Reassessing the strength of a class of Wigner’s friend no-go theorems

Elias Okon
Universidad Nacional Autónoma de México, Mexico City, Mexico.

Two recent, prominent theorems—the “no-go theorem for observer-independent
facts” and the “Local Friendliness no-go theorem”—employ so-called extended
Wigner’s friend scenarios to try to impose novel, non-trivial constraints on
the possible nature of physical reality. While the former is argued to entail
that there can be no theory in which the results of Wigner and his friend can
both be considered objective, the latter is said to place on reality stronger
constraints than the Bell and Kochen-Specker theorems. Here, I conduct a
thorough analysis of these theorems and show that they suffer from a list of
shortcomings that question their validity and limit their strength. I conclude
that the “no-go theorem for observer-independent facts” and the “Local
Friendliness no-go theorem” fail to impose significant constraints on the
nature of physical reality.

##  1 Introduction

Recent years have seen a renewed interest in the Wigner’s friend paradox. In
particular, so-called extended Wigner’s friend scenarios (EWFS) have been in
vogue as means to display limits regarding the level of _objectivity_ one can
make use of while interpreting quantum theory. These EWFS consist of settings
in which entangled systems are shared by sets of Wigner’s friend arrangements;
that is, they are made up of Bell-type experimental settings, in which the
components of entangled systems are sent to different labs, but with such labs
enhanced with Wigner’s friend arrangements, where the friend first measures
the received component and then a ‘superobserver’ measures the whole lab.

The current surge in interest in EWFS has been somehow unsystematic, making it
difficult to assess the real value of the different efforts and the relations
among them. One useful way to classify them is as follows. On the one hand,
there are those proposals which leave fixed the measurement settings of all
observers involved, both friends and superobservers. The idea is to design the
experiments in such a way that one employs, on a single run, measurement
settings that, in a standard Bell-type scenario, would correspond to different
possibilities. One then tries to employ the correlations among the results of
all observers to set limits on possible interpretations of quantum theory.
Efforts in this direction include the Pusey-Masanes (Pusey,, 2016) and
Żukowski-Markiewicz (Żukowski and Markiewicz,, 2021) arguments, which try to
challenge the objectivity of quantum measurement outcomes, and the Frauchiger-
Renner theorem (Frauchiger and Renner,, 2018), which allegedly shows that
quantum theory does not apply at all scales. The problem with this type of
arguments, though, is that the rely on a mistaken assumption regarding the
correlations between the results of the friends and those of superobservers,
rendering them invalid (see Okon, (2022) for details).

On the other hand, there are proposals which, as in the standard Bell
scenario, do allow for the observers to choose what measurement to perform. In
particular, in these settings, on each run of the experiment, each
superobserver only performs one of a list of possible measurements. Moreover,
these proposals build their results only employing correlations among the
results of the superobservers. This group includes the “no-go theorem for
observer-independent facts” in Brukner, (2018), as well as the “Local
Friendliness no-go theorem” in Bong et al., (2020) (see also Cavalcanti and
Wiseman, (2021)). These theorems are the main subject of this assessment.

Building on Brukner, (2015), the first (to my knowledge) to employ an EWFS,
Brukner, (2018) presents a “no-go theorem for observer-independent facts”. To
do so, he considers two Wigner’s friend arrangements, sharing an entangled
pair, and a particular sequence of measurements performed by the friends and
the superobservers. Then, by positing four assumptions—‘Universal validity of
quantum theory’, ‘Locality’, ‘Freedom of choice’ and ‘Observer-independent
facts’—he derives a theorem interpreted as showing that there can be no theory
in which the results of both the superobservers and the friends can jointly be
considered as objective properties of the world.

Bong et al., (2020), however, argues that Brukner’s ‘Observer-independent
facts’ assumption is equivalent to Kochen-Specker non-contextuality (KSNC) and
that Brukner’s result can be obtained from ‘Freedom of choice’ and KSNC alone
(no ‘Locality’ required). Moreover, by arguing that the Kochen-Specker theorem
already establishes that ‘Freedom of choice’ and KSNC lead to a contradiction
with quantum mechanics, it is concluded that Brukner’s theorem fails to set
constraints on the objectivity of observations. Nevertheless, Bong et al.,
(2020) maintains that there is something of value in Brukner’s arrangement and
that, in fact, it can be used to derive a theorem by retaining his ‘Freedom of
choice’ and ‘Locality’ assumptions, but replacing his ‘Observer-independent
facts’ by what they call the ‘Absoluteness of Observed Events’ assumption.
Since it is argued that the new set of assumptions is strictly weaker that
those of Bell and Kochen-Specker theorems, the new result is advertised as
placing strictly stronger constraints on physical reality than these old
theorems.

Here, I perform a thorough analysis of the “no-go theorem for observer-
independent facts” of Brukner, (2018) and the “Local Friendliness no-go
theorem” of Bong et al., (2020). Regarding the former, I show that it depends
on a non-trivial hidden assumption and that it relies on a mistaken
application of standard quantum mechanics to the EWFS considered. As for the
latter, I argue that it is not true that its assumptions are strictly weaker
that those of Bell and that it relies on the same mistaken application of
standard quantum mechanics. I conclude that the theorems fail to set
interesting constraints on the possible nature of physical reality.

My manuscript is organized as follows. In section 2, I present an overview of
the “no-go theorem for observer-independent facts” of Brukner, (2018) and the
“Local Friendliness no-go theorem” of Bong et al., (2020). Then, in section 3,
I offer my assessment of such theorems. Finally, in section 4, I state my
conclusions.

##  2 The theorems

We start with an overview of the “no-go theorem for observer-independent
facts” in Brukner, (2018) and the “Local Friendliness no-go theorem” in Bong
et al., (2020).

###  2.1 The no-go theorem for observer-independent facts

Elaborating on Brukner, (2015)—the first to consider a Bell-type scenario
supplemented with Wigner’s friend arrangements—Brukner, (2018) asks whether
there exists a theory, possibly different from quantum theory, in which the
results of both Wigner and his friend can jointly be considered observer-
independent, objective facts of the world. The question is given a negative
answer with the introduction of a “no-go theorem for observer-independent
facts”.

To construct the theorem, Brukner considers an EWFS with two Wigner’s friend
arrangements—lab 1 with Charlie inside and Alice outside and lab 2 with Debbie
inside and Bob outside—each receiving a particle of an entangled pair. Inside
their labs, Charlie and Debbie perform spin measurements along direction $z$.
Let’s denote by $\ket{Z_{\pm}}_{i}$ the state of lab $i$ after the friend
measures the spin along $z$ and finds the result $\pm$. Then, Alice and Bob
perform, on their respective labs, one of two measurements: a “Z measurement”
in basis $\\{\ket{Z_{+}}_{i},\ket{Z_{-}}_{i}\\}$ or an “X measurement” in
basis
$\\{\ket{X_{+}}_{i}=\frac{1}{\sqrt{2}}\left(\ket{Z_{+}}_{i}+\ket{Z_{-}}_{i}\right),\ket{X_{-}}_{i}=\frac{1}{\sqrt{2}}\left(\ket{Z_{+}}_{i}-\ket{Z_{-}}_{i}\right)\\}$.
That is, on each run of the experiment, each superobserver only performs one
of these two possible measurements.

