2}=\frac{2}{d}\sum_{i=1}^{\frac{d-2}{2}}\left|l_{i}(0)\right|^{2k}\left|l_{i}(1)\right|^{2(n-k)}+\frac{1}{d}\left(l_{\frac{d}{2}}(0)\right)^{2k}\left(l_{\frac{d}{2}}(1)\right)^{2(n-k)}.$
(73)

The $\frac{d}{2}$ term has a different factor from the rest. However, this
does not make any difference, since we are only interested in the bounds in
the asymptotic limit of $n\rightarrow\infty$, and the summation over $k$ has
the same form as for the rest of the terms
$i\in\left\\{1,2,\dots,\frac{d-2}{2}\right\\}$:

$\displaystyle\sum_{k=0}^{n}\binom{n}{k}\left(l_{\frac{d}{2}}(0)\right)^{2k}\left(l_{\frac{d}{2}}(1)\right)^{2(n-k)}=\left(\left(l_{\frac{d}{2}}(0)\right)^{2}+\left(l_{\frac{d}{2}}(1)\right)^{2}\right)^{n}.$
(74)

##  Appendix B Proof of Result 2

This section of the Appendix includes technical derivation of the proof of
Result 2. First part is devoted to identifying the set $\mathrm{Q}_{1}$ from
Ref. [32] and to precise description of the corresponding moment matrix
$\Gamma$. In the second part, we use the symmetries of the derived
inequalities to transform $\Gamma$ to $\tilde{\Gamma}$ and then block-
diagonalize it.

###  B.1 Macroscopic Locality and the corresponding $\Gamma$ matrix

Ref. [32] defines a hierarchy, often referred to as the NPA hierarchy, of
necessary conditions for the observed correlations
$\\{\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)\\}_{k,l,i,j}$ to have a quantum-
mechanical realization in terms of positive operator-valued measure (POVM)
elements of Alice $\\{\mathcal{A}^{i}_{k}\\}_{i,k}$, and Bob
$\\{\mathcal{B}^{j}_{l}\\}_{j,l}$ and a bipartite quantum state $\rho_{AB}$.
This hierarchy is also sufficient for the systems of finite dimension. For
every level of this hierarchy, the condition is expressed in terms of the so-
called moment matrix $\Gamma$, in which some of the terms are specified by
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, and which is required to be positive
semidefinite. Generally, to define $\Gamma$, one considers a set of operators
$\mathcal{O}$, which includes the POVM elements
$\\{\mathcal{A}^{i}_{k}\\}_{i,k}$, $\\{\mathcal{B}^{j}_{l}\\}_{j,l}$ and their
products. Fixing an order of operators $\mathcal{O}$, one constructs a moment
matrix $\Gamma$, such that an element on $i$-th row and $j$-th column of
$\Gamma$ is associated to the expectation value of the product of $i$-th and
$j$-th operators in $\mathcal{O}$.

In the first level of the hierarchy of Ref. [32], which is equivalent to
Macroscopic Locality, applied to our problem, the set of operators
$\mathcal{O}$ is simply

$\mathcal{O}=\\{\mathds{1}\\}\cup\\{\mathcal{A}^{i}_{0},\dots,\mathcal{A}^{i}_{d-1}\\}_{i=0}^{d-1}\cup\\{\mathcal{B}^{j}_{0},\dots,\mathcal{B}^{j}_{d-1}\\}_{j=0}^{1},$
(75)

where $\mathds{1}$ is the identity operator defined in the same space as the
POVM elements $\\{\mathcal{A}^{i}_{k}\\}_{i,k}$,
$\\{\mathcal{B}^{j}_{l}\\}_{j,l}$. It is most convenient to identify the
elements of matrix $\Gamma$ by the corresponding operators, i.e., if
$\mathcal{O}_{i}$, $\mathcal{O}_{j}$ are the $i$-th and $j$-th elements of the
set $\mathcal{O}$, then the element of the matrix $\Gamma$ on $i$-th row and
$j$-th column we denote as $\Gamma(\mathcal{O}_{i},\mathcal{O}_{j})$. For the
set of operators $\mathcal{O}$ in Eq. (75), the square matrix $\Gamma$ is
$(d+1)^{2}\times(d+1)^{2}$.

