# Monogamy relations of entropic non-contextual inequalities and their
experimental realization

Dileep Singh<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education & Research Mohali, Sector 81 SAS Nagar,
Manauli PO 140306 Punjab, India.  Jaskaran Singh<EMAIL_ADDRESS>Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla,
Spain  Kavita Dorai<EMAIL_ADDRESS>Department of Physical Sciences,
Indian Institute of Science Education & Research Mohali, Sector 81 SAS Nagar,
Manauli PO 140306 Punjab, India.  Arvind<EMAIL_ADDRESS>Department
of Physical Sciences, Indian Institute of Science Education & Research Mohali,
Sector 81 SAS Nagar, Manauli PO 140306 Punjab, India.  Vice Chancellor,
Punjabi University Patiala, 147002, Punjab, India

###### Abstract

We develop a theoretical framework based on a graph theoretic approach to
analyze monogamous relationships of entropic non-contextuality (ENC)
inequalities. While ENC inequalities are important in quantum information
theory and are well studied, theoretical as well as experimental demonstration
of their monogamous nature is still elusive. We provide conditions for ENC
inequalities to exhibit a monogamous relationship and derive the same for
general scenarios. We show that two entropic versions of the Bell-CHSH
inequality acting on a tripartite scenario exhibit a monogamous relationship,
for which we provide a theoretical proof as well as an experimental validation
on an NMR quantum information processor. Our experimental technique to
evaluate entropies has been designed to obtain information about entropies via
measurement of only the expectation values of observables.

##  I Introduction

It is well known that measurements on quantum states exhibit correlations
which defy understanding based on classical theories. The outcomes of
measurements performed on spatially separated or even on single indivisible
systems may exhibit correlations greater than their apparent classical values.
These correlations are termed as Bell non-local Brunner  _et al._ (2014);
Clauser  _et al._ (1969); BELL (1966); Collins  _et al._ (2002) or contextual
Kochen and Specker (1975); Acín  _et al._ (2015); Araújo  _et al._ (2013);
Amaral  _et al._ (2014); Cabello  _et al._ (2014a) for bipartite space-like
separated and for single indivisible systems respectively, and are studied by
means of inequalities, a violation of which implies non-classical behavior
Larsson  _et al._ (2014); Giustina  _et al._ (2015); Shalm  _et al._ (2015);
Dogra  _et al._ (2016); Kirchmair  _et al._ (2009). The most widely studied of
these inequalities include the Bell-CHSH Clauser  _et al._ (1969) and the KCBS
inequality Klyachko  _et al._ (2008), quantifying Bell non-local and
contextual correlations, respectively.

Quantum correlations that go beyond the classical allowed values have found
immense application in quantum information processing tasks, including self-
testing Popescu and Rohrlich (1992); Braunstein  _et al._ (1992), quantum key
distribution Diamanti  _et al._ (2016); Scarani  _et al._ (2009), device
independent quantum key distribution Acín  _et al._ (2006a); Vazirani and
Vidick (2014) and randomness certification Acín and Masanes (2016), to name a
few. A common ground of most of these applications is that they utilize the
property of monogamy of correlations Acín  _et al._ (2006b); Pawłowski (2010);
Saha and Ramanathan (2017); Li  _et al._ (2018). Such monogamy relations
dictate that when two parties are violating a particular inequality
(contextual or Bell type), then it is not possible for one of the parties to
violate a second similar inequality with a third party on a tripartite state
shared amongst all three of them. These monogamy relations have been well
studied in the case of Bell inequalities and contextuality Pawłowski and
Brukner (2009); Ramanathan  _et al._ (2012) Interestingly, monogamous
relations also exist between Bell non-locality and contextuality and these
have been verified experimentally Kurzyński  _et al._ (2014); Zhan  _et al._
(2016); Saha and Ramanathan (2017).

Offering a different perspective on quantum correlations, Braunstein and Caves
introduced an information theoretic approach to understand non-local
correlations Braunstein and Caves (1988); Chaves and Budroni (2016) studied in
the form of inequalities. These inequalities are eponymously termed as
entropic inequalities. In this formalism, the joint Shannon entropies carried
by observables must satisfy an inequality in order to have a local description
of correlations. This makes the inequalities non-linear functions of
probabilities, unlike standard Bell inequalities which are linear. These
inequalities are one-way statements; only when an inequality is violated, can
it be implied that a joint probability distribution over the observables under
consideration cannot exist. However, if the inequality is satisfied, no
definite conclusion can be drawn. Subsequently, the work was extended to
include non-contextual scenarios Chaves (2013); Chaves and Fritz (2012) as
well, and the inequalities are generally termed as entropic non-contextuality
(ENC) inequalities. Since their inception, these inequalities have been
studied extensively and have also been realized experimentally Cao  _et al._
(2016a); Katiyar  _et al._ (2013); Cao  _et al._ (2016b); Zhan  _et al._
(2017); Qu  _et al._ (2020). Experimental implementations of state dependent
Zhan  _et al._ (2017) as well as state independent Qu  _et al._ (2020) ENC
inequalities have been performed on optical systems.

These ENC inequalities offer an added advantage of being independent of the
number of outcomes in a measurement, unlike Bell scenarios where the number of
outcomes of a measurement play an important rule in determining the local
value. This property of ENC inequalities makes them suitable candidates for
applications in non-locality distillation Chaves and Fritz (2012) and
bilocality scenarios, among many other applications. Due to their non-linear
nature, a general description of monogamous relations for all possible ENC
scenarios is a nontrivial task. While there exist scenarios where it is
possible to derive a monogamous relationship Chaves and Budroni (2016), a
generalization of this recipe would be too complex and require involved
calculations. There have been attempts in this direction Kurzyński and
Kaszlikowski (2014) but based on a different approach than ours. Furthermore,
while there are a few experimental studies verifying monogamy of ENC
inequalities, all of them are based on optical methods, which are capable of
directly yielding probabilities of outcomes of a measurement in the form of
frequency of photon detections. However, to the best of our knowledge, there
is currently no experimental verification of monogamy of ENC inequalities till
date.

Providing a fresh perspective in this paper, we describe an elegant
theoretical recipe to evaluate monogamous relations of ENC inequalities and
verify the results on an NMR quantum information processor. We develop a
theoretical framework, based on graph theoretic formalism, to study monogamous
relations of ENC inequalities in arbitrary no-signalling scenarios, apply our
formalism to the entropic Bell-CHSH scenario, and derive a monogamous relation
between three parties: Alice, Bob and Charlie. We show that if Alice and Bob
violate the entropic Bell-CHSH inequality, then Alice and Charlie cannot do so
(and vice versa). An added advantage is that our technique also holds for
arbitrary $m$ ENC distributed over $n$ parties.