The setting described above is used by Brukner to argue for the mutual
incompatibility of four assumptions: (1) ‘Universal validity of quantum
theory’, establishing that quantum predictions hold at any scale; (2)
‘Locality’, requiring that the choice of measurement on one side has no
influence on the outcomes of the other; (3) ‘Freedom of choice’, stipulating
statistical independence between measurement settings and the rest of the
experiment and (4) ‘Observer-independent facts’, demanding for there to exist
a joint assignment of truth values to propositions about the outcomes of all
observers. In particular, according to Brukner, this last assumption implies
that the statements he calls $A_{1}$: “the pointer of Charlie’s apparatus
points to result $+$” and $A_{2}$: “after performing an X measurement, the
pointer of Alice’s apparatus points to result $X_{-}$”, have well-defined
truth values. That is, all such statements are assumed to have truth values,
regardless of what measurements the superobservers decide to perform.

The incompatibility among Brukner’s assumptions is shown as follows. He
denotes by $A_{1}$ and $A_{2}$ the two measurement settings of Alice, which
correspond to the observational statements Charlie and Alice can make about
their respective outcomes (analogously with $B_{1}$ and $B_{2}$ for Bob and
Debbie). Then, it is noted that assumptions (2), (3) and (4) imply the
existence of local hidden variables, which determine $A_{1}$, $A_{2}$, $B_{1}$
and $B_{2}$ (with values $\pm 1$). Moreover, it is noted that these
assumptions imply the existence of a joint probability distribution,
$p(A_{1},A_{2},B_{1},B_{2})$, whose marginals (e.g.,
$p(A_{1},B_{2})=\sum_{A_{2},B_{1}}p(A_{1},A_{2},B_{1},B_{2}))$ satisfy the
CHSH inequality:
$E(A_{1},B_{1})+E(A_{1},B_{2})+E(A_{2},B_{1})-E(A_{2},B_{2})\leq 2$ (with,
e.g., $E(A_{1},B_{2})=\sum_{A_{1},B_{2}}A_{1}B_{2}p(A_{1},B_{2}))$. The
problem is that there exist entangled states to be distributed among the two
labs, such as

$\frac{1}{2}\left[\sin(\pi/8)\ket{+_{z},+_{z}}+\cos(\pi/8)\ket{+_{z},-_{z}}-\cos(\pi/8)\ket{-_{z},+_{z}}+\sin(\pi/8)\ket{-_{z},-_{z}}\right],$
(1)

which lead to quantum predictions that violate the CHSH inequality. Since the
assumptions are shown to be inconsistent, the conclusion reached by Brukner is
that there can be no theoretical framework where the results of different
observers can jointly be considered objective facts of the world.

###  2.2 The Local Friendliness no-go theorem

The starting point of the “Local Friendliness no-go theorem” presented in Bong
et al., (2020) (see also Cavalcanti and Wiseman, (2021, section 4)), is an
analysis of Brukner’s “no-go theorem for observer-independent facts”.
According to Bong et al., (2020), Brukner’s ‘Observer-independent facts’
assumption demands propositions about all observables _that might be measured_
, by an observer or a superobserver, to be assigned a truth value
independently of which measurements the superobservers perform. That is, they
read Brukner’s assumption as implying, for instance, that even when Alice
performs a Z measurement, statements regarding the result of Alice’s
_unperformed_ X measurement have well-defined truth values.

Given this reading, they argue that Brukner’s ‘Observer-independent facts’
assumption is, in fact, equivalent to Kochen-Specker non-contextuality (KSNC).
Moreover, they argue that such an assumption, together with ‘Freedom of
choice’, are sufficient to derive Brukner’s result. Since the Kochen-Specker
theorem already establishes that ‘Freedom of choice’ and KSNC are incompatible
with quantum mechanics, they conclude that the ‘Observer-independent facts’
assumption plays no role in the theorem so, in the end, it fails in its
mission to place limits on the objectivity of facts.

Still, according to Bong et al., (2020), Brukner’s arrangement is valuable
and, in fact, can be used to derive a strong result. To do so, echoing
Brukner, they consider two perfectly isolated vaults, one containing Charlie
and the other Debbie, each receiving a particle of an entangled pair. Charlie
and Debbie then perform spin measurements on a fixed basis, obtaining outcomes
$C$ and $D$. Then, Alice and Bob perform measurements $X$ and $Y$, selected
out of a list of possible measurements, on Charlie’s and Debbie’s vaults,
respectively. Their outcomes are denoted $A$ and $B$. The possible
measurements by Alice are described as follows. If $X=1$, Alice opens
Charlie’s vault, asks him what he observed and sets her result equal to that
of Charlie. If $X\neq 1$, Alice performs a measurement on the contents of the
whole vault, including Charlie, in a basis incompatible with the one used by
Charlie. This can be done, it is argued, by reversing the unitary evolution
that entangled Charlie and his apparatus with his particle, and performing a
measurement on the particle alone on a different basis (the possible
measurements by Bob are obtained by substituting $X$ for $Y$ and Charlie for
Debbie).

To analyze this setup, Bong et al., (2020) introduces three assumptions:
‘Absoluteness of Observed Events’ (AOE), ‘No-Superdeterminism’ (NSD) and
‘Locality’ (L). The conjunction of these three assumptions is then called
‘Local Friendliness’ (LF). These assumption are described as follows.

The idea behind AOE is that every observed event exists absolutely, not
relatively. That is, that performed experiments have observer-independent,
absolute results. For the experiment under consideration, this implies that,
in each run, there exists a well-defined value for the result observed by each
observer; i.e., in each run, there is a well-defined value for $A$, $B$, $C$
and $D$. This, in turn, implies the existence of a joint probability
distribution, $P(ABCD|XY)$, from which the joint probability for the results
of Alice and Bob, $P(AB|XY)$, can be obtained. Moreover, the joint probability
distribution must ensure consistency between the outcomes of friends and
superobservers when $X,Y=1$. Given all this, for the experiment considered,
AOE is formalized as follows:

  * •

$\exists$ P(ABCD|XY) such that

i.

$P(AB|XY)=\sum_{C,D}P(ABCD|XY)\quad\forall\quad A,B,X,Y$

ii.

$P(A|CD,X=1,Y)=\delta_{AC}\quad\forall\quad A,C,D,Y$

iii.

$P(B|CD,X,Y=1)=\delta_{BD}\quad\forall\quad B,C,D,X$ .

The motivation for NSD is the idea that the experimental settings can be
chosen freely. It is argued to be a formalization of the ‘Freedom of choice’
assumption used to derive Bell’s theorem, and is stated as follows: “any set
of events on a space-like hypersurface is uncorrelated with any set of freely
chosen actions subsequent to that space-like hypersurface”. In the experiment
under consideration, and assuming AOE, NSD is taken to imply that $C$ and $D$
are independent of the choices $X$ and $Y$; that is:

  * •

$P(CD|XY)=P(CD)\quad\forall\quad C,D,X,Y$ .

