# Events in quantum mechanics are maximally non-absolute

George Moreno  International Institute of Physics, Federal University of Rio
Grande do Norte, 59078-970, Natal, Brazil  Departamento de Computação,
Universidade Federal Rural de Pernambuco, 52171-900, Recife, Pernambuco,
Brazil<EMAIL_ADDRESS>Ranieri Nery  International Institute of
Physics, Federal University of Rio Grande do Norte, 59078-970, Natal, Brazil
<EMAIL_ADDRESS>Cristhiano Duarte  International Institute of
Physics, Federal University of Rio Grande do Norte, 59078-970, Natal, Brazil
School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United
Kingdom<EMAIL_ADDRESS>Rafael Chaves  International Institute of
Physics, Federal University of Rio Grande do Norte, 59078-970, Natal, Brazil
School of Science and Technology, Federal University of Rio Grande do Norte,
Natal, Brazil<EMAIL_ADDRESS>
(August 16, 2022)

###### Abstract

The notorious quantum measurement problem brings out the difficulty to
reconcile two quantum postulates: the unitary evolution of closed quantum
systems and the wave-function collapse after a measurement. This problematics
is particularly highlighted in the Wigner’s friend thought experiment, where
the mismatch between unitary evolution and measurement collapse leads to
conflicting quantum descriptions for different observers. A recent no-go
theorem has established that the (quantum) statistics arising from an extended
Wigner’s friend scenario is incompatible when one try to hold together three
innocuous assumptions, namely no-superdeterminism, parameter independence and
absoluteness of observed events. Building on this extended scenario, we
introduce two novel measures of non-absoluteness of events. The first is based
on the EPR2 decomposition, and the second involves the relaxation of the
absoluteness hypothesis assumed in the aforementioned no-go theorem. To prove
that quantum correlations can be maximally non-absolute according to both
quantifiers, we show that chained Bell inequalities (and relaxations thereof)
are also valid constraints for Wigner’s experiment.

##  1 Introduction

In spite of the undeniable success of quantum mechanics as a physical theory,
its foundations remain unsettled by a number of thorny issues. At the center
of the debate is the measurement problem [1, 2].The measurement problem makes
clear the apparent mismatch between the unitary, reversible and deterministic
evolution of closed quantum systems and the non-unitary, irreversible,
probabilistic behavior after a measurement in a collapse perspective. The
problem arises from a reductionist perspective where macroscopic phenomena,
like irreversible measurements and their consequent collapse, should arise
from the behavior of its microscopic descriptions, like the unitary evolution
described by the Schrödinger’s equation [3].

The conceptual difficulties in the measurement problem become apparent in the
gedanken experiment known as "Wigner’s Friend" [4]. The scenario put forward
by the thought experiment involves a quantum system in a superposition of
states, an observer (Wigner’s friend) performing measurements on it and a
super-observer (Wigner) who observes the friend and the friend’s measurements
on the quantum system. According to the standard collapse descriptions,
Wigner’s friend describes the measurement as an irreversible process, by the
collapse of the wave function describing the quantum system into one of the
eigenstates of the observable being measured. In turn, Wigner might consider
his friend and the system the friend is interacting with as a joint quantum
system, so that from Wigner’s perspective, the measurement process is
described by a global unitary evolution generating an entangled state between
both. In this case, Wigner traces out the state of his friend to obtain the
final state of the system. Depending on who is observing, Wigner or Wigner’s
friend, we arrive then at two possible distinct descriptions for the same
physical process, an apparent contradiction between the friend and Wigner’s
perspective, since the latter does not ascribe a well-defined value to the
outcome associated with his friend’s well defined observation.

Eugene Wigner himself thought this gedanken experiment supported that
consciousness was a necessary ingredient to describe quantum measurements [4].
Contrastly, Everett and his many-worlds interpretation [5], reconciled the
irreversible nature of measurements with an unitary behavior, since each of
the possible outcomes observed by Wigner’s friend would happen, only in
different worlds. Resolutions based on hidden-variable models would required
either faster-than-light signals [6], superdeterminism [7, 8] or
retrocausality [9]. To the Copenhagen [10], Relational [11] or Quantum
Bayesian interpretations [12], there is no issue brought about by the thought
experiment, as the quantum state should be seen as relative to the observer –
a solution that might imply the rejection of the idea that measurement
outcomes are absolute and agent-independent. Alternatively, one can introduce
extra mechanisms in quantum theory, such as non-unitary quantum dynamics [13]
or the gravity-induced collapse of the wave function [14, 15], which would
rule out the existence of macroscopic superpositions.

More recently, the interest in the measurement problem and its implications
for the Wigner’s friend experiment were reignited by new results [16, 17, 18,
19, 20, 21, 22, 23, 24, 25, 26, 27, 28] showing how the assumptions of the
universality of quantum theory (its applicability to describe micro as well
macroscopic phenomena) and the absoluteness of events (globally well-defined
and observer-independent) imply experimentally testable constraints. Of
particular relevance to us is the no-go theorem in Ref. [25], that building up
on the earlier results of Brukner [16, 17] shows that there are quantum
predictions that are incompatible with the conjunction of no-superdeterminism
(NSD), parameter independence (PI) and absoluteness of observed events (AOE).
Similar to Bell’s theorem [29], such assumptions imply constraints on the
statistics arising out of that scenario, called local friendliness (LF)
inequalities in [25], that must be obeyed by any correlations compatible with
them.

Leveraging the scenario established in Ref. [25], our aim in this paper is
two-fold. First, we introduce two different measures of non-absoluteness of
observed events. Second, we prove that quantum correlations can reach the
maximum of such measures, thus proving that quantum events can be maximally
non-absolute. The first measure we consider is based on the EPR2 decomposition
[30], the fraction of events of a given observed probability distribution that
cannot be considered to be absolute. The second measure is based on the
relaxation of the assumption that events should be absolute. How much should
we give up on absoluteness in order to explain quantum correlations? With that
aim, we prove that the a relaxed version of the chained Bell inequalities [31]
constrains the set of correlations compatible with absoluteness of events.

The paper is organized as follows. In Sec. 2 we review the EWFS introduced in
[25]. In Sec. 3 we introduce the two measures of non-absoluteness employed in
the paper. In Sec. 4 we prove that the chained inequality is also a valid LF
inequality and generalize it to the case where the AOE assumption is relaxed.
Based on this inequality we prove our main result, that quantum correlations
can lead to events that are maximally non-absolute. In Sec. 5 we discuss our
findings and point out interesting future research directions.

##  2 The extended Wigner’s Friend Scenario

The bipartite extended Wigner’s friend scenario (EWFS), as introduced in Ref.
[25] consists of two observers (the friends), Charlie and Debbie, and two
superobservers Alice and Bob (the Wigners). Charlie and Debbie share a source
of correlations on which they always perform some fixed measurement each
inside a completely isolated laboratory, assumed to be space-like separated,
obtaining the outcomes $c,\;d\in\\{0,\dots,k-1\\}$ for Charlie and Debbie
respectively. Alice observes the laboratory containing Charlie and applies a
measurement $x\in\\{0,\dots,m-1\\}$ obtaining the outcome $a\in\\{0,1\\}$ in
the following way: if $x=m-1$ then $a=c$, which is equivalent to opening
Charlie’s laboratory and ask about the outcome Charlie has obtained; if $x\neq
m-1$ Alice performs an arbitrary measurement on the laboratory. The same is
done by Bob and Debbie. See Fig. 1 for an illustration of the EWFS.

To test the objectiveness of the friends’s observations, that is, the fact
that what the friend observes is "out there" and that its truth value is not
subjective to the agent observing it, the EWFS relies on three assumptions,
the conjunction of which the authors in [25] call local friendliness (LF). The
first of these assumptions, referred to as _absoluteness of observed events_
(AOE), states the existence of a joint distribution $\mathcal{P}(a,b,c,d|x,y)$
such that the experimentally obtained joint probability distribution
$p(a,b|x,y)$ satisfies:

$\displaystyle\left\\{\begin{array}[]{l}p(a,b|x,y)=\sum_{c,d}\mathcal{P}(a,b,c,d|x,y),\\\
p(a=c|x=m-1,y)=1,\\\ p(b=d|x,y=m-1)=1.\end{array}\right.$ (4)

Notice that we choose to write this condition in a slightly different way than
reference [25]. This makes the problem more approachable from the numerical
perspective while remaining completely equivalent to the original definition.

The AOE assumption assigns values only to observed outcomes. Since we deal
with the probability distribution $p(a,b|x,y)$ observed in a given experiment
realized by the two Wigners (Alice and Bob), we do not impose that
measurements that are not performed do have results. This is quite different
from the local hidden variable model entering in Bell’s theorem, that
according to Fine’s theorem [32] implies that even the outcomes of non-
performed measurements should have a well defined probability distribution. In
the EWFS, this is only the case for Alice’s and Bob’s outcome associated with
measurements $x=m-1$ and $y=m-1$. More precisely, while a behavior
$p(a,b|x,y)$ compatible with the Bell scenario can be associated with a
distribution $p(a_{0},a_{1},\dots,a_{m-1},b_{0},b_{1},\dots,b_{m-1}$, a
behavior $p(a,b|x,y)$ compatible with LF can only be associated with a
distribution $p(a,b,a_{m-1},b_{m-1}|x,y)$ or just $p(a,b,c,d|x,y)$. Under the
AOE assumption, these special outcomes have values even when the corresponding
measurements are not performed, that is, $x\neq m-1$ and $y\neq m-1$. That
follows from the fact that the outcome of Alice associated with the
measurement $x=m-1$ is encoded in the outcome $c$, which is measured in every
run of the experiment by Charlie (a similar argument holding for Bob and
Debbie).

Figure 1: Pictorial illustration of the extended Wigner’s friend experiment.
The observers Charlie and Debbie measure their parts of an entangled system,
obtaining measurement outcomes $c$ and $d$, respectively. In turn, the super-
observers Alice and Bob input $x$ and $y$ (respectively) and then observe
their friends and measurement devices, obtaining outcomes $a$ and $b$,
respectively.