Next we turn to the experimental realization of our results by applying the
entropic Bell-CHSH inequality to two different cases. In the first case, all
the parties share a mixed tripartite state, such that Alice and Bob share a
maximally entangled state with probability $p$, and Alice and Charlie share
the same with probability $1-p$. In the second case, all the parties together
share an entangled tripartite pure state. In both the cases, we observe that
the monogamous relation is satisfied at all times, while only one of the
inequalities is capable of exhibiting a violation. It should be noted that
experiments on an NMR quantum information processor only yield an expectation
value of the observable, while probabilities of various outcomes cannot be
addressed directly. This makes the task of evaluating entropic quantities on
this system difficult. To circumvent this difficulty, we provide a novel
recipe to evaluate the entropy of an observable on an NMR quantum information
processor. In the experiment, all the pure and mixed states were prepared with
an experimental fidelity greater than $0.92$ and $0.96$ respectively, and the
experimental results match well with the theoretical predictions.

The paper is organized as follows: in Sec. II.1 we begin by giving a brief
description of entropic inequalities. In Sec. II.2 we describe our main
theoretical result concerning monogamy of ENCs. In Sec. III.1 we provide an
experimental demonstration using a mixed state, while in Sec. III.2 we use a
pure tripartite state for the experiment. Sec. IV contains some concluding
remarks.

##  II Entropic inequalities and their monogamy

In this section we describe our theoretical results, which will be used later
in the paper. We begin by providing a brief review of ENC inequalities and
focus particularly on the entropic version of the Bell-CHSH inequality.

###  II.1 The entropic inequalities

An experiment corresponding to a contextuality inequality can be represented
by a graph Cabello  _et al._ (2014b). Mathematically, a graph $G$ is defined
by a set of vertices $V$ and and a set of edges $E$, such that $G=(V,E)$. In
relation to the contextuality inequality, the set $V$ can represent either
events, projectors or observables depending upon the scenario, while the set
$E$ represents a relationship between different elements of $V$, such as
orthogonality, exclusivity or commutativity Cabello  _et al._ (2014b). In this
paper we use commutativity graphs to study ENC inequalities, in which the set
of vertices corresponds to observables and two vertices are connected by an
edge if they commute with each other Kurzyński  _et al._ (2012).

Consider an $n$-cycle commutation graph in which the $n$ vertices represent
observables $X_{i}$ and the existence of an edge indicates that the
corresponding observables commute. An example of one such graph for $n=4$ is
given in Fig. 2 (top panel).

We assume that a non-contextual joint probability distribution exists over the
entire set of observables considered, even though most of them do not commute.
It is our aim to construct a condition based on the preceding assumption, for
which a violation would indicate that such a non-contextual joint probability
distribution does not exist.

The existence of a non-contextual joint probability distribution over the
observables $X_{i}$ implies that it is possible to define a joint Shannon
entropy $H(X_{0},...,X_{n-1})$ of them. We can then write

$H(X_{0},X_{n-1})\leq H(X_{0},...,X_{n-1}),$ (1)

where the relationship $H(X)\leq H(X,Y)$ is physically motivated by the fact
that two random variables cannot contain less information than a single one of
them. Furthermore, with repeated application of the chain rule
$H(X,Y)=H(X|Y)+H(Y)$, where $H(X|Y)$ denotes the conditional entropy of
observable $X$ given information about observable $Y$, the right hand side of
the inequality can be re-written as,

$\displaystyle H(X_{0},...,X_{n-1})$ $\displaystyle\leq
H(X_{0}|X_{1},...,X_{n-1})+H(X_{1}|X_{2},...X_{n-1})+...+H(X_{n-2}|X_{n-1})+H(X_{n-1})$
(2) $\displaystyle\leq
H(X_{0}|X_{1})+H(X_{1}|X_{2})+...+H(X_{n-2}|X_{n-1})+H(X_{n-1}).$

The latter inequality in Eq. (2) is a consequence of the relationship
$H(X|Y)\leq H(X)$, which implies that conditioning cannot increase the
information content of a random variable. Plugging Eq. (2) in Eq. (1) and
using $H(X_{0}|X_{n-1})=H(X_{0},X_{n-1})-H(X_{n-1})$, we finally get the
required entropic non-contextuality inequality,

$H_{K_{1}}:H(X_{0}|X_{n-1})\leq H(X_{0}|X_{1})+...H(X_{n-2}|X_{n-1}).$ (3)

A violation of Eq. (3) would then indicate that a non-contextual joint
probability distribution over the corresponding set of observables does not
exist.

It should be noted that no assumptions have been made regarding the nature of
the observables $X_{i}$. For all intents and purposes they can correspond to
projective measurements or POVMs with any number of outcomes. Furthermore, the
observables could correspond to local scenarios or non-local, in which case
the entropic inequality would be termed as non-contextual or Bell non-local.
Since we consider generalized scenarios, we term all entropic inequalities as
non-contextual as it subsumes the non-local scenarios as well.

One of the many interesting cases arises for $n=4$ observables, which
corresponds to the well known Bell-CHSH scenario. Consider two parties, Alice
and Bob, each having two observables labelled $\\{A_{0},A_{1}\\}$ and
$\\{B_{0},B_{1}\\}$ respectively, such that observables of any one party do
not commute, while observables of different parties commute. Differing from
the traditional Bell-CHSH scenario, it is also assumed that a measurement of
the observables can have an arbitrary number of outcomes. The corresponding
entropic inequality is written as,

$H_{K_{1}}:H(A_{1}|B_{1})-H(A_{1}|B_{0})-H(B_{0}|A_{0})-H(A_{0}|B_{1})\leq 0.$
(4)

A violation of the above inequality implies a non-existence of a joint
probability distribution over all the observables $A_{x}$ and $B_{y}$ $\forall
x,y\in\\{0,1\\}$. In the next section, we show that ENC inequalities admit a
monogamous relationship which can be derived using a graph theoretic
formalism. We particularly focus on the entropic Bell-CHSH scenario described
above and show that it admits a monogamous relationship.

Figure 1: The chordal subgraph where the vertices represents observables
$X_{i}$ with $i=1,2,3,4$ and the edge indicate commutativity relationship
between the observables.

###  II.2 Monogamy of entropic inequalities

We now elucidate the formalism to check and derive a monogamous relationship
for any arbitrary scenario using the graph theoretic formalism. It should be
remembered that a violation of the inequality (4) implies that a joint
probability distribution cannot exist over the given observables.

######  Proposition 1.

A monogamous relationship for a set of $n$ observables $X_{i}$, corresponding
to $m$ non-contextuality scenarios exists if their joint commutation graph can
be vertex decomposed into $m$ chordal subgraphs such that all edges appearing
in the original $m$ non-contextuality graphs must appear just once in the
decomposition. The monogamous relationship then reads as,

$H_{K_{1}}+H_{K_{2}}+H_{K_{3}}+....+H_{K_{m}}\leq 0,$ (5)

where $H_{K_{i}}$ denotes the $i$th ENC inequality.