Some of the terms of $\Gamma$ are fixed by
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, in particular

$\begin{split}\Gamma(\mathds{1},\mathds{1})&=1,\\\
\Gamma(\mathds{1},\mathcal{A}^{i}_{k})&=\Gamma(\mathcal{A}^{i}_{k},\mathcal{A}^{i}_{k})=\mathrm{P}(A=k|\alpha=i),\quad\forall
i,k\in[d],\\\
\Gamma(\mathds{1},\mathcal{B}^{j}_{l})&=\Gamma(\mathcal{B}^{j}_{l},\mathcal{B}^{j}_{l})=\mathrm{P}(B=l|\beta=j),\quad\forall
l\in[d],j\in[2],\\\
\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{A}^{i}_{k^{\prime}})&=0,\quad\forall
k^{\prime}\neq k,\forall i\in[d],\\\
\Gamma(\mathcal{B}^{j}_{l}\,,\mathcal{B}^{j}_{l^{\prime}})&=0,\quad\forall
l^{\prime}\neq l,\forall j\in[2],\\\
\Gamma(\mathcal{A}^{i}_{k}\,,B^{j}_{l})&=\mathrm{P}(A=k,B=l|\alpha=i,\beta=j),\quad\forall
i,k,l\in[d],j\in[2],\end{split}$ (76)

while the rest, namely
$\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{A}^{i^{\prime}}_{k^{\prime}})$, for
$i^{\prime}\neq i$, and
$\Gamma(\mathcal{B}^{j}_{l}\,,\mathcal{B}^{j^{\prime}}_{l^{\prime}})$ for
$j^{\prime}\neq j$, are the variables of optimization. If one can find an
assignment to all
$\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{A}^{i^{\prime}}_{k^{\prime}})$ and
$\Gamma(\mathcal{B}^{j}_{l}\,,\mathcal{B}^{j^{\prime}}_{l^{\prime}})$ such
that $\Gamma\geq 0$, then the correlations
$\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$ satisfy the Macroscopic Locality
principle [28].

In the $d2dd$ scenario, which we consider in this paper, the moment matrix
$\Gamma$ for the first level of NPA hierarchy can be written as

$\Gamma=\begin{pmatrix}1&\langle P_{A}^{0}|&\dots&\langle P_{A}^{d-1}|&\langle
P_{B}^{0}|&\langle P_{B}^{1}|\\\
|P_{A}^{0}\rangle&V^{0,0}&\dots&V^{0,d-1}&P^{0,0}&P^{0,1}\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
|P_{A}^{d-1}\rangle&(V^{0,d-1})^{T}&\dots&V^{d-1,d-1}&P^{d-1,0}&P^{d-1,1}\\\
|P_{B}^{0}\rangle&(P^{0,0})^{T}&\dots&(P^{d-1,0})^{T}&W^{0,0}&W^{0,1}\\\
|P_{B}^{1}\rangle&(P^{0,1})^{T}&\dots&(P^{d-1,1})^{T}&(W^{0,1})^{T}&W^{1,1}\end{pmatrix},$
(77)

where vectors $|P^{i}_{A}\rangle$, $|P^{j}_{B}\rangle\in\mathds{R}^{d}$ are
defined as

$\begin{split}|P^{i}_{A}\rangle&=\sum_{k=0}^{d-1}\mathrm{P}(A=k|\alpha=i)\,|k\rangle,\quad\forall
i\in[d],\qquad|P^{j}_{B}\rangle=\sum_{l=0}^{d-1}\mathrm{P}(B=l|\beta=j)\,|l\rangle,\quad\forall
j\in[2].\end{split}$ (78)

The matrices $V^{i,i^{\prime}},P^{i,j}$, and $W^{j,j^{\prime}}$, which are all
$d\times d$, are defined below

$\begin{split}P^{i,i^{\prime}}&=\sum_{k,l}^{d-1}\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)|k\rangle\langle
l|,\quad\forall i\in[d],j\in[2],\\\
V^{i,i^{\prime}}&=\sum_{k,k^{\prime}=0}^{d-1}v^{i,i^{\prime}}_{k,k^{\prime}}|k\rangle\langle
k^{\prime}|,\quad\forall i<i^{\prime},\qquad
V^{i,i}=\sum_{k=0}^{d-1}\mathrm{P}(A=k|\alpha=i)|k\rangle\langle
k|,\quad\forall i\in[d],\\\
W^{0,1}&=\sum_{l,l^{\prime}}^{d-1}w_{l,l^{\prime}}|l\rangle\langle
l^{\prime}|,\quad
W^{j,j}=\sum_{l=0}^{d-1}\mathrm{P}(B=l|\beta=j)|l\rangle\langle
l|,\quad\forall j\in[2].\end{split}$ (79)