The second assumption, is that of _no-superdeterminism_ (NSD), expressed as

$p(c,d|x,y)=p(c,d).$ (5)

stating that $c$ and $d$ are independent of $x$ and $y$, or equivalently that
the choices of Alice and Bob cannot affect the results of Charlie and Debbie
and the correlations shared among them. Notice that this is reminiscent of the
free-will or measurement independence assumption in Bell’s theorem [33, 34,
35, 36].

The third assumption, which we call _parameter independence_ (PI) (named
locality assumption in [25]), states that

$\left\\{\begin{array}[]{l}p(a|c,d,x,y)=p(a|c,d,x),\;\;\forall\;a,c,d,x,y,\\\
p(b|c,d,x,y)=p(b|c,d,y),\;\;\forall\;b,c,d,x,y.\end{array}\right.$ (6)

Notice that the conditions above imply the no-signalling constraints [37] over
the observed probability distribution $p(a,b|x,y)$, that is,

$\left\\{\begin{array}[]{l}p(a|x,y)=p(a|x),\;\;\forall\;a,x,y,\\\
p(b|x,y)=p(b|y),\;\;\forall\;b,x,y.\end{array}\right.$ (7)

The conjunction of assumptions (4), (5) and (6) define the set of _local
friendliness_ (LF) correlations that we call $\mathcal{S}^{LF}$.

##  3 Quantifying the absoluteness of events

It is clear that if we drop the key conditions $p(a|c,x=m-1,y)=\delta_{a,c}$
and $p(b|d,x,y=m-1)=\delta_{b,d}$ from the AOE assumption, then the set of
correlations compatible with (4), (5) and (6) is equal to the set of non-
signalling correlations defined by (7) and that we denote by
$\mathcal{S}^{NS}$. From this observation we can introduce two natural
measures quantifying the absoluteness of events.

The first one is inspired by the EPR2 decomposition [30], and quantifies the
fraction of events associated with a given probability distribution that
cannot be described by a local friendly model. Quite generally, as
$\mathcal{S}^{LF}$ is a convex polytope contained in $\mathcal{S}^{NS}$, any
observed distribution on the EWFS can be decomposed as

$p(a,b|x,y)=q\cdot p_{LF}(a,b|x,y)\\\ +(1-q)\cdot p_{NS}(a,b|x,y),$ (8)

where $p_{LF}(a,b|x,y)\in\mathcal{S}^{LF}$ and
$p_{NS}(a,b|x,y)\in\mathcal{S}^{NS}$. In other words, for a certain
probability distribution $p(a,b|x,y)$, the convex decomposition of eq. (8)
explores the fractions $q$ of local-friendliness and $1-q$ of non-signaling
that $p(a,b|x,y)$ admits. Put another way, one may want to see a particular
decomposition of $p(a,b|x,y)$ as in eq. (8) as a signature of the fraction $q$
of local-friendliness an event might have. In this sense, our first measure of
absoluteness of events, that we call _non-absoluteness fraction_
$\mathcal{A}_{f}$, is then defined as the minimal fraction of non-absolute
events weight over all possible decompositions as in (8), that is,

$\mathcal{A}_{f}(p(a,b|x,y))=\min_{p_{LF},p_{NS}}(1-q),$ (9)

an optimization that can be performed using a linear programming [38]. Notice
that LF correlations have $\mathcal{A}_{f}=0$ while correlations with
maximally non-absolute events have $\mathcal{A}_{f}=1$. A possible
interpretation of $\mathcal{A}_{f}$ can be given as follows. If one considers
the that joint probabilities have a frequentist meaning, in a statistical
experiment, repeated many times, the non-absoluteness fraction
$\mathcal{A}_{f}$ can be understood as the fraction of runs of the experiment
where the observed measurements outcomes can be deemed to be non-absolute, in
the sense that they are incompatible with the conditions $p(a=c|x=m-1,y)=1$
and $p(b=d|x,y=m-1)=1$.

Notice that any LF inequality provides a lower bound to the non-absoluteness
fraction. Given a generic LF inequality of the form
$I(p(a,b|x,y))=\sum_{a,b,x,y}\omega_{a,b,x,y}p(a,b|x,y)\leq\Omega_{LF}$ where
$\Omega_{LF}$ is the maximum value achievable by a LF correlation, it follows
that

$\mathcal{A}_{f}\geq
1-\frac{\Omega_{NS}-\Omega_{Q}}{\Omega_{NS}-\Omega_{LF}},$ (10)

where $\Omega_{NS}$ is the maximum value achievable by no-signalling
correlations and $\Omega_{Q}$ is the value corresponding to a given quantum
probability distribution under test. This bound follows from the fact that
$I(p)=\Omega_{Q}=q\cdot I(p_{LF})+(1-q)\cdot I(p_{NS})\leq
q\cdot\Omega_{LF}+(1-q)\cdot\Omega_{NS}$. In particular, notice that if
$\Omega_{Q}=\Omega_{NS}$, that is, the quantum value achieves the maximum no-
signalling violation of the LF inequality, then it follows that
$\mathcal{A}_{f}=1$.

As mentioned, the mathematical expression of $\mathcal{A}_{f}$ is similar to
the so-called non-local fraction $\mathcal{N}_{f}$, introduced in Ref. [30],
that quantifies the non-locality of a given distribution $p(a,b|x,y)$ in a
Bell scenario. The key difference, however, is the fact that in the Bell case
the probability $p_{LF}$ should be substituted by $p_{Local}$, that is, by
probabilities compatible with local hidden variable models. Since the local
friendly set of correlations is in general bigger than the local set, we have
that, for a given $p(a,b|x,y)$, $\mathcal{N}_{f}\geq\mathcal{A}_{f}$—recall
that we are minimizing the function $(1-q)$. In particular, we can have
situations where $\mathcal{N}_{f}>0$ but $\mathcal{A}_{f}=0$. Regarding the
bound in Eq. (10) the same idea holds. In the case of a Bell scenario we would
need to replace $\Omega_{LF}$ by $\Omega_{Local}$, the bound of a given
inequality respected by Bell local correlations.

As a second measure of absoluteness of events we consider a relaxation of the
condition in (4), that we call _$\epsilon-$ AOE_. Namely, we now allow that

$\displaystyle\left\\{\begin{array}[]{l}p(a,b|x,y)=\sum_{c,d}\mathcal{P}(a,b,c,d|x,y),\\\
p(a=c|x=m-1,y)\geq 1-\epsilon,\\\ p(b=d|x,y=m-1)\geq 1-\epsilon.\\\
\end{array}\right.$ (14)

The conjunction of (14), (5) and (6) define the set of _relaxed local
friendliness_ (RLF) correlations denoted as $\mathcal{S}^{RLF}_{\epsilon}$.
Clearly

$\mathcal{S}^{LF}\subseteq\mathcal{S}^{RLF}_{\epsilon}\subseteq\mathcal{S}^{RLF}_{\epsilon^{\prime}}\subseteq\mathcal{S}^{NS}.$
(15)

whenever $\epsilon\leq\epsilon^{\prime}$. See Fig. 2 for a pictorial
illustration.

From this we can define the _non-absoluteness coefficient_ $\mathcal{A}_{c}$,
given by

$\mathcal{A}_{c}(p(a,b|x,y))=\min 2\epsilon\text{ s.t.
}p(a,b|x,y)\in\mathcal{S}^{RLF}_{\epsilon}.$ (16)

Observed distributions compatible with LF have $\mathcal{A}_{c}=0$ while
maximally non-absolute events are those where $\mathcal{A}_{c}=1$, thus
implying that the measurement outcome of Alice when her input is $x=m-1$ can
be completely uncorrelated from that of her friend Charlie (similarly for Bob
and Debbie).

Notice that the non-absoluteness coefficient $\mathcal{A}_{c}$, as opposed to
the non-absolutness fraction $\mathcal{A}_{f}$, has no analogue in a Bell
scenario, since it is defined via the relaxation of the absoluteness of
events, an assumption that plays absolutely no role in Bell’s theorem. In our
view, it is precisely the relaxation of the absoluteness of events (AOE) that
makes clear the distinction between the extended Wigner scenario (local
friendliness) and the Bell scenario (local realism). As for the interpretation
of the non-absolutness coefficient $\mathcal{A}_{c}$, notice that is does not
take the conditions $p(a=c|x=m-1,y)=1$ and $p(b=d|x,y=m-1)=1$ for granted.
Quite the contrary, asks how much such conditions have to be relaxed, in the
form $p(a=c|x=m-1,y)\geq 1-\epsilon$ and $p(b=d|x,y=m-1)=1-\epsilon$, such
that the observed data admits an explanation. In this sense, the measure
$\mathcal{A}_{c}$ is related to a correlation set that contains the local
friendly set and tends to the non-signalling set as
$\mathcal{A}_{c}\rightarrow 1$.

Figure 2: Pictorial illustration of the $\mathcal{S}^{LF}$, its relaxed
version and $\mathcal{S}^{NS}$. As stated in the main text
$\mathcal{S}^{LF}\subseteq\mathcal{S}^{RLF}_{\epsilon}\subseteq\mathcal{S}^{RLF}_{\epsilon^{\prime}}\subseteq\mathcal{S}^{NS}$
whenever $\epsilon\leq\epsilon^{\prime}$. Furthermore
$\mathcal{S}^{LHV}\subseteq\mathcal{S}^{LF}$ where $\mathcal{S}^{LHV}$ refers
to the set of Bell local correlations. A point over the hypersurface limiting
$\mathcal{S}^{RLF}_{\epsilon}$ has $\mathcal{A}_{c}=2\epsilon$ and
$\mathcal{A}_{f}=(1-q)$.

Our first result shows the relation between both measures.

######  Result 1.

The non-absoluteness coefficient is upper bounded by the non-absoluteness
fraction, that is

$\mathcal{A}_{c}\leq\mathcal{A}_{f}.$ (17)

###### Proof.

First, we prove that $\mathcal{A}_{c}$ is a convex function: Let
$p(a,b|x,y)=q\,p_{A}(a,b|x,y)+(1-q)\,p_{B}(a,b|x,y)$, with $0\leq q\leq 1$.
Assume that $\mathcal{A}_{c}(p_{A})=2\epsilon_{A}$ and
$\mathcal{A}_{c}(p_{B})=2\epsilon_{B}$ and assume that these optimal values
are attainable with global distributions $\mathcal{P}_{A}(a,b,c,d|x,y)$ and
$\mathcal{P}_{B}(a,b,c,d|x,y)$, respectively.