###### Proof.

A chordal graph is a graph in which all cycles of four or more vertices have a
chord. A chord is an edge that is not part of the cycle but which connects two
of the vertices. It should be noted that the definition of chordal subgraphs
implies that these subgraphs must have an edge connecting two vertices, such
that any induced cycles in the subgraph have a length equal to three. Any
induced cycles of length greater than $3$ should then have an edge such that
the resultant induced cycle satisfies the definition.

For these $3$-cycle induced subgraphs, it is possible to write down the
corresponding entropic inequality. Since it is a $3$-cycle graph, a joint
probability distribution over the observables always exists Ramanathan  _et
al._ (2012) and the entropic inequality is never violated. Therefore, we can
add all the entropic inequalities obtained in this fashion to obtain

$\sum_{l}\sum_{i,j=1}^{n}g_{l}(H^{\prime}(X_{i}|X_{j})),$ (6)

where $H^{\prime}(X_{i}|X_{j})=H(X_{i}|X_{j})$ only if the vertices $i$ and
$j$ belong to an edge and zero otherwise, $g(X)=\pm X$ depending on the $3$
cycle chosen and $l$ is the number of edges appearing in the $m$ chordal
graphs. It should be noted that Eq. (6) is a linear combination of
$H(X_{i}|X_{j})$ and since we require that all the terms appearing in the
individual $m$ ENC inequalities appear in the decomposition, it is possible to
obtain the form in Eq. (5) via linear manipulation of the ENC inequalities of
the $3$-cycle induced graphs. It should be noted that each term appearing in
Eq. (3) corresponds to an edge in the commutativity graph and missing even a
single edge in the chordal decomposition, would make one of the non-
contextuality inequalities incomplete and Eq. (5) unachievable. Furthermore,
the linear manipulations required correspond to choosing a suitable form of
$g(X)$ such that the $m$ ENC inequalities can be obtained by grouping certain
terms together. This manipulation depends on the commutation graph of the
interested scenario.

As an example of our technique consider the chordal graph given in Fig. 1
which is formed by two $3$-cycle graphs. We assume that this is one of the
subgraphs obtained via vertex decomposition of a joint commutativity graph
such that the term $H(X_{2}|X_{4})$ corresponding to the edge $(X_{2},X_{4})$
does not appear in Eq. (5), while the terms corresponding to the other edges
do appear. In order to eliminate this term, we consider the ENC inequality for
the $3$-cycle graph with vertices $(X_{1},X_{2},X_{4})$, written in a cyclic
form as,

$H(X_{1}|X_{4})-H(X_{1}|X_{2})-H(X_{2}|X_{4})\leq 0,$ (7)

while for the other $3$-cycle graph formed by the vertices
$(X_{2},X_{3},X_{4})$ we use an anti-cyclic form of the ENC inequality as,

$H(X_{2}|X_{4})-H(X_{2}|X_{3})-H(X_{3}|X_{4})\leq 0.$ (8)

Since all $3$-cycle graphs admit a joint probability distribution, the
aforementioned inequalities are always satisfied and can therefore be added to
give,

$H(X_{1}|X_{4})-H(X_{1}|X_{2})-H(X_{2}|X_{3})-H(X_{3}|X_{4})\leq 0,$ (9)

which is the ENC inequality over the graph in Fig 1 without the edge
$(X_{2},X_{4})$. We use this technique to derive our monogamous relationships.
In the case when such a vertex decomposition cannot be found, we cannot
guarantee that a monogamous relation will exist.

Figure 2: The joint commutation graph of Alice-Bob-Eve (top) and its chordal
decomposition (below) according to Proposition 1, where solid lines indicate
commutativity between observables of Alice and Bob, while dashed lines
indicate commutativity between Alice (Bob) and Charlie.

We now apply this technique to derive a monogamous relationship of two
entropic CHSH inequalities. Consider a standard monogamous relationship
between two entropic non-contextuality scenarios, $K_{1}$ and $K_{2}$ which
should dictate,

$H_{K_{1}}+H_{K_{2}}\leq 0,$ (10)

where $H_{K_{1(2)}}\leq 0$ denotes the existence of a joint probability
distribution over the $n$ observables. Since the monogamy relationship must be
obeyed at all times, Eq. (10) then physically implies that a joint probability
distribution over all the observables appearing in the new entropic
inequality, $H=H_{K_{1}}+H_{K_{2}}$ exists.

As explained, in order to achieve a form of Eq. (10), the joint graph of the
non-contextuality scenarios must be decomposable into chordal subgraphs, which
admit a joint probability distribution. However, we also require an additional
feature that all edges of the individual non-contextuality graphs must appear
at least once in any of the chordal subgraphs.

Consider the CHSH scenario in which two parties Alice and Bob can perform a
measurement of the observables $A_{0}$, $A_{1}$ and $B_{0}$, $B_{1}$
respectively. We assume a third party Charlie with observables $E_{0}$ and
$E_{1}$, which commute with the observables of Alice and Bob. The scenario is
illustrated by a joint graph as shown in Fig. 2, where vertices represent
observables and edges indicate the commutation relationship. Without loss of
generality we assume that Charlie would like to violate the entropic CHSH
inequality with Alice. The two corresponding Alice-Bob and Alice-Charlie
entropic CHSH inequalities are,

$\displaystyle H_{K_{1}}:$ $\displaystyle H(A_{0}|B_{0})-H(A_{0}|B_{1})$ (11)
$\displaystyle-H(B_{1}|A_{1})-H(A_{1}|B_{0})\leq 0,$

$\displaystyle H_{K_{2}}:$ $\displaystyle H(A_{0}|E_{0})-H(A_{0}|E_{1})$ (12)
$\displaystyle-H(E_{1}|A_{1})-H(A_{1}|E_{0})\leq 0.$

The joint commutation graph is decomposed into two chordal graphs while
keeping the edges appearing in the individual Alice-Bob and Alice-Charlie CHSH
scenarios intact in the decomposition, as shown in Fig. (2). The corresponding
ENC inequalities for each $3$-cycle graph in every chordal subgraph are given
as,

$\displaystyle H(A_{0}|E_{0})-H(A_{0}|B_{1})-H(B_{1}|E_{0})\leq 0,$ (13)
$\displaystyle H(B_{1}|E_{0})-H(B_{1}|A_{1})-H(A_{1}|E_{0})\leq 0,$ (14)
$\displaystyle H(A_{0}|B_{0})-H(A_{0}|E_{1})-H(E_{1}|B_{0})\leq 0,$ (15)
$\displaystyle H(E_{1}|B_{0})-H(E_{1}|A_{1})-H(A_{1}|B_{0})\leq 0,$ (16)

Being cyclic and chordal, all the above inequalities are a necessary and
sufficient condition for a joint probability distribution to exist.
Furthermore, we have carefully chosen the terms with positive and negative
coefficients so as to achieve a final form according to Eq. (10). Adding the
above inequalities and grouping the terms according to $H_{K_{1}}$ and
$H_{K_{2}}$, we obtain

$H_{K_{1}}+H_{K_{2}}\leq 0,$ (17)

which is the required monogamy relationship. We note that this monogamy
relationship was also derived in Chaves and Budroni (2016), albeit in a
different manner and specifically for the entropic CHSH inequality. However,
our formalism can be readily generalized to $n$ observables distributed among
$m$ parties. ∎

The derived monogamy relationship (17) imposes severe restrictions on the
violation of Alice-Charlie entropic CHSH inequality.