In the above formulas, we denoted the variables of optimization as
$v^{i,i^{\prime}}_{k,k^{\prime}}$ and $w_{l,l^{\prime}}$, which correspond to
the following elements of matrix $\Gamma$

$v^{i,i^{\prime}}_{k,k^{\prime}}=\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{A}^{i^{\prime}}_{k^{\prime}}),\quad
w_{l,l^{\prime}}=\Gamma(\mathcal{B}^{0}_{l}\,,\mathcal{B}^{1}_{l^{\prime}}),\quad\forall
i,i^{\prime},k,k^{\prime},l,l^{\prime}\in[d],\;i<i^{\prime}.$ (80)

Due to the relations $\sum_{k=0}^{d-1}\mathcal{A}^{i}_{k}=\mathbbm{1}$,
$\forall i\in[d]$, $\sum_{l=0}^{d-1}\mathcal{B}^{j}_{l}=\mathbbm{1}$, $\forall
j\in[2]$ on POVM elements there are further constraints on
$v^{i,i^{\prime}}_{k,k^{\prime}}$ and $w_{l,l^{\prime}}$, namely

$\begin{split}\sum_{k=0}^{d-1}v^{i,i^{\prime}}_{k,k^{\prime}}&=\mathrm{P}(A=k^{\prime}|\alpha=i^{\prime}),\quad\sum_{k^{\prime}=0}^{d-1}v^{i,i^{\prime}}_{k,k^{\prime}}=\mathrm{P}(A=k|\alpha=i),\quad\forall
i,i^{\prime},k,k^{\prime}\in[d],\;i<i^{\prime}.\\\
\sum_{l=0}^{d-1}w_{l,l^{\prime}}&=\mathrm{P}(B=l^{\prime}|\beta=1),\quad\sum_{l^{\prime}=0}^{d-1}w_{l,l^{\prime}}=\mathrm{P}(B=l|\beta=0),\quad\forall
l,l^{\prime}\in[d].\end{split}$ (81)

We are interested in characterizing the set of ML correlations in the space of
probabilities $\\{p(e|j)\\}_{e,j}$ from Eq. (2). Here, for the convenience we
restate the form of these probabilities,

$p(e|j)=\frac{1}{d}\sum_{i=0}^{d-1}\mathrm{P}(A\oplus B=i\cdot j\oplus
e|\alpha=i,\beta=j),\quad\forall e\in[d],j\in[2].$ (82)

Since the probabilities $\\{p(e|j)\\}_{e,j}$ form a subspace of the entire
space of correlations $\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)$, we can reduce
the size of the problem by considering the symmetries present in
$\\{p(e|j)\\}_{e,j}$.

###  B.2 Symmetrization and block-diagonalization of $\Gamma$

In our proof, we consider the cyclic group of permutations and not the whole
symmetric group $S_{d}$. Let $\pi_{A}$ denote the generating element of cyclic
permutation of outcomes of Alice, and similarly $\pi_{B}$ for permutation of
outcomes of Bob, and $\pi_{\alpha}$ for permutation of settings of Alice. We
define the action of these permutations on POVM elements of Alice and Bob as
follows,

$\pi_{A}^{r}.\mathcal{A}^{i}_{k}=\mathcal{A}^{i}_{k\oplus
r},\quad\pi^{r}_{B}.\mathcal{B}^{j}_{l}=\mathcal{B}^{j}_{l\oplus
r},\quad\pi^{r}_{\alpha}.\mathcal{A}^{i}_{k}=\mathcal{A}^{i\oplus
r}_{k},\quad\forall r,i,k,l\in[d],j\in[2],$ (83)

where $\pi_{A}^{r}$ is the $r$-th power of the permutation $\pi_{A}$, and the
same notation is used for $\pi_{B}$ and $\pi_{\alpha}$. The order of each of
these elements is $d$, hence the inverse of e.g., $\pi_{A}^{r}$ is
$\pi_{A}^{(d-r)}$. Finally, let $\pi_{AB}=\pi_{A}\cdot\pi_{B}^{-1}$ be the
composition of permutation of outcomes of Alice with the inverse permutation
of outcomes of Bob, and $\pi_{\alpha B}=\pi_{\alpha}\cdot\pi^{j}_{B}$ be the
composition of permutation of Alice’s settings combined with permutation of
Bob’s outputs, conditioned on $j$. We can now define the commutative symmetry
group

