# Wigner’s friend scenarios: on what to condition and how to verify the
predictions

Flavio Del Santo  Group of Applied Physics, University of Geneva, 1211 Geneva,
Switzerland  Constructor University, Geneva, Switzerland  Vienna Center for
Quantum Science and Technology (VCQ), Faculty of Physics, Boltzmanngasse 5,
University of Vienna, Vienna A-1090, Austria.  Institute for Quantum Optics
and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse
3, A-1090 Vienna, Austria.  Gonzalo Manzano  Institute for Cross-Disciplinary
Physics and Complex Systems, (IFISC, UIB-CSIC), Campus Universitat de les
Illes Balears E-07122, Palma de Mallorca, Spain  Časlav Brukner  Vienna Center
for Quantum Science and Technology (VCQ), Faculty of Physics, Boltzmanngasse
5, University of Vienna, Vienna A-1090, Austria.  Institute for Quantum Optics
and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse
3, A-1090 Vienna, Austria.

(August 28, 2024)

###### Abstract

Wigner’s friend experiment and its modern extensions display the ambiguity of
the quantum mechanical description regarding the assignment of quantum states.
While the friend applies the state-update rule to the system upon observing an
outcome of her measurement in a quantum system, Wigner describes the friend’s
measurement as a unitary evolution, resulting in an entangled state for the
composite system of the friend and the system. In this respect, Wigner is
often referred to as a “superobserver” who has the supreme technological
ability to keep the friend’s laboratory coherent. As such, it is often argued
that he has the “correct” description of the state. Here we show that the
situation is more symmetrical than is usually thought: there are different
types of information that each of the observers has that the other
fundamentally cannot have - they reside in different “bubbles” (in
Calvalcanti’s terminology). While this can explain why the objectivity of the
state assignment is only relative to the bubble, we consider more elaborated
situations in form of a game in which the players can switch between bubbles.
We find that, in certain circumstances, observers may be entitled to adopt and
verify the state assignment from another bubble if they condition their
predictions on all information that is in principle available to them.

##  I Introduction

The notorious quantum measurement problem stems from an ambiguity about two
alternative dynamics that are present in quantum theory. In the words of
Baumann and Wolf [1], “in what is called standard quantum mechanics, the
measurement-update rule, commonly associated with a collapse, is a break with
the otherwise unitary evolution governed by the Schrödinger equation. The
formalism, however, provides no indication about when to apply this rule: It
does not state what qualifies some interactions as measurements but not others
(the measurement problem).”

The problem is most notably exemplified by the Wigner’s friend thought
experiment [2], whose scope has been enlarged in recent years [3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. This takes the
ambiguity about which dynamical law to apply to the extreme by featuring an
observer –historically referred to as the friend– who takes both the roles of
an observer performing a quantum experiment and of a quantum system observed
by another observer, Wigner. More specifically, the friend performs a
measurement on a quantum system in her laboratory, while Wigner is able to
perform quantum measurement on the whole laboratory, thus treating both the
friend and the particle as a joint quantum system.

What is the state of the system after the friend’s measurement? Since the
friend observes a definite outcome, it seems legit to say that the state
should be updated accordingly, thus applying the “collapse” rule. On the other
hand, from Wigner’s perspective, the friend and the system are merely two
interacting subsystems of an isolated system, thus he seems entitled to
describe the same physical process unitarily. The question then arises: is
there a single, absolute, “correct” state? By this we mean that _every_
hypothetical observer ought to assign the same state in order to make
predictions that turn out to be empirically correct.

The answer in the positive view has been recently challenged by a series of
no-go theorems [3, 5, 4, 11, 12, 19, 15, 18, 20], in so far as it is possible
that there exist no “single reality” that can be shared between the friend and
Wigner, and they instead live in different “bubbles” [12]. 111This is
reminiscent of the philosophical position of _fragmentalism_ [23, 24], which
upholds “that the world is not a monolithic whole constructed from mutually
compatible facts, but rather a collection of fragments, with each fragment
containing mutually compatible facts, while different fragments are
incompatible” [25]. A bubble is an information-theoretic concept and is to be
understood as a “region” in which certain information about, e.g., a
measurement result, is (in principle) available. Every hypothetical observer
who has in principle access to the same relevant classical information –i.e.,
that can be copied, broadcasted, etc.– lives in the same bubble. In formal
terms, these no-go theorems state that, under certain assumptions, there is no
joint probability for the results obtained by the friend and Wigner such that
the respective probabilities for their results can be understood as marginals
of it. This resolves the ambiguity of the state assignment, insofar as a state
can be considered objective only relatively to a specified bubble, i.e., each
and every hypothetical observer within the same bubble will be entitled to
assign the same state within their respective bubble. However, the states will
in general be different for different bubbles.

The strategies for rational prediction are less clear when the observers can
“switch” bubbles, for example, when a measurement is performed in another
bubble, but the player finds a way to verify the outcomes of this measurement
and thus their previous predictions. In such cases, if an observer who resides
in one bubble has access to a quantum state assignment from another bubble,
should they adopt it for the purpose of predicting the probabilities for the
occurrence of the measurement outcomes? Imagine, for example, that the friend
knows the state in which Wigner describes her laboratory, and she also knows
which measurement of her laboratory Wigner will make next. On whose knowledge
should she condition her prediction of the result of this measurement, her own
or Wigner’s? Given that the measurement will affect her memory, will she even
be able to evaluate her previous prediction after the measurement has been
completed?

In this paper we address these questions by considering a game played between
the friend and Wigner. We will argue that, in certain circumstances, both
players can be persuaded that their state assignment, based solely on the
information available in their individual bubbles, is not optimal for
predicting future measurements, but that of the other player is. In some
cases, the players will even be entitled to use the information that is
available in other bubbles to make predictions about the measurements made in
those bubbles, in addition to their own perspective when making predictions
about measurements made in their own bubble. To this end, we discuss to what
extent one can assume the same identity of an observer playing the game, i.e.
making predictions, undergoing measurements of their own bubble including
their memories, and verifying their previous predictions.

The first step towards investigating situation when an observer may “switch”
the bubbles was taken in Ref. [8], but only for the case where the friend was
placed in the situation. Here we aim at the situations more symmetrical for
the two observers,. Hence, we also analyse the game when the knowledge of the
two observers is not only maximally incompatible (i.e. when the two
measurements in the respective laboratories are maximally non-commutative),
but partially overlapping, thus covering the whole spectrum from full
compatibility to maximal incompatibility. In that sense our work amounts to
the task of finding information-theoretic conditions that allow to univocally
decide which state assignment is rational to adopt in cases when an observer
can “switch” from one bubble to another and when some information is allowed
to be exchanged between bubbles.