A possible attempt for solving $\mathcal{A}_{c}(p)$ is thus provided by
$q\,\mathcal{P}_{A}(a,b,c,d|x,y)+(1-q)\,\mathcal{P}_{B}(a,b,c,d|x,y)$, which
enables us to ensure that, at least,
$p(a,b|x,y)\in\mathcal{S}^{RLF}_{\epsilon}$ for
$\epsilon=q\,\epsilon_{A}+(1-q)\,\epsilon_{B}$. Since the proposed solution is
not necessarily optimal, we must have $\mathcal{A}_{c}(p)\leq
q\,2\epsilon_{a}+(1-q)\,2\epsilon_{B}$, which, by definition, is the same as

$\mathcal{A}_{c}(q\,p_{A}+(1-q)\,p_{B})\leq
q\,\mathcal{A}_{c}(p_{A})+(1-q)\,\mathcal{A}_{c}(p_{B}).$

Consider now a possible realization for $\mathcal{A}_{f}(p)$, given by
$p_{LF}$ and $p_{NS}$, i.e. $p=q^{*}\,p_{LF}+(1-q^{*})\,p_{NS}$, where
$1-q^{*}=\mathcal{A}_{f}(p)$. Using the convexity of $\mathcal{A}_{c}$ upon
this decomposition leads to

$\mathcal{A}_{c}(p)\leq
q^{*}\,\mathcal{A}_{c}(p_{LF})+(1-q^{*})\,\mathcal{A}_{c}(p_{NS}),$

but since $\mathcal{A}_{c}(p_{LF})=0$, we identify the rhs of the above
expression with $\mathcal{A}_{f}(p)\,\mathcal{A}_{c}(p_{NS})$, leading to the
inequality

$\mathcal{A}_{c}(p)\leq\mathcal{A}_{f}(p)\,\mathcal{A}_{c}(p_{NS}),$

for a given $p_{NS}$ participating in an optimal decomposition of $p(a,b|x,y)$
for $\mathcal{A}_{f}(p)$. We could tighten the inequality by performing a
linear program minimizing $\mathcal{A}_{c}(p_{NS})$ within every possible
decomposition for $p$ that has a fixed weight $1-q=\mathcal{A}_{f}(p)$. Going
on the opposite direction, however, we may loosen the inequality by
considering the general bound $\mathcal{A}_{c}(p_{NS})\leq 1$, which leads to
the general lower bound for $\mathcal{A}_{f}$:

$\mathcal{A}_{c}(p)\leq\mathcal{A}_{f}(p).$ (18)

∎

To illustrate both measures of non-absoluteness in a concrete scenario, we
first consider relaxations of the complete set of LF inequalities derived in
[25] for scenario where $m=3$ and $k=2$. This case is completely characterized
by 6 classes of non-trivial LF inequalities given by

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle-\langle
A_{2}\rangle-\langle A_{1}\rangle-\langle B_{2}\rangle-\langle
B_{1}\rangle-\langle A_{2}B_{2}\rangle$ $\displaystyle-$ $\displaystyle
2\langle A_{2}B_{1}\rangle-2\langle A_{1}B_{2}\rangle+2\langle
A_{1}B_{1}\rangle-\langle A_{1}B_{0}\rangle$ $\displaystyle-$
$\displaystyle\langle A_{0}B_{1}\rangle-\langle
A_{0}B_{0}\rangle\overset{\Omega_{1}}{\leq}6,$ $\displaystyle I_{2}$
$\displaystyle=$ $\displaystyle-\langle A_{2}\rangle-\langle
A_{1}\rangle-\langle A_{0}\rangle-\langle B_{2}\rangle-\langle
A_{2}B_{2}\rangle$ $\displaystyle-$ $\displaystyle\langle
A_{1}B_{2}\rangle-\langle A_{0}B_{2}\rangle-2\langle A_{2}B_{1}\rangle+\langle
A_{1}B_{1}\rangle$ $\displaystyle+$ $\displaystyle\langle
A_{0}B_{1}\rangle-\langle A_{1}B_{0}\rangle+\langle
A_{0}B_{0}\rangle\overset{\Omega_{2}}{\leq}5,$ $\displaystyle I_{3}$
$\displaystyle=$ $\displaystyle-\langle A_{2}\rangle+\langle
A_{1}\rangle+\langle B_{2}\rangle-\langle B_{1}\rangle+\langle
A_{2}B_{2}\rangle$ $\displaystyle-$ $\displaystyle\langle
A_{2}B_{1}\rangle-\langle A_{2}B_{0}\rangle-\langle A_{1}B_{2}\rangle+\langle
A_{1}B_{1}\rangle$ $\displaystyle-$ $\displaystyle\langle
A_{1}B_{0}\rangle-\langle A_{0}B_{0}\rangle-\langle
A_{0}B_{1}\rangle\overset{\Omega_{3}}{\leq}4,$ $\displaystyle I_{4}$
$\displaystyle=$ $\displaystyle-\langle A_{1}\rangle-\langle
A_{0}\rangle-\langle B_{1}\rangle-\langle B_{0}\rangle-\langle
A_{2}B_{1}\rangle$ $\displaystyle+$ $\displaystyle\langle
A_{2}B_{0}\rangle-\langle A_{1}B_{2}\rangle-\langle A_{1}B_{1}\rangle-\langle
A_{1}B_{0}\rangle$ $\displaystyle+$ $\displaystyle\langle
A_{0}B_{2}\rangle-\langle A_{0}B_{1}\rangle-\langle
A_{0}B_{0}\rangle\overset{\Omega_{4}}{\leq}4,$ $\displaystyle I_{5}=\langle
A_{2}B_{2}\rangle-\langle A_{2}B_{0}\rangle+\langle A_{1}B_{2}\rangle+\langle
A_{1}B_{0}\rangle\overset{\Omega_{5}}{\leq}2,$ $\displaystyle I_{6}=\langle
A_{2}B_{1}\rangle-\langle A_{2}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle
A_{0}B_{0}\rangle\overset{\Omega_{6}}{\leq}2,$

where $\langle A_{x}B_{y}\rangle=\sum_{a,b=0,1}(-1)^{a+b}p(a,b|x,y)$ is the
expectation value of the measurement outcomes of $a$ and $b$ Alice and Bob
given the inputs $x$ and $y$, respectively (similarly for the marginal
expectation values $\langle A_{x}\rangle$ and $\langle B_{y}\rangle$).

Using the linear program described in the Appendix, we can see how these LF
inequalities change if we allow for relaxations of the AOE - our condition
(14). It follows that the LF bound $\Omega_{i}$ for each inequality $I_{i}$ is
modified to $\Omega_{i}^{\epsilon}$ as

$\displaystyle\Omega_{1}^{\epsilon}=6+8\epsilon,$
$\displaystyle\Omega_{2}^{\epsilon}=5+8\epsilon,$
$\displaystyle\Omega_{3}^{\epsilon}=\Omega_{4}^{\epsilon}=4+8\epsilon,$
$\displaystyle\Omega_{5}^{\epsilon}=\Omega_{6}^{\epsilon}=2+4\epsilon.$

It follows immediately that a probability distribution $p(a,b|x,y)$ achieving
a value of $I_{i}$ given by $\Omega_{i}^{\epsilon}$ has
$\mathcal{A}_{c}=2\epsilon$. As we argue next, this is the same value for
$\mathcal{A}_{f}$, thus implying that $\mathcal{A}_{f}=\mathcal{A}_{c}$ in
those cases. To see that, notice that if $I_{i}=\Omega_{i}^{\epsilon}$ then
the lower bound given by eq. (10) implies that $\mathcal{A}_{f}\geq
2\epsilon$. Further we notice that any decomposition $p(a,b|x,y)=q\cdot
p_{LF}(a,b|x,y)+(1-q)\cdot p_{NS}(a,b|x,y)$ provides an upper bound given by
$\mathcal{A}_{f}\leq(1-q)$. In particular, if we choose $p_{NS}$ as a no-
signalling (NS) distribution achieving the maximum $\Omega^{\epsilon=1/2}_{i}$
of inequality $I_{i}$, $p_{LF}$ as a LF distribution achieving
$I_{i}=\Omega_{i}$, and make $q=1-2\epsilon$, we obtain that $p(a,b|x,y)$
leads to $I_{i}=\Omega^{\epsilon}_{i}$ with $1-q=2\epsilon$ and thus
$\mathcal{A}_{f}\leq 2\epsilon$. As the lower and upper bounds for
$\mathcal{A}_{f}$ coincide, it follows that
$\mathcal{A}_{c}=\mathcal{A}_{f}=2\epsilon$.

There are cases, however, where this equivalence between $\mathcal{A}_{c}$ and
$\mathcal{A}_{f}$ does not hold any longer. To show that, we consider a
tripartite scenario. As described in the Appendix, the following symmetry of
Mermin’s inequality [39] is also a valid LF inequality:

$M=\langle A_{3}B_{3}C_{2}\rangle+\langle A_{1}B_{1}C_{2}\rangle\\\ +\langle
A_{1}B_{3}C_{1}\rangle-\langle A_{3}B_{1}C_{1}\rangle\leq\beta_{M}=2.$ (19)

If we allow for a relaxation $\epsilon$ of the AOE assumption, it follows that
the LF bound is changed as $\Omega_{M}^{\epsilon}=2+8\epsilon$. This implies
that a probability distribution achieving $M=\Omega_{M}^{\epsilon}$ requires
$\mathcal{A}_{c}=2\epsilon$. In turn, using the same approach described above
for the bipartite case, we can obtain lower and upper bounds for
$\mathcal{A}_{f}$ that coincide and lead to $\mathcal{A}_{f}=4\epsilon$. That
is, in this case, $\mathcal{A}_{c}=\mathcal{A}_{f}/2$.