##  III Experimental demonstration

In this section, we experimentally demonstrate the monogamy relationship
derived for the entropic Bell-CHSH scenario (17) on an NMR quantum information
processor, using two different set of states. We show that for both the sets
of states, the entropic Bell-CHSH obeys the monogamy relationship we derived
above.

###  III.1 Implementation using a mixed tripartite state

We experimentally implement the monogamy inequality (17) using a mixed
tripartite state which is a classical mixture of two pure maximally entangled
states given as:

$\rho=p(|\psi_{1}\rangle\langle\psi_{1}|)+(1-p)(|\psi_{2}\rangle\langle\psi_{2}|),$
(18)

where

$\displaystyle|\psi_{1}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|001\rangle+|111\rangle),$ (19)
$\displaystyle|\psi_{2}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(|010\rangle+|111\rangle),$

and $p\in\left[0,1\right]$. As can be seen, the state $\rho$ physically
implies that Alice and Bob share a maximally entangled state with probability
$p$ while Charlie is separable, and with probability $1-p$, Alice and Charlie
share a maximally entangled state while Bob is separable.

The observables of Alice, Bob and Charlie are assumed to lie in the $X-Z$
plane and correspond to Pauli spin measurements along the unit vectors
$\bm{a},\bm{a^{\prime}},\bm{b},\bm{b^{\prime}}$ and $\bm{e},\bm{e^{\prime}}$
respectively. The vectors $\bm{a},\bm{b^{\prime}},\bm{a^{\prime}}$ and
$\bm{b}$ are successively separated by an angle $\frac{\theta}{3}$, while the
vectors $\bm{e}$ and $\bm{e^{\prime}}$ are taken to be the same as Bob’s. The
corresponding Pauli observables are then given as:

$\displaystyle A_{0}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix},\quad$
$\displaystyle
A_{1}=\begin{bmatrix}\cos\frac{2\theta}{3}&-\sin\frac{2\theta}{3}\\\
\sin\frac{2\theta}{3}&\cos\frac{2\theta}{3}\end{bmatrix}$ (20) $\displaystyle
B_{0}=\begin{bmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{bmatrix},\quad$ $\displaystyle
B_{1}=\begin{bmatrix}\cos\frac{\theta}{3}&-\sin\frac{\theta}{3}\\\
\sin\frac{\theta}{3}&\cos\frac{\theta}{3}\end{bmatrix},$

while the observables of Charlie ($E_{0},E_{1}$) are the same as Bob’s but
acting in a different Hilbert space. All of the aforementioned observables
have eigenvalues $a_{i},b_{i},e_{i}\in\\{-1,+1\\}$ and follow the
commutativity conditions as shown in Fig. 2.

For the set of observables given in Eq. (20), the maximum violation of the
Alice-Bob entropic CHSH inequality $H_{K1}=0.237$ bits is found to be at
$\theta=0.457$ radians when the parties share the state $\ket{\psi_{1}}$. The
same also holds for Alice-Charlie entropic CHSH inequality when the parties
share the state $\ket{\psi_{2}}$.

We experimentally implemented the monogamy relation given in Eq. (17) on an
eight-dimensional quantum system. We used a molecule of 13C -labeled diethyl
fluoromalonate dissolved in acetone-D6, with the 1H, 19F and 13C spin-1/2
nuclei being encoded as ‘qubit one’, ‘qubit two’ and ‘qubit three’,
respectively (see Fig. 3(a)). The system was initialized in the pseudopure
state (PPS) $|000\rangle$ using the spatial averaging technique Mitra  _et
al._ (2007) and the corresponding NMR spectra and experimental tomographs are
given in Fig. 3(a),(b).

Experiments were performed on a Bruker Avance III 600-MHz FT-NMR spectrometer
equipped with a QXI probe. Local unitary operations were achieved by RF pulses
of suitable amplitude, phase, and duration and nonlocal unitary operations
were achieved by free evolution under the system Hamiltonian. The $T_{1}$ and
$T_{2}$ relaxation times of 1H, 19F, 13C spin-1/2 nuclei range from 4.16 sec
to 7.16 sec and 0.99 sec to 3.56 sec, respectively. The duration of the
$\frac{\pi}{2}$ pulses for 1H, 19F, and 13C nuclei are 9.5 $\mu$s at 18.14 W
power level, 22.2 $\mu$s at a power level of 42.27 W, and 15.2 $\mu$s at a
power level of 179.47 W, respectively. At room temperature, NMR experiments
are only sensitive to the deviation density matrix and the initial state is
prepared from the thermal equilibrium into a PPS:

$\rho_{000}=\frac{1-\epsilon}{2^{3}}I_{8}+\epsilon|000\rangle\langle 000|$
(21)

where $\epsilon\approx 10^{-6}$ and $I_{8}$ is $8\times 8$ identity operator.
The system is initialized in the PPS state i.e.$|000\rangle$ using the spatial
averaging technique Mitra  _et al._ (2007), which is based on dividing the
system in sub-ensembles which can be accessed independently in NMR by using a
combination of RF pulses and pulsed magnetic gradients. The state tomography
was performed using the least square optimization techniqueGaikwad  _et al._
(2021) with an experimental state fidelity of 0.98.

Figure 3: (a) Molecular structure of $\rm{}^{13}C$ \- labeled diethyl
fluoromalonate. $\rm{}^{1}H$, $\rm{}^{19}F$, $\rm{}^{13}C$ are used as three
qubits in the experiment. NMR pulse sequence for the initialization of the
state into the PPS $|000\rangle$ state. The value of the flip angle $\alpha$
is kept at $57.87^{\circ}$. The phases and the flip angles are written with
each pulse. The $U_{12},U_{23},U_{13}$ are evolution operators with time
intervals $\tau_{12}$, $\tau_{23}$, $\tau_{13}$ equal to $\frac{1}{2J_{HF}}$,
$\frac{1}{2J_{HC}}$, and $\frac{1}{2J_{FC}}$ respectively, shown by red dotted
lines. (b) The NMR spectra corresponding to the PPS $|000\rangle$. The
horizontal scale represents the chemical shifts in ppm. (c) Real (left) and
imaginary (right) parts of the experimentally reconstructed tomographs of the
state $|000\rangle$ with an experimental state fidelity of 0.98.