##  II Interacting “bubbles”

Let us consider the qubit system (S) and the friend’s (F) and Wigner’s (W)
environments or laboratories. At the beginning the situations of the friend
and of Wigner are fully symmetrical and similar to the one experienced by our
experimental colleagues in everyday practice. In particular, we assume that
they each possess reliable experimental devices and act as rational agents.
Moreover, the interaction between the quantum system and the environment in
each laboratory results in environmentally-induced decoherence and a
preferred environmental basis in which measurement outcomes are redundantly
recorded [26, 27, 28, 29]. The records may include “click” in a detector,
laboratory notes or even the observer’s memory. Nevertheless, for the purposes
of our argument, we will assume that the preferred bases of the friend’s and
Wigner’s environments are not identical, and that there is a “boundary region”
between two bubbles, in which a unitary transformation maps one basis into the
other.

We assume that (macroscopic) degrees of freedom of the environments, among
other states, can take two distinct states, $|\phi_{0}\rangle_{F}$ or
$|\phi_{1}\rangle_{F}$ in the preferred basis of friend’s and
$|\omega_{0}\rangle_{W}$ or $|\omega_{1}\rangle_{W}$ in the preferred basis of
Wigner’s laboratory. The preferred bases in the respective laboratories will
contain these pairs of states. The probabilities for finding these pairs of
states will be denoted as $p^{F}_{\phi_{i}}$ $(i=0,1)$ in the friend’s and
$p^{W}_{\omega_{j}}$ $(j=0,1)$ in Wigner’s laboratories.

We now consider Wigner’s friend though experiment. Consider the following
initial state

$|\Psi_{0}\rangle=|\psi\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W},$
(1)

where $|\psi\rangle_{S}$ is a state of the (qubit) system and
$|\phi_{0}\rangle_{F}$, $|\omega_{0}\rangle_{W}$ are no further specified
“ready” states of the friend’s and Wigner’s environment, rspectively. We
assume for simplicity
$|\psi\rangle_{S}=\frac{1}{\sqrt{2}}(|\uparrow\rangle_{S}+|\downarrow\rangle_{S})$
where $\\{|\uparrow\rangle_{S},|\downarrow\rangle_{S}\\}$ is the orthonormal
basis of the qubit in which the friend measures, say the $z$-direction of the
spin.

The friend then performs her measurement. In Wigner’s description, this
corresponds to the application of a unitary operator $U_{\mathrm{meas}}$ that
correlates the degrees of freedom of the system and of friend’s environment
(which includes her memory). We assume, for simplicity, that
$U_{\mathrm{meas}}$ implements a quantum controlled-NOT operation such that if
the spin is in state $|\uparrow\rangle_{S}$, the environmental state
$|\phi_{0}\rangle_{F}$ remains unchanged, and if it is
$|\downarrow\rangle_{S}$, it flips the state to $|\phi_{1}\rangle_{F}$. After
the measurement the overall state of the system and the friend’s environment
becomes perfectly correlated:

$|\Psi_{1}^{W}\rangle=U_{\mathrm{meas}}|\Psi_{0}\rangle=\frac{1}{\sqrt{2}}\Big{(}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}+|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}\Big{)}|\omega_{0}\rangle_{W},$
(2)

where the superscript $W$ indicates that this is the state assigned by Wigner.
Note that the presence of Wigner’s environment doesn’t play any role yet at
this stage since he has not performed any measurement.

It is important to note that we are assuming that there are degrees of
freedom, not explicitly denoted in Eq. (2), in the friend’s lab that do not
get correlated with the outcome. It is these degrees of freedom that can and
will be freely used in communication with Wigner’s bubble without decohering
the state.

On the other hand, the friend performing the measurement on the qubit system
enforces the measurement (state-update) postulate by the application of
projectors $\Pi_{\uparrow}=|\uparrow\rangle_{S}\langle\uparrow|$ and
$\Pi_{\downarrow}=|\downarrow\rangle_{S}\langle\downarrow|$ over
$|\psi_{s}\rangle$ in Eq. (1). An equivalent perspective on the measurement
process for the friend is to first adopt the state assignment (2) for the
qubit and her environment, while concurrently applying the projector
$\Pi^{F}_{0}=|\phi_{0}\rangle_{F}\langle\phi_{0}|$ and
$\Pi^{F}_{1}=|\phi_{1}\rangle_{F}\langle\phi_{1}|$ over $|\Psi^{W}_{1}\rangle$
in Eq. (2)222Provided that for the friend only reading out the states in the
preferred basis of the environment is permitted, two friend’s descriptions
either by applying the state update rule to the quantum system in state (1) or
by applying the state update rule to his environment in state (2) are
equivalent. Indeed, when performing experiments, our experimental colleagues
apply the state update rule to the measured system. Nevertheless, they may
still employ description (2) to analyse the decoherence process induced by the
measurement. Note that in either case, the state of the Wigner’s environment
is irrelevant and can be traced out.. This leads to classically correlating
the friend environment to the measurement result in the following states,
depending on the friend’s measurement outcome:

$|\Psi_{1}^{F}\rangle=\begin{cases}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W},&\text{for
outcome }\phi_{0}\\\ \vspace{0.1cm}\\\
|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{0}\rangle_{W},&\text{for
outcome }\phi_{1}\end{cases}$ (3)

each of them occurring with probabilities
$p^{F}_{\phi_{0}}=p^{F}_{\phi_{1}}=1/2$. Here the superscript $F$ stands for
the friend’s state assignment and Wigner’s environment is added just for
completeness but is unaccessible to her.

Let us consider now that a hypothetical physical process at the boundary of
the two labs couples the two environments as illustrated in Fig. 1 and both
observers knows the unitary. This coupling is described by a unitary operator
$U_{\mathrm{int}}$ acting over the system and the two environments, defined
such that the only elements on which it acts non-trivially are:

$\begin{split}{U_{\text{int}}}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}&\leavevmode\nobreak\
{=}\leavevmode\nobreak\
\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle_{SF}|\omega_{0}\rangle_{W}+|\Phi^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),\\\
{U_{\text{int}}}|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{0}\rangle_{W}&\leavevmode\nobreak\
{=}\leavevmode\nobreak\
\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle_{SF}|\omega_{0}\rangle_{W}-|\Phi^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),\end{split}$
(4)

where
$|\Phi^{\pm}\rangle_{SF}=(|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}\pm|\downarrow\rangle_{S}|\phi_{1}\rangle_{F})/\sqrt{2}$
are two of the standard Bell’s states over the system’s and friend’s Hilbert
space. Therefore, by applying $U_{\text{int}}$ to Eq. (2) we obtain for
Wigner’s description:

$\displaystyle|\Psi_{2}^{W}\rangle=U_{\text{int}}|\Psi_{1}^{W}\rangle=|\Phi^{+}\rangle_{SF}|\omega_{0}\rangle_{W}.$
(5)

Figure 1: Wigner and his friend assign in general different states to the same
physical systems, i.e., they live in different “bubbles”. In the friend’s
bubble a measurement of a qubit is performed and an outcome is obtained. We
assume that there exists an interface at the “boundary” of the two bubbles
where the environmental states of the friend’s and Wigner’s bubble unitarily
interact. Depending on the unitary $\hat{U}_{\text{int}}$, information about
the results of the friend’s measurements may or may not be encoded in the
Wigner’s environment, making the descriptions of the friend and the Wigner
compatible (hence merging the two bubbles in one) or not. In the extreme case
of the unitary (4), there is no information about the friend’s outcome in
Wigner’s environment.