##  4 Quantum events are maximally non-absolute

Our main objective is to prove that quantum correlations can be maximally non-
absolute according to both measures introduced above, that is, that
$\mathcal{A}_{f}(p_{Q})=1=\mathcal{A}_{c}(p_{Q})$ for some non-local joint
probability distribution $p_{Q}(a,b|x,y)$ in the quantum set [40]. To do so,
we first prove a general inequality for the EWFS with dichotomic outcomes but
arbitrary measurements and an AOE relaxation as in eqs.(14).

######  Result 2.

For an EWFS with $k=2$ (2 outcomes), arbitrary $m$ (measurement inputs), and
with the AOE relaxed as in eqs.(14) the following inequality

$\displaystyle C^{(m-1)}$ $\displaystyle:=$ $\displaystyle\langle
A_{m-1}B_{m-1}\rangle-\langle A_{0}B_{m-1}\rangle$ $\displaystyle+$
$\displaystyle\sum_{l=0}^{m-2}\left(\langle A_{l}B_{l}\rangle+\langle
A_{l+1}B_{l}\rangle\right)$ $\displaystyle\leq$ $\displaystyle
2(m-1)+4\epsilon$

tightly bounds the set $S^{RLF}_{\epsilon}$.

Before proceeding to our main result, it is worthy mentioning that if
$\epsilon=0$, that is, in the usual EWFS, the inequality (2) is one of the
symmetries of the so-called chained Bell inequalities [31].

Next we state our main result.

######  Result 3.

Quantum mechanics predicts probabilities distributions $p_{Q}(a,b|x,y)$ in the
extended Wigner’s friend scenario that are maximally non-absolute according to
the non-absoluteness coefficient as well as the non-absoluteness fraction.
That is, for which $\mathcal{A}_{f}(p_{Q})=1$ and $\mathcal{A}_{c}(p_{Q})=1$.

###### Proof.

The maximal value of $C^{(m-1)}$ allowed in quantum mechanics is [42]

$\displaystyle C^{(m-1)}_{Q_{\mbox{max}}}=2m\cos\left(\frac{\pi}{2m}\right),$

which can be reached by using a two-qubit system in the entangled state
$|\Psi\rangle$,

$\displaystyle|\Psi\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right),$

shared between Alice and Bob. Alice measures one of the operators
$A_{j}=r_{j}\sigma_{x}+s_{j}\sigma_{z}$ and Bob
$B_{j}=r_{j}^{\prime}\sigma_{x}+s_{j}^{\prime}\sigma_{z}$ for
$j\in\\{0,\dots,m-1\\}$. Here, $\sigma_{x}$ and $\sigma_{z}$ standing for the
Pauli matrices, and

$\displaystyle r_{j}$ $\displaystyle=$
$\displaystyle\sin\left(\frac{j\pi}{m}\right),\;\;\;s_{j}=\cos\left(\frac{j\pi}{m}\right),$

and

$\displaystyle r_{j}^{\prime}$ $\displaystyle=$
$\displaystyle\sin\left(\frac{(2j+1)\pi}{m}\right),\;\;\;s_{j}^{\prime}=\cos\left(\frac{(2j+1)\pi}{m}\right).$

Based on the fact that quantum correlations allow for maximal violations of
the chained inequality (2), we will first prove that $\mathcal{A}_{f}=1$.
Using the lower bound in eq. (10) it follows that if $\Omega_{Q}=\Omega_{NS}$
then $\mathcal{A}_{f}=1$ and that it is exactly the case of the chained
inequality (2), since the maximum quantum violation is equal to the algebraic
maximum (and thus also to $\beta_{NS}$) in the limit $m\rightarrow\infty$.
This proves that quantum correlations can be maximally non-absolute according
to the measure $\mathcal{A}_{f}$.

The proof that $\mathcal{A}_{c}=1$ also follows a simple argument. Notice the
maximal quantum violation of the chained inequality reaches $C^{m-1}=2m$ a
result that requires a relaxation $\epsilon=1/2$ in (2) and thus implies
maximal non-absoluteness of quantum correlations also regarding the measure
$\mathcal{A}_{c}$. ∎

##  5 Discussion

The interest in the quantum measurement problem and its formulation via
Wigner’s thought experiment have been recently reignited by a number of
results and no-go theorems [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. Similarly
to what Bell’s theorem [29] did for the EPR experiment [43], these results
establish testable constraints of the assumptions – absoluteness of observed
events, no-superdeterminism and parameter independence – underlying such
Wigner’s scenarios. Holding on the assumptions of no-superdeterminism and
parameter independence, the violation of some constraints, called local
friendliness inequalities, are thus experimental witnesses of the non-
absoluteness of events. Likewise to the Bell case [34, 44, 45, 46, 47], it is
natural then to quantify the degree of non-absoluteness of events implied by
the violation of a LF inequality.

With that aim in mind, we have introduced two different ways to quantify non-
absoluteness. The first measure, called the non-absoluteness fraction
$\mathcal{A}_{f}$, is based on the EPR2 decomposition [30] and measures the
fraction of measurement outcomes of given experiment that cannot be considered
absolute. The second measure, called non-absoluteness coefficient
$\mathcal{A}_{c}$, considers an explicit relaxation of the assumption of
absoluteness of observed events and thus allows the predictions of two
observers about a given measurement outcome to differ. We have established
that $\mathcal{A}_{f}\leq\mathcal{A}_{c}$, and have also detailed particular
cases where the measures coincide and others where they do not coincide, which
shows that both quantities are not trivial.

Following that, we show that a given symmetry 111Any Bell inequality has a
number of symmetries, other valid inequalities obtained via the permutation of
measurement outcomes, inputs or even the exchange of parties. Those
operations, however, are not generally valid when we consider an extended
Wigner friend scenario, as now the parties are not on par with each other. For
instance, the absoluteness of events constraint specifies ”special” inputs,
for which the corresponding measurement outcome should be the same as that
obtained by the related friend. This would break the symmetry of the
inequality related to the permutation of inputs. It is in this sense that we
say that even though a given Bell inequality can also be a LF inequality, not
all symmetries of that same inequality will have the same property. of the
chained Bell inequality [31] is still a valid LF inequality, and also
generalized it to the case where relaxations of AOE are allowed. With that, we
were able to conclude that quantum correlations can be maximally non-absolute
according to the two considered measures, that is,
$\mathcal{A}_{f}=\mathcal{A}_{c}=1$.

Notice that any local friendliness inequality is also valid Bell inequality.
Theoretically, that is no issue, since the assumptions and the physical
scenario is quite different in both cases. This is made clear by the
relaxation of AOE that we consider here, a relaxation that simply would make
no sense in a Bell scenario. Experimentally, however, the issue is trickier.
From a device-independent perspective, the finer details of the experiment and
the fact the systems that Alice and Bob measure are actually themselves
observers are irrelevant, since Charlie, Debbie, their labs and quantum
systems should be treated as a black-box and are thus indistinguishable from a
simple common source in a Bell scenario. How can one then decide whether the
experimental violation of an LF inequality is not just witnessing the standard
Bell nonlocality rather than non-absoluteness of events? One option is to move
to a semi-device-independent description where we can distinguish the
observers inside the box from a simple source of correlations. Another option,
would be to find experimentally testable differences between Bell and the
extended Wigner scenarios. For instance, could it be that free-operations for
Bell nonlocality can nonetheless increase the level of non-absoluteness of a
given observed correlation (see Fig 3)? If that would be the case, then one
could argue that an experiment where such operations are realized and an
increase in the violation of an LF inequality is observed, would be a clear
signature for non-absoluteness instead of the good old Bell nonlocality. For
that, a resource theory of local friendliness will be needed and we hope our
work might motivate attempts in this direction.

Figure 3: Pictorial representation of the set of correlations such that
$\mathcal{S}^{LV}\subset\mathcal{S}^{RLF}\subset\mathcal{S}^{NS}$. Notice that
if the set of free operations for non-locality and non-absoluteness differ, it
might be possible to use free operations for non-locality that nonetheless
might activate the violation of a LF inequality.

##  6 Acknowledgements

We thank Noah Chaves Leniger for the help with the figures. This work was
supported by The John Templeton Foundation via the grant Q-CAUSAL No 61084
(the opinions expressed in this publication are those of the author(s) and do
not necessarily reflect the views of the John Templeton Foundation), by the
Serrapilheira Institute (Grant No. Serra-1708-15763), by the Simons Foundation
(Grant Number 884966, AF), the Brazilian National Council for Scientific and
Technological Development (CNPq) via the National Institute for Science and
Technology on Quantum Information (INCT-IQ) and Grants No. 406574/2018-9 and
307295/2020-6, the Brazilian agencies MCTIC, CAPES and MEC. This research was
also supported by the Fetzer Franklin Fund of the John E. Fetzer Memorial
Trust and by grant number FQXi-RFP-IPW-1905 from the Foundational Questions
Institute and Fetzer Franklin Fund, a donor advised fund of Silicon Valley
Community Foundation.

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##  Appendix A The linear program formulation for the relaxation of
absoluteness of Observed Events

A numerical approach to assess the effects of a relaxation on the AOE
condition is possible by writing the problem as a linear program. To see this,
first we show that the conditions PI and NSD are equivalent to the following
condition,

$\displaystyle\left\\{\begin{array}[]{lll}p(a,c,d|x,y)&=&p(a,c,d|x),\;\;\forall\;\;a,c,d,x,y,\\\
p(b,c,d|x,y)&=&p(b,c,d|y),\;\;\forall\;\;b,c,d,x,y.\end{array}\right.$ (23)

Indeed, (5) and (6) imply (23),

$\displaystyle p(a|c,d,x,y)$ $\displaystyle=$ $\displaystyle p(a|c,d,x)$
$\displaystyle\implies p(a|c,d,x,y)p(c,d|x,y)$ $\displaystyle=$ $\displaystyle
p(a|c,d,x)p(c,d|x,y)$ $\displaystyle\implies p(a|c,d,x,y)p(c,d|x,y)$
$\displaystyle=$ $\displaystyle p(a|c,d,x)p(c,d|x)$ $\displaystyle\implies
p(a,c,d|x,y)$ $\displaystyle=$ $\displaystyle p(a,c,d|x).$

Also, (23) implies (5),

$\displaystyle p(a,c,d|x,y)=p(a,c,d|x)\implies p(c,d|x,y)=p(c,d|x),$ (24)

and

$\displaystyle p(b,c,d|x,y)=p(b,c,d|y)\implies p(c,d|x,y)=p(c,d|y).$ (25)

Finally, (23) implies (6),

$\displaystyle p(a,c,d|x,y)$ $\displaystyle=$ $\displaystyle p(a,c,d|x)$
$\displaystyle\implies p(a|c,d,x,y)p(c,d|x,y)$ $\displaystyle=$ $\displaystyle
p(a|c,d,x)p(c,d|x)$ $\displaystyle\implies p(a|c,d,x,y)p(c,d|x)$
$\displaystyle=$ $\displaystyle p(a|c,d,x)p(c,d|x)$ $\displaystyle\implies
p(a|c,d,x,y)$ $\displaystyle=$ $\displaystyle p(a|c,d,x).$

Notice that the same approach works by replacing $a$ by $b$.

