††thanks: The first three authors contributed equally.

# Generalized probability rules from a timeless formulation of Wigner’s friend
scenarios

Veronika Baumann  Institute for Quantum Optics and Quantum Information (IQOQI-
Vienna) of the Austrian Academy of Sciences, Boltzmanngasse 3, A-1090, Vienna,
Austria  Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090
Vienna, Austria  Faculty of Informatics, Università della Svizzera italiana,
Via G. Buffi 13, CH-6900 Lugano, Switzerland  Flavio Del Santo  Institute for
Quantum Optics and Quantum Information (IQOQI-Vienna) of the Austrian Academy
of Sciences, Boltzmanngasse 3, A-1090, Vienna, Austria  Faculty of Physics,
University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria  Alexander R.
H. Smith  Department of Physics and Astronomy, Dartmouth College, Hanover, New
Hampshire 03755, USA  Flaminia Giacomini  Institute for Quantum Optics and
Quantum Information (IQOQI-Vienna) of the Austrian Academy of Sciences,
Boltzmanngasse 3, A-1090, Vienna, Austria  Faculty of Physics, University of
Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria  Perimeter Institute for
Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
Esteban Castro-Ruiz  Institute for Quantum Optics and Quantum Information
(IQOQI-Vienna) of the Austrian Academy of Sciences, Boltzmanngasse 3, A-1090,
Vienna, Austria  Faculty of Physics, University of Vienna, Boltzmanngasse 5,
A-1090 Vienna, Austria  QuIC, Ecole polytechnique de Bruxelles, C.P. 165,
Université libre de Bruxelles, 1050 Brussels, Belgium  C̆aslav Brukner
Institute for Quantum Optics and Quantum Information (IQOQI-Vienna) of the
Austrian Academy of Sciences, Boltzmanngasse 3, A-1090, Vienna, Austria
Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna,
Austria

###### Abstract

The quantum measurement problem can be regarded as the tension between the two
alternative dynamics prescribed by quantum mechanics: the unitary evolution of
the wave function and the state-update rule (or “collapse”) at the instant a
measurement takes place. The notorious Wigner’s friend gedankenexperiment
constitutes the paradoxical scenario in which different observers (one of whom
is observed by the other) describe one and the same interaction differently,
one –the Friend– via state-update and the other –Wigner– unitarily. This can
lead to Wigner and his friend assigning different probabilities to the outcome
of the same subsequent measurement. In this paper, we apply the Page-Wootters
mechanism (PWM) as a timeless description of Wigner’s friend-like scenarios.
We show that the standard rules to assign two-time conditional probabilities
within the PWM need to be modified to deal with the Wigner’s friend
gedankenexperiment. We identify three main definitions of such modified rules
to assign two-time conditional probabilities, all of which reduce to standard
quantum theory for non-Wigner’s friend scenarios. However, when applied to the
Wigner’s friend setup each rule assigns different conditional probabilities,
potentially resolving the probability-assignment paradox in a different
manner. Moreover, one rule imposes strict limits on when a joint probability
distribution for the measurement outcomes of Wigner and his Friend is well-
defined, which single out those cases where Wigner’s measurement does not
disturb the Friend’s memory and such a probability has an operational meaning
in terms of collectible statistics. Interestingly, the same limits guarantee
that said measurement outcomes fulfill the consistency condition of the
consistent histories framework.

##  1 Introduction

Standard quantum theory features two dynamical processes: the so-called
“collapse of the wave function” that occurs during a measurement and the
unitary evolution describing the propagation of the wave function in the
absence of measurements. However, quantum theory itself does not specify when
each process should apply, leading to the well-known quantum measurement
problem111In the vast literature devoted to this topic, the “quantum
measurement problem” does not have a unique, clear definition. Intuitively, it
can be thought of as the problem of _how_ , _when_ and _under what
circumstances_ definite values of physical variables are obtained [1].
However, in this paper we focus on the ambiguity of the application between
unitary and “collapse” dynamics. [2, 3].

This situation is illustrated by the gedankenexperiment of Wigner’s friend
[4]. An observer $F$ (the Friend) performs a measurement on a quantum system
$S$ in a closed laboratory, the outcome of which is recorded by the click of a
measuring apparatus and/or as a definite record in the Friend’s memory inside
the laboratory. After the measurement, $F$ assigns a state to $S$ conditioned
on her measurement outcome (i.e., state update). Simultaneously, another
observer $W$ (Wigner) situated outside of the laboratory, would describe the
entire measurement by $F$ as unitary, resulting in him assigning a specific
entangled state to $F$ and $S$. This leads to the paradoxical situation that
the two observers $F$ and $W$ assign different states after $F$’s measurement
and, hence, different probabilities to the outcomes of subsequent
measurements. As shown in Refs. [5, 6, 7], the seemingly natural assumptions
that quantum theory is universal, that different observers can each apply
either of the two dynamical processes of the theory with respect to their
description, and that the reasoning of different agents about one another can
be freely combined to make statements about an objective reality lead to
contradictions for Wigner’s friend setups. These contradictions can be
regarded as resulting from an observer dependent application of the state-
update rule and involve inferences about the outcomes different observers
obtain at different times.

We begin by formulating the Wigner’s friend gedankenexperiment in terms of the
a priori timeless formulation of quantum theory put forward by Page and
Wootters [8]. In such an approach time evolution emerges from quantum
correlations between a clock and a system whose dynamics the clock will track.
The advantage of the timeless approach in the context of the Wigner’s friend
gedankenexperiment is that it assigns a timeless state from which one can
compute probabilities of an event conditioned on another, regardless of their
temporal order as indicated by the clock. This allows us to ask the following
two questions:

  1. 1.