$G=\langle\\{\pi_{AB},\pi_{\alpha B}\\}|\pi_{AB}\cdot\pi_{\alpha
B}=\pi_{\alpha B}\cdot\pi_{AB}\rangle.$ (84)

The action of the symmetry group $G$ can be extended to elements of the moment
matrix $\Gamma$, since we enumerate its rows and columns by the POVM elements.
More precisely, we write

$\begin{split}(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathds{1},\mathds{1})&=\Gamma(\mathds{1},\mathds{1}),\quad\forall
r,s\in[d],\\\ (\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathds{1},\mathcal{A}^{i}_{k})&=\Gamma(\mathds{1},\mathcal{A}^{i\oplus
s}_{k\oplus r}),\quad\forall r,s,i,k\in[d],\\\
(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathds{1},\mathcal{B}^{j}_{l})&=\Gamma(\mathds{1},\mathcal{B}^{j}_{l\ominus
r\oplus s\cdot j}),\quad\forall r,s,l\in[d],j\in[2],\\\
(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{A}^{i^{\prime}}_{k^{\prime}})&=\Gamma(\mathcal{A}^{i\oplus
s}_{k\oplus r}\,,\mathcal{A}^{i^{\prime}\oplus s}_{k^{\prime}\oplus
r}),\quad\forall r,s,i,i^{\prime},k,k^{\prime}\in[d],\\\
(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathcal{B}^{j}_{l}\,,\mathcal{B}^{j^{\prime}}_{l^{\prime}})&=\Gamma(\mathcal{B}^{j}_{l\ominus
r\oplus s\cdot j}\,,\mathcal{B}^{j^{\prime}}_{l^{\prime}\ominus r\oplus s\cdot
j^{\prime}}),\quad\forall r,s,l,l^{\prime}\in[d],j,j^{\prime}\in[2],\\\
(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\Gamma(\mathcal{A}^{i}_{k}\,,\mathcal{B}^{j}_{l})&=\Gamma(\mathcal{A}^{i\oplus
s}_{k\oplus r}\,,\mathcal{B}^{j}_{l\ominus r\oplus s\cdot j}),\quad\forall
r,s,i,k,l\in[d],j\in[2].\end{split}$ (85)

The notation $(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha B}).\Gamma$ should be
understood as the action of $(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha B})$ on each
element of the matrix $\Gamma$, as defined by Eq. (85). Since we already
identified probabilities $\mathrm{P}(A=i,B=j|\alpha=k,\beta=l)$ with elements
$\Gamma(\mathcal{A}^{k}_{i},\mathcal{B}^{l}_{j})$ of matrix $\Gamma$, the
action of $G$ extends to the former in the obvious way:

$(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha
B}).\mathrm{P}(A=k,B=l|\alpha=i,\beta=j)=\mathrm{P}(A=k\oplus r,B=l\ominus
r\oplus s\cdot j|\alpha=i\oplus s,\beta=j),$ (86)

$\forall r,s,i,k,l\in[d],j\in[2].$ Form the above Eq. it must be clear that
$G$ preserves the probabilities $p(e|j)$ in Eq. (82).

In our proof, we define the following matrix

$\tilde{\Gamma}=\frac{1}{d}\sum_{r,s=0}^{d-1}(\pi_{AB}^{r}\cdot\pi^{s}_{\alpha}).\Gamma,$
(87)

and as follows from Ref. [44], the condition of $\tilde{\Gamma}\geq 0$ is
equivalent to the condition $\Gamma\geq 0$ if one is concerned with bounding
the space of probabilities $\\{p(e|j)\\}_{e,j}$ in Eq. (82).

In Eq. (85) we determined what happens to the entries of $\Gamma$ matrix when
we apply the permutation from $G$. These relations can be used to determine
the orbits corresponding to each element of the moment matrix, and
subsequently to find the orbit mean values, when we sum over $r$ and $s$ in
Eq. (87).