Using the relation 14, the problem of finding the upper bound for
$I=\sum_{a,b,x,y}\omega_{a,b,x,y}p(a,b|x,y)$ in the presence of a relaxation
$\epsilon$, i.e., by a behavior featuring $\mathcal{A}_{c}=2\epsilon$, can be
formally expressed as a linear program:

$\displaystyle\begin{split}\underset{\mathbf{q}\in\mathbbm{R}^{k^{4}*m^{2}}}{\max}\quad&\langle\mathbf{I},\Pi_{a,b,x,y}q\rangle\\\
\text{s.t.}\quad&\mathbf{q}\geq 0,\\\ &N\mathbf{q}=\bm{1}^{(m^{2})},\\\
&(\Pi_{a,c,d,x,y}-\Pi_{a,c,d,x,y^{\prime}})\mathbf{q}=\bm{0}^{(k^{2}m)}\;\;\;\;\forall
a,c,d,x,y,y^{\prime},\\\
&(\Pi_{b,c,d,x,y}-\Pi_{b,c,d,x^{\prime},y})\mathbf{q}=\bm{0}^{(k^{2}m)}\;\;\;\;\forall
a,c,d,x,x^{\prime},y,\\\ &\Pi_{a=c,x=0,y}\geq(1-\epsilon)\bm{1}^{(m)},\\\
&\Pi_{b=d,x,y=0}\geq(1-\epsilon)\bm{1}^{(m)},\end{split}$ (26)

in which $\bm{1}^{(\cdot)}$ is a column vector of the dimension indicated
inside the brackets, the entries of which are always $1$ and the vector
$\bm{0}^{(\cdot)}$ is defined analogously. The matrix $N$ is given by,

$\displaystyle
N=\left[\begin{array}[]{cccc}\bm{1}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}\\\
\bm{0}^{(k^{4}*m^{2})}&\bm{1}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}\\\
\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{1}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}\\\
\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{0}^{(k^{4}*m^{2})}&\bm{1}^{(k^{4}*m^{2})}\\\
\end{array}\right]^{T}.$ (31)

The matrix $\Pi_{a,b,x,y}$ projects the vector associated with a distribution
$p(a,b,c,d|x,y)$ into the vector associated with its marginal distribution
$p(a,b|x,y)$. The matrices $\Pi_{a,c,d,x,y=0}$, $\Pi_{a,c,d,x,y=1}$,
$\Pi_{b,c,d,x=0,y}$, and $\Pi_{b,c,d,x=1,y}$ are defined in the same way,
while $\Pi_{a=c,x=0,y}$ ($\Pi_{b=d,x,y=0}$) returns a vector containing the
probability that $a=c$ ($b=d$) for different values of $y$ ($x$) as entries.

Finally, in equation (26), $\mathbf{I}$ is a vector such that
$\langle\mathbf{I},\mathbf{p}\rangle=\sum_{a,b,x,y}\omega_{a,b,x,y}p(a,b|x,y)$,
for $\mathbf{p}$ being the vector associated with the distribution
$p(a,b|x,y)$.

This analysis can be directly extended into the n-partite case, by
generalizing relaxation (14), and conditions (6) and (5) accordingly. By
performing this analysis in the tripartite case, we observe that some
symmetries of the Mermin inequality are also valid LF inequalities. For
instance given a relaxation of AOE by an amount $\epsilon$ we have that,

$\displaystyle M=\langle A_{3}B_{3}C_{2}\rangle+\langle
A_{1}B_{1}C_{2}\rangle+\langle A_{1}B_{3}C_{1}\rangle-\langle
A_{3}B_{1}C_{1}\rangle\leq\Omega_{M}^{\epsilon}=2+8\epsilon.$ (32)

in which, $\Omega_{M}^{\epsilon}=2+8\epsilon$.

##  Appendix B Proof of Result2

###### Proof.

We begin noticing that our $\epsilon-$AOE relaxation is analogous to allowing
a certain degree of indeterminism in the scenario described in ref. [41].

Consider relaxation 14, and define the quantities $\alpha_{c,d}$,
$\beta_{c,d}$, and $\gamma_{c,d}$,

$\displaystyle\gamma^{(c,d)}_{x,y}$ $\displaystyle:=$ $\displaystyle
p(a=0,b=0|c,d,x,y),$ $\displaystyle\alpha^{(c,d)}_{x,y}$ $\displaystyle:=$
$\displaystyle p(a=0|c,d,x,y),$ $\displaystyle\beta^{(c,d)}_{x,y}$
$\displaystyle:=$ $\displaystyle p(b=0|c,d,x,y).$

Positivity implies that

$\displaystyle\max\\{0,\alpha^{(c,d)}_{x,y}+\beta^{(c,d)}_{x,y}-1\\}\leq\gamma^{(c,d)}_{x,y}\leq\min\\{\alpha^{(c,d)}_{x,y},\beta^{(c,d)}_{x,y}\\}.$
(33)

Now define the _conditional correlators_ $\langle
A_{x}B_{y}\rangle_{c,d}=\sum_{a,b}(-1)^{a+b}p(a,b|c,d,x,y)$. It holds that,

$\displaystyle\langle A_{x}B_{y}\rangle_{c,d}$ $\displaystyle=$ $\displaystyle
1+4\gamma^{(c,d)}_{x,y}-2\alpha^{(c,d)}_{x,y}-2\beta^{(c,d)}_{x,y},$

which, combined with inequalities in eq. (33), and using the fact that
$2\max\\{r,s\\}=r+s+|r-s|$ for any real $r$ and $s$, leads to

$\displaystyle
2\left|\alpha^{(c,d)}_{x,y}+\beta^{(c,d)}_{x,y}-1\right|-1\leq\langle
A_{x}B_{y}\rangle_{c,d}\leq
1-2\left|\alpha^{(c,d)}_{x,y}-\beta^{(c,d)}_{x,y}\right|.$

Hence the quantity $C_{c,d}^{m-1}=\langle A_{m-1}B_{m-1}\rangle_{c,d}-\langle
A_{0}B_{m-1}\rangle_{c,d}+\sum_{l=0}^{m-2}\left(\langle
A_{l}B_{l}\rangle_{c,d}+\langle A_{l+1}B_{l}\rangle_{c,d}\right)$ is bounded,

$\displaystyle C_{c,d}^{m-1}$ $\displaystyle\leq$ $\displaystyle
2-2\left|\alpha^{(c,d)}_{m-1,m-1}-\beta^{(c,d)}_{m-1,m-1}\right|-2\left|\alpha^{(c,d)}_{0,m-1}+\beta^{(c,d)}_{0,m-1}-1\right|+\sum_{l=0}^{m-2}\left(2-2\left|\alpha^{(c,d)}_{l,l}-\beta^{(c,d)}_{l,l}\right|\right.$
$\displaystyle-$
$\displaystyle\left.2\left|\alpha^{(c,d)}_{l+1,l}-\beta^{(c,d)}_{l+1,l}\right|\right).$

Using condition 6, the above expression can be simplified replacing
$\alpha^{(c,d)}_{x,y}$ ($p(a=0|c,d,x,y)$) and $\beta^{(c,d)}_{x,y}$
($p(b=0|c,d,x,y)$), by $\alpha^{(c,d)}_{x}$ ($p(a=0|c,d,x)$) and
$\beta^{(c,d)}_{y}$ ($p(b=0|c,d,y)$) respectively,

$\displaystyle C_{c,d}^{m-1}$ $\displaystyle\leq$ $\displaystyle 2m-2J_{c,d},$

in which

$\displaystyle J_{c,d}$ $\displaystyle=$
$\displaystyle\left|\alpha^{(c,d)}_{m-1}-\beta^{(c,d)}_{m-1}\right|+\left|\alpha^{(c,d)}_{0}+\beta^{(c,d)}_{m-1}-1\right|+\sum_{l=0}^{m-2}\left(\left|\alpha^{(c,d)}_{l}-\beta^{(c,d)}_{l}|+|\alpha^{(c,d)}_{l+1}-\beta^{(c,d)}_{l}\right|\right).$

Notice that:

$\displaystyle\left|\alpha_{l}^{(c,d)}-\beta_{l}^{(c,d)}\right|+\left|\alpha_{l+1}^{(c,d)}-\beta_{l}^{(c,d)}\right|\geq\left|\alpha_{l}^{(c,d)}-\alpha_{l+1}^{(c,d)}\right|,$

with the equality holding true for $\beta_{l}^{(c,d)}$ verifying

$\displaystyle\beta_{l}^{(c,d)}=\frac{1}{2}\left(\alpha_{l}^{(c,d)}+\alpha_{l+1}^{(c,d)}\right)\;\;\mbox{for
}l\leq m-2.$ (34)

Which leads to:

$\displaystyle J_{c,d}$ $\displaystyle\geq$
$\displaystyle\left|\alpha^{(c,d)}_{m-1}-\beta^{(c,d)}_{m-1}\right|+\left|\alpha^{(c,d)}_{0}+\beta^{(c,d)}_{m-1}-1\right|+\sum_{l=0}^{m-2}\left(\left|\alpha^{(c,d)}_{l}-\alpha^{(c,d)}_{l+1}\right|\right),$

equality being reached via equation (34).