The additional unitary operation does not affect the initial correlations
between the state of the qubit and the friend’s environment. Hence, looking at
his environmental degree of freedom, Wigner can only access the information
about the system and the friend being perfectly correlated in state
$|\Phi^{+}\rangle$, while the actual information about “which-outcome”
$\uparrow$ or $\downarrow$ of the qubit measurement is inaccessible. In
Appendix A we consider more general unitaries where partial information about
friend’s measurement outcome is leaked into Wigner’s environment.

On the other hand, applying the unitary $U_{\text{int}}$ to the friend’s post-
measurement states in Eq. (3) we get the states:

$|\Psi_{2}^{F}\rangle=\begin{cases}\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle_{SF}|\omega_{0}\rangle_{W}+|\Phi^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),\\\
\\\
\frac{1}{\sqrt{2}}(|\Phi^{+}\rangle_{SF}|\omega_{0}\rangle_{W}-|\Phi^{-}\rangle_{SF}|\omega_{1}\rangle_{W})\end{cases}$
(6)

associated respectively to $\phi_{0}$ and $\phi_{1}$ results. The above state
can be equivalently written as

$|\Psi_{2}^{F}\rangle=\begin{cases}\frac{1}{\sqrt{2}}(|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{+}\rangle_{W}+|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{-}\rangle_{W}),\\\
\\\
\frac{1}{\sqrt{2}}(|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{-}\rangle_{W}+|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{+}\rangle_{W}),\end{cases}$
(7)

with
$|\omega_{\pm}\rangle_{W}=(|\omega_{0}\rangle_{W}\pm|\omega_{1}\rangle_{W})/\sqrt{2}$.
From Eq. (6), it can be seen that Wigner’s environmental states
$\\{|\omega_{0}\rangle_{W},|\omega_{1}\rangle_{W}$ } are perfectly correlated
with the Bell states of the system and the friend’s environment. This implies
that no information about the outcome of the friend’s measurement can be
inferred from the preferred environmental basis in Wigner’s laboratory.
Moreover, from Eq. (7), it can be seen that for any of the two outcomes of the
qubit measurement and following the action of $U_{\mathrm{int}}$, the qubit
system and the friend’s laboratory become entangled in a state with two
“branches”. In each branch, the qubit state and the friend’s laboratory
records are found to be perfectly correlated for any measurement result: if
the system is $\uparrow$, the state of the record is $\phi_{0}$, and if it is
$\downarrow$, the record is in $\phi_{1}$ (and it is never so that e.g.
outcome $\uparrow$ is correlated with $\phi_{1}$). This demonstrates that the
correlation between the spin and the laboratory’s record is still maintained
after the unitary interaction. We will analyze the meaning of this state in
greater detail in Section V.

For the purpose of the game, we will assume that after the first measurement
of the qubit by the friend, one of two possible measurements will be performed
next. Either the friend will repeat the same measurement ($\text{M}_{F}$) of
her qubit, measuring again the spin along the $z$ direction, or alternatively,
Wigner will perform a measurement (MW) of the entire friend’s laboratory in
the preferred basis red of his environment, i.e., using projectors
$\Pi_{0}^{W}=|\omega_{0}\rangle\langle\omega_{0}|_{W}$ and
$\Pi_{1}^{W}=|\omega_{1}\rangle\langle\omega_{1}|_{W}$. In the game, the two
observers will be required to provide their predictions regarding the outcomes
of one or the other measurement.

##  III The rules of the game

Next we will introduce the rules of the game. The game involves two players,
Wigner (W) and the friend (F), as well as two referees, RW and RF, located in
Wigner’s and the friend’s bubbles, respectively.

1. Each player knows the state in their respective bubble. In particular, F knows the definite outcome of the first measurement (and can also compute state $|\Psi_{2}^{F}\rangle$), and W knows $|\Psi_{2}^{W}\rangle$. 2. The next measurement, either MF or MW, is chosen at random. 3. The players submit their predictions regarding the probabilities of measurement outcomes to the referee in the bubble where the measurement will be conducted, to RF in case of MF, or RW in case of MW. 4. The measurement is performed. 5. The referee, situated within the bubble in which the measurement was performed, awards the player whose prediction is confirmed by the measurement.

It is important to note that all forms of communication, including the
submission of predictions and the sharing of random choices regarding the next
measurement, will not contain any information about the outcome of the first
friend’s measurement and will not affect the states as assigned in the
bubbles. In this sense, a bubble is an information theoretic concept, rather
defined as the kind of information in principle available than to an amount of
leaked information (i.e., there is no need of complete isolation between two
regions for them to be two bubbles).

We will now proceed to comment on the rules of the game. The players need not
rely on the honesty of the referees in step 5. in evaluating their
predictions. Instead, they can inspect their previous predictions in form of
permanent records (e.g., signed pieces of paper on which their prediction is
written) that they handed to the responsible referee. Also, for case $M_{W}$
the friend’s memory will be subjected to a measurement thereby erasing that
(tiny) part of their memory which recorded the outcome of the previous
measurement. This may give rise to the concern that the “forgetful” observer
may not agree that they ever made any prediction, or at least not the one
which they will be faced with at the end of the game. However, since the
players may still be presented with a record of their previous predictions
–the signed piece of paper with their signature– this may provide them with
sufficient evidence that the prediction was indeed made by them.

Finally, in order to avoid potential difficulties arising from the so-called
self-reference problem [30], namely, the circumstance that the friend might
have to measure her own state, let us append to the already considered
scenario a further auxiliary memory system (A) of the same dimension than $F$,
prepared in an initial “ready” state $|R\rangle_{A}$. Applying then a unitary
swap operation, $U_{\mathrm{SWAP}}$ acting over friend and the auxiliary
system, the memory of the friend is coherently transferred to the auxiliary
system (such that it swaps the indices of the friend and the auxiliary
system). Under the action of $U_{\mathrm{SWAP}}$, the states in Eq. (5) and
(6) transforms as

$\displaystyle U_{\mathrm{SWAP}}|R\rangle_{A}|\Psi^{W}_{2}\rangle$
$\displaystyle=|\Psi^{W}_{3}\rangle|R\rangle_{F},$ (8) $\displaystyle
U_{\mathrm{SWAP}}|R\rangle_{A}|\Psi^{F}_{2}\rangle$
$\displaystyle=|\Psi^{F}_{3}\rangle|R\rangle_{F},$ (9)

respectively. This SWAP operation can be performed by Wigner himself or, if
the friend does not trust him, by a hypothetical machine that the friend
programs beforehand for carrying out this task. In such a way, the friend
would be able to measure a state containing a record of what she observed
without having to measure her own state. By applying the SWAP operation, the
content of the friend’s memory will be erased, and the comments previously
made about the loss of her memory can be applied to this step as well. Note
that Wigner’s memory is not affected by this operation.