Finally, L prohibits the influence of local settings on distant outcomes. That
is, it is the assumption usually called ‘Parameter independence’. In Bong et
al., (2020), it is enunciated as follows: “the probability of an observable
event $e$ is unchanged by conditioning on a space-like-separated free choice
$z$, even if it is already conditioned on other events not in the future
light-cone of $z$”. For the experiment in question, and assuming AOE, L is
said to imply:

  * •

$P(A|CDXY)=P(A|CDX)\quad\forall\quad A,C,D,X,Y$

  * •

$P(B|CDXY)=P(B|CDY)\quad\forall\quad B,C,D,X,Y$ .

The construction of the theorem then proceeds in two steps. On the one hand,
it is shown that the LF assumptions imply a set of constraints on the joint
probability for the results of Alice and Bob, $P(AB|XY)$. Such constraints are
expressed in the form of the “LF inequalities”. On the other hand, it is
argued that if a superobserver can perform arbitrary quantum operations on an
observer and its environment—that is, if quantum evolution (including quantum
measurement) is controllable on such a scale—then quantum mechanics predicts
the violation of the LF inequalities. In other words, if the quantum
operations required by the experiment can, in principle, be performed, then
quantum mechanics predicts the violation of the LF inequalities.

With all this in mind, by defining a ‘physical theory’ as “any theory that
correctly predicts the correlations between the outcomes observed by the
superobservers Alice and Bob”, the LF theorem is stated in Bong et al., (2020)
as follows:

> LF Theorem: If a superobserver can perform arbitrary quantum operations on
> an observer and its environment, then no physical theory can satisfy the LF
> assumptions.

Regarding the LF inequalities, it is noted that, for the specific experiment
considered by Brukner—in which the superobservers only have two binary-outcome
measurement options—the set of LF correlations is identical to those allowed
by Bell. This, however, is not the case in general. In more complicated
scenarios, in which, for instance, the superobservers have more than two
measurement choices or measurements with more than two possible outcomes, the
set of LF correlations can be larger than the set of Bell correlations. In
fact, it is argued that, for a given scenario, the set of LF correlations
strictly contains the set of Bell correlations. This implies that, for a given
scenario, it is possible for quantum correlations to violate a Bell
inequality, while satisfying all of the LF inequalities (but not the other way
around).

An important claim in Bong et al., (2020) is that their theorem “places
strictly stronger constraints on physical reality than Bell’s theorem”. To
reach such a conclusion, they note that the derivation of the Bell
inequalities requires NSD and ‘Factorizability’. The latter, in turn, follows
from the conjunction of ‘Parameter independence’ (here called L) and ‘Outcome
independence’. Moreover, they argue that the AOE assumption is necessary,
although often left implicit, for the derivation of Bell’s theorem. From this,
they conclude that the LF assumptions are strictly weaker than the assumptions
needed to derive Bell’s theorem, which means that a violation of the LF
inequalities would have stronger implications than violations of the Bell
inequalities. Finally, it is pointed out that, as Bell’s, the LF theorem is
theory-independent, in the sense that its conclusions hold for any theory, as
long as the quantum predictions are realized in the laboratory.

Regarding the realization of such a test, Bong et al., (2020) reports the
violation of the LF inequalities in an experiment in which photon’s paths play
the role of observers. It is readily acknowledged that such an experiment is
simply a proof of concept, and that a true test of the inequalities, with
observers of adequate complexity, is well beyond present technology. However,
it is argued that, if the friend were an artificial intelligence algorithm,
simulated in a quantum computer, then there would be good reason to think that
quantum mechanics would allow control of the type required.

##  3 Assessing the theorems

I’m finally in position to explore these theorems in detail. I start with an
evaluation of Brukner’s work.

###  3.1 Evaluation of the no-go theorem for observer-independent facts

The first observation I make is that the discussion in Brukner, (2018) of the
key ‘Observer-independent facts’ assumption is somehow ambiguous and allows
for two different interpretations. To begin with, Brukner, (2018) uses the
symbols $A_{1},A_{2},B_{1}$ and $B_{2}$ for three different things: i) the
settings of Alice and Bob, ii) their results and iii) certain observational
statements, such as “after performing an X measurement, the pointer of Alice’s
apparatus points to result $X_{-}$” (which is called $A_{2}$ in Brukner,
(2018)). On top of that, it is not actually clear whether what Brukner calls
the observational statement $A_{1}$ (“The pointer of Wigner’s friend’s
apparatus points to result $z+$”), refers to an observational statement
uttered by Charlie, or by Alice when she performs a Z measurement. To see that
Brukner, (2018) is, in fact, ambiguous, we note that, on the one hand, he
writes statements such as “$A_{1}$ and $A_{2}$ correspond to the observational
statements Charlie and Alice can make about their respective outcomes”, in
which case it seems clear that $A_{1}$ corresponds to an observational
statement made by Charlie. However, since Brukner, (2018) also uses $A_{1}$
and $A_{2}$ for Alice’s settings, it contains statements such as “Alice
chooses between two measurement settings $A_{1}$ and $A_{2}$”, in which case
$A_{1}$ no longer seems to be a statement made by Charlie, but a claim
associated with Alice.

Why is this (apparently pedantic) issue important? The point is that these two
possible readings regarding the statement $A_{1}$ lead to radically different
contents for the ‘Observer-independent facts’ assumption. Let’s start with the
second option, namely, that $A_{1}$ is a statement made by Alice. In that
case, since the ‘Observer-independent facts’ assumption requires “an
assignment of truth values to statements $A_{1}$ and $A_{2}$ independently of
which measurement Wigner performs”, then the assumption would demand, not only
for all performed experiments to have well-defined results, but also for (at
least some) unperformed experiments to do so.

This, in fact, seems to be the interpretation of the ‘Observer-independent
facts’ assumption offered in Bong et al., (2020) where, as we saw, it is taken
as demanding propositions about all observables _that might be measured_ to be
assigned a truth value, independently of whether they where actually performed
or not. That is, they read Brukner’s assumption as implying that, even when
Alice performs a Z measurement, a statement regarding the result of her
_unperformed_ X measurement has to have a well-defined value. This is why, in
Bong et al., (2020), it is concluded that the ‘Observer-independent facts’
assumption is equivalent to KSNC. However, even with this reading of Brukner’s
assumption, it seems to me that this identification is illegitimate. KSNC is
an assumption demanding for the results of all experiments to be independent
of the _context_ of the measurement and, in particular, independent of which
other measurements were performed simultaneously. I fail to see how, by
reading $A_{1}$ as a statement by Alice, the ‘Observer-independent facts’ is
equivalent to demanding non-contextuality.

That is, KSNC would demand, for instance, for the result of Alice’s experiment
to be independent of what Bob decides to measure but, by reading $A_{1}$ as a
statement by Alice, the ‘Observer-independent facts’ demands, say, for the
result of Alice’s $A_{1}$ measurement to be well-defined, even if Alice
measures $A_{2}$. Moreover, the KSNC is a demand for all results, of all
possible experiments. It is not clear to me that the particular demand by
Brukner, made for a specific experiment, could straightforwardly be read as
setting such a strong demand for all possible all observables, in all possible
experiments. In any case, it is important to clearly distinguish between a
demand for experiments to yield observer-independent, absolute results—the
motivation behind ‘Observer-independent facts’—and a demand for all
measurements to passively, context-independently, reveal previously possessed
well-defined values—the motivation behind KSNC. Of course, the latter is a
much stronger requirement, already ruled out by the Kochen-Specker theorem.