We began by preparing the mixed tripartite state given in Eq. (18) for
different values of $p$. In order to achieve this, we utilized the temporal
averaging technique Oliveira  _et al._ (2007); Knill  _et al._ (1998). Using
this technique, it is possible to prepare arbitrary mixed states on an NMR
quantum information processor, by applying suitable unitary transformations on
some common initial PPS. The different experiments are performed on common
initial states, the results of which are independently stored. Finally, these
results are combined to produce an average state which simulates the behavior
of a mixed state.

In our case, we prepared the mixed state given in Eq. (18), which is a mixture
of two pure states $|\psi_{1}\rangle$, $|\psi_{2}\rangle$. We first prepared
these two states by applying suitable unitaries on the initial state
$|000\rangle\langle 000|$ in two different and independent experiments Singh
_et al._ (2019a). The states of these two experiments are then added with
appropriate probabilities to achieve the desired mixed state given in Eq.
(18). To demonstrate the monogamy relation, we chose different values of $p$
to experimentally prepare the desired mixed state with experimental state
fidelities $\geq 0.956$. The tomograph of one such experimentally prepared
state with $p=1$ is shown in Fig. 4.

Figure 4: Real and imaginary parts of the tomographs of the theoretically and
experimentally reconstructed density operator for the tripartite state $\rho$
with $p=1$ and an experimental state fidelity of $0.97$.

After experimentally preparing the states, we measured the desired
probabilities in order to calculate the entropies involved in the inequality
given in Eq. (17). In order to calculate the probabilities, we transformed the
required probabilities in terms of expectation values. This is necessary
because experiments on an NMR quantum information processor yield only
expectation values of the observables. These expectation values are evaluated
by decomposing the observables in terms of linear combinations of Pauli
operators which can be mapped to the single-qubit Pauli $Z$ operator. This
mapping is particularly useful in the context of an NMR experimental setup
where the expectation value of the $Z$ operator is easily accessible and
corresponds to the observed $z$ magnetization of a nuclear spin in a
particular quantum state. The normalized experimental intensities of the NMR
signal then provide an estimate of the expectation value of the Pauli $Z$
operator in that quantum state Singh  _et al._ (2019b).

The probabilities can be written as
$P(A_{i}=a_{i},B_{j}=b_{j})=\text{tr}(\rho|a_{i}\rangle\langle
a_{i}|\otimes|b_{j}\rangle\langle b_{j}|\otimes I)$ and
$P(A_{i}=a_{i},E_{j}=e_{j})=\text{tr}(\rho|a_{i}\rangle\langle a_{i}|\otimes
I\otimes|e_{j}\rangle\langle e_{j}|)$ where $|a_{i}\rangle$, $|b_{j}\rangle$,
$|e_{j}\rangle$ are the eigenvector of the observables corresponding to Alice,
Bob and Charlie, respectively. The observables ($|a_{i}\rangle\langle
a_{i}|\otimes|b_{j}\rangle\langle b_{j}|\otimes I$) , ($|a_{i}\rangle\langle
a_{i}|\otimes I\otimes|e_{i}\rangle\langle e_{i}|)$ are decomposed in terms of
linear combinations of Pauli operators and details are given in Appendix-A.
The idea is to unitarily map the state $\rho$ to another state
$\rho^{\prime}$, such that $\langle X\rangle_{\rho}=\langle
I_{iz}\rangle_{\rho^{\prime}}$ where $X$ is the observable to be measured in
the state $\rho$ and $I_{iz}$ is the z-spin angular momentum of the qubit.
This can be achieved by measuring the $I_{iz}$ on the state $\rho^{\prime}$.
For example, one can find the expectation values of
$\langle\sigma_{x}\otimes\sigma_{x}\otimes I\rangle_{\rho}$ and
$\langle\sigma_{x}\otimes I\otimes\sigma_{x}\rangle_{\rho}$ which are involved
in the evaluation of probabilities (see Appendix-A), by using the quantum
circuit and corresponding NMR pulse sequence given in Fig.5(a) and Fig.5(b)
respectively, where the implementation is followed by a measurement of the
spin magnetization of the second and third qubits, respectively.

Figure 5: Quantum circuit and corresponding NMR pulse sequence to map the
state $\rho$ to the state $\rho^{\prime}$ such that (a)
$\langle\sigma_{x}\otimes\sigma_{x}\otimes I\rangle_{\rho}=\langle
I_{2z}\rangle_{\rho^{\prime}}$, (b) $\langle\sigma_{x}\otimes
I\otimes\sigma_{x}\rangle_{\rho}=\langle I_{3z}\rangle_{\rho^{\prime}}$ . In
the NMR pulse sequences, the filled rectangles are $\frac{\pi}{2}$ RF pulses
while unfilled rectangles denotes the $\pi$ RF pulses. The phase of each pulse
is written over the respective pulse and bar over the phase denotes the
negative phase. The free evolution of time is denoted by
$\tau_{12},\tau_{13}$.

We experimentally calculated the ENC inequalities $H_{K1}$, $H_{K2}$ and the
ENC monogamy relation $H_{K1}+H_{K2}$ for the tripartite mixed state given in
Eq. (18) for different values of $p$. Experimental values of $H_{K1}$,
$H_{K2}$, $H_{K1}+H_{K2}$ with respect to various values of $p$ are plotted in
Fig. 6. It can be seen that $H_{K1}$ is violated for the tripartite state with
$p=1$, while $H_{K2}$ is violated for the state with $p=0$. The monogamy
relation $H_{K1}+H_{K2}$ is never violated for any value of $p$ and the
results are in good agreement with the theoretical predictions.

Figure 6: The experimentally calculated values for the monogamy inequality Eq.
(17) denoted by $f^{\prime}(K_{1},K_{2})$ for the tripartite mixed states
given in Eq. (18) for different values of $p$. The black dotted line is the
maximum non-contextual value that can be achieved by inequalities $H_{K1}$,
$H_{K2}$ and $H_{K1}+H_{K2}$ to follow the ENC theories and monogamy
relationship, respectively. The red color line and the red points with error
bars represent the theoretical and experimental calculated values for the
inequality $H_{K1}$, respectively. Similar representations for the
inequalities $H_{K2}$ and $H_{K1}+H_{K2}$ are used.

It is seen that the experimental values are always lower than the
corresponding theoretical values. This is due to the fact that the maximum
value of the inequality is always achieved for a pure state and any addition
of noise makes it a mixed state. Therefore, the value is always observed to be
lower than the theoretical one. The difference between them also increases
with increasing values of $p$ upto $p=0.5$. For this region, the state
prepared for $p=0.0$ is the dominant one which is shown to violate
$H_{K_{1}}$. Furthermore, the corresponding inequalities are logarithmic in
nature, which also compounds the errors for states till $p=0.5$. However,
after this value the state prepared for $p=1.0$ dominates and the errors again
follow a similar trend. It should be noted that the experimental curve follows
the same trend as the theoretical curve.