In what follows we will demonstrate that, in accordance with the established
rules of the game, players will be led to conclude that it is preferential to
adopt the state assignment of the other bubble when the measurement is
performed in that other bubble, while maintaining their own for all
measurements performed within their own bubble. In certain instances (for the
friend), the state assignment from the other bubble may be accessible in
principle, whereas in others, it may not be (Wigner).

##  IV Measurement in Wigner’s bubble

Let us suppose that the random choice determines that measurement MW in which
Wigner measures in the preferred basis of his environment
{$|\omega_{0}\rangle_{W},|\omega_{1}\rangle_{W}\\}$ should be made next. On
the basis of his state assignment (5), Wigner makes a prediction that the two
outcomes will occur with probabilities
$p^{W}_{\omega_{0}}=1,p^{W}_{\omega_{1}}=0$. He then hands this prediction to
referee RW. The aforementioned prediction is equally applicable also in the
case when the SWAP operation is applied prior to the MW measurement.

On the other hand, when applying the SWAP operation the state of the friend is
just Eq. (6) where index F have been replaced by A. She predicts that
probabilities for observing one or the other outcome of MW are
$p^{F}_{\omega_{0}}=p^{F}_{\omega_{1}}=1/2$ (not to be confused with the
friend’s probabilities of the first qubit measurement). These numbers can be
encoded in a degree of freedom of her laboratory that did not get correlated
to the outcome of the previous qubit measurement. Since the message containing
the prediction also carries no information about the outcome of friend’s qubit
measurement, it factorises out from her laboratory and can be transmitted to
Wigner’s laboratory to the responsible referee RW as illustrated in Fig. 2.

Figure 2: The schematic illustration of the game protocol in the case that
measurement MW in Wigner’s bubble is chosen to be performed. On the basis of
quantum state assignments in their respective laboratories, the two players
make their predictions and submit them to the responsible referee RW in
Wigner’s bubble. The probabilistic prediction of the friend contains no
information about the outcome of the qubit observed in her laboratory.
Consequently, its communication to the referee has no effect on the state of
the laboratory as described by Wigner. The outcome of the MW measurement is in
accordance with the prediction of Wigner, and in repeated trials, a
discrepancy from the prediction of the friend can be observed.

The measurement is then performed in Wigner’s bubble and the result
$\omega_{0}$ is observed. Since the friend and Wigner predictions differ in
the estimated probabilities, one would have to perform a sufficient number of
runs to verify the difference between the two predictions with a certain
degree of confidence, always keeping the records of both the measurement
outcomes and the two predictions of the previous runs. The observed relative
frequency will eventually show a significant deviation from the friend’s
prediction as given e.g. by Wald’s sequential probability ratio test [31]
333One can also construct a scenario in which Wigner’s assigns probability one
to a specific outcome, while the friend assigns a probability that is
arbitrarily close to zero, see Ref. [8]. This would render the argument
arbitrarily close to a deterministic one. and consequently the referee RW will
award Wigner. Interestingly, this situation is maintained even in the cases
where partial information about the qubit measurement outcomes are leaked into
Wigner’s environment, as explicitly discussed in Appendix B.

Note that the measurement procedure affects the friend’s memory in that the
content of the memory that contained the information about the outcome have
been swapped with the ancilla state. While this allowed the friend to avoid
the problem of self-reference, her memory of the outcome was lost in the
process. One might ask, then, whether a partial loss of the memory would alter
the entire game to such an extent that the friend’s identity as an observer
would be lost, and she could no longer be regarded as the same player who
started the game. We leave this point for the discussion, and remind the
reader at this point that there is still a record (possibly with her
signature) of the friend’s previous prediction, which she can inspect and
accept as valid evidence of her previous prediction.

The result of the game shows us that the friend could have made a more
accurate prediction if she adopted the Wigner state assignment whenever the
measurement MW is chosen. In fact, she can do this, since there are no
fundamental restrictions as to why this information should be inaccessible to
her. In fact, Wigner could simply send this information to her bubble, or the
friend herself could first construct the experimental situation from Wigner’s
perspective and then enter the sealed laboratory that becomes her bubble.

##  V Measurement in the friend’s bubble

Next we consider the case where the decision is made to take the measurement
MF. This means that the measurement of the qubit along the z-direction in the
friend bubble using projectors $\Pi_{\uparrow}$ and $\Pi_{\downarrow}$ is
repeated. The responsible referee is RF. The game is illustrated in Fig. 3.

Figure 3: The schematic illustration of the game protocol in the case that
measurement MF in the friend’s bubble is chosen to be performed. The two
players make their predictions on the basis of their state assignments and
submit them to the responsible referee RF in the friend’s bubble. The
prediction of Wigner can be sent inside the friend’s laboratory with no
additional effects. The outcome of the measurement is consistent with the
prediction of the friend, and in repeated trials, a discrepancy between the
observed relative frequencies and the prediction of Wigner can be observed.

Which state is the friend going to use as a basis for her prediction? After
the action of the unitary interaction $U_{\text{int}}$ the friend’s state
becomes Eq. (6). It is important to note that this is not the state that the
friend would normally adopt in order to make predictions about measurements in
her laboratory. Rather, it is the state that the friend estimates she should
use for predicting results of measurement in Wigner’s bubble, such as MW,
after the action of the unitary and conditional on knowledge of her
measurement outcome. In fact, as discussed below Eq. (7), from the perspective
of the friend, the state of her memory (or of the ancilla after the SWAP)
remains in a definite state perfectly correlated with that of the spin even
after the action of the interaction unitary. For all subsequent measurements
in her lab, all that matters for predicting future results is the last record
in her lab. And that will be either definitely $\uparrow$ or definitely
$\downarrow$444It may happen, however, that the action of the unitary induces
a flip in the friend’s memory, but even then the friend will still have a
definite record of the first measurement in her bubble. All her subsequent
predictions can then be conditioned on that record. Alternatively, in “many-
worlds” terms, one might say that unitary evolution “splits” the world into
two branches, so that in each branch a friend sees a definite (and different)
outcome. Also in this view, in order to predict the outcome of any future
measurement within her branch, the friend should condition her prediction only
on the last record in her branch.. Thus, the friend’s description of the
relevant degrees of freedom for predicting future measurement results in her
laboratory remains to be given by state (3), even after the action of the
unitary interaction.

If the friend repeats the same measurement, she can use an additional degree
of freedom F’ in her laboratory, which was initially sufficiently isolated
that it did not get correlated with the previous result, and can be used to
store the outcome of the repeated measurement. To formally describe it, one
could start from state (7) obtained after the unitary interaction, add the
degree of freedom F’ and apply the quantum controlled-NOT operation acting
between the spin and degree F’, as a purified description of the repeated
measurement. The final state would be similar to Eq. (7), with the difference
that states $|\phi_{0}\rangle_{F}$ and $|\phi_{1}\rangle_{F}$ of a single
record in the two branches would be replaced by states
$|\phi_{0}\rangle_{F}|\phi_{0}\rangle_{F^{\prime}}$ and
$|\phi_{1}\rangle_{F}|\phi_{1}\rangle_{F^{\prime}}$ of two copies of the
record. However, as we have argued, only that part of the state that features
the correlations between the spin and laboratory degrees of freedom are
relevant for predicting the outcomes of future measurements in the friend’s
laboratory. The friend could just as well discard the action of the unitary
interaction and describe her repetition of the measurement simply by starting
from Eq. (3), adding the degree of freedom F’, and applying the quantum
controlled NOT to the spin and F’.