What is the probability that $W$ measures outcome $w$ at clock time $t_{2}$,
given that $F$ has measured outcome $f$ at a previous clock time $t_{1}$?

  2. 2.

What is the probability that $F$ measures outcome $f$ at clock time $t_{1}$,
conditioned on the fact that $W$ will measure outcome $w$ at a later clock
time $t_{2}$?

We will show that there exist more than one way to consistently assign
conditional probabilities using the Page-Wootters mechanism (PWM), all of
which reduce to standard quantum conditional probabilities for non-Wigner’s
friend scenarios. Depending on the choice of conditional-probability rule,
however, the Wigner’s friend gedankenexperiment leads to different
probabilities for the results of Wigner and his Friend, thus resolving the
paradox of the ambiguous probability assignments in different manners. Using
the PWM all observers will agree on the overall state and assuming they all
use the same probability rule they will agree in their probability
assignments. The first rule gives probabilities that correspond to applying
the state-update rule after the Friend’s measurement and, hence, suggests
collapse dynamics for all observers. The other two rules are in accordance
with unitary evolutionand give conditional probabilities different from the
first rule. Moreover, they give well-defined probabilities only under certain
conditions implying that the above two questions can only be meaningfully
answered for specific settings. In case of the last rule, these conditions
single out those settings for which Wigner’s measurement does not disturb the
Freind’s memory and, hence, the respective probabilities correspond to
collectible statistics for the Friend.

##  2 The Page and Wootters Mechanism

We now summarize the Page-Wootters mechanism [8, 9], which begins by
considering a clock and system whose dynamics the clock tracks. The clock and
system are described respectively by the Hilbert spaces $\mathcal{H}_{C}\simeq
L^{2}(\mathbb{R})$ and $\mathcal{H}_{S}$. The joint state
$\left|\left.\Psi\right>\right>$ assigned to the clock and system is a
solution to a Wheeler-DeWitt-like equation

$\displaystyle\hat{H}\ket{\Psi}\rangle=\left(\hat{H}_{C}+\hat{H}_{S}\right)\ket{\Psi}\rangle=0,$
(1)

where $\hat{H}_{S}$ is the system Hamiltonian and $H_{C}=\hat{P}_{C}$ is the
clock Hamiltonian, which is taken to be the momentum operator $\hat{P}_{C}$ on
$\mathcal{H}_{C}$. In general, a Wheeler-DeWitt-like equation can in some
cases be interpreted as arising from the canonical quantization of a gauge
theory with a Hamiltonian constraint. General relativity is an example of such
a gauge theory and it is for this reason that Page and Wootters first put
forward their formulation of quantum theory to address the problem of time
[10, 11]. Solutions to Eq. (1) can be obtained by acting on states in the
_kinematical_ Hilbert space
$\mathcal{K}\simeq\mathcal{H}_{C}\otimes\mathcal{H}_{S}$ with the operator

$\displaystyle P^{\rm
ph}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\int_{\mathbb{R}}ds\,e^{-is\hat{H}},$
(2)

that is, supposing $\ket{\psi}\in\mathcal{K}$, then
$\left|\left.\Psi\right>\right>=P^{\rm ph}\ket{\psi}$ is a solution to Eq.
(1). Physical states of the theory,
$\left|\left.\Psi\right>\right>\in\mathcal{H}$, are elements of the _physical_
space $\mathcal{H}$, which is not a subspace of the kinematical Hilbert space
$\mathcal{K}$. This is because the spectrum of $\hat{H}$ is continuous around
zero, from which it follows that physical states are not normalizable in the
kinematical inner product [12, 13]. Thus a new inner product must be used to
normalize the physical states, which in turn defines the physical Hilbert
space $\mathcal{H}$; such an inner product is defined in Eq. (6) and motivated
in what follows.

First, one defines the time read by the clock as the measurement outcome of a
time observable $\hat{T}_{C}$ that is covariant [14, 15] with respect to the
clock Hamiltonian $\hat{H}_{C}$. By covariant it is meant that supposing the
spectral decomposition of the time observable on $\mathcal{H}_{C}$ is

$\displaystyle\hat{T}_{C}=\int_{\mathbb{R}}dt\,t\ket{t}\\!\bra{t},$ (3)

where $\ket{t}$ are eigenkets of $\hat{T}_{C}$ associated with the eigenvalue
$t\in\mathbb{R}$, the eigenkets are connected to one another by the unitary
generated by the clock Hamiltonian,
$\ket{t^{\prime}}=e^{-i\hat{H}_{C}(t^{\prime}-t)}\ket{t}$. In this case,
enforcing the covariance condition implies that $\hat{T}_{C}$ is canonically
conjugate to the clock Hamiltonian, $[\hat{T}_{C},\hat{H}_{C}]=i$, and thus is
equivalent to the position operator on $\mathcal{H}_{C}$.

Now consider a system observable $\hat{M}=\sum_{m}m\Pi_{m}$, where
$\\{\Pi_{m},\ \forall\,m\in\operatorname{Spec}{\hat{M}}\\}$ defines a
projective valued measure on $\mathcal{H}_{S}$. The probability that $\hat{M}$
takes the value $m$ when the clock reads time $t$ is given by the Born rule

$\displaystyle P\left(\hat{M}=m\ \mbox{when}\ \hat{T}_{C}=t\right)$
$\displaystyle\qquad=\frac{\left<\left<\Psi\right.\right|\big{(}|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\Pi_{m}\big{)}\left|\left.\Psi\right>\right>}{\left<\left<\Psi\right.\right|\big{(}|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\mathds{1}_{S}\big{)}\left|\left.\Psi\right>\right>}.$
(4)

In what follows, we will use an abbreviated expression to refer to these
probabilities in which we omit the operators, i.e., $P(\hat{M}=m\ \mbox{when}\
\hat{T}_{C}=t)\equiv P(m\ \mbox{when}\ t)$.