At any rate, I agree with Bong et al., (2020) that, by reading the ‘Observer-
independent facts’ assumption this way, Brukner’s no-go theorem fails to set
relevant limits on the objectivity of facts. The point is that, by
interpreting $A_{1}$ as a statement by Alice (and $B_{1}$ as a statement by
Bob), Brukner, at best, sets constraints on framework demanding objectivity,
not only of performed experiments, but also of unperformed ones. It seems
clear to me that such a demand goes well beyond what is usually required of a
framework which seeks to maintain objectivity, so letting go of such a strong
demand does not seem problematic at all.

What about opting for the other interpretation of $A_{1}$, namely, taking
$A_{1}$ as a statement uttered by Charlie? I begin by pointing out that, in
spite of the ambiguities described above, this is by far the most natural
reading of Brukner’s ‘Observer-independent facts’ assumption and the one in
line with Brukner’s conclusions. As we saw, he takes the theorem to show that
there can be no framework where the results of Wigner and his friend can
jointly be considered objective facts of the world. Therefore, it seems clear
that what he has in mind is that the observational statement made by Charlie
has a well-defined truth value, even if Alice decides to perform a Z
measurement. Let’s explore the import of Brukner’s theorem, given this
interpretation of $A_{1}$.

The starting point in the construction of the theorem is the observation that
the ‘Observer-independent facts’ assumption implies the existence of the joint
probability distribution $p(A_{1},A_{2},B_{1},B_{2})$. It is important to
note, though, that, given the interpretation of $A_{1}$ (and $B_{1}$) under
consideration, this would mean the existence of a joint probability
distribution for the results of all four observers, Charlie, Alice, Debbie and
Bob, in a run in which Alice and Bob decide to perform an X measurement—as
opposed to a distribution for two possible results of Alice and two of Bob.
That is, the ‘Observer-independent facts’ assumption implies the reality of
those particular four results of all four observers, so there must be a joint
probability distribution for them. Since this is so, the marginals,
expectation values and the CHSH inequality constructed out of
$p(A_{1},A_{2},B_{1},B_{2})$ must also refer to these four particular
measurements.111This is, in fact, the exact construction employed in the
Pusey-Masanes argument, (Pusey,, 2016; Okon,, 2022). The problem, though, is
that Brukner proceeds to compare such results with the quantum predictions,
not for those four measurements, but for the two possible results of Alice and
the two possible results of Bob. That is, even though, at the beginning, he
interprets $A_{1}$ and $B_{1}$ as observational statements made by Charlie and
Debbie, in the end, he takes $A_{1}$ and $B_{1}$ to be the results of Alice
and Bob, when they chose to perform a Z measurement.

There is, then, a _hidden assumption_ behind the derivation of the theorem in
Brukner, (2018), namely, that the results of Alice and Bob, when they perform
a Z measurement, necessarily coincide with the results previously obtained by
Charlie and Debbie, respectively. This might seem like a completely trivial
assumption but, in fact, it is not. For instance, it is not even true for
pilot-wave theory, the paradigmatic observer-independent framework. We already
saw that, for the construction of the theorem in Bong et al., (2020), such an
assumption is made explicit (see items ii and iii of the AOE assumption).
Therefore, I postpone a detailed discussion of the issue until I explore the
LF theorem. For now, I just note the presence of this implicit, non-trivial
assumption in Brukner, (2018), without which the theorem cannot be derived.

The next observation I make is related with the last step in the derivation of
the theory, namely, the comparison with quantum predictions. According to
Brukner, (2018), there are entangled states to be distributed among the two
labs, such as that in Eq. (1), which lead to quantum predictions that violate
the CHSH inequality. While this, of course, is the case for a standard Bell-
type experiment, this does not automatically mean that this is also the case
for the novel EWFS considered by Brukner, containing four agents and, in
particular, two superobservers performing measurements over whole labs. One
must be careful, then, to have in mind what is the correct _physical
interpretation_ of the expectation values in a given CHSH inequality—and not
because the quantum predictions violate such an inequality for a given
experimental scenario, it means that they do so for others.

So what about the quantum predictions for Brukner’s scenario? The problem is
that, since it involves intermediate measurements by Charlie and Debbie, such
predictions crucially depend on what exactly happens during measurements. The
issue, of course, is that the standard interpretation is hopelessly vague in
that respect (Bell,, 1990). The upshot is that _standard quantum mechanics is
simply unable to make concrete predictions for the scenario considered_ —the
measurement problem gets in the way. That is, the Wigner’s friend scenarios
under consideration are precisely the type of settings for which one cannot
get away with pressing on, adopting an operational stance and ignoring the
conceptual limitations of the standard framework.

It is often (implicitly or explicitly) assumed that a correct application of
standard quantum mechanics to a Wigner’s friend scenario means that Wigner
must describe the laboratory, and all of its contents, as evolving
unitarily—allegedly in accordance with the quantum rule for the evolution of
isolated systems. This, presumably, is what Brukner has in mind with his
‘Universal validity of quantum theory’ assumption. It is not clear to me,
though, that standard quantum mechanics contains such a rule and, if so, that
this would be a correct application of it. In any case, one could simply
_stipulate_ that the laboratory and all of its contents evolve unitarily, even
during the measurements of the friends. The issue is that Brukner, (2018) also
wants to assume that measurements yield objective results, but we know from
Maudlin, (1995) that those two assumptions, unitarity and objective outcomes,
are incompatible with another assumption inherent to the standard framework:
that the physical description given by the quantum state is complete.
Therefore, on pain of inconsistency, it is simply impossible to assume
unitarity and objective outcomes, and to employ the standard framework to make
predictions. One can, of course, employ a framework which solves the
measurement problem to make predictions. The issue is that different ways of
doing so lead to different predictions for the scenario in question. For
instance, while pilot-wave does predict a violation of the CHSH inequality,
objective collapse models to not (see Okon, (2022) for details).

A final comment regarding the conclusions in Brukner, (2018). We saw that the
theorem is interpreted as showing that there can be no theoretical framework
where the results of different observers can jointly be considered objective
facts of the world. However, even ignoring all the problems described above,
and taking the result as valid, it is not clear why would one be forced to
discard the ‘Observer-independent facts’ assumption, as Brukner suggests. At
best, the theorem would demand for (at least) one of its assumptions to be
abandoned, without pointing to any one of them in particular.

Taking stock, the no-go theorem for observer-independent facts in Brukner,
(2018) depends on a non-trivial, hidden assumption demanding for the Z
measurement results of Alice and Bob to coincide with those of Charlie and
Debbie. Moreover, there is no such thing as the correct quantum predictions
for the proposed experiment, so there is no quantum benchmark with which to
compare the predictions of models satisfying the imposed constrains—namely,
the four assumptions in Brukner, (2018) plus the hidden one mentioned above.
That is, the constraints imposed on frameworks satisfying these assumptions
are not really in conflict with quantum predictions. Below I will show that
the theorem in Bong et al., (2020) suffers from quite similar issues. I delay
a full assessment of the impact of these limitations on the validity,
significance and strength of these theorems until after the detailed
evaluation of the “Local Friendliness no-go theorem”, to which we turn next.