###  III.2 Implementation using a pure tripartite state

In this subsection, we experimentally test the monogamy relation Eq. (17) for
a pure tripartite state, having two parameters which we can vary. We show that
the monogamy relation holds and is in good agreement with theoretical
predictions.

We take a pure tripartite state of the form,

$|\phi\rangle=N(p_{1}|001\rangle+p_{2}|010\rangle+(p_{1}+p_{2})|111\rangle),$
(22)

where $N=\frac{1}{\sqrt{p_{1}^{2}+p_{2}^{2}+(p_{1}+p_{2})^{2}}}$ is the
normalization factor. We experimentally prepared five different states
corresponding to various values of $p_{1}$ and $p_{2}$. The quantum circuit
and the corresponding NMR pulse sequence is given in Fig. 7(a), (b). Different
pure states corresponding to various values of $p_{1}$ and $p_{2}$ were
generated by suitably choosing the values of $\theta_{1},\theta_{2}$ and
$\theta_{3}$. Tomograph of one such experimentally prepared state with
$p_{1}=0.25$ and $p_{2}=0.50$ is depicted in Fig. 8, with an experimental
state fidelity of $0.93$. After preparing the states, we measured the desired
probabilities by mapping the state onto the Pauli basis operators in order to
calculate the entropies involved in the inequality Eq. (17) as discussed
earlier in Sec. III.1.

Figure 7: (a) Quantum circuit for state preparation for the tripartite pure
state $|\phi\rangle$ and (b) corresponding NMR pulse sequence. The filled
rectangles denote $\frac{\pi}{2}$ RF pulses while the unfilled rectangles
denote $\pi$ RF pulses. The phase of each pulse is written over the respective
pulse, and bar over the phase denotes a negative phase.
$\tau_{12},\tau_{23},\tau_{13}$ are the free time evolutions. All the CNOTS
are separated by red dotted lines. Figure 8: Real and imaginary part of
tomograph of the theoretically and experimentally reconstructed density
operator for the tripartite pure state
$\rho^{\prime}=|\phi\rangle\langle\phi|$ with $p_{1}=0.25,p_{2}=0.50$ with an
experimental state fidelity of $0.93$.

We experimentally evaluate $H_{K_{1}}$, $H_{K_{2}}$ and $H_{K_{1}}+H_{K_{2}}$
with respect to the different values of $p_{1}$ and $p_{2}$ and the results
are given in Table-1. We see that the inequality $H_{K_{1}}$ is violated for
$p_{1}=1$ and $p_{2}=0$ and $H_{K_{2}}$ is violated for $p_{1}=0$ and
$p_{2}=1$, while the monogamy relationship $H_{K_{1}}+H_{K_{2}}$ is never
violated for any value of $p_{1}$ and $p_{2}$. This shows that the monogamy
relation between the ENC inequalities is obeyed for all such possible states.

Table 1: Theoretically computed and experimentally measured values of the monogamy inequalities for a pure state. |  | $H_{K_{1}}$ | $H_{K_{2}}$ | $H_{K_{1}}+H_{K_{2}}$
---|---|---|---|---
$p_{1}$ | $p_{2}$ | Theory | Experiment | Theory | Experiment | Theory | Experiment
1.00 | 0.00 | 0.236 | 0.156$\pm$0.032 | -1.436 | -1.522$\pm$0.035 | -1.200 | -1.366$\pm$0.034
0.50 | 0.25 | -0.492 | -0.606$\pm$0.021 | -1.338 | -1.413$\pm$0.027 | -1.830 | -2.019$\pm$0.024
0.50 | 0.50 | -1.017 | -1.103$\pm$0.022 | -1.017 | -1.082$\pm$0.030 | -2.034 | -2.185$\pm$0.026
0.25 | 0.50 | -1.338 | -1.397$\pm$0.021 | -0.492 | -0.598$\pm$0.024 | -1.830 | -1.995$\pm$0.023
0.00 | 1.00 | -1.436 | -1.523$\pm$0.028 | 0.236 | 0.149$\pm$0.025 | -1.200 | -1.374$\pm$0.027

##  IV Concluding Remarks

In this paper we develop a theoretical framework to analyze when a scenario
will exhibit a monogamous relationship. We perform our analysis of the
monogamy relations expected in a general tripartite scenario for the entropic
Bell-CHSH inequality and give the first theoretical study of the monogamy of
entropic inequalities based on the graph theoretic formalism. We also give the
first experimental demonstration of monogamy of entropic correlations using
three NMR qubits. Evaluating entropy on an NMR quantum processor is quite hard
and has not been performed earlier. We are able to obtain information about
entropies using measurements of only the expectation values of observables. We
experimentally show the monogamy of entropic inequality for a pure tripartite
state as well as for the mixed state. This indicates with certainty that our
formalism holds in general.

It should be noted that due to limitations of access to individual events, the
NMR implementation of such scenarios involves use of Born rule to interpret
the results of measurements. This is not the ideal way of carrying out such
experiments and a more refined test would evaluate the requisite probabilities
without assuming the Born rule. This is possible on optical systems Zhan  _et
al._ (2017); Qu  _et al._ (2020), where the probabilities are calculated by
estimating the frequency clicks of the photodetectors rather than the Born
rule.

The experimental implementation of monogamy inequalities is important for
quantum information processing tasks and our results are a step forward in
this direction.

###### Acknowledgements.

All the experiments were performed on a Bruker Avance-III 600 MHz FT-NMR
spectrometer at the NMR Research Facility of IISER Mohali. A. acknowledges
financial support from DST/ICPS/QuST/Theme-1/2019/Q-68. K .D. acknowledges
financial support from DST/ICPS/QuST/Theme-2/2019/Q-74. JS acknowledges
support by Universidad de Sevilla Project Qdisc (Project No. US-15097), with
FEDER funds, MCINN/AEI Projet No. PID2020-113738GB-I00, and QuantERA grant
SECRET, by MCINN/AEI (Project No. PCI2019-111885-2).

##  Appendix A Observable decomposition in terms of Pauli operators

To experimentally test the inequality on an NMR quantum processor, we have
calculated the entropies involved in the inequality by measuring the
probabilities for an experimentally prepared state. This can be achieved by
decomposing the projectors $|a_{\pm 1}\rangle\langle a_{\pm 1}|\otimes|b_{\pm
1}\rangle\langle b_{\pm 1}|$ as linear combinations of Pauli operators, which
can then be mapped to a single-qubit Pauli $Z$ operator. In the tripartite
scenario, the probabilities can be written as $P(A_{i}=\pm 1,B_{j}=\pm
1)=\text{tr}(\rho|a_{\pm 1}\rangle\langle a_{\pm 1}|\otimes|b_{\pm
1}\rangle\langle b_{\pm 1}|\otimes I)$ and $P(A_{i}=\pm 1,E_{j}=\pm
1)=\text{tr}(\rho|a_{\pm 1}\rangle\langle a_{\pm 1}|\otimes I\otimes|e_{\pm
1}\rangle\langle e_{\pm 1}|)$ for the Alice-Bob and Alice-Charlie entropic
CHSH inequalities. To test the monogamy inequality Eq.(17), we measure the
probabilities to calculate the entropies $H(A_{i}|B_{j})$ $H(A_{i}|E_{j})$
involved in the inequalities $H_{K1}$ and $H_{K2}$.