Hence, if she repeats the same measurement, she will always confirm the record
of the previous measurement result (which she had after the action of the
unitary): she finds the outcome $\uparrow$ with probability $1$ and
$\downarrow$ with probability $0$ if the record of her previous observation
was $\uparrow$, and similarly she finds $\downarrow$ with probability $1$ and
$\uparrow$ with probability $0$ if the record of her first measurement was
$\downarrow$. Hence, she predicts $p^{F}_{\phi_{0}}=1$ (or 0),
$p^{F}_{\phi_{1}}=0$ (or 1) and gives this prediction to referee RF. Formally,
the state assigned by the friend after the repeated measurement is completed
is given by:

$|\Psi^{F}_{4}\rangle=\begin{cases}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\phi_{0}\rangle_{F^{\prime}}|\omega_{0}\rangle_{W}&\text{for
outcome, }\phi_{0}\\\ \vspace{0.1cm}\\\
|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\phi_{1}\rangle_{F^{\prime}}|\omega_{0}\rangle_{W}&\text{for
outcome }\phi_{1}.\end{cases}$ (10)

From Wigner’s point of view, a repetition of the friend’s measurement is a
repeated application of the measurement unitary $U_{\text{meas}}$, which acts
as a quantum controlled-NOT operation between the system and the additional
degree of freedom $F^{\prime}$. This implies that after the repeated
measurement the state assigned by Wigner is

$\displaystyle|\Psi^{W}_{4}\rangle$
$\displaystyle=|\mathrm{GHZ}\rangle_{SFF^{\prime}}|\omega_{0}\rangle_{W}$
$\displaystyle=(\hat{U}_{\mathrm{meas}}\otimes\mathds{1}_{F})|\Phi^{+}\rangle_{SF}|\phi_{0}\rangle_{F^{\prime}}|\omega_{0}\rangle_{W},$
(11)

where

$|\mathrm{GHZ}\rangle=\frac{1}{\sqrt{2}}\Big{(}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\phi_{0}\rangle_{F^{\prime}}+|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\phi_{1}\rangle_{F^{\prime}}\Big{)}$
(12)

is the Greenberger-Horne-Zeilinger (GHZ) state. The probabilities for the
outcomes $\uparrow$ and $\downarrow$ are both
$p^{W}_{\phi_{0}}=p^{W}_{\phi_{1}}=$1/2 in this state. Wigner sends this
prediction to referee RF inside the friend’s bubble. As in the case of the
measurement in Wigner’s bubble, the friend’s and Wigner’s predictions differ
in their estimated probabilities, which can be compared to a desired degree of
confidence if a sufficient number of runs were made, always keeping track of
both the measurement results and the two predictions from the previous runs.
The observed relative frequency will eventually show a significant deviation
from Wigner’s prediction and the friend will be awarded by the referee.
Similar conclusions are obtained in the case of partial leakage of which
outcome information into Wigner’s environment (see Appendix B).

We conclude that the friend’s state assignment was more accurate than Wigner’s
for a measurement carried out in her laboratory. However, unlike the case of
the MW measurement, where the friend could adopt Wigner’s prediction before
the measurement was carried out and thus estimate the results of the
measurement as well as Wigner, this is not true in the present case since any
leakage of the information about the result of the first friend’s measurement
to Wigner would correspondingly collapse his state.

##  VI (In)compatibility between different bubbles

In the case where the information about which outcome the friend has observed
cannot be unambiguously retrieved from the Wigner’s environment, Wigner and
the friend give different, irreconcilable state assignments. None of them is
“right” and the other “wrong”. They live in different “bubbles” [12] and each
of the state assignment is “right” relative to the measurements performed in
their respective bubbles. In these cases both Wigner and her friend might be
tempted to think that the other party is making a wrong assignment. On the one
hand, Wigner may think that the operation performed by the friend was not
actually a “proper measurement”, since, following his description, unambiguous
information about the outcomes is not detectable anywhere in his bubble. On
the other hand, the friend –who has evidences that a proper measurement
actually happened– may think that Wigner’s unitary description was incorrect,
since it did not include the spread of the information about the outcomes
actually generated in her measurement. Nevertheless, the assumption that no
observer is preferred and that observers in other bubbles may experience
definite outcomes, just as they do in their own, may prompt them to consider
the possibility of bubbles with fundamentally incompatible outcomes.

On the other hand, as explicitly shown in Appendix B, in the special case in
which information about “which-outcome” is completely leaked into Wigner’s
environment (i.e. each outcome is correlated to an accessible distinguishable
state of the environment), the measurement update rule can be safely applied
by both observers. One can say that if both observers share the information
about the outcomes then they live in the same “bubble”. However, it is enough
to have an arbitrarily small uncertainty in Wigner’s environmental states
about the result of the qubit measurement in friend’s lab to create an (in
principle) detectable incompatibility between Wigner and the friend’s state
assignment. Interestingly, the number of runs needed to determine the validity
of Wigner (friend) predictions versus the alternative one is a monotonically
increasing function of the encoding parameter representing the amount of
leaked information (see Appendix B): it increases as the amount of information
leaked into Wigner’s environment increasing and diverges when the two
descriptions are compatible.

It is important to notice that the assumption usually adopted that the friend
must be in a perfectly close lab is unwarranted, but the distinction lies in
what kind of information is leaked outside: Any unsubstantial information
about the measurement procedure, like e.g. the information that the system and
the friend are entangled for Wigner, or about whether forecast, a film
critique etc, Wigner and the friend can communicate without affecting their
state assignments. In a similar manner, Wigner and his friend can communicate
their predictions, share the random choice determining the subsequent
measurement, or use communication to ascertain that their descriptions are
incompatible.

##  VII Conclusions

The fundamental principles of textbook quantum mechanics indicate that an
observer may apply two distinct forms of evolution to a given quantum state,
contingent upon whether the evolution is between measurements or after the
measurement. In the former case, a unitary transformation is applied, whereas
in the latter, projections are employed (in the ideal case). However, the
textbooks are silent on the conditions that specify which evolution should be
applied in a concrete situation. This ambiguity can be attributed to a
fundamental issue in defining the characteristics distinguishing between
laboratory devices that serve as measuring instruments and those that
implement unitaries – an issue known as “the quantum measurement problem”.