On the other hand, we can perform a similar analysis considering
$\alpha_{l}^{(c,d)}$ for $l\in\\{1,\dots,m-2\\}$. Similarly to the previous
case, we can use the triangle inequality successively in the terms within the
summation in Eq. (B) to conclude that

$\sum_{l=0}^{m-2}\left|\alpha_{l}^{(c,d)}-\alpha_{l+1}^{(c,d)}\right|\geq\left|\alpha_{0}^{(c,d)}-\alpha_{m-1}^{(c,d)}\right|.$
(35)

We may saturate this lower bound by setting $\alpha_{l}^{(c,d)}$ as

$\alpha_{l}^{(c,d)}=\frac{l}{m-1}\alpha_{m-1}^{(c,d)}+\frac{m-1-l}{m-1}\alpha_{0}^{(c,d)},$
(36)

for $l\in\\{1,\ldots,m-2\\}$, which can easily be verified, for instance by
noticing that
$\alpha_{l+1}^{(c,d)}-\alpha_{l}^{(c,d)}=(\alpha_{m-1}^{(c,d)}-\alpha_{0}^{(c,d)})/(m-1)$.
It should be noted that this choice of $\alpha_{l}^{(c,d)}$ is equivalent to
$\alpha_{l}^{(c,d)}=(\alpha_{l-1}^{(c,d)}+\alpha_{l+1}^{(c,d)})/2$, which
follows from the same reasoning as used for $\beta_{l}^{(c,d)}$.

Using the triangle inequality again, we can find the optimal value of
$\alpha^{(c,d)}_{0}$:

$\displaystyle\left|\alpha^{(c,d)}_{0}-\left(1-\beta^{(c,d)}_{m-1}\right)\right|+\left|\alpha^{(c,d)}_{m-1}-\alpha^{(c,d)}_{0}\right|\geq\left|\beta^{(c,d)}_{m-1}-1+\alpha^{(c,d)}_{m-1}\right|,$
(37)

with the equality holding true when,

$\displaystyle\alpha^{(c,d)}_{0}=\frac{1}{2}\left(1-\beta_{m-1}^{(c,d)}+\alpha^{(c,d)}_{m-1}\right).$
(38)

From which we conclude that,

$\displaystyle J_{c,d}$ $\displaystyle\geq$
$\displaystyle\left|\alpha^{(c,d)}_{m-1}-\beta^{(c,d)}_{m-1}\right|+\left|\alpha^{(c,d)}_{m-1}+\beta^{(c,d)}_{m-1}-1\right|.$

Using the triangle inequality one last time, we can see that

$\displaystyle\left|\alpha^{(c,d)}_{m-1}-\beta^{(c,d)}_{m-1}\right|+\left|\alpha^{(c,d)}_{m-1}+\beta^{(c,d)}_{m-1}-1\right|\geq\left|2\alpha^{(c,d)}_{m-1}-1\right|,$

which can be achieved by setting $\beta^{(c,d)}_{m-1}=\alpha^{(c,d)}_{m-1}$,
leading to the following tight relation:

$\displaystyle J_{c,d}$ $\displaystyle\geq$
$\displaystyle\left|2\alpha^{(c,d)}_{m-1}-1\right|$

We highlight that the values for which the above equation becomes an equality
are always well-defined. From equation (38), one gets that
$\alpha^{(c,d)}_{0}\in[0,1]$ as long as $0\leq\alpha_{m-1}^{(c,d)}\leq 1$ and
$0\leq 1-\beta_{0}^{(c,d)}\leq 1$ – which is always the case. From equation
(36), for $1\leq l\leq m-2$, $\alpha_{l}^{(c,d)}\in[0,1]$ as long as
$0\leq\alpha_{m-1}^{(c,d)}\leq 1$ and $0\leq\alpha^{(c,d)}_{0}\leq 1$ – which
is always the case. The same holds for $\beta_{l}^{(c,d)}$ in the case $l>0$.

The above result leads to the following condition

$\displaystyle C^{m-1}=\sum_{c,d}p(c,d)C^{m-1}_{c,d}\leq 2m-2J,$ (39)

$J$ being a convex combination of $J_{c,d}$,

$\displaystyle J$ $\displaystyle=$ $\displaystyle\sum_{c,d}p(c,d)J_{c,d}$ (40)
$\displaystyle\geq$
$\displaystyle\sum_{c,d}p(c,d)\left|2\alpha^{(c,d)}_{m-1}-1\right|.$

This means that the minimum value of $J$ is obtained by setting
$p(c^{*},d^{*})=1$, in which $c^{*}$ and $d^{*}$ are such that
$\left|2\alpha^{(c^{*},d^{*})}_{m-1}-1\right|\geq\left|2\alpha^{(c,d)}_{m-1}-1\right|$,
for all $c$ and $d$. Which implies that

$\displaystyle J$ $\displaystyle\geq$
$\displaystyle\left|2p(a=0,c^{*},d^{*}|x=m-1)-p(c^{*},d^{*})\right|$ (41)
$\displaystyle=$ $\displaystyle\left|2p(a=0,c^{*},d^{*}|x=m-1)-1\right|$
$\displaystyle=$ $\displaystyle\left|2p(a=0,c^{*}|x=m-1)-1\right|$
$\displaystyle=$ $\displaystyle\left|2p(a=0,c^{*}|x=m-1,y)-1\right|,$

in which we have set $p(c^{*},d^{*})=1$.

If $c^{*}=0$, then $p(a=0,c^{*}|x=m-1)\in[0,1-\epsilon]$ and $J\geq
1-2\epsilon$, on the other hand, the case in which $c^{*}=1$ imply that
$p(a=0,c^{*}|x=m-1)\in[0,\epsilon]$, which also lead to $J\geq 1-2\epsilon$.
Thus concluding that the following tight bound holds

$\displaystyle C^{m-1}\leq 2(m-1)+4\epsilon.$ (42)

∎

##  Appendix C Alternative proof that the Chained Bell inequality is also a
valid LF inequality

######  Result 4.

The EWFS with $k=2$ (2 outcomes) and arbitrary $m$ (measurement inputs) is
bounded by the following LF inequality:

$\displaystyle C^{(m-1)}_{j}=\langle A_{m-1}B_{m-1}\rangle-\langle
A_{j}B_{m-1}\rangle+\sum_{l=j}^{m-2}\left(\langle A_{l}B_{l}\rangle+\langle
A_{l+1}B_{l}\rangle\right)\leq 2((m-j)-1)$ (43)

for j=0.

###### Proof.

Before proceeding with the proof, it is worthy mentioning that for each $j$
and $m$ the inequality above is one of the symmetries of the so-called chained
Bell inequalities [31]. That said, we write this inequality in a slightly
different way of how it is typically presented (see for instance [48]), which
can be recovered from our expression by setting $j=0$ and, by defining
$\langle A_{m}B_{m-1}\rangle=-\langle A_{0}B_{m-1}\rangle$.

Now consider the following symmetries of the chained inequality, which bound
the Bell scenario for $k=2$ and arbitrary $m$,

$\displaystyle C^{(m-1)}_{j}=\langle A_{m-1}B_{m-1}\rangle-\langle
A_{j}B_{m-1}\rangle+\sum_{l=j}^{m-2}\left(\langle A_{l}B_{l}\rangle+\langle
A_{l+1}B_{l}\rangle\right)\leq 2((m-j)-1)$ (44)

Putting the term with $l=j$ outside the summation we obtain

$\displaystyle C^{(m-1)}_{j}$ $\displaystyle=$ $\displaystyle\langle
A_{m-1}B_{m-1}\rangle-\langle A_{j}B_{m-1}\rangle+\langle
A_{j}B_{j}\rangle+\langle A_{j+1}B_{j}\rangle+\sum_{l=j+1}^{m-2}\left(\langle
A_{l}B_{l}\rangle+\langle A_{l+1}B_{l}\rangle\right),$

and adding $0=\langle A_{j+1}B_{m-1}\rangle-\langle A_{j+1}B_{m-1}\rangle$ to
it we get

$\displaystyle C^{(m-1)}_{j}$ $\displaystyle=$ $\displaystyle\langle
A_{j+1}B_{m-1}\rangle-\langle A_{j+1}B_{m-1}\rangle+\langle
A_{m-1}B_{m-1}\rangle-\langle A_{j}B_{m-1}\rangle+\langle
A_{j}B_{j}\rangle+\langle A_{j+1}B_{j}\rangle$ $\displaystyle+$
$\displaystyle\sum_{l=j+1}^{m-2}\left(\langle A_{l}B_{l}\rangle+\langle
A_{l+1}B_{l}\rangle\right).$

Now using the very definition of $C^{(m-1)}_{j}$ we obtain

$\displaystyle C^{(m-1)}_{j}$ $\displaystyle=$ $\displaystyle\langle
A_{j+1}B_{m-1}\rangle-\langle A_{j}B_{m-1}\rangle+\langle
A_{j}B_{j}\rangle+\langle A_{j+1}B_{j}\rangle+C^{(m-1)}_{j+1}$
$\displaystyle=$ $\displaystyle\tilde{C}^{(m-1)}_{j}+C^{(m-1)}_{j+1},$

in which

$\displaystyle\tilde{C}_{j}^{(m-1)}=\langle A_{j+1}B_{m-1}\rangle-\langle
A_{j}B_{m-1}\rangle+\langle A_{j}B_{j}\rangle+\langle A_{j+1}B_{j}\rangle,$
(45)

is also a symmetry of the chained inequality with only two inputs.