The form of Eq. (4) suggests defining the conditional state of the system
given the clock reads the time $t$ as

$\displaystyle\ket{\psi_{S}(t)}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\big{(}\bra{t}\otimes\mathds{1}_{S}\big{)}\left|\left.\Psi\right>\right>.$
(5)

Further, demanding that the conditional state remains normalized for all $t$
implies that the physical state $\left|\left.\Psi\right>\right>$ must be
normalized with respect to the physical inner product [16]

$\displaystyle\braket{\braket{\Psi}{\Psi}}{}{\rm
PW}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\braket{\braket{\Psi}{\big{(}|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\mathds{1}_{S}\big{)}}{\Psi}}{=}{1},$
(6)

for all $t\in\mathbb{R}$, from which it follows that
$\braket{\psi_{S}(t)}{\psi_{S}(t)}=1$. Given Eqs. (5) and (6), the probability
in Eq. (4) may be expressed as

$\displaystyle P\left(m\ \mbox{when}\ t\right)$
$\displaystyle=\braket{\psi_{S}(t)}{\Pi_{m}}{\psi_{S}(t)}.$ (7)

Since Eq. (7) is identical to the probability assigned to the measurement
outcome $m$ given the state of the system is $\ket{\psi_{S}(t)}$ by the Born
rule in the standard formulation of quantum theory, we are justified in
identifying the conditional state in Eq. (5) as the standard time-dependent
wave function. Further, the covariance condition satisfied by the eigenstates
of $\hat{T}_{C}$ implies that the conditional state satisfies the Schrödinger
equation [9].

We note that because the eigenstates of $\hat{T}_{C}$ can be used to form a
resolution of the identity on the clock Hilbert space, $\mathds{1}_{C}=\int
dt\,|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|$, physical states may be
expressed as

$\displaystyle\left|\left.\Psi\right>\right>=\int_{\mathbb{R}}dt\,\ket{t}\ket{\psi_{S}(t)}.$
(8)

From the above equation one can immediately see that in general physical
states describe an entangled state of the clock and the system. It is the
correlations between the clock and system described by this entangled state
that are responsible for the relative evolution of the system with respect to
the clock.

##  3 Timeless formulation of the Wigner’s friend experiment

Figure 1: The circuit representation of the Wigner’s friend experiment as
encoded in the physical state $\left|\left.\Psi\right>\right>$ given in Eq.
(11). This can be thought of as the perspective of a hypothetical higher level
observer whose experimental powers are even beyond that of Wigner. As
described in the main text, at time $t_{F}$ according to clock $C$ the Friend
performs a measurement and her memory gets entangled with the system $S$. At
another time $t_{W}>t_{F}$ Wigner measures both $S$ and $F$ and his memory
gets entangled with the joint system.

We now present a description of the Wigner’s friend experiment depicted in
Fig. 1 in terms of the Page and Wootters mechanism.

Consider a two-level system ($S$) associated with the Hilbert space
$\mathcal{H}_{S}\simeq\mathbb{C}^{2}$ and a projective valued measurement of
$S$ that yields two possible outcomes, $\\{\uparrow,\downarrow\\}$. Such a
measurement is described by the set of projectors
$\\{\Pi_{\uparrow},\Pi_{\downarrow}\\}\subset\mathcal{E}(\mathcal{H}_{S})$,
where $\Pi_{\uparrow}+\Pi_{\downarrow}=\mathds{1}_{S}$ and
$\mathcal{E}(\mathcal{H})$ denotes the space of effect operators on the
Hilbert space $\mathcal{H}$. The Friend ($F$) makes such a measurement of $S$
at the time $t_{F}$ and the outcome is recorded by $F$’s measuring device and
eventually in her memory, which we associate with the Hilbert space
$\mathcal{H}_{F}\simeq\mathbb{C}^{3}$. Situated outside of $F$’s laboratory is
Wigner ($W$), who makes a joint measurement of $S$ and $F$ with two possible
outcomes, $\\{\mbox{yes},\mbox{no}\\}$, at a time $t_{W}>t_{F}$. This
measurement corresponds to the set of projectors $\\{\Pi_{\rm yes},\Pi_{\rm
no}\\}\subset\mathcal{E}(\mathcal{H}_{S}\otimes\mathcal{H}_{F})$ with
$\Pi_{\rm yes}+\Pi_{\rm no}=\mathds{1}$, and its result is recorded in $W$’s
memory, the state of which is encoded in the Hilbert space
$\mathcal{H}_{W}\simeq\mathbb{C}^{3}$. The times $t_{F}$ and $t_{W}$ are
associated with a clock system $C$ described by the Hilbert space $H_{C}\simeq
L^{2}(\mathbb{R})$ as discussed in Sec. 2.

Following Hellmann _et al._ [17] and Giovannetti _et al._ [18], we use a von
Neumann measurement model to describe $F$’s and $W$’s measurement within the
PWM. The physical states describing the Wigner’s friend experiment in Fig. 1
are solutions to

$\displaystyle\left(\\!\hat{H}_{C}\\!+\\!\delta(\hat{T}\\!-\\!t_{F})\hat{K}_{SF}\\!+\\!\delta(\hat{T}\\!-\\!t_{W})\hat{K}_{SFW}\\!\right)\\!\left|\left.\Psi\right>\right>\\!=\\!0,$
(9)

where $\hat{K}_{SF}$ and $\hat{K}_{SFW}$ are the interaction Hamiltonians
coupling respectively $S$ and $F$ during $F$’s measurement of $S$ and $S$,
$F$, and $W$ during $W$’s measurement of $SF$:

$\displaystyle
e^{-i\hat{K}_{SF}}\ket{\psi_{S}}\ket{R_{F}}=\sum_{f\in\\{\uparrow,\downarrow\\}}\Pi_{f}\ket{\psi_{S}}\ket{f_{F}},$
$\displaystyle e^{-i\hat{K}_{SFW}}\ket{\phi_{SF}}\ket{R_{W}}=\sum_{w\in\\{{\rm
yes},\,{\rm no}\\}}\Pi_{w}\ket{\phi_{SF}}\ket{w_{W}},$

where $\ket{\psi_{S}}$ is the state of the system for clock readings $t<t_{F}$
and $\ket{R_{F}}\in\mathcal{H}_{F}$, $\ket{R_{W}}\in\mathcal{H}_{W}$ are the
pre-measurement “ready” states of the memories of $F$ and $W$. The sets
$\\{\ket{R_{F}},\ket{\uparrow_{F}},\ket{\downarrow_{F}}\\}$ and
$\\{\ket{R_{W}},\ket{{\rm yes}_{W}},\ket{{\rm no}_{W}}\\}$ form orthonormal
bases for $\mathcal{H}_{F}$ and $\mathcal{H}_{W}$ respectively. Using this
measurement scheme we will calculate the probabilities of result $m$ by
projecting onto the memory entries of the respective observer. Hence, the
formula in Eq. (4) for arbitrary time $t$ becomes

$\displaystyle P\left(m\ \mbox{when}\ t\right)$
$\displaystyle=\frac{\left<\left<\Psi\right.\right|\big{(}|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\Pi^{m}\big{)}\left|\left.\Psi\right>\right>}{\left<\left<\Psi\right.\right|\big{(}|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\mathds{1}\big{)}\left|\left.\Psi\right>\right>},$
(10)

where $\Pi^{m}$ now acts on $\mathcal{H}_{M}$, the Hilbert space of the
apparatus or memory storing the result, and $\left|\left.\Psi\right>\right>$
is a solution to Eq. (9). For simplicity we have assumed no free dynamics of
$S$, $F$, and $W$ in Eq. (9) in addition to their interaction with one another
during the measurements. The solutions take the form

$\displaystyle\ket{\Psi}\rangle$
$\displaystyle=\int_{-\infty}^{t_{F}}dt\,\ket{t}\ket{\psi_{S}}\ket{R_{F}}\ket{R_{W}}$
$\displaystyle\quad+\int_{t_{F}}^{t_{W}}dt\,\ket{t}\sum_{f\in\\{\uparrow,\downarrow\\}}\Pi_{f}\ket{\psi_{S}}\ket{f_{F}}\ket{R_{W}}$
$\displaystyle\quad+\int_{t_{W}}^{\infty}dt\,\ket{t}\\!\\!\\!\sum_{\begin{subarray}{c}f\in\\{\uparrow,\,\downarrow\\}\\\
w\in\\{{\rm yes},\,{\rm
no}\\}\end{subarray}}\\!\\!\\!\Pi_{w}\Pi_{f}\ket{\psi_{S}}\ket{f_{F}}\ket{w_{W}}\\!,$
(11)

for arbitrary states $\ket{\psi_{S}}$ of the system. When expressed in the
clock basis like in Eq. (11) the physical state encodes the complete history
of system $S$ and the measurements performed on it. In what follows we will
explicitly consider states

$\displaystyle\ket{\psi_{S}}=a\ket{\uparrow_{S}}+be^{i\phi_{S}}\ket{\downarrow_{S}},$
(12)

where $a,b,\phi_{S}\in\mathbb{R}$ and

$\displaystyle\ket{{\rm
yes}_{SF}}=\alpha\ket{\uparrow_{S}}\ket{\uparrow_{F}}+\beta
e^{i\phi_{SF}}\ket{\downarrow_{S}}\ket{\downarrow_{F}},$ (13)

where $\alpha,\beta,\phi_{SF}\in\mathbb{R}$, with $\Pi_{\rm yes}=|{\rm
yes}\vphantom{\rm yes}\rangle\langle{\rm yes}\vphantom{\rm yes}|$ and
$\Pi_{\rm no}=\mathds{1}-|{\rm yes}\vphantom{\rm yes}\rangle\langle{\rm
yes}\vphantom{\rm yes}|$, moreover, we will use
$\phi\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\phi_{S}-\phi_{SF}$.
Note that a measurement by Wigner as implied by Eq. (13) will in general
affect the Friend’s memory state and potentially alter her perceived result.

##  4 Different definitions of conditional probabilities, different solutions
of the Wigner’s friend paradox

We want to express the dynamical content of a theory in a manifestly
operational way, that is, in terms of probabilities of measurement outcomes.
As emphasized by Kuchař [10], _the fundamental question in any dynamical
theory_ is: if one measures an observable at the time $t_{1}$ and obtains
outcome $a$, what is the probability of a different measurement yielding the
outcome $b$ at the time $t_{2}>t_{1}$? For non-Wigner’s friend scenarios there
have been two proposals for defining probability rules within the Page-
Wootters mechanism [19, 18] in order to answer this fundamental question. Both
approaches recover the standard quantum formalism for the situation of two
observables measured at different times inquired above.

In order to deal with Wigner’s friend scenarios (in particular with the setup
depicted in Fig. 1), however, one needs to generalize the standard definition
of the conditional-probability rule. The rationale behind this generalization
is that the state of the Friend’s memory is –except for particular cases–
modified by Wigner’s measurement. Hence, it is not any longer possible to
simply read out the information in $F$’s memory at the end of the protocol in
order to learn what outcome she obtained. Thus, we will be interested in
calculating two-time conditional probabilities at times $t_{1}$ (in between
$F$’s and $W$’s measurement) and $t_{2}$ (after $W$’s measurement), i.e.,
$t_{F}<t_{1}<t_{W}<t_{2}$. We propose three main alternative conditional
probability definitions, two of which are inspired by Refs. [19] and [18]
respectively, as well as a third novel one. While all of these definitions
give the standard quantum probabilities for non-Wigner’s friend cases (as
shown explicitly in Appendix A), they give different conditional probabilities
for the measurement results of Wigner and his Friend.