###  3.2 Evaluation of the Local Friendliness no-go theorem

I start with a simple, intuitive proof of the LF inequality, for the
particular case when Alice and Bob only have two binary-outcome measurement
options—in which case the LF inequality coincides with the CHSH inequality.
Consider the LF setting and an ensemble of runs for which Alice and Bob choose
$X,Y=2$. By AOE, there is a joint probability distribution for the results of
all observers and, by Fine’s theorem (Fine,, 1982), the expectation values of
products of results, calculated with the marginals of such a joint probability
distribution, satisfy the CHSH inequality

$\left\langle C_{2}D_{2}\right\rangle+\left\langle
C_{2}B_{2}\right\rangle+\left\langle A_{2}D_{2}\right\rangle-\left\langle
A_{2}B_{2}\right\rangle\leq 2$ (2)

with, e.g., $\left\langle C_{2}B_{2}\right\rangle$, the expectation value of
the product of the results of Charlie and Bob, when $X,Y=2$.

Next, we employ the LF assumptions to transform this inequality, involving
results of all four observers, into the CHSH inequality for the results of
Alice and Bob. In particular, we note that

$\left\langle C_{2}D_{2}\right\rangle\overset{\text{NSD}}{=}\left\langle
C_{1}D_{1}\right\rangle\overset{\text{AOE}}{=}\left\langle
A_{1}B_{1}\right\rangle$ (3)

and that

$\left\langle C_{2}B_{2}\right\rangle\overset{\text{L,NSD}}{=}\left\langle
C_{1}B_{2}\right\rangle\overset{\text{AOE}}{=}\left\langle
A_{1}B_{2}\right\rangle$ (4) $\left\langle
A_{2}D_{2}\right\rangle\overset{\text{L,NSD}}{=}\left\langle
A_{2}D_{1}\right\rangle\overset{\text{AOE}}{=}\left\langle
A_{2}B_{1}\right\rangle.$ (5)

That is, we see that NSD and L allow the identification of certain expectation
values with the same observers, but different settings, and that AOE and, in
particular, conditions ii and iii, allow us to substitute, under the right
circumstances, Alice for Charlie and Bob for Debbie. Putting everything
together,

$\left\langle A_{1}B_{1}\right\rangle+\left\langle
A_{1}B_{2}\right\rangle+\left\langle A_{2}B_{1}\right\rangle-\left\langle
A_{2}B_{2}\right\rangle\leq 2.$ (6)

Next, I point out that, while standard proofs of Bell’s theorem explicitly
allow for the probabilities to change from run to run (through their
dependence on what is usually denoted by $\lambda$), Bong et al., (2020) does
not allow for such a change—they explicitly take the results of the friends to
play the role of hidden variables, but do not allow for additional variables
that could alter the joint probability distribution from run to run. This
might seem like a strong and problematic assumption, and it is, but the issue
is relatively easy to remedy.

One can think of Bell’s $\lambda$ as codifying whatever characterizes the
complete state of the system to be measured in each run. If, as in original
formulations of Bell’s theorem, in all runs one prepares the same quantum
state (e.g., a singlet), then $\lambda$ summarizes whatever else changes
besides the preparation of the initial state. In contrast, if as in Bong et
al., (2020), one does not want to impose the same preparation for all runs,
then $\lambda$ would codify the preparation employed in each run, plus
whatever else changes in that run. In any case, the assumption of no variation
of the joint probability distribution with $\lambda$, seems problematic.
However, one can fix the problem by adding a dependency of the joint
probability distributions on $\lambda$ and averaging over it. It is important
to point out, though, that for this procedure to work, one must assume, on top
of the LF assumptions, the so-called ‘Settings independence’ assumption—which
is another version of what they call ‘No-Superdeterminism’. Such an assumption
demands, for the distribution of $\lambda$ over the whole ensemble of runs, to
be statistically independent of the settings:
$\rho(\lambda|X,Y)=\rho(\lambda)$. Note that $\rho(\lambda)$ is the actual
distribution of physical states $\lambda$ over the ensemble of measured
systems, and not a probability distribution, so question regarding the
existence or convergence of $\rho(\lambda)$ are easily answered in the
affirmative.

The upshot is that, given settings independence, it is enough for the
expectation values calculated with the joint probability distribution of each
possible value of $\lambda$ to satisfy the inequality, for the average of
them, over $\lambda$, to do so. In other words, even if you allow for
different runs to have different joint probability distributions, labeled by
different $\lambda$, if each joint probability distribution satisfies the
inequality, which it will if the LF assumptions are imposed, and settings
independence is the case, then the predictions of these models, averaged over
$\lambda$, will also satisfy the inequality. Let’s move on the evaluation of
the theorem.

####  3.2.1 The ‘Absoluteness of Observed Events’ assumption

Above we saw that the starting point of the LF theorem is a critique of
Brukner’s ‘Observer-independent facts’ assumption, which leads to its
substitution by LF’s AOE. Such an assumption demands for all performed
experiments to have observer-independent results which, for the experiment
under consideration, implies the existence of a joint probability
distribution, $P(ABCD|XY)$. Moreover, such a joint probability distribution is
demanded to ensure consistency between the outcomes of friends and
superobservers when $X,Y=1$.

I argued, however, that the critique in Bong et al., (2020) seems to depend on
a particular, arguably not very natural, reading of an ambiguous element in
Brukner’s presentation. In particular, they read Brukner, (2018) as demanding
all measurements that might be performed to have well-defined values, but it
is more natural to read it as demanding all performed experiments to have
well-defined values, which seems equivalent to what AOE demands. We also saw
that Brukner’s derivation of the theorem depends on a hidden assumption, to
the effect that the results of Alice and Bob, when they perform a Z
measurement, necessarily coincide with the results previously obtained by
Charlie and Debbie. It seems, then, that this hidden assumption is identical
to the demand of consistency between friends and superobservers imposed by
conditions ii and iii in AOE. I conclude that, by adopting the most natural
interpretation of Brukner’s claims, AOE is exactly equivalent to the
conjunction of Brukner’s ‘Observer-independent facts’ with his hidden
assumption.

All this makes it clear that the AOE assumption contains two _independent_
parts: 1) the claim that there is a joint probability distribution for all
results and 2) conditions ii and iii, which have to do with the particular
case in which Wigner measures in the same basis as his friend. Moreover, I
want to argue that, while Bong et al., (2020) take both parts as necessary
demands in order to ensure for observed events to be considered observer-
independent or absolute, it is only the first part that is required to enforce
absoluteness of observed events. The key issue to have in mind is that, even
when Wigner and his friend measure “in the same basis” (e.g., $X,Y=1$), it is
never the case that what they are comparing are the result of the exact same
measurement (I use the quotations to emphasize the fact that friends and
superobservers perform measurements on different systems so, strictly
speaking, the bases cannot be the same). As a result, a demand for
absoluteness of observed events does not necessarily imply a demand for these
two measurements to yield the exact same result. Let’s discuss this issue with
some care.