Using this recurrence relation we can rewrite

$\displaystyle C^{(m-1)}_{0}$ $\displaystyle=$
$\displaystyle\tilde{C}^{(m-1)}_{0}+C^{(m-1)}_{1}$ $\displaystyle=$
$\displaystyle\tilde{C}^{(m-1)}_{0}+\tilde{C}^{(m-1)}_{1}+C^{(m-1)}_{2}$
$\displaystyle\vdots$ $\displaystyle=$ $\displaystyle
C^{(m-1)}_{m-2}+\sum_{j=0}^{m-3}\tilde{C}^{m-1}_{j}.$

By the results of reference [25], we know that in the EWFS, certain symmetries
of the CHSH inequality are a valid LF inequalities. In particular it holds
that

$\displaystyle\tilde{C}^{(m-1)}_{j}\leq 2\;\;\;\;\forall\;j,$ (46)

and

$\displaystyle C^{(m-1)}_{m-2}\leq 2,$ (47)

which implies that

$\displaystyle C^{(m-1)}_{0}$ $\displaystyle\leq$ $\displaystyle
2+\sum_{j=0}^{m-3}2$ (48) $\displaystyle=$ $\displaystyle 2(m-1),$

concluding the proof. ∎


11institutetext: CASOS, Institute for Software Research
Carnegie Mellon University, Pittsburgh, PA 15213
11email: {huixiann, icruicks<EMAIL_ADDRESS>
# Coordinating Narratives and the Capitol Riots on Parler ††thanks: The
research for this paper was supported in part by the Knight Foundation and the
Office of Naval Research Grant (N000141812106) and an Omar N. Bradley
Fellowship, and by the center for Informed Democracy and Social-cybersecurity
(IDeaS) and the center for Computational Analysis of Social and Organizational
Systems (CASOS) at Carnegie Mellon University. The views and conclusions are
those of the authors and should not be interpreted as representing the
official policies, either expressed or implied, of the Knight Foundation,
Office of Naval Research or the US Government.

Lynnette Hui Xian Ng ID 11 Iain Cruickshank ID 11 Kathleen M. Carley ID 11

###### Abstract

Coordinated disinformation campaigns are used to influence social media users,
potentially leading to offline violence. In this study, we introduce a general
methodology to uncover coordinated messaging through analysis of user parleys
on Parler. The proposed method constructs a user-to-user coordination network
graph induced by a user-to-text graph and a text-to-text similarity graph. The
text-to-text graph is constructed based on the textual similarity of Parler
posts. We study three influential groups of users in the 6 January 2020
Capitol riots and detect networks of coordinated user clusters that are all
posting similar textual content in support of different disinformation
narratives related to the U.S. 2020 elections.

###### Keywords:

Coordinated Messaging Parler Disinformation Natural Language Processing

##  1 Introduction

The free speech social network Parler has been under the spotlight for its
role as a preparatory medium for the 6 January 2020 Capitol riots, where
supporters of then-president Donald Trump stormed the Capitol building. The
United States have previously seen protests like the BlackLivesMatter and the
StopAsianHateMovements fuelled by disinformation narratives on social media,
which can lead to offline violence [19, 10].

A key part of the field of social cybersecurity is the detection of
disinformation on social media platforms. Disinformation, in this context is a
deliberately manipulated narrative or distortion of facts, to serve a
particular purpose [7]. In particular, we are interested in the coordination
of spreading disinformation by online actors; coordinated operations can pose
threats to the social fabric [4]. Within the realm of coordinated
disinformation campaigns, we aim to identify key actors that spread harmful
narratives. In particular, this study seeks to adress which actors, or classes
of actors, are involved in disinformation spread on Parler? For this, we
analyzed the activity of the users on Parler, with a special focus on three
account types: military, patriot and QAnon accounts. We then combined textual
and user-level information to analyze coordinated messaging and find
influential accounts amon those identified classes of users.

The main contribution of our work are as follows:

  1. 1.

We identify three classes of accounts, namely those that identify as military
or “patriot” or QAnon that are very influential on Parler after the Januray
6th Capitol riot and active in perpetrating disinformation themes.

  2. 2.

We propose a new technique to find coordinated messaging among users using
features of their parleys. This text-coordination technique reduces complex
networks into core members, providing leads for further analysis.

##  2 Background

Parler positions itself as a free speech social network, where its users can
“speak freely and express yourself openly”. A significant portion of its user
base are Donald Trump supporters, conservatives, QAnon conspiracy theorists
and far-right extremists [1]. Additionally, the platform has been in the spot
light for being a coordination platform for the January 6 2021 Capitol riots
[13], where hundreds of individuals stormed the US Capitol Hill building,
emboldened by calls from then-president Donald Trump to stop the certification
the 2020 U.S. election. The social network was shut down on January 11.

Prior work analyzed the Parler’s role in shaping the narratives of different
contexts. Parleys between August 2018 and January 2021 revealed a huge
proportion of hashtags supporting Trump on the elections [1]. Analysis of the
platform on the COVID-19 vaccine disproportionately slants towards anti-
vaccine opinions [2]. Specific to the capitol riots, a comparative study
between Twitter and Parler shows that a huge proportion of Parler activity was
in support of undermining the validity of the 2020 US Presidential elections,
while the Twitter conversation disapproved Trump [6].

One point of concern is the detection of coordinated campaigns used to
influence users, which may therefore lead to real-world violence like the
Capitol riot. Several techniques of coordination detection on social media
have been previously researched: using network community detection algorithms
[14], statistical analysis of retweets, co-tweets, hashtags and other
variables [18], and constructing a synchronized action framework These works,
however, depend largely on a network-based structure, which is not dominant in
Parler. Another way of assesing coordinated behavior between social media
websites is comparing the textual content that users are sharing. Pachecho et
al. have recently utilized a textually-based user comparison as a means of
overcoming limitations in finding coordinated user behavior with just explicit
user-to-user networks [16].

##  3 Data and Methodology

In this section, we detail the data used in our study along with the analysis
techniques for understanding the actors who spread content on Parler and the
narratives present within that content. We first detail the data used and how
it was processed for analysis. We then detail how we analyzed the textual
content for disinformation themes and coordinating propagation of
disinformation.

###  3.1 Data Collection and Processing

The data used for this study comes from the Parler data leak following the
January 6th Capitol Riot. After the Capitol riots, Amazon Web Services banned
Parler from being hosted on its website. Shortly before that, an independent
researcher and other internet users sought to preserve the data by performing
a scrape of its posts, or “parleys” [9]. In particular, we use a partial HTML
scrape of the posts111https://ddosecrets.com/wiki/Parler. The data set
consists of 1.7 million posts from 290,000 unique users and has 98% of its
posts from January 3rd to January 10th of 2021.

We developed a parsing tool to parse the useful elements from the HTML: users,
text, external website URLs, echo counts, impression counts and so on 222Code
for parsing Parler HTML is available at: https://github.com/ijcruic/Parse-
Parler-Data. Table 1 presents some terminology used by the Parler platform.

Term |  Definition
---|---
Parley |  A post on the Parler platform, up to a length of 1000 text characters
Echo |  A re-posting of a Parley, similar to Twitter’s re-tweet feature
Hashtag |  A word or phrase prefixed with the pound (#) sign, usually identifying topics in the text
Comment |  A reply to a Parley, up to a length of 1000 text characters
Upvotes |  An indication of approval of the Parley, similar to Twitter’s “like” feature
Impressions |  Appearance of the Parley on another user’s feed
Table 1: Terminology used in Parler

###  3.2 User Class Analysis

For the users, we wanted to focus on those users that are most likely to be
involved with disinformation or who had an outsize influence in the January
6th Capitol Riot. In particular, we coded for three classes of users based
upon news of those being involved in the January 6th events: those who openly
display a military or veteran affiliation, those who identify themselves with
the term “patriot” and those who use QAnon related terms in their names [15].
To find these users, we used regex string matching on their “author names” and
their “author user names”. Each user had both an “author user names” which is
his actual account handle, or @-name, and an “author names” which is a short
free-text description the user writes themselves. Table 2 summarizes the terms
used for finding accounts belonging to these three classes of authors. We then
analyzed the social media usage and artifacts of these three classes of users.
Using these three user classes, we analyzed the textual content of these three
classes of users and identified coordinated messaging among the user classes.

User Category |  Terms used to define that user category
---|---
Military/Veteran |  army, navy, air force, airforce, marine, veteran, military, servicemember, coastguard, coast guard, soldier, infantry, sergeant
Patriot |  patriot
QAnon |  qanon, wwg1wga, Q, thegreatawakening, thestorm, theplan
Table 2: Terms used to heuristically define three main user classes

###  3.3 Identification of Coordinated Messaging

Finally, we use the textual information to evaluate user coordination in
spreading messages themes. The Parler data shows a low proportion of parley
echoes, which means users are not actively sharing other user’s parleys. In
contrast, Twitter data has a huge proportion of retweets [6]. This renders
using methods of identifying user coordination through retweets and co-tweets
inadequate, hence we turn to analyzing similar texts.

Similar to previous work, we create user-to-user graphs based on users’
textual content to analyze user accounts with very similar textual messages
[16]. However, instead of only creating an edge if there is at least one pair
of posts between two users that are above an arbitrary similarity threshold,
we use a text-to-text graph to induce a user-to-user graph, and set the
threshold of which we filter edges through a statistical analysis of all graph
edge weights.

We consider only original parleys and not echoed-parleys in order to highlight
authors who are consciously writing similar texts, but may be trying to hide
their affiliation by not explicitly echoing each other. We first begin by
creating a user-to-text binary graph $P$, which represents the users that
wrote each parley. Next, we create a text-to-text graph $A$ through a
k-Nearest Neighbor (kNN) representation of the parley texts, with
$k=log_{2}{N}$ where $N$ is the number of parleys [11]. To do so, we perform
BERT vectorization on each parley text create contextualized embeddings into a
768-dimensional latent semantic space [5]. We make use of the FAISS library to
index the text vectors and perform an all-pairs cosine similarity search to
determine the top k closest vectors to each parley vector [8]. We then
symmetrize this kNN-graph via $P^{\prime}=\frac{P+P^{T}}{2}$ to produce a
symmetric kNN graph of the posts, as this tends to better maintain meso-
structures from the data, like clusters, in the graph [12], [17], [3]. The
edges of the graph are weighted between $[0,1]$ by the cosine similarity of
the two parleys.