$\begin{split}\sum_{r,s=0}^{d-1}\Gamma(\mathds{1},\mathds{1})&=d^{2},\quad\sum_{r,s=0}^{d-1}\Gamma(\mathcal{B}^{0}_{l\ominus
r}\,,\mathcal{B}^{1}_{l^{\prime}\ominus r\oplus s})=1,\quad\forall
l,l^{\prime}\in[d],\\\ \sum_{r,s=0}^{d-1}\Gamma(\mathcal{A}^{i\oplus
s}_{k\oplus r}\,,\mathcal{A}^{i^{\prime}\oplus s}_{k^{\prime}\oplus
r})&=\sum_{r,s=0}^{d-1}v^{s,i^{\prime}\ominus i\oplus s}_{r,k^{\prime}\ominus
k\oplus r},\quad\forall i<i^{\prime},\;i,k,k^{\prime}\in[d],\\\
\sum_{r,s=0}^{d-1}\Gamma(\mathds{1},\mathcal{A}^{i\oplus s}_{k\oplus
r})&=d,\quad\sum_{r,s=0}^{d-1}\Gamma(\mathcal{A}^{i\oplus s}_{k\oplus
r}\,,\mathcal{A}^{i\oplus s}_{k\oplus r})=d,\quad\forall i,k\in[d],\\\
\sum_{r,s=0}^{d-1}\Gamma(\mathds{1},\mathcal{B}^{j}_{l\ominus r\oplus s\cdot
j})&=d,\quad\sum_{r,s=0}^{d-1}\Gamma(\mathcal{B}^{j}_{l\ominus r\oplus s\cdot
j}\,,\mathcal{B}^{j}_{l\ominus r\oplus s\cdot j})=d,\quad\forall
l\in[d],j\in[2],\end{split}$ (88)

and most importantly,

$\sum_{r,s=0}^{d-1}\Gamma(\mathcal{A}^{i\oplus s}_{k\oplus
r}\,,\mathcal{B}^{j}_{l\ominus r\oplus s\cdot
j})=\sum_{r,s=0}^{d-1}\mathrm{P}(A=r,B=l\oplus k\ominus r\oplus
j\cdot(s\ominus i)|\alpha=s,\beta=j)=dp(l\oplus k\ominus i\cdot j|j),$ (89)

where the last Eq. is valid for all $i,k,l\in[d],j\in[2]$.

Similar to the way we defined $\Gamma$ in Eq. (77), we can define
$\tilde{\Gamma}$ in terms of $d\times d$ blocks. Using the notation
$|+\rangle=\sum_{i=0}^{d-1}|i\rangle$, we can express the new matrix as
follows

$\tilde{\Gamma}=\begin{pmatrix}d&\langle+|&\langle+|&\dots&\langle+|&\langle+|&\langle+|\\\
|+\rangle&\mathbbm{1}&\tilde{V}^{1}&\dots&\tilde{V}^{d-1}&\tilde{P}^{0,0}&\tilde{P}^{0,1}\\\
|+\rangle&(\tilde{V}^{1})^{\text{T}}&\mathbbm{1}&\dots&\tilde{V}^{d-2}&\tilde{P}^{1,0}&\tilde{P}^{1,1}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
|+\rangle&(\tilde{V}^{d-1})^{T}&(\tilde{V}^{d-2})^{T}&\dots&\mathbbm{1}&\tilde{P}^{d-1,0}&\tilde{P}^{d-1,1}\\\
|+\rangle&(\tilde{P}^{0,0})^{T}&(\tilde{P}^{1,0})^{T}&\dots&(\tilde{P}^{d-1,0})^{T}&\mathbbm{1}&\mathbbm{J}/d\\\
|+\rangle&(\tilde{P}^{0,1})^{T}&(\tilde{P}^{1,1})^{T}&\dots&(\tilde{P}^{d-1,1})^{T}&\mathbbm{J}/d&\mathbbm{1}\end{pmatrix}.$
(90)

The sub-matrices $\tilde{V}^{\Delta i}$, where $1\leq\Delta i\leq d-1$ are
defined as follows,

$\tilde{V}^{\Delta i}=\sum_{\Delta k}\tilde{v}^{\Delta i}_{\Delta
k}{X}^{\Delta k},\quad\mbox{ where }\quad\tilde{v}^{\Delta i}_{\Delta
k}=\frac{1}{d}\sum_{r,s=0}^{d-1}v^{s,\Delta i\oplus s}_{r,\Delta k\oplus r}.$
(91)

Finally, we consider $\tilde{P}^{i,j}$.