Probabilities are calculated by decomposing the observables in terms of linear
combination of Pauli operators as:

P(A${}_{1}=+1$)
$\displaystyle=0.286\text{b${}_{16}$}+0.410\text{b${}_{48}$}+0.500I,$ (23)
P(B${}_{1}=+1$)
$\displaystyle=0.476\text{b${}_{12}$}+0.149\text{b${}_{4}$}+0.500I,$
P(A${}_{0}=+1$, B${}_{0}=+1$)
$\displaystyle=0.152\text{$b_{12}$}+0.197\text{b${}_{4}$}+0.250\text{b${}_{48}$}$
$\displaystyle+0.197\text{b${}_{52}$}+0.152\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=+1$, B${}_{0}=-1$)
$\displaystyle=-0.152\text{b${}_{12}$}-0.197\text{b${}_{4}$}+0.250\text{b${}_{48}$}$
$\displaystyle-0.197\text{b${}_{52}$}-0.152\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=-1$, B${}_{0}=+1$)
$\displaystyle=0.152\text{b${}_{12}$}+0.197\text{b${}_{4}$}-0.250\text{b${}_{48}$}$
$\displaystyle-0.197\text{b${}_{52}$}-0.152\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=-1$, B${}_{0}=-1$)
$\displaystyle=-0.152\text{b${}_{12}$}-0.197\text{b${}_{4}$}-0.250\text{b${}_{48}$}$
$\displaystyle+0.197\text{b${}_{52}$}+0.152\text{b${}_{60}$}+0.250\text{I},$
P(A${}_{0}=+1$, B${}_{1}=+1$)
$\displaystyle=0.238\text{b${}_{12}$}+0.074\text{b${}_{4}$}+0.250\text{b${}_{48}$}$
$\displaystyle+0.074\text{b${}_{52}$}+0.238\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=+1$, B${}_{1}=-1$)
$\displaystyle=-0.238\text{b${}_{12}$}-0.074\text{b${}_{4}$}+0.250\text{b${}_{48}$}$
$\displaystyle-0.074\text{b${}_{52}$}-0.238\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=-1$, B${}_{1}=+1$)
$\displaystyle=0.238\text{b${}_{12}$}+0.074\text{b${}_{4}$}-0.250\text{b${}_{48}$}$
$\displaystyle-0.074\text{b${}_{52}$}-0.238\text{b${}_{60}$}+0.250I,$
P(A${}_{0}=-1$, B${}_{1}=-1$)
$\displaystyle=-0.238\text{b${}_{12}$}-0.074\text{b${}_{4}$}-0.250\text{b${}_{48}$}$
$\displaystyle+0.074\text{b${}_{52}$}+0.238\text{b${}_{60}$}+0.250I,$

P(B${}_{1}=+1$, A${}_{1}=+1$)
$\displaystyle=0.205\text{b${}_{12}$}+0.074\text{b${}_{16}$}+0.042\text{b${}_{20}$}$
(24)
$\displaystyle+0.061\text{b${}_{28}$}+0.143\text{b${}_{4}$}+0.238\text{b${}_{48}$}$
$\displaystyle+0.136\text{b${}_{52}$}+0.195\text{b${}_{60}$}+0.250I,$
P(B${}_{1}=+1$, A${}_{1}=-1$)
$\displaystyle=-0.205\text{b${}_{12}$}+0.074\text{b${}_{16}$}-0.042\text{b${}_{20}$}$
$\displaystyle-0.061\text{b${}_{28}$}-0.143\text{b${}_{4}$}+0.238\text{b${}_{48}$}$
$\displaystyle-0.136\text{b${}_{52}$}-0.195\text{b${}_{60}$}+0.250I,$
P(B${}_{1}=-1$, A${}_{1}=+1$)
$\displaystyle=0.205\text{b${}_{12}$}-0.074\text{b${}_{16}$}-0.042\text{b${}_{20}$}$
$\displaystyle-0.061\text{b${}_{28}$}+0.143\text{b${}_{4}$}-0.238\text{b${}_{48}$}$
$\displaystyle-0.136\text{b${}_{52}$}-0.195\text{b${}_{60}$}+0.250I,$
P(B${}_{1}=-1$, A${}_{1}=-1$)
$\displaystyle=-0.205\text{b${}_{12}$}-0.074\text{b${}_{16}$}+0.042\text{b${}_{20}$}$
$\displaystyle+0.061\text{b${}_{28}$}-0.143\text{b${}_{4}$}-0.238\text{b${}_{48}$}$
$\displaystyle+0.136\text{b${}_{52}$}+0.195\text{b${}_{60}$}+0.250I,$

P(A${}_{1}=+1$, B${}_{0}=+1$)
$\displaystyle=0.152\text{b${}_{12}$}+0.143\text{b${}_{16}$}+0.113\text{b${}_{20}$}$
(25)
$\displaystyle+0.078\text{b${}_{28}$}+0.197\text{b${}_{4}$}+0.205\text{b${}_{48}$}$
$\displaystyle+0.162\text{b${}_{52}$}+0.125\text{b${}_{60}$}+0.250I,$
P(A${}_{1}=+1$, B${}_{0}=-1$)
$\displaystyle=-0.152\text{b${}_{12}$}+0.143\text{b${}_{16}$}-0.113\text{b${}_{20}$}$
$\displaystyle-0.078\text{b${}_{28}$}-0.197\text{b${}_{4}$}+0.205\text{b${}_{48}$}$
$\displaystyle-0.162\text{b${}_{52}$}-0.125\text{b${}_{60}$}+0.250I,$
P(A${}_{1}=-1$, B${}_{0}=+1$)
$\displaystyle=0.152\text{b${}_{12}$}-0.143\text{b${}_{16}$}-0.113\text{b${}_{20}$}$
$\displaystyle-0.078\text{b${}_{28}$}+0.197\text{b${}_{4}$}-0.205\text{b${}_{48}$}$
$\displaystyle-0.162\text{b${}_{52}$}-0.125\text{b${}_{60}$}+0.250I,$
P(A${}_{1}=-1$, B${}_{0}=-1$)
$\displaystyle=-0.152\text{b${}_{12}$}-0.143\text{b${}_{16}$}+0.113\text{b${}_{20}$}$
$\displaystyle+0.078\text{b${}_{28}$}-0.197\text{b${}_{4}$}-0.205\text{b${}_{48}$}$
$\displaystyle+0.162\text{b${}_{52}$}+0.125\text{b${}_{60}$}+250I.$