First we propose a conditional-probability rule similar to that put forward by
Dolby [19], where measurement operators
$|{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\operatorname*{\otimes}\Pi_{m}$
together with the operator $P^{\rm ph}$ (defined in Eq. (2)) are used to
calculate two-time conditional probabilities. Specifically, after the
application of one operator
$|{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\operatorname*{\otimes}\Pi_{m}$
to the physical state $\left|\left.\Psi\right>\right>$ the constraint in Eq.
(1) is in general no longer fulfilled, i.e.,
$\hat{H}(|{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\operatorname*{\otimes}\Pi_{m}\left|\left.\Psi\right>\right>)\neq
0$. As proposed in [19] one uses the operator $P^{\rm ph}$ to obtain again a
physical state, $\hat{H}\big{(}P^{\rm
ph}|{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\operatorname*{\otimes}\Pi_{m}\left|\left.\Psi\right>\right>\big{)}=0$,
before applying a second measurement operator
$|{t_{2}}\vphantom{t_{2}}\rangle\langle{t_{2}}\vphantom{t_{2}}|\operatorname*{\otimes}\Pi_{n}$.
In order to describe Wigner’s friend scenarios we account for the measurements
as in Eqs. (9)-(11).

Definition 1

(two-time “collapse”). The conditional probability of result $n$ at time
$t_{2}$ given result $m$ at time $t_{1}$ is

$\displaystyle P_{1}\left(n\textrm{ when }t_{2}\,|\,m\textrm{ when
}t_{1}\right)=\frac{\left<\left<\Psi\right.\right||{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\\!\otimes\\!\Pi^{m}P^{\rm
ph}|{t_{2}}\vphantom{t_{2}}\rangle\langle{t_{2}}\vphantom{t_{2}}|\otimes\Pi^{n}P^{\rm
ph}\ket{t_{1}}\\!\bra{t_{1}}\\!\otimes\\!\Pi^{m}|\Psi\rangle\rangle}{\langle\langle\Psi|\ket{t_{1}}\\!\bra{t_{1}}\\!\otimes\\!\Pi^{m}|\Psi\rangle\rangle},$
(14)

where $\Pi^{m}$ and $\Pi^{n}$ are now the projectors on the respective states
of the apparatus or memories in $\mathcal{H}_{M}$ and $\mathcal{H}_{N}$, and
$P^{\rm ph}$ was defined in Eq. (2).

Intuitively, the numerator in Eq. (14), in the fashion of Ref. [19], is the
modulus square of the physical state $\left|\left.\Psi\right>\right>$ which is
first projected on $\Pi^{m}$ at time $t_{1}$, then brought back into the
physical space by applying $P^{\rm ph}$ and finally projected on $\Pi^{n}$ at
time $t_{2}$. Definition 1 always gives well-defined probabilities, and for
the results of Wigner and his Friend this leads to the conditional
probabilities displayed in Table 1. We label this definition “two-time
collapse” because any observer will agree on these probabilities assignments
which correspond to applying the state-update rule (“collapse”) after every
measurement. Moreover, it is genuinely two-time, since the expression
explicitly depends on both the times $t_{1}$ and $t_{2}$. Hence, this resolves
the probabilistic paradox for Wigner’s friend scenarios insofar as it suggests
that both Wigner and the Friend should describe $F$’s measurement with
“collapse” dynamics. Note, however, that this rule does not give rise to a
well-defined joint probability since $P_{1}\left(n\textrm{ when
}t_{2}\,|\,m\textrm{ when }t_{1}\right)\cdot P(m\textrm{ when }t_{1})\neq
P_{1}\left(m\textrm{ when }t_{1}\,|\,n\textrm{ when }t_{2}\right)\cdot
P(n\textrm{ when }t_{2})$. This reflects the fact that the numerator in Eq.
(14) is not invariant under the exchange of
$|{t_{1}}\vphantom{t_{1}}\rangle\langle{t_{1}}\vphantom{t_{1}}|\otimes\Pi^{m}$
and
$|{t_{2}}\vphantom{t_{2}}\rangle\langle{t_{2}}\vphantom{t_{2}}|\otimes\Pi^{n}$.

$\bm{P_{1}\left(w\ {\rm when}\ t_{2}\,|\,f\ {\rm when}\ t_{1}\right)}$
---
$f$ $w$ |  $\rm{yes}$ |  $\rm{no}$
$\uparrow$ |  $\alpha^{2}$ |  $\beta^{2}$
$\downarrow$ |  $\beta^{2}$ |  $\alpha^{2}$
$\bm{P_{1}\left(f\ {\rm when}\ t_{1}\,|\,w\ {\rm when}\ t_{2}\right)}$
$f$ $w$ |  $\rm{yes}$ |  $\rm{no}$
$\uparrow$ |  $\alpha^{2}$ |  $\beta^{2}$
$\downarrow$ |  $\beta^{2}$ |  $\alpha^{2}$
Table 1: The conditional probabilities of Wigner seeing result $w$ at time
$t_{2}$ given that the Friend saw result $f$ at time $t_{1}$ and of the Friend
seeing $f$ at $t_{1}$ given that Wigner will see $w$ at $t_{2}$ according to
definition 1. Note that the two conditional probabilities are equal and hence,
applying Bayes’ rule will in general _not_ give rise to _one_ joint
probability expression.