I start by noting that there are alternative ways to describe a measurement by
Wigner when he measures “in the same basis” as his friend, and that some of
them may elicit incorrect impressions. For instance, according to Brukner,
(2018), Alice and Bob might perform a “Z measurement” in basis
$\\{\ket{Z_{+}}_{i},\ket{Z_{-}}_{i}\\}$ with $\ket{Z_{\pm}}_{i}$ the state of
whole lab $i$ after the friend measures the spin along $z$ and finds the
result $\pm$. However, according to Bong et al., (2020), if $X=1$, Alice opens
Charlie’s vault, asks him what he observed and sets her result equal to that
of Charlie. The issue is that, portraying the measurement of Alice as her
simply opening the door and asking Charlie the result, incorrectly conveys the
idea that the measurement in question is quite mundane while, of course, it is
not. The experiments considered assume that Charlie’s vault is absolutely
isolated from the outside world, so any intuitions we might have regarding
everyday experiences of opening doors and asking colleges about results
obtained in previous experiments, should not be interpreted as reasonable
constraints on the set of models considered. In particular, in the same way
that a measurement of a superobserver in a different basis might fully erase
all records (including memories) of a previous measurement, a measurement “in
the same basis” might also do so.

Another way to see this, is to point out that, when the measurements by Alice
and Bob are described in Bong et al., (2020), they are treated differently,
depending on whether they measure “in the same basis” than the friend or not.
As we saw, when they do, the measurements are described as they simply opening
the door. However, when they measure in a different basis, it is argued that
the measurement can be thought of as one in which Alice and Bob undo the
measurement of the friend, and measure the particle again directly. For
starters, it is clear that this description is available only by assuming that
the measurements are _reversible_ , which might not be the case for some
models. Moreover, if measurements are reversible, even when Alice and Bob
measure “in the same basis” as the friend, one could think of the measurement
as them undoing the friend’s measurement, thereby fully erasing all records of
the friends’ results (including their memories) and then measuring the
particle again in the same basis. Put like this, it is not unreasonable to
think of models in which the results of Alice and Bob do not necessarily
coincide with the friend’s, even when they measure in the same basis.

In fact, as mentioned above, this non-necessary coincidence between the
measurements of, say, Bob and Debbie, when they measure in the same direction,
is actually the prediction of pilot-wave theory, the preeminent, realist,
objective quantum framework. As shown in Okon, (2022, Appendix A), if one
models the experimental arrangement in Bong et al., (2020) employing pilot-
wave theory, one finds the correlation between the results of, say, results of
Bob and Debbie along the same direction, to be a function of the measurement
setting employed by Alice—with possible settings of Alice completely erasing
the correlation between Bob and Debbie. Then, according to pilot-wave, it is
not the case that Bob and Debbie necessarily obtain the same result when they
measure along the same direction. That is, pilot-wave theory violates
conditions ii and iii of AOE. Does this mean that such a framework is not
sufficiently objective? I certainly do not think so.

In sum, the AOE assumption contains two independent parts, one demanding for
there to be a joint probability distribution for all results and another
demanding consistency between the results of Wigner and his friend when they
both measure “in the same basis”. And, while the first part does seem to be a
reasonable demand to make in order to attain objectivity of observed results,
the second part seems like too strong a demand, not even true in existing
frameworks widely regarded as realist and objective.

As I mentioned in section 2.2, an important, far-reaching claim in Bong et
al., (2020) is that their theorem “places strictly stronger constraints on
physical reality than Bell’s theorem”. To arrive at this claim, first it is
argued out that, although usually left implicit, the AOE assumption is
necessary for the derivation of Bell’s theorem. Then, it is pointed out that,
while the derivation of the LF inequalities require AOE, NSD and L, the
derivation of the Bell inequalities requires AOE, NSD, L and ‘Outcome
independence’. Since, according to this reasoning, the LF assumptions are a
subset of Bell’s assumptions, it is concluded that it is logically impossible
to construct a model that allows violation of the LF inequality, but does not
allow violation of Bell’s inequalities.

I contend, though, that the AOE assumption is _not_ necessary for the
derivation of Bell’s theorem. Therefore, it is not the case that the LF
theorem is strictly stronger than Bell’s theorem. As we saw, the AOE
assumption involves two independent parts; and, while it seems true that
demanding for results to be objective is, in fact, a necessary (implicit)
supposition behind Bell’s result, a demand of consistency between the results
of friends and superobservers, is not required for the derivation of Bell’s
theorem. In fact, Bell’s experimental scenario does not even involve friends
and superobservers, so any condition constraining the relation between the
results of friends and superobservers, seems fully irrelevant.

One could argue that Bell’s experiment indeed involves sequences of
observations, in the sense that it contains any number of “intermediate”
measurements. For instance, a detector measures a particle, the detector is
measured by a computer, the computer is measured by an observer, etc.
Moreover, it could be argued that, if at any stage in that chain, the
measurement does not simply reproduce the preexisting data, then a model
satisfying all of Bell’s assumptions would be able to violate his inequality
at some stage. That is, it could be argued that we always assume that the
records we read on a computer are accurate records of the detector
measurements at some earlier time and that, without such an assumption, one
could have a model satisfying Bell’s assumptions, but breaking the inequality.
The conclusion, then, is that conditions ii and iii in AOE are necessary to
prove Bell’s theorem.

This line of reasoning is mistaken. First of all, the step from, say, the
computer recording the result, to an observer reading the result from the
computer, does not seem to constitute an instance in which conditions ii and
iii of AOE apply, as such conditions only deal with measurements by
superobservers—i.e., situations in which decoherence is fully kept under
control. More importantly, models satisfying Bell’s assumptions are simply
unable break the inequality at any stage, even if one allows for changes in
the chain of observations. That is, a model satisfying Bell’s assumptions will
not display quantum correlations for the first measurements in the chain, and
any number of _local_ jumps in the chain on each side will _not_ be able to
produce quantum correlations in later steps of the chain. We conclude that,
contrary to what is stated in Bong et al., (2020), conditions ii and iii in
AOE are not necessary to prove Bell’s theorem.

The idea that conditions ii and iii in AOE are necessary to prove Bell’s
theorem might be thought to be related with the so-called “collapse locality
loophole” described in Kent, (2005). The observation behind such a loophole is
that, while Alice and Bob’s measurements in Bell experiments are supposed to
be performed at spacelike separation, in real experiments, there is some
leeway as to where in spacetime the measurement outcomes actually arise. In
particular, if one maintains, for instance, that measurements end until a
human learns the result, then the measurements in all existing experiments are
not spacelike separated. As a result, Kent points out that, if one considers
models satisfying Bell’s assumptions, but stipulating that human observation
is required for measurements to end, then, in spite of all existing Bell
experiments, one can argue that the inequalities have not been violated
experimentally—opening the door for local models to be empirically adequate.

The point I want to make is that one must not confuse playing with the time at
which measurements occur, which, following Kent, allows for experiments
claiming violations of Bell’s inequality to be questioned, with violations of
conditions ii and iii, which do not modify when measurements happen, but allow
for a variation of results in subsequent measurements. Consider, then, a model
in which Bell’s assumptions are satisfied, conditions ii and iii are violated,
and in which it is stipulated that measurements fully end as soon as particles
travel through the detector. Such a model is simply unable to break Bell’s
inequality and since, by construction, does not employ Kent’s loophole, it is
ruled out by existing Bell experiments.