Having found a latent graph of the textual content of the posts we then induce
a user-to-user graph. We do this through matrix Cartesian product formula of:

$U=PA^{\prime}P^{T}$ (1)

Where $U$ is the user-to-user graph, $A$ is a user-to-text bipartite graph
where an edge indicates that a user posted a given parley, and $P^{\prime}$ is
the text-to-text kNN graph as previously defined. The resulting graph, $U$ has
edges that represent the strength of textual similarity between two users,
given how close in similarity their posts are in a latent semantic space. $U$
better accounts for not only having multiple, similar posts but better respect
the textual data manifold when measuring between two users as compared at only
identifying the most similar posts between two users [16].

To sieve out the core structure of the graph, we further prune the graph $U$
based on link weights, forming $U^{\prime}$, keeping only the links that
weight greater than one standard deviation away from the mean link weight.
This leaves behind users that strongly resemble each other in terms of the
semantic similarity of their parley texts.

##  4 Results

In this section, we detail the results of our analysis. We begin with
presenting the social media usage statistics of the three classes of users. We
then look at the textual content being spread for disinformation narratives.
Finally, we present the results of analyzing users and textual content.

###  4.1 Social Media Usage and Artifacts

After having identified classes of users (i.e. military/veteran, patriot,
QAnon) using regex searches within their author names, we then analyzed the
differences in their activity on Parler. For the military/veteran users we
identified 975 accounts (0.3% of all accounts) that posted 14,745 Parleys
(0.8% of all Parleys). For the patriot users we identified 5,472 accounts
(1.9% of all accounts) that posted 81,293 Parleys (4.7% of all Parleys). For
the QAnon users we identified 3,400 accounts (1.2% of all accounts) that
posted 71,998 Parleys (4.1% of all Parleys). So, while these accounts make up
a small portion of the users, less than 5% of the all of the user accounts,
they account for close to 10% of all of the posts by volume. Table
LABEL:tab:social_media_artifacts summarizes the social media engagement and
use of various social media artifacts across all of the accounts.

Table 3: Social media usage patterns across the three user classes. Most users have very few echos and use very few (or none) social media artifacts, like hashtags or mentions. Users that identify with QAnon terms use more social media artifacts and see greater engagement. | All Users |  | Military/
---
Veteran
Patriot | QAnon
Avg. echos by users | 6.33+/-15.49 | 2.05+/-6.92 | 2.65+/-6.96 | 2.65+/-6.88
Avg. users echoed | 13.35+/-736.96 | 18.81+/-203.26 | 20.31+/-242.05 | 38.66+/-518.36
| Avg. posts per user with
---
a mention
0.09+/-0.8 | 0.16+/-0.9 | 0.10+/-0.49 | 0.16+/-0.86
| Avg. posts per user with
---
a hashtag
0.20+/-1.14 | 0.24+/-0.84 | 0.34+/-1.27 | 0.34+/-1.2
| Avg. posts per user with
---
a URL
0.73+/-2.83 | 0.87+/-2.46 | 0.91+/-2.89 | 1.18+/-4.64
| Avg. posts per user with
---
media
0.47+/-2.02 | 0.88+/-2.77 | 0.62+/-1.61 | 1.02+/-3.24
| Avg. comments received
---
per post
4.28+/-195.77 | 3.66+/-26.24 | 3.07+/-24.27 | 6.96+/-103.20
| Avg. upvotes received
---
per post
29.53+/-1416.6 | 28.57+/-213.84 | 32.81+/-329.84 | 67.93+/-874.82
| Avg. impressions received
---
per post
| 4312.42+/-
---
241116
| 3946.72+/-
---
36358
| 3002.06+/-
---
29649
| 6666.75+/-
---
76551
| Mode impressions received
---
per post
10 | 11 | 24 | 12

Generally, we observe that the social engagement is different for the accounts
that identify with QAnon and patriot terms, as compared to the overall Parler
population. The military, patriot and QAnon accounts are all echoed more than
the population average. This trend of having posts be highly echoed is
especially true for the QAnon accounts which, on average, see three times as
much echoing on their posts than the average user does. Additionally, the
QAnon user category also sees more impressions, upvotes, and comments than
does any other category of user. The military, patriot, and QAnon accounts
also use more social media artifacts, like hashtags or mentions than do the
average account. So, those accounts that openly identify with QAnon terms tend
to see more social engagement with their posts than any other class of
accounts and all of the special types of accounts use more social media
artifacts than average users.

Additionally, the verbiage used by these accounts tends toward those
associated with disinformation campaigns. The top words by their counts within
the textual content of all the posts of all three accounts includes the
following words: trump, pence, patriot, people, antifa, country, president,
dc, echo, election, need, today, wwg1wga, and stopthesteal. While some of
these terms are common throughout the data set, like ‘trump’, ‘need’,
‘president’, or ‘people’, others show up more prominently in the three classes
of accounts, like ‘wwg1wga’ and ‘antifa’, than they do across all posts. These
usage of these words are almost uniquely associated with disinformation
campaigns like the 2020 U.S. Election being fraudulent or Antifa being
responsible for the January 6th Capitol Riot. Thus, we observe that the three
classes of users also frequently employ verbiage associated with
disinformation.

###  4.2 Coordinated Messaging between Users

Figure 1 presents the graph $U^{\prime}$, the core structure of the induced
user-to-user graph $U$. We first observe that this graph has a distinct core
structure which is due to the fact that almost all of the conversation
surrounds the 2020 U.S. Election. This graph is annotated with clusters of
coordinated messaging between users derived from the analysis of Parley
textual similarity, where the strength of the coordination is represented by
the link weights. We segment the graph $U^{\prime}$ into subgraphs (a) to (g)
for better clarity of each portion of the graph. The messaging of each
subgraph are consistent, showing the effectiveness of our approach in
uncovering coordination between users as well as distinct disinformation
themes spread by those users.

Subgraph (a) represents Patriot users who have every conviction they will
fight for their Republic and president. Subgraph (b) calls for freedom in
French, with many users having handles that end with the different Canadian
provinces. The users associate themselves with different user groups. Subgraph
(c) sieves out users in the center of the graph that strongly coordinate with
each other in disputing the electoral vote count. These are predominantly
patriot users and a handful of military users disputing the vote.

Subgraph (d) in which the core patriot in the center of the graph strongly
coordinates with each other. This user group mainly advocates that the truth
about the elections will come to pass and the storm is upon us. These messages
express support for the disinformation narratives of the 2020 U.S. Election
being fraudulent as well as QAnon conspiracy theories relating to then
President Donald Trump seizing power in the U.S.

Subgraph (e) represents QAnon users, who mostly purport that Trump will win,
and provide updates about the capitol situation in terms of spurring the group
on (“PLS SPREAD TrumpVictory FightBack ProtectPatriots”) and information
updates (“Protesters are now in Nancy Pelosi‘s office”). Some of these
narratives are blaming BLM and a deliberately designed terrorist Antifa who
infiltrated protesters who stormed the Capitol.

Subgraph (f) sieves out users at the fringe of the graph that call for justice
and revolution and not to be let down. Subgraph (g) represents military users
that hope for a free world and generally call to Stop the Steal of the 2020
elections. “Stop the Steal” has been a recurring phrase throughout Trump’s
political career, peaking in the Capitol riots as a movement to overturn the
2020 US election results and has been a key phrase associated with the
disinformation campaign trying to de-legitimize the 2020 U.S. election.

Figure 1: Core structure of user-to-user graph $U$’ representing textual
coordination between groups of users. Military users are colored blue; patriot
users red and QAnon users green. The thickness of the links represent the
strength of the coordination. Nodes are sized by total degree centrality
value.

##  5 Discussion and Conclusion

In this work, we analyzed actors involved in disinformation spread on Parler
during the January 6th event. We looked at three main user groups based upon
unusual online activity and their salience in stories associated with the
January 6th riots: users that openly present a military or veteran
affiliation, user that use the moniker ‘patriot’, and users that identify with
QAnon-related terms. We found that the posts of these three groups of users
are echoed more and receive more impressions than the population average, with
the QAnon users having the most of all. These user groups also employ typical
tactics of using more hashtags and mentions than the average accounts to
spread their messages, as well as key verbiage relating to disinformation
campaigns. So, we find that certain users that self-identify with particular
identities typically revered by the politically right in the U.S. (i.e.
military service, patriotism, etc.) tend be more involved in guiding the
conversation within right-leaning social media spheres.

Having identified these influential accounts, we then looked at textual
coordination between actors. Identifying the strength of user-to-user
coordination over link weights presents coordinated activity as a spectrum
rather than a binary state of shared activity [18]. This shows which users
typically post similar texts, allow identification of how similar user
accounts are to each other. Our approach of inducing a user-to-user graph by
their textual similarity produced distinct clusters of users that had similar
textual themes. These themes include known disinformation themes like the
dispute of the electoral vote count and that Trump will win the 2020 US
elections. So, within Parler, we observed distinct groups of users, typically
that identify as patriot or QAnon that did engage in posting the same textual
content, that aligns to known 2020 election narratives.

There are some important limitations to this study. First, the data collected
was only a partial scrape of the parleys made on the platform surrounding the
Capitol riots event. As the platform was shut down, we cannot conclude this
was the entire set of parleys related to the event. Nonetheless, we believe
that the huge volume of parleys should provide sufficient generalizability of
the online discourse around this event. Second, we have not explored all of
the means of properly learning the textual similarity graph or sieving out the
main graph structure from the induced graph. So, we do not know if we have
found the optimal graph representation.

Future work include looking at a multimodal coordination effort on the
platform in contributing to the spread of disinformation, rather than only the
text field, taking into account the multitude of images and videos posted on
Parler, resulting in the offline violence. Another important future direction
is to study dynamic coordination networks, based upon temporal information to
characterize the change in coordinated messaging among users.

In this study we used a general approach to sieve out the core structure of a
user-to-user coordination graph which relies textual content of a social media
data. This provides a more robust filtering approach as compared to work that
relies on an arbitrary link weight threshold value, below which the links are
discarded. Our approach can further help to identify disinformation campaigns
where automated groups of user accounts post a consistent similar message, and
can overcome shortcomings with just trying to find coordination among
disinformation actors by their explicit networks (i.e. echoing on Parler). We
hope the techniques here can be used to better analyze social media platforms
and streamline network structures into core components for further
investigation. The early detection of coordinated effort to organize riots and
political violence online may present the possibility of stopping that
violence in the real world.

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