Alternatively, we consider the conditional probability rule proposed by
Giovanetti _et al._ [18] in the context of Wigner’s friend experiments. The
authors of [18] include generalized measurements as described in Eq. (9)-(11)
in the constraint Hamiltonian and apply projection operators on the
measurement apparatus or memories at some time $t$ after both measurements in
question have been performed, i.e.,
$|{t}\vphantom{t}\rangle\langle{t}\vphantom{t}|\operatorname*{\otimes}\Pi^{m}\operatorname*{\otimes}\Pi^{n}$.
This equates measurements at different times with one joint measurement of the
records of the obtained results at a final single time, which is no longer a
trivial thing to do when considering Wigner’s friend scenarios. The original
formulation in [18] states the following rule for calculating conditional
probabilities.

Definition 2a

(two-time unitary, uninterpreted conditions). The conditional probability (in
the cases where it is well-defined) of result $n$ at time $t_{2}$ given result
$m$ at time $t_{1}$ is

$\displaystyle P_{2a}\left(n\textrm{ when }t_{2}\,|\,m\textrm{ when
}t_{1}\right)$
$\displaystyle\qquad=\frac{\langle\langle\Psi\ket{t_{2}}\bra{t_{2}}\otimes\Pi^{n}\otimes\Pi^{m}\left|\left.\Psi\right>\right>}{\langle\langle\Psi\ket{t_{1}}\bra{t_{1}}\operatorname*{\otimes}\Pi^{m}\left|\left.\Psi\right>\right>}.$
(15)

The intuition behind this definition is to read out both $F$’s and $W$’s
memories at the final time $t_{2}$ and take the modulus square to define a
joint probability, and use that to construct a conditional probability.
Namely, to take the expectation value of the operators $\Pi^{m}$ and $\Pi^{n}$
at time $t_{2}$ with the physical state $\left|\left.\Psi\right>\right>$
(numerator). This is then divided by the one-time probability of $F$ finding
outcome $m$ at time $t_{1}$ (denominator), i.e., before $W$’s measurement
takes place. Equation (15) depends on both measurement outcomes and on both
times $t_{1}$ and $t_{2}$, thus amounting to a genuine two-time probability.
However, as shown in Appendix B, in the Wigner’s friend scenario of Fig. 1,
these expressions are not always proper probabilities, for they are normalized
only if

$\displaystyle
2\alpha^{2}\beta^{2}\left(\frac{b}{a}\right)^{2}+2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{b}{a}$
$\displaystyle=2\alpha^{2}\beta^{2}\left(\frac{a}{b}\right)^{2}-2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{a}{b}$
$\displaystyle=1-\alpha^{4}-\beta^{4}.$ (16)

Under these conditions, definition 2a gives the probabilities stated in Table
2. While these expressions are formally probabilities when Eqs. (16) are
satisfied, their operational meaning is admittedly not clear to us, except for
special cases (see Sec. 5).

$\bm{P_{2a}\left(w\ {\rm when}\ t_{2}\,|\,f\ {\rm when}\ t_{1}\right)}$
---
$f$ $w$ | $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\frac{1+2\alpha^{2}+(\beta^{4}-\alpha^{4})\cos(2\phi)\pm\chi}{4}$ | $\frac{1+2\beta^{2}+(\alpha^{4}-\beta^{4})\cos(2\phi)\mp\chi}{4}$
$\downarrow$ | $\frac{1+2\beta^{2}+(\alpha^{4}-\beta^{4})\cos(2\phi)\pm\chi}{4}$ | $\frac{1+2\alpha^{2}+(\beta^{4}-\alpha^{4})\cos(2\phi)\mp\chi}{4}$
with
$\chi\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}2\cos(\phi)\sqrt{1-(\alpha^{2}-\beta^{2})^{2}\sin^{2}(\phi)}$

Table 2: The conditional probabilities of Wigner seeing result $w$ at time
$t_{2}$ given that the Friend saw result $f$ at time $t_{1}$ according to
definition 2a. The different signs correspond to the different solutions of
the quadratic equations in Eqs.(16). Note that there is no sensible definition
of $P_{2a}\left(f\ {\rm when}\ t_{1}\,|\,w\ {\rm when}\ t_{2}\right)$, since
the numerator in Eq.(15) does not depend on $t_{1}$ and
$\langle\langle{\Psi}\ket{t_{1}}\bra{t_{1}}\Pi^{w}\operatorname*{\otimes}\left|\left.\Psi\right>\right>=0$.

Considering the general arguments in [17] in favor of equating measurements at
different times with one collective measurement at a final time, we propose a
straightforward modification of definition 2a for conditional probabilities.

Definition 2b

(one-time unitary). The conditional probability of result $n$ given $m$ is

$\displaystyle P_{2b}\left(n\textrm{ when }t_{2}\,|\,m\textrm{ when
}t_{2}\right)$
$\displaystyle\quad=\frac{\langle\langle\Psi\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{n}\otimes\Pi^{m}\left|\left.\Psi\right>\right>}{\langle\langle\Psi\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{m}\left|\left.\Psi\right>\right>},$
(17)

where $t_{2}$ is some time after the second measurement has been performed.

These one-time probabilities are always well-defined and shown in Table 3.
They correspond to jointly observable statistics after full unitary evolution
(i.e., without state-update) and are equivalent to the proposal in [20]. This
resolves the paradox of the ambiguous probability assignment in Wigner’s
friend scenarios insofar as it corresponds to all observers describing all
measurements unitarily up to that point where they can all compare their
records, that is, the entries in $F$’s and $W$’s memories after both
measurements took place. It is important to note that $F$’s memory will in
general be altered by $W$’s measurement and reading it out at the end will no
longer correspond to what was encoded after $F$’s measurement. Note that for
non-Wigner’s friend scenarios definitions 2a and 2b are equivalent, since in
this case
$\langle\langle{\Psi}\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{m}\left|\left.\Psi\right>\right>=\langle\langle{\Psi}\ket{t_{1}}\bra{t_{1}}\operatorname*{\otimes}\Pi^{m}\left|\left.\Psi\right>\right>$
(see Appendix A). This is due to the fact that the memory, in which the result
$m$ is stored at some time $t_{1}$, undergoes no further evolution until time
$t_{2}$ and after.