In sum, conditions ii and iii of the AOE assumption are _not_ necessary for
the derivation of Bell’s theorem. Therefore, in opposition to what is defended
in Bong et al., (2020), it is perfectly possible to construct a model, which
allows for a violation of the LF inequality, but does not allow for a
violation of Bell’s inequality. That is, it is perfectly possible to construct
a non-local model that breaks conditions ii and iii of AOE, such that: it
satisfies Bell’s inequality in a standard Bell experiment, but breaks the LF
inequality in Brukner’s experiment. All one needs is a model, which yields
quantum correlations for measurements of superobservers, but not for those of
the friends. The fact that ii and iii are not satisfied by such a model, makes
this perfectly possible.

In reply, it could be argued that it is not possible to perform Brukner’s
experiment without, at the same time, performing a Bell experiment. The point
is that, in the end, the public data concerns the correlations $P(A,B|X,Y)$
between Alice and Bob, and this is, by construction, a Bell scenario. It is
very important to recognize, though, that if one adopts this fully
_operational_ definition of what a Bell experiment consists of, one must also
acknowledge that a model might behave very differently in different _concrete
realizations_ of a Bell scenario. For instance, one can consider Brukner’s
experiment and contrast it with what I would call a standard Bell experiment,
i.e., one in which the two spin-$\frac{1}{2}$ particles of a singlet are sent
to Alice and Bob for them to measure spin. Clearly, a model does not have to
perform equally in both scenarios. For instance, as pointed out in Bong et
al., (2020), objective collapse models break the inequality in the standard
scenario, but satisfy it in Brukner’s EWFS. Moreover, I take it that, when it
is argued that it is logically impossible to construct a model that allows
violations of the LF inequality, but does not allow violations of Bell’s
inequality, what is meant is that it is impossible to construct a model which
violates the Bell inequality, in a standard Bell scenario, but which does not
allow for violations of the LF inequalities, in Brukner’s scenario. My claim
is that it is perfectly possible to construct such a model.

####  3.2.2 A toy model

Consider a Brukner EWFS in which, for concreteness, the two entangled
particles are prepared in a singlet. The toy model I have in mind is such that
the state it assigns to the two particles is given by the standard quantum
state associated with a singlet, together with a pair of randomly selected
angles $\theta_{1}$ and $\theta_{2}$. Moreover, according to this model, when
Charlie and Debbie measure the spin of their particles, three things happen.
First, they obtain a result according to the following probabilities

$P(C=1)=\cos^{2}\theta_{1},\qquad P(C=-1)=\sin^{2}\theta_{1}$ (7)
$P(D=1)=\cos^{2}\theta_{2},\qquad P(D=-1)=\sin^{2}\theta_{2}.$ (8)

Clearly, the results of Charlie and Debbie will not be correlated. Second, the
values of $\theta_{1}$ and $\theta_{2}$ are updated according to the result
obtained. Third, the state of Charlie and Debbie’s labs gets entangled, though
a purely unitary evolution, with the state of the incoming particles. Finally,
according to this model, when Alice and Bob measure the state of Charlie and
Debbie’s labs, the model behaves exactly as an objective collapse model. That
is, the measurement on one side causes a non-local collapse to an
(approximate) eigenstate of the measured observable, leading to standard Bell-
like quantum correlations for the results of Alice and Bob.

More generally, the proposed toy model is such that 1) the state of the
prepared entangled pair is given by the quantum state plus a random pair of
angles $\theta_{1}$ and $\theta_{2}$, 2) the results of direct measurements of
the spin of the particles are governed by Eqs. (7) and (8), 3) after a
measurement of spin, the angles are updated according to the result obtained,
4) such a measurement also causes the state of the particles to get entangled
with the macroscopic measurement apparatus (and environment) and 5) when the
particles get entangled with macroscopic objects, the results of Wigner’s
friend-type measurements on such macroscopic objects behave analogously to an
objective collapse model. Above we saw that the measurements of the
superobservers can also be thought of as they undoing the measurements
performed by the friends and measuring the particles again directly. It could
be objected that the proposed model would have trouble dealing with this way
of performing the experiment. Note, however, that the idea of undoing the
measurements of the friends presupposes for those measurements to be
reversible. Since, in this model, the measurements of the friends are
probabilistic, this alternative way of thinking about the measurements of the
superobservers is simply unavailable.

What assumptions does this model satisfy? First, it is such that all
experiments have observer-independent, absolute results. Therefore, in the
context of Brukner’s EWFS, there is a joint probability distribution for the
results of all four observers. It is not the case, however, that the results
of friends and superobservers must coincide when $X,Y=1$ so, even tough
observed results are absolute, conditions ii and iii and AOE are violated.
Regarding NSD, since the probabilities for $C$ and $D$ are given by Eqs. (7)
and (8), it is clear that they are independent of $X$ and $Y$, so such an
assumption is straightforwardly satisfied. As for L, since the results of
Alice and Bob are governed in analogy with an objective collapse model, and
since such models satisfy the ‘Parameter independence’ assumption (called L in
Bong et al., (2020)), then this model also does so. Note, however, that
objective collapse models do not satisfy the ‘Outcome independence’
assumption, so this model also fails to do so. This, of course, renders the
the model non-local in Bell’s sense.

What about the behaviour of the model in experiments? In a standard Bell
experiment, the results are independently governed by $\theta_{1}$ and
$\theta_{2}$, so they will not be able to break the inequality. In contrast,
in an EWFS, when the superobservers measure, the results mimic an objective
collapse model, so they certainly would violate the LF inequality. It is
possible, then, to construct a model which satisfies Bell’s inequality but
violates the LF inequalities. Another way to put all this, is that a model
violating or not the LF inequalities (in an EWFS) is independent of it
violating or not Bell’s inequality (in a standard Bell experiment). For
instance, knowing that a model violates the LF inequality does not settle the
question as to how it would perform in a standard Bell experiment. It is not
the case, then, that the LF theorem “places strictly stronger constraints on
physical reality than Bell’s theorem”.

####  3.2.3 The wording of the theorem

The next issue I would like to analyze has to do with the wording of the LF
theorem. As we saw in section 2.2, the construction of the theorem has two
parts. First, it is shown that models obeying the LF assumptions satisfy the
LF inequalities. Second, it is argued for the following conditional: if the
quantum operations required by the proposed experiment can, in principle, be
performed, then the quantum predictions violate the LF inequalities. Putting
everything together, by defining a ‘physical theory’ as “any theory that
correctly predicts the correlations between the outcomes observed by the
superobservers Alice and Bob”, the LF theorem is stated as a conditional: if a
superobserver can perform arbitrary quantum operations on an observer and its
environment, then no physical theory can satisfy the LF assumptions.