In this work, we do not address the measurement problem. Rather, we accept the
fact that through everyday praxis, an observer acquires the experience of
various devices and can sort them into those that serve as measurement
apparata and those that implement unitary evolutions. This is the starting
point for our discussion. Nevertheless, Wigner’s thought experiment and its
modern extensions present an additional challenge to the consistency of the
textbook quantum mechanics. This is because different observers may assign
different quantum states to the same system. One possible way to overcome this
challenge is to adopt the view that the fundamental concepts of quantum
mechanics, such as quantum state, unitary evolution, or updated state, can
only be defined relative to a bubble. Within a single bubble, all hypothetical
observers may share their knowledge, thereby establishing intersubjectivity
within the bubble. Since a bubble is defined only in terms of the (kind of)
information that is in principle available in a certain region, we call this
“relative objectivity”. In fact, according to this view there is in general
not a universal correct state, but once the physical conditions are specified
(that is, the bubble is defined) this fixes a state. This is reminiscent of
relativity theory, where for example simultaneity between space-like separated
events is not absolutely defined, but this relation is fixed within any chosen
foliation. It would be desirable to find analogous ways to formally transform
the description of a bubble into another by developing the analogue of Lorentz
transformations.

Here, we examine scenarios in which observers are able to transition between
bubbles. The observers make predictions about measurements that will be made
on their own bubbles, including their own memory. After the measurement, they
can end up in another bubble and check the accuracy of their previous
predictions. Our findings necessitate a refinement of the concept of
objectivity relative to the bubbles. We showed that in certain circumstances,
observers are eligible to adopt the state assignment from other bubbles from
which measurements are performed and make predictions accordingly. In fact,
the friend can have information about her state as seen by Wigner without
decohering her state, but informing Wigner about her spin state would collapse
the state for him. While in this paper we consider the two players in a more
symmetric role, this difference still differentiates between them.

If we consider the possibility of a hierarchy of a large numbers of bubbles,
each of which can be measured from the ones “higher” it in the hierarchy, we
may conclude that a complete description of the physical situation of each
observer would also require a hierarchy of joint state assignments: the
friend’s state for measurements in the friend’s bubble, Wigner’s state for
measurements in Wigner’s bubble, and so on. It is important to reiterate that
an observer cannot have the state assignment of any bubbles “below” themselves
in the hierarhy without decohering them. For instance, Wigner cannot have the
state assignment of the friend without collapsing it. But it is this asymmetry
that allows the “lower” observer in the hierarchy to be more successful in
certain tasks than the “higher” level observer.

Finally, we comment on the memory loss of the aforementioned friend whose
laboratory, including his memory, underwent the measurement procedure. It is
sometimes argued that because of the loss of memory, the person should not be
considered the same throughout the experiment (see [14] and references
therein). In such a case, the individual would not be considered the same
“friend” or “player” in our game. Consequently, they might not accept that
they made a prediction in the past that led to them losing the game. The
conventional notion of personal identity is defined by the distinctive ways in
which an individual defines themselves, encompassing aspects such as self-
image, cultural references, political credos, or other interests. Since the
loss of memory in the measurement solely affects the recollection of the
outcome of the very last measurement, these fundamental aspects of identity
are expected to remain unaltered. Moreover, it is not uncommon for individuals
to forget events from the past. However, when presented with compelling
evidence that an event occurred in the past, they tend to accept that it did
occur and that they simply forgot about it. Similarly, in our game, after
players are presented with the signed paper containing their prediction, they
may accept it as legitimate. Finally, one might approach the problem of
identity in the present case pragmatically: Would you enter the game under the
risk of losing the memory of one bit of information, if the award were one
million coins?

## Acknowledgements

FDS and ČB acknowledge the discussion with Veronika Baumann. This research was
funded in whole or in part by the Austrian Science Fund (FWF) 10.55776/COE1
(Quantum Science Austria), [10.55776/F71] (BeyondC) and [10.55776/RG3]
(Reseacrh Group 3). For Open Access purposes, the authors have applied a CC BY
public copyright license to any author accepted manuscript version arising
from this submission. This publication was made possible through the financial
support of the ID 62312 grant from the John Templeton Foundation, as part of
The Quantum Information Structure of Spacetime (QISS) Project (qiss.fr). The
opinions expressed in this publication are those of the authors and do not
necessarily reflect the views of the John Templeton Foundation. We also
acknowledge support from CoQuSy project PID2022-140506NB-C21, and the María de
Maeztu project CEX2021-001164-M for Units of Excellence, funded by
MICIU/AEI/10.13039/501100011033/FEDER, UE. GM acknowledges the Ramón y Cajal
program RYC2021-031121-I funded by MICIU/AEI/10.13039/501100011033 and
European Union NextGenerationEU/PRTR. This research was also supported by the
FWF (Austrian Science Fund) through an Erwin Schrödinger Fellowship (Project J
4699)

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##  Appendix A General encoding for the transmitted information between labs

In this Appendix we consider generic scenarios allowing that information about
the qubit measurement results in the friend’s laboratory becomes accessible to
Wigner. More precisely, we consider unitary interactions that partially
correlate the accessible environmental states in the friend’s laboratory
$\\{|\phi_{0}\rangle_{F},|\phi_{1}\rangle_{F}\\}$ (containing univocal
information about the outcome of the qubit measurement) with those in Wigner’s
laboratory $\\{|\omega_{0}\rangle_{W},|\omega_{1}\rangle_{W}\\}$. This can be
achieved by generalizing the unitary that couples the system, the friend’s lab
and Wigner’s environment defined through Eq. (4) by allowing it to depend on a
parameter $\theta$, i.e. $U_{\mathrm{int}}(\theta)$ with $\theta\in[0,1]$, so
that its relevant action over the states of the system, the friend and
Wigner’s environments is now:

$\displaystyle{U_{\rm int}}(\theta)$
$\displaystyle|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}$
(13) $\displaystyle{=}\leavevmode\nobreak\
\frac{1}{\sqrt{2}}(|\Phi_{\theta}^{+}\rangle_{SF}|\omega_{0}\rangle_{W}+|\Phi_{\theta}^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),$
$\displaystyle{U_{\rm int}}(\theta)$
$\displaystyle|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{0}\rangle_{W}$
$\displaystyle{=}\leavevmode\nobreak\
\frac{1}{\sqrt{2}}(|\varphi_{\theta}^{+}\rangle_{SF}|\omega_{0}\rangle_{W}-|\varphi_{\theta}^{-}\rangle_{SF}|\omega_{1}\rangle_{W}).$

Here we introduced the following parameterized states of the system and the
friend’s lab:

$\displaystyle|\Phi_{\theta}^{\pm}\rangle_{SF}$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}[1\pm\sin\left(\frac{\pi\theta}{2}\right)]|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\pm\cos\left(\frac{\pi\theta}{2}\right)|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}\Big{)},$
(14) $\displaystyle|\varphi_{\theta}^{\pm}\rangle_{SF}$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}\cos\left(\frac{\pi\theta}{2}\right)|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\pm[1\mp\sin\left(\frac{\pi\theta}{2}\right)]|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}\Big{)},$
(15)

which reduce to the two standard Bell states given below Eq. (4) for
$\theta=0$. Henceforth it is easy to check that the above unitary
$U_{\mathrm{int}}(\theta)$ reduces to Eq. (4) for the case $\theta=0$. On the
other hand, for $\theta>0$ such a unitary generically encodes information
about the qubit measurement results in friend’s lab into superpositions of the
visible states of Wigner’s environment with different amplitudes that depend
on $\theta$. In the extreme case $\theta=1$ we have
$|\Phi_{1}^{+}\rangle_{W}=|\uparrow_{0}\rangle_{S}|\phi_{0}\rangle_{F}$ and
$|\varphi_{1}^{-}\rangle_{W}=|\downarrow_{0}\rangle_{S}|\phi_{1}\rangle_{F}$
(while $|\Phi_{1}^{-}\rangle=|\varphi_{1}^{+}\rangle=0$) so that Wigner’s
accessible lab states become perfectly correlated with the measured qubit
states and friend’s lab results.