$\tilde{P}^{i,j}=\sum_{e=0}^{d-1}S_{(e+i\cdot j)}p(e|j),$ (92)

where we remind the reader that
$S_{E}=\sum_{k=0}^{d-1}|k\rangle\langle\bar{k}\oplus E|$. In order to further
simplify $\tilde{\Gamma}$, we apply a unitary transformation $U$, specified
below.

$\displaystyle\Gamma^{\prime}=U^{\dagger}\tilde{\Gamma}U$
$\displaystyle=\begin{pmatrix}1\\\ &H^{\dagger}\\\ &&\ddots\\\
&&&H^{\dagger}\\\ &&&&H^{\dagger}S^{\dagger}_{d-1}\\\
&&&&&H^{\dagger}S^{\dagger}_{d-1}\end{pmatrix}\cdot\tilde{\Gamma}\cdot\begin{pmatrix}1\\\
&H\\\ &&\ddots\\\ &&&H\\\ &&&&S_{d-1}H\\\ &&&&&S_{d-1}H\end{pmatrix}$ (93)
$\displaystyle=$ $\displaystyle\begin{pmatrix}d&\sqrt{d}\langle
0|&\dots&\sqrt{d}\langle 0|&\sqrt{d}\langle 0|&\sqrt{d}\langle 0|\\\
\sqrt{d}|0\rangle&\mathbbm{1}&\dots&H^{\dagger}\tilde{V}^{(d-1)}H&H^{\dagger}\tilde{P}^{0,0}S_{d-1}H&H^{\dagger}\tilde{P}^{0,1}S_{d-1}H\\\
\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
\sqrt{d}|0\rangle&H^{\dagger}(\tilde{V}^{(d-1)})^{T}H&\dots&\mathbbm{1}&H^{\dagger}\tilde{P}^{d-1,0}S_{d-1}H&H^{\dagger}\tilde{P}^{d-1,1}S_{d-1}H\\\
\sqrt{d}|0\rangle&H^{\dagger}S_{d-1}^{\dagger}(\tilde{P}^{0,0})^{T}H&\dots&(\tilde{P}^{d-1,0})^{T}H&\mathbbm{1}&\frac{1}{d}H^{\dagger}S_{d-1}^{\dagger}\mathbbm{J}S_{d-1}H\\\
\sqrt{d}|0\rangle&H^{\dagger}S_{d-1}^{\dagger}(\tilde{P}^{0,1})^{T}H&\dots&(\tilde{P}^{d-1,1})^{T}H&\frac{1}{d}H^{\dagger}S_{d-1}^{\dagger}\mathbbm{J}S_{d-1}H&\mathbbm{1}\end{pmatrix}.$
(94)

We can separately analyze the resulting blocks and see that the unitary action
$U$ diagonalizes all of them,
$\frac{1}{d}H^{\dagger}S^{\dagger}_{d-1}\mathbbm{J}S_{d-1}H=\frac{1}{d}H^{\dagger}\mathbbm{J}H=|0\rangle\langle
0|$ and

$\displaystyle H^{\dagger}\tilde{V}^{\Delta i}H=$ $\displaystyle
H^{\dagger}\left(\sum_{\Delta k}\tilde{v}^{\Delta i}_{\Delta k}{X}^{\Delta
k}\right)H=\sum_{\Delta k}\tilde{v}^{\Delta i}_{\Delta k}H^{\dagger}X^{\Delta
k}H=\sum_{\Delta k}\tilde{v}^{\Delta i}_{\Delta k}Z^{\Delta
k}\equiv\sum_{k=0}^{d-1}\nu_{m}^{i}|m\rangle\langle m|.$ (95) $\displaystyle
H^{\dagger}\tilde{P}^{i,j}S_{d-1}H=$ $\displaystyle
H^{\dagger}\left(\sum_{e=0}^{d-1}S_{(e+i\cdot
j)}p(e|j)\right)S_{d-1}H=\sum_{e=0}^{d-1}p(e|j)H^{\dagger}\left(S_{(e+i\cdot
j)}S_{d-1}\right)H$ (96)
$\displaystyle=\sum_{e=0}^{d-1}p(e|j)H^{\dagger}X^{(e+i\cdot
j+1)}H=\sum_{e=0}^{d-1}p(e|j)Z^{(e+i\cdot
j+1)}\equiv\sum_{k=0}^{d-1}\omega^{m\cdot(i\cdot j)}p_{m}^{j}|m\rangle\langle
m|,$ (97)