I start by inspecting the definition of ‘physical theory’. As we just saw, in
2.2, that notion is reserved for any theory that “correctly predicts” the
correlations between Alice and Bob. But what are these “correct predictions”?
Since, as already was established, standard quantum mechanics cannot produce
actual predictions for the experiment in question, these “correct predictions”
cannot be the quantum predictions. Therefore, the “correct predictions” must
mean those that coincide with the actual, experimentally observed
correlations, whatever they turn out to be. Of course, for all we know, these
actual correlations can be anything. Therefore, in order to constrain them,
2.2 employs a conditional to the effect that, if a superobserver can perform
arbitrary quantum operations on an observer and its environment, then the
quantum predictions violate the LF inequalities. The idea being that, while,
in general, it is true that there are no correct quantum predictions for the
proposed experiment, if the antecedent of the conditional is fulfilled, i.e.,
if a superobserver can, in fact, perform arbitrary quantum operations on an
observer and its environment, then unambiguous predictions can be produced.
Moreover, the idea is that, in that case, such predictions violate the LF
inequalities. The problem is that such a conditional is false.

To begin with, I find the operational tone of the antecedent of the
conditional vague and inadequate. The antecedent in question is formulated in
Bong et al., (2020) in a couple of different ways, one general and one
particular to the EWFS in question. The general form, as we saw, asks for
superobservers to be able to perform arbitrary quantum operations on an
observer and its environment—that is, for quantum evolution (including quantum
measurement) to be controllable on such a scale. The particular formulation
demands for the quantum operations required by the proposed experiment to be
performable, at least in principle. Either way, what is it exactly that the
condition demands? Presumably, the idea is something like this. If, for
instance, the measurement of one of the friends objectively breaks unitarity,
then the superobserver would not be able to perform a quantum operation that
maintains the friend (and her environment) in an entangled superposition
corresponding to different observational states. That is, the fact that
unitary is broken, is supposed to restrict the operations available to the
superobserver.

It seems to me, however, that the fact that a measurement of a friend breaks
unitarity is better understood as a feature of the internal dynamics of the
model in question, and not really as a constraint on what operations the
superobservers can or cannot perform. In other words, even if the measurements
of the friends break unitarity, the superobservers are perfectly capable of
performing the quantum operations required by the proposed experiment, namely,
certain measurements on the corresponding labs—i.e., the breakdown of
unitarity does not restrict the measurements available to the superobservers.
It seems, then, that what the antecedent of the conditional needs to capture
is a constraint on how macroscopic systems evolve during measurements. And,
even though Bong et al., (2020) shies away from using these terms, what it
seems to be actually demanded is for the evolution during those measurements
to be purely unitary.

More importantly, even if the measurements of the friends involve purely
unitarity evolution, and the superobservers are able to perform quantum
operations that maintain the friends in coherent superpositions, that does not
mean that the correlations predicted are going to break the LF inequalities.
That is, even if the states of the labs, after the friends measure, are given
by what quantum mechanics calls a coherent superposition of different
observational states, it is not necessarily the case that the LF inequalities
would be broken. As we saw already, standard quantum mechanics cannot make
predictions for the EWFS in question. Therefore, to make predictions for them,
it is necessary to come up with an alternative, non-standard framework,
capable of doing so. The point I want to make is that the fact that one of
these frameworks stipulates purely unitary evolution during measurements in no
way implies that such a model predicts violations of the LF inequalities. As I
said, these models must be non-standard alternatives to the standard framework
and, as such, they can possess all sorts of non-standard features, including
predictions that do not break the LF inequalities, even if superobservers are
able to fully control the quantum states of the friends. We must not forget
that the assumption that results are objective, together with the assumption
of purely unitary evolution, implies the the existence of additional variables
(Maudlin,, 1995). And this additional variables can play a non-trivial role in
the calculation of predictions, allowing for predictions that do not break the
LF inequalities.

In sum, Bong et al., (2020) looks for a condition that would guarantee
predictions that break the LF inequality and it tries to formulate it in terms
of the operations the superobservers are able to perform. The problem is that
such a proposal does not work and, more generally, there seems to be no simple
condition that would achieve what they are looking for. The truth is that, to
make predictions for these scenarios, alternatives to the standard framework
are required and different models will make different predictions, without
there being a single condition that could determine on which side of the LF
inequalities the predictions of a model would land. Of course, what Bong et
al., (2020) does show is that models satisfying the LF assumptions do not
break the inequalities, but what is lacking is a way to stipulate what models
are able to break them.

We can contrast all this with Bell’s theorem. In that case, one has 1) that
models satisfying Bell’s assumptions satisfy Bell’s inequality and 2) that
standard quantum mechanics unambiguously predicts violations of the
inequality. The theorem can then be paraphrased as “the predictions of local
models are incompatible with those of quantum mechanics”. In the LF case, in
contrast, one shows that models satisfying the LF assumptions satisfy the LF
inequalities, but one cannot show that quantum mechanics predicts violations
of the inequality—and it is not even clear how to characterize models that
violate it. One is then only left with the connection between the LF
assumptions and the LF inequality.

Now, it is of course true that, once Bell experiments have been performed, and
clear violations of the inequalities have been observed empirically, the fact
that standard quantum mechanics violates Bell’s inequality becomes almost
irrelevant. With experimental violations of the inequality, what we now have
is a much stronger claim, namely, that “the predictions of local models are
incompatible with actual experiments”—a claim which is completely independent
of quantum mechanics. Couldn’t one do the same with the LF inequalities? The
issue is that, unlike Bell’s experiment, the one required to experimentally
probe the LF inequalities, while presumably doable in principle, is
unrealizable in practice (and will be so for the foreseeable future). It is
true, then, that such experiments would set constraints on empirically viable
models, but the fact that we are unable to actually perform them, greatly
diminishes the force behind the LF result.

Summing up, in Bong et al., (2020) it is shown that models satisfying the LF
assumptions satisfy the LF inequalities. However, contrary to what it is
argued there, it is not the case that such assumptions are strictly weaker
that those of Bell. Moreover, since quantum mechanics is unable to make
predictions for the EWFS considered, there is no standard violating the LF
inequalities, with which to contrast models satisfying the LF assumptions.
Finally, the proposed experiment is, at this stage, no more than a
gedankenexperiment, so the prospects of experimentally constraining LF models
seem, at least for a long time, unattainable.

##  4 Conclusions

The “no-go theorem for observer-independent facts” and the “Local Friendliness
no-go theorem” are taken by their authors to impose non-trivial constraints on
physical reality. However, such theorems suffer from a list of shortcomings
that question their validity and limit their strength. In this work, I have
shown that the theorem in Brukner, (2018) depends on a non-trivial, implicit
assumption, regarding the relation between the results of friends and
superobservers, and that it relies on the mistaken notion that standard
quantum mechanics is able to make predictions for the EWFS considered.

As for the theorem Bong et al., (2020), I have shown that, contrary to what is
alleged, it does not rest on assumptions that are strictly weaker that those
of Bell. Moreover, it relies on the same mistaken application of standard
quantum mechanics to the EWFS in question, so there is no standard with which
to contrast models satisfying the LF assumptions. Finally, I note that the
proposed EWFS is unrealizable in practice, a fact that greatly diminishes the
force behind the LF result.

In the end, all these theorems offer is the fact that models satisfying a
certain group of assumptions satisfy certain inequalities. However, without
theoretical or experimental benchmarks with which to compare those models, the
established fact sharply looses impact and relevance. I conclude that the
theorems fully fail in imposing significant constraints on the possible nature
of physical reality.

## Acknowledgments

I acknowledge support from CONACYT grant 140630.

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