Applying $U_{\mathrm{int}}(\theta)$ to Wigner’s state in Eq. (2) we obtain
[c.f. Eq. (5)] :

$\displaystyle|\Psi_{2}^{W}\rangle$
$\displaystyle=\frac{1}{2}\big{[}(|\Phi_{\theta}^{+}\rangle_{SF}+|\varphi_{\theta}^{+}\rangle_{SF})|\omega_{0}\rangle$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+(|\Phi_{\theta}^{-}\rangle_{SF}-|\varphi_{\theta}^{-}\rangle_{SF})|\omega_{1}\rangle\big{]}.$
(16)

Using the explicit form of the states $|\Phi_{\theta}^{\pm}\rangle$ and
$|\varphi_{\theta}^{\pm}\rangle$ that state can be rewritten as

$\displaystyle|\Psi_{2}^{W}\rangle$
$\displaystyle=\frac{1}{2\sqrt{2}}\Big{(}\big{[}1+\sin\left(\frac{\pi\theta}{2}\right)+\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}$
$\displaystyle+\big{[}1-\sin\left(\frac{\pi\theta}{2}\right)+\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{0}\rangle_{W}$
$\displaystyle+\big{[}1-\sin\left(\frac{\pi\theta}{2}\right)-\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{1}\rangle_{W}$
$\displaystyle+\big{[}1+\sin\left(\frac{\pi\theta}{2}\right)-\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{1}\rangle_{W}\Big{)}.$
(17)

Therefore, depending on the value of $\theta$, Wigner’s environmental states
$|\omega_{0}\rangle_{E}$ and $|\omega_{1}\rangle_{E}$ can respectively get
correlated to the system the and friend’s lab states
$|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}$ and
$|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}$. In particular, the correlation
is stronger, the larger the value of $\theta$ is, while for $\theta=0$ the two
last lines of the state above disappear and we recover Eq. (5).

In the case $\theta=1$ the state in Eq. (A) reduces to:

${|\Psi_{2}^{W}\rangle}=\frac{1}{\sqrt{2}}\Big{(}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}+|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{1}\rangle_{W}\Big{)}$
(18)

corresponding to perfect correlations between the measurement results and both
friend’s and Wigner’s environmental states. In this case by looking at the
preferred basis of Wigner’s environment we obtain with $1/2$ probabilities the
states $|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}$ and
$|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{1}\rangle_{W}$.

On the other hand, applying $U_{\mathrm{int}}(\theta)$ to the friend’s
(collapse-rule) state assignment in Eq. (2) gives us the two following states:

$|\Psi_{2}^{F}\rangle=\begin{cases}\frac{1}{\sqrt{2}}(|\Phi_{\theta}^{+}\rangle_{SF}|\omega_{0}\rangle_{W}+|\Phi_{\theta}^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),\\\
\vspace{0.1cm}\\\
\frac{1}{\sqrt{2}}(|\varphi_{\theta}^{+}\rangle_{SF}|\omega_{0}\rangle_{W}-|\varphi_{\theta}^{-}\rangle_{SF}|\omega_{1}\rangle_{W}),\end{cases}$
(19)

for outcomes $\phi_{0}$ and $\phi_{1}$, respectively, to be compared with the
state $|\Psi_{2}^{F}\rangle$ in Eq. (6). It should be noticed here that in
this case the environmental states $|\omega_{0}\rangle$ and
$|\omega_{1}\rangle$ are partially correlated with
$|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}$ and
$|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}$ as long as $\theta>0$. If
$\theta=1$ we obtain:

$|\Psi_{2}^{F}\rangle=\begin{cases}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\omega_{0}\rangle_{W}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for\leavevmode\nobreak\
outcome}\leavevmode\nobreak\ \phi_{0},\\\ \vspace{0.1cm}\\\
|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\omega_{1}\rangle_{W}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for\leavevmode\nobreak\
outcome}\leavevmode\nobreak\ \phi_{1},\end{cases}$ (20)

with perfect correlations between the preferred basis states of the friend’s
and Wigner lab degrees of freedom. Therefore the information about the
friend’s measurement outcomes is completely accessible from Wigner’s
environmental states without ambiguity in this case.

##  Appendix B Wigner’s and friend’s predictions for partial leakage of
information

Let’s now discuss the results of the checking procedure presented in Secs. IV
and V for measurements in Wigner’s and friend’s bubbles, respectively, in the
case of general encoding of information presented in Appendix A.

In the case of measurement $M_{W}$ in Wigner’s preferred basis
$\\{|\omega_{0}\rangle,|\omega_{1}\rangle\\}$, based on state assignment in
Eq. (A) the prediction of Wigner for the two outcomes of the measurement are:

$|\Psi_{3}^{W}\rangle=\begin{cases}\frac{1}{\sqrt{2}}(|\Phi_{\theta}^{+}\rangle_{SF}+|\varphi_{\theta}^{+}\rangle_{SF})|\omega_{0}\rangle_{W}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for\leavevmode\nobreak\
outcome}\leavevmode\nobreak\ \omega_{0},\\\ \vspace{0.1cm}\\\
\frac{1}{\sqrt{2}}(|\Phi_{\theta}^{-}\rangle_{SF}-|\varphi_{\theta}^{-}\rangle_{SF})|\omega_{1}\rangle_{W}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm for\leavevmode\nobreak\
outcome}\leavevmode\nobreak\ \omega_{1},\end{cases}$ (21)

with respective probabilities $p_{\omega_{0}}^{W}=[1+\cos(\pi\theta/2)]/2$ and
$p_{\omega_{1}}^{W}=[1-\cos(\pi\theta/2)]/2$, which are submitted to the
referee $R_{W}$. Notice that now the prediction of Wigner is not
deterministic, but biased in general towards $\omega_{0}$, as $1/2\leq
p_{\omega_{0}}^{W}\leq 1$ and $0\leq p_{\omega_{1}}^{W}\leq 1/2$. On the other
hand, the friend’s prediction of the probabilities for observing one or the
other outcome of MW, basing on state assignment Eq. (19), are again
$p^{F}_{\omega_{0}}=1/2$ and $p^{F}_{\omega_{1}}=1/2$, which are also
submitted to $R_{W}$. Notice that if no information about which outcome is
present in the encoding, i.e. $\theta=0$, this reduces to the case discussed
in Sec. IV. However, we observe that even if a part of the “which-outcome”
information is leaked from the friend’s to Wigner’s environment, their
predictions of the estimated probabilities of the two observers will still
differ and hence the referee $R_{W}$ by performing the measurement $M_{W}$ can
detect this mismatch performing a sufficient number of runs. Interestingly
Wigner’s and the friend’s predictions become closer to each other as more
information about the qubit measurement is leaked in the interaction between
labs (which corresponds to increasing values of $\theta$) reaching the case of
equal predictions for $\theta=1$.