where $p^{j}_{m}=\sum_{e=0}^{d-1}\omega^{m\cdot(1+e)}p(e|j)$. Moreover, from
the normalization of optimization variables and of probabilities we have that
$\nu_{0}^{i}=1$ and $p_{0}^{j}=1$ for all $i\in[d],j\in[2]$. Having
diagonalized all the matrices, we can rearrange $\Gamma^{\prime}$ in a block-
diagonal form. The first block can be constructed by concatenating nonzero
entries of first row and first column and by picking only the first $(1,1)$
entry of each $d\times d$ sub-block of $\Gamma^{\prime}$. The resulting block
can be expressed as follows,

$\Gamma_{0}=\begin{pmatrix}d&\sqrt{d}&\sqrt{d}&\dots&\sqrt{d}&\sqrt{d}&\sqrt{d}\\\
\sqrt{d}&1&\nu_{0}^{1}&\dots&\nu_{0}^{d-1}&p_{0}^{0}&p_{0}^{1}\\\
\sqrt{d}&(\nu_{0}^{1})^{\dagger}&1&\ddots&\vdots&p_{0}^{0}&p_{0}^{1}\\\
\vdots&\vdots&\ddots&\ddots&\nu_{0}^{1}&\vdots&\vdots\\\
\sqrt{d}&(\nu_{0}^{d-1})^{\dagger}&\dots&(\nu_{0}^{1})^{\dagger}&1&p_{0}^{0}&p_{0}^{1}\\\
\sqrt{d}&(p_{0}^{0})^{\dagger}&\dots&&(p_{0}^{0})^{\dagger}&1&1\\\
\sqrt{d}&(p_{0}^{1})^{\dagger}&\dots&&(p_{0}^{1})^{\dagger}&1&1\end{pmatrix}=\begin{pmatrix}d&\sqrt{d}&\sqrt{d}&\dots&\sqrt{d}&\sqrt{d}&\sqrt{d}\\\
\sqrt{d}&1&1&\dots&1&1&1\\\ \sqrt{d}&1&1&\ddots&\vdots&1&1\\\
\vdots&\vdots&\ddots&\ddots&1&\vdots&\vdots\\\ \sqrt{d}&1&\dots&1&1&1&1\\\
\sqrt{d}&1&\dots&1&1&1&1\\\ \sqrt{d}&1&\dots&1&1&1&1\end{pmatrix}.$ (98)

Matrix $\Gamma_{0}\geq 0$ (is positive semidefinite). The next block,
$\Gamma_{1}$ can be constructed by picking only the second diagonal entry
$(2,2)$ from each sub-block of $\Gamma^{\prime}$. And more generally,
$(m-1)$-th block can be constructed by picking only the diagonal entry $(m,m)$
from each sub-block of $\Gamma^{\prime}$ for $1\leq m\leq d+1$:

$\Gamma_{m}=\begin{pmatrix}1&\nu_{m}^{1}&\dots&\nu_{m}^{d-1}&p_{m}^{0}&p_{m}^{1}\\\
(\nu_{m}^{1})^{\dagger}&1&\ddots&\vdots&p_{m}^{0}&\omega^{m}p_{m}^{1}\\\
\vdots&\ddots&\ddots&\nu_{m}^{1}&\vdots&\vdots\\\
(\nu_{m}^{d-1})^{\dagger}&\dots&(\nu_{m}^{1})^{\dagger}&1&p_{m}^{0}&\omega^{(d-1)\cdot
m}p_{m}^{1}\\\ (p_{m}^{0})^{\dagger}&\dots&&(p_{m}^{0})^{\dagger}&1&0\\\
(p_{m}^{1})^{\dagger}&\dots&&(\omega^{(d-1)\cdot
m}p_{m}^{1})^{\dagger}&0&1\end{pmatrix}.$ (99)

We have, thus, reduced checking compatibility with ML to positivity of block-
diagonal matrices.

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