P(A${}_{1}=+1$)
$\displaystyle=0.286\text{b${}_{16}$}+0.410\text{b${}_{48}$}+0.500I,$ (26)
P(E${}_{1}=+1$)
$\displaystyle=0.149\text{b${}_{1}$}+0.476\text{b${}_{3}$}+0.500I,$
P(A${}_{0}=+1$, E${}_{0}=+1$)
$\displaystyle=0.197\text{b${}_{1}$}+0.152\text{b${}_{3}$}+0.250\text{b${}_{48}$}$
$\displaystyle+0.197\text{b${}_{49}$}+0.152\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=+1$, E${}_{0}=-1$)
$\displaystyle=-0.197\text{b${}_{1}$}-0.152\text{b${}_{3}$}+0.250\text{b${}_{48}$}$
$\displaystyle-0.197\text{b${}_{49}$}-0.152\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=-1$, E${}_{0}=+1$)
$\displaystyle=0.197\text{b${}_{1}$}+0.152\text{b${}_{3}$}-0.250\text{B${}_{48}$}$
$\displaystyle-0.197\text{b${}_{49}$}-0.152\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=-1$, E${}_{0}=-1$)
$\displaystyle=-0.197\text{b${}_{1}$}-0.152\text{b${}_{3}$}-0.250\text{b${}_{48}$}$
$\displaystyle+0.197\text{b${}_{49}$}+0.152\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=+1$, E${}_{1}=+1$)
$\displaystyle=0.074\text{b${}_{1}$}+0.238\text{b${}_{3}$}+0.250\text{b${}_{48}$}$
$\displaystyle+0.074\text{b${}_{49}$}+0.238\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=+1$, E${}_{1}=-1$)
$\displaystyle=-0.074\text{b${}_{1}$}-0.238\text{b${}_{3}$}+0.250\text{b${}_{48}$}$
$\displaystyle-0.074\text{b${}_{49}$}-0.238\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=-1$, E${}_{1}=+1$)
$\displaystyle=0.074\text{b${}_{1}$}+0.238\text{b${}_{3}$}-0.250\text{b${}_{48}$}$
$\displaystyle-0.074\text{b${}_{49}$}-0.238\text{b${}_{51}$}+0.250I,$
P(A${}_{0}=-1$, E${}_{1}=-1$)
$\displaystyle=-0.074\text{b${}_{1}$}-0.238\text{b${}_{3}$}-0.250\text{b${}_{48}$}$
$\displaystyle+0.074\text{b${}_{49}$}+0.238\text{b${}_{51}$}+0.250I,$

P(E${}_{1}=+1$, A${}_{1}=+1$)
$\displaystyle=0.143\text{b${}_{1}$}+0.074\text{b${}_{16}$}+0.042\text{b${}_{17}$}$
(27)
$\displaystyle+0.061\text{b${}_{19}$}+0.205\text{b${}_{3}$}+0.238\text{b${}_{48}$}$
$\displaystyle+0.136\text{b${}_{49}$}+0.195\text{B${}_{51}$}+0.250I,$
P(E${}_{1}=+1$, A${}_{1}=-1$)
$\displaystyle=-0.143\text{b${}_{1}$}+0.074\text{b${}_{16}$}-0.042\text{b${}_{17}$}$
$\displaystyle-0.061\text{b${}_{19}$}-0.205\text{b${}_{3}$}+0.238\text{b${}_{48}$}$
$\displaystyle-0.136\text{b${}_{49}$}-0.195\text{b${}_{51}$}+0.250I,$
P(E${}_{1}=-1$, A${}_{1}=+1$)
$\displaystyle=0.1430\text{b${}_{1}$}-0.074\text{b${}_{16}$}-0.042\text{b${}_{17}$}$
$\displaystyle-0.061\text{b${}_{19}$}+0.205\text{b${}_{3}$}-0.238\text{b${}_{48}$}$
$\displaystyle-0.136\text{b${}_{49}$}-0.195\text{b${}_{51}$}+0.250I,$
P(E${}_{1}=-1$, A${}_{1}=-1$)
$\displaystyle=-0.143\text{b${}_{1}$}-0.074\text{b${}_{16}$}+0.042\text{b${}_{17}$}$
$\displaystyle+0.061\text{b${}_{19}$}-0.205\text{b${}_{3}$}-0.238\text{b${}_{48}$}$
$\displaystyle+0.136\text{b${}_{49}$}+0.195\text{b${}_{51}$}+0.250I,$

P(A${}_{1}=+1$, E${}_{0}=+1$)
$\displaystyle=0.197\text{b${}_{1}$}+0.143\text{b${}_{16}$}+0.113\text{b${}_{17}$}$
(28)
$\displaystyle+0.078\text{b${}_{19}$}+0.152\text{b${}_{3}$}+0.205\text{b${}_{48}$}$
$\displaystyle+0.162\text{b${}_{49}$}+0.125\text{b${}_{51}$}+0.250I,$
P(A${}_{1}=+1$, E${}_{0}=-1$)
$\displaystyle=-0.197\text{b${}_{1}$}+0.143\text{b${}_{16}$}-0.113\text{b${}_{17}$}$
$\displaystyle-0.078\text{b${}_{19}$}-0.152\text{b${}_{3}$}+0.205\text{b${}_{48}$}$
$\displaystyle-0.162\text{b${}_{49}$}-0.125\text{b${}_{51}$}+0.250I,$
P(A${}_{1}=-1$, E${}_{0}=+1$)
$\displaystyle=0.197\text{b${}_{1}$}-0.143\text{b${}_{16}$}-0.113\text{b${}_{17}$}$
$\displaystyle-0.078\text{b${}_{19}$}+0.152\text{b${}_{3}$}-0.205\text{b${}_{48}$}$
$\displaystyle-0.162\text{b${}_{49}$}-0.125\text{b${}_{51}$}+0.250I,$
P(A${}_{1}=-1$, E${}_{0}=-1$)
$\displaystyle=-0.197\text{b${}_{1}$}-0.143\text{b${}_{16}$}+0.113\text{b${}_{17}$}$
$\displaystyle+0.078\text{b${}_{19}$}-0.152\text{b${}_{3}$}-0.205\text{b${}_{48}$}$
$\displaystyle+0.162\text{b${}_{49}$}+0.125\text{b${}_{51}$}+0.250I.$

where $\text{b}{{}_{i}}=\text{tr}(\rho\text{B}{{}_{i}})$ with
$\text{B}{{}_{i}}$ are the Pauli operators. The order of $\text{B}{{}_{i}}$ is
in the four-base subscript and the base-four notation, 0, 1, 2, 3 can be
directly mapped to either identity or Pauli $x$, $y$, and $z$ matrices. For
example $\text{B}{{}_{3}}$ has the form $I\otimes I\otimes\sigma_{z}$ where
$I,\sigma_{z}$ are identity and Pauli $z$ matrices, respectively. Similarly we
can find the forms of other $\text{B}{{}_{i}}$ (details are given in Singh
_et al._ (2018)).

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