Figure 4: Top panel: Two examples of Wald’s sequential probability ratio test
for $\theta=0$ (case in the main text). The blue dots represents the log-
likelihood ratio for sequence of consecutive $\omega_{0}$ outcomes, reaching
the acceptance threshold of Wigner’s prediction (dashed line) for
$\epsilon=10^{-3}$ at run $n=10$. The green dots represent instead the log-
likelihood ratio in the eventual case of a sequence of 4 consecutive
$\omega_{0}$ values followed by $\omega_{1}$, leading to the acceptance of the
friend’s prediction at run $n=5$. Bottom panel: Minimum number of runs $n_{0}$
leading to outcomes $\omega_{0}$ in the measurement $M_{W}$ needed to validate
Wigner’s prediction as a function of the encoding parameter $\theta$ in the
range $[0,1]$ for two different values of the failure probability $\epsilon$.

In particular, assuming Wald’s sequential probability ratio test [31], the
referee $R_{W}$ may collect statistics from consecutive runs of the game and
construct the cumulative log-likelihood ratio between Wigner’s and friend
predictions. After $n$ runs obtaining outcomes $\\{r_{1},r_{2},...,r_{n}\\}$
the cumulative log-likelihood ratio would read:

$\displaystyle\log\Lambda(r_{1},...,r_{n}):=\log\frac{p_{r_{1}}^{W}p_{r_{2}}^{W}...p_{r_{n}}^{W}}{p_{r_{1}}^{F}p_{r_{2}}^{F}...p_{r_{n}}^{F}}$
(22)
$\displaystyle=n_{0}\log[1+\cos\left(\frac{\pi\theta}{2}\right)]+n_{1}\log[1-\cos\left(\frac{\pi\theta}{2}\right)],$

where $p^{W}_{r_{i}}$ ($p^{F}_{r_{i}}$) are the probability assignments from
Wigner (friend) perspective for outcome $r_{i}=\\{\omega_{0},\omega_{1}\\}$,
and in the second line $n_{0}$ stands for the number of outcomes $\omega_{0}$
obtained and $n_{1}$ for the number of outcomes $\omega_{1}$. That quantity
increases as outcomes $r_{i}=\omega_{0}$ in the sequence are accumulated in
consecutive runs, and decreases when outcomes $r_{i}=\omega_{1}$ are detected
(see Fig. 4). The referee will award Wigner as soon as $\log\Lambda$ first
reaches the upper threshold $\log[(1-\epsilon)/\epsilon]>0$, where
$\epsilon>0$ is the (typically small) error of test failure. On the contrary,
the referee will award the friend as soon as $\log\Lambda$ reaches the lower
threshold given by $\log[\epsilon/(1-\epsilon)]<0$. For the case $\theta=0$
the lower thresholds will be automatically reached as soon as the first
outcome $r_{i}=\omega_{1}$ is detected since it is fully incompatible with
Wigner’s prediction. On the other hand, for $\theta=1$ where the predictions
of Wigner and her friend become equal, $\log\Lambda=0$ for any sequence and
hence awarding one or the other is not possible.

The minimum number of runs leading to outcome $\omega_{0}$ needed to validate
Wigner’s prediction (occurring in the case when no outcome $\omega_{1}$ is
obtained) would be:

$n_{0}=\frac{\log[(1-\epsilon)/\epsilon]}{\log[1+\cos\left(\frac{\pi\theta}{2}\right)]},$
(23)

which increases monotonically with $\theta$ (see Fig.4). In the limit
$\theta\rightarrow 1$ where the “which-outcome” information is unambiguously
leaked to Wigner’s environment, the number of necessary runs diverges and both
descriptions (unitary evolution and state collapse) perfectly agree.

A similar situation is obtained in the case of performing the measurement
$M_{F}$ in the friend’s bubble. Similarly to the situation described in Sec. V
when a second measurement of the qubit state is performed in the friend’s lab
the results will confirm the initial measurement state assignment: she
predicts $p^{F}_{\phi_{0}}=1$ (or 0), $p^{F}_{\phi_{1}}=0$ (or 1) and gives
this prediction to referee RF. On the other hand, from the point of view of
Wigner, the state after the repeated measurement using unitary $U_{\rm meas}$
between system and an extra degree of freedom of the friend’s lab $F^{\prime}$
reads:

$\displaystyle|\Psi^{W}_{4}\rangle=(\hat{U}_{\mathrm{meas}}\otimes\mathds{1}_{F})|\Psi^{W}_{2}\rangle|\phi_{0}\rangle_{F^{\prime}}$
(24)
$\displaystyle=\frac{1}{2\sqrt{2}}\Big{(}\big{[}1+\sin\left(\frac{\pi\theta}{2}\right)+\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\phi_{0}\rangle_{F^{\prime}}|\omega_{0}\rangle_{W}$
$\displaystyle+\big{[}1-\sin\left(\frac{\pi\theta}{2}\right)+\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\phi_{1}\rangle_{F^{\prime}}|\omega_{0}\rangle_{W}$
$\displaystyle+\big{[}1-\sin\left(\frac{\pi\theta}{2}\right)-\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\uparrow\rangle_{S}|\phi_{0}\rangle_{F}|\phi_{0}\rangle_{F^{\prime}}|\omega_{1}\rangle_{W}$
$\displaystyle+\big{[}1+\sin\left(\frac{\pi\theta}{2}\right)-\cos\left(\frac{\pi\theta}{2}\right)\big{]}|\downarrow\rangle_{S}|\phi_{1}\rangle_{F}|\phi_{1}\rangle_{F^{\prime}}|\omega_{1}\rangle_{W}\Big{)}.$

The probabilities for the outcomes $\uparrow$ and $\downarrow$ from Wigner’s
perspective are $p^{W}_{\phi_{0}}=(1+\sin(\pi\theta/2)\cos(\pi\theta/2))/2$
and $p^{W}_{\phi_{1}}=(1-\sin(\pi\theta/2)\cos(\pi\theta/2))/2$ from the above
state, which are sent to referee RF inside the friend’s bubble. As we can see
in this situation the Referee $R_{F}$ can also see a difference between Wigner
and friend’s predictions for the measurement $M_{F}$ whenever $\theta<1$.
Again, as expected, when perfect “which-outcome” information is available in
Wigner’s environment $(\theta=1)$ both descriptions becomes equivalent.


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