$\bm{P_{2b}\left(w\ {\rm when}\ t_{2}\,|\,f\ {\rm when}\ t_{2}\right)}$
---
$f$ $w$ | $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\frac{\frac{\alpha^{2}}{\beta^{2}}+2\frac{b\alpha}{a\beta}\cos(\phi)+\frac{b^{2}}{a^{2}}}{N_{\uparrow}}$ | $\frac{\frac{\beta^{2}}{\alpha^{2}}-2\frac{b\beta}{a\alpha}\cos(\phi)+\frac{b^{2}}{a^{2}}}{N_{\uparrow}}$
$\downarrow$ | $\frac{\frac{\beta^{2}}{\alpha^{2}}+2\frac{a\beta}{b\alpha}\cos(\phi)+\frac{a^{2}}{b^{2}}}{N_{\downarrow}}$ | $\frac{\frac{\alpha^{2}}{\beta^{2}}+2\frac{a\alpha}{b\beta}\cos(\phi)+\frac{a^{2}}{b^{2}}}{N_{\downarrow}}$
with
$N_{\uparrow}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}+2\frac{b}{a}\cos(\phi)\left(\frac{\alpha}{\beta}-\frac{\beta}{\alpha}\right)+2\frac{b^{2}}{a^{2}}$
and
$N_{\downarrow}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\frac{\alpha^{2}}{\beta^{2}}+\frac{\beta^{2}}{\alpha^{2}}+2\frac{a}{b}\cos(\phi)\left(\frac{\beta}{\alpha}-\frac{\alpha}{\beta}\right)+2\frac{a^{2}}{b^{2}}$
$\bm{P_{2b}\left(f\ {\rm when}\ t_{2}\,|\,w\ {\rm when}\ t_{2}\right)}$
$f$ $w$ | $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\alpha^{2}$ | $\beta^{2}$
$\downarrow$ | $\beta^{2}$ | $\alpha^{2}$

Table 3: The conditional probabilities for results $w$ of Wigner and $f$ of
the Friend according to definition 2b. The joint probability is well-defined
and corresponds to the numerator in Eq. (17).

The fact that the Friend’s memory is the _only_ record of her observed result
and is in general affected by Wigner’s measurement raises the question whether
it is operationally meaningful to assign a joint probability to the results
$f$ and $w$. By construction of the setup these two results are in general not
jointly observable by any single observer. However, as discussed in Ref. [21],
in the special case of the joint state of the quantum system and the Friend
being in an eigenstate of Wigner’s measurement, $F$’s memory is not affected
(a _non-disturbance measurement_) and she can, in principle, learn both
results. In fact, in this case Wigner can send his observed outcomes to the
Friend who can combine them with her own (left undisturbed by Wigner’s
measurement) and construct the joint probability distribution. We, thus,
propose a third rule for calculating conditional two-time probabilities, which
agrees with standard quantum theory for non-Wigner’s friend scenarios and
gives well-defined probabilities for Wigner’s friend setups, in the cases
where these probabilities are operationally meaningful.

Definition 3

(two-time unitary, consistency conditions). The conditional probability (in
the cases where it is well-defined) of result $n$ at time $t_{2}$ given result
$m$ at time $t_{1}$ is

$\displaystyle P_{3}\left(n\textrm{ when }t_{2}\,|\,m\textrm{ when
}t_{1}\right)=$
$\displaystyle\frac{\langle\langle\Psi|(\ket{t_{2}}\bra{t_{2}}\otimes\Pi^{n})P^{\rm
ph}(\ket{t_{1}}\bra{t_{1}}\otimes\Pi^{m})\left|\left.\Psi\right>\right>}{\langle\langle\Psi\ket{t_{1}}\bra{t_{1}}\otimes\Pi^{m}\left|\left.\Psi\right>\right>}.$
(18)

$\bm{P_{3}\left(w\ {\rm when}\ t_{2}\,|\,f\ {\rm when}\ t_{1}\right)}$
---
$f$ $w$ |  $\rm{yes}$ |  $\rm{no}$
$\uparrow$ |  $1$ (or $0$) |  $0$ (or $1$)
$\downarrow$ |  $1$ (or $0$) |  $0$ (or $1$)
$\bm{P_{3}\left(f\ {\rm when}\ t_{1}\,|\,w\ {\rm when}\ t_{2}\right)}$
$f$ $w$ |  $\rm{yes}$ |  $\rm{no}$
$\uparrow$ |  $\alpha^{2}$ (or $0$) |  0 (or $\beta^{2}$)
$\downarrow$ |  $\beta^{2}$ (or $0$) |  0 (or $\alpha^{2}$)

Table 4: The conditional probabilities of Wigner seeing result $w$ at $t_{2}$
and the Friend seeing result $f$ at time $t_{1}$ (above) and of the Friend
seeing $f$ at $t_{1}$ given that Wigner will see $w$ at $t_{2}$ (below),
according to definition 3. the probabilities are well-defined only in the case
where Wigner’s measurement is non-disturbing, i.e. $a=\alpha$ and $b=\beta$
(or $a=\beta$ and $b=-\alpha$).

This rule gives well-defined probabilities under the condition that the
measurement operators commute on the physical state when compared at the same
instant of time, i.e.

$\displaystyle\left[\mathcal{U}(t_{1},t_{2})\Pi^{m}\mathcal{U}^{\dagger}(t_{1},t_{2}),\Pi^{n}\right]\left|\left.\Psi\right>\right>=0,$
(19)

where $\mathcal{U}(t_{2},t_{1})=\bra{t_{2}}P^{\rm ph}\ket{t_{1}}$.
We note, that condition (19) ensures that $m$ and $n$ are part of a family of
consistent quantum histories [22] as shown in Appendix C. According to the
consistent histories framework only within such a family can one meaningfully
assign a set of probabilities to the properties encoded in the respective
histories; in the case considered here, the results of Wigner and his Friend.
This agrees with the fact that definition 3 singles out the non-disturbing
measurements by Wigner in the setup depicted in Fig. 1 (see Appendix D for
explicit calculations), namely

$\displaystyle\phi=n\pi\qquad\text{ and
}\qquad\frac{a}{\alpha}=\frac{b}{\beta}.$ (20)

The probabilities given by definition 3 are listed in Table 4 and coincide
with those given by definitions 2a and 2b (i.e., without the application of
the state-update). In contrast to definition 2b, however, definition 3
describes genuine two-time probabilities, which, contrary to those given by
definition 2a, have a clear interpretation and operational meaning. However,
note that collecting these statistics is not straight-forward, for also in the
non-disturbance case the Friend cannot simply send out her observed results to
Wigner without changing the probabilities for Wigner’s result (see Ref. [21]).
Yet, Wigner can in principle send inside the laboratory his measured data,
from which the Friend can construct a joint (and conditional) probability
distribution.

##  5 Discussion and outlook

We have presented three generalizations of the standard rule to assign
probabilities to consecutive quantum measurements in a Wigner’s friend
scenario using the Page-Wootters mechanism. We showed how these probability
rules potentially remove, in different manners, the ambiguity between the
application of unitary dynamics and the state-update rule (“collapse”). While
these represent a formal resolution (in fact a few distinct ones) of the
probability-assignment paradox in Wigner’s friend-like experiments, a number
of key issues remain open: There are potentially more than just the three
probability rules we presented, possibly leading to other resolutions of the
paradox. Is there a way of classifying all of them? Furthermore, what is the
operational meaning of each of the expressions that formally give
probabilities?

Concerning the first question, we ought to stress that the only necessary
condition that we have required from our definitions of two-time probabilities
was, in fact, that they have to reduce to the standard Born rule in the non-
Wigner’s friend case, and that they give well-defined probabilities for at
least certain cases of a Wigner’s friend scenario. However, one can imagine
further possible definitions that comply with these requirements, hence, a
more systematic study would be desirable. In particular, one extension of the
original proposal of Page and Wootters put forward by Gambine _et al_. is to
compute conditional probabilities between averages over ‘evolving constants of
motion’,222More precisely, one would construct a family of Dirac observables,
which commute with the constraint in Eq. (1), comprised of a partial
observables associated with a clock and observables on Wigner’s and his
friend’s memories. Such a family of Dirac observables is parametrized by the
times read by the clock. Conditional probabilities would be computed as in Eq.
(4), with the important difference that the numerator and denominator are
averaged over all readings of the clock. See Ref. [23] for more detail. which
would be interesting to examine in Wigner’s friend setups.

Figure 2: Suggestive graphical representation of the probabilities according
to the different definitions in a Wigner’s friend scenario (when they are
well-defined). For non Wigner’s friend scenarios all the definitions recover
standard quantum probabilities, thus they fully overlap. The two planes
distinguish between joint probabilities that are in accordance with applying
(collapse) or not applying (unitary) the state-update rule after the Friend’s
measurement. In the non-disturbance case (i.e., $\phi=n\pi$ and $a/\alpha=\pm
b/\beta$) definition 2a, definition 2b and definition 3 give the same unitary,
conditional probabilities, that are different from the collapse ones given by
definition 1. There is, however, at least one instance where definitions 1,
2a, and 2b coincide (i.e., $\phi=(n+1/2)\pi$, $a=b=1/\sqrt{2}$ and arbitrary
$\alpha$.

|  standard QT |  positive |  normalized |  two-time |  sym. joint |  Conditions
---|---|---|---|---|---|---
Def. 1 |  ✓ |  ✓ |  ✓ |  ✓ |  ✗ |
|  |  |  |  |  |
Def. 2a |  ✓ |  ✓ |  ✗ |  ✓ |  ✓ |  normalized, if
$\alpha^{4}+\beta^{4}+2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{b}{a}+2\alpha^{2}\beta^{2}\left(\frac{b}{a}\right)^{2}=1$
and
$\alpha^{4}+\beta^{4}-2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{a}{b}+2\alpha^{2}\beta^{2}\left(\frac{a}{b}\right)^{2}=1$.
Def. 2b |  ✓ |  ✓ |  ✓ |  ✗ |  ✓ |
|  |  |  |  |  |
Def. 3 |  ✓ |  ✗ |  ✓ |  ✓ |  ✗ |  positive for non-disturbance measurement: $a=\alpha,b=\beta$ or $a=\beta,b=-\alpha$. And $\phi=n\pi$.

Table 5: Comparison of the different proposed conditional probability rules
for describing Wigner’s friend experiments. From the left to the right, the
columns indicate whether each definition: (i) reduces to standard quantum
theory for non-Wigner’s friend scenarios; (ii) is positive; (iii) is
normalized; (iv) is a genuine two-time expression; (v) gives rise to a well-
defined joint probability distribution (i.e. $P(A)P(B|A)=P(B)P(A|B)$). Note,
in fact, that definitions 2a and 3 become probabilities only under certain
conditions, the blue ✗ indicates that the respective property is satisfied
under certain conditions but not in general. The last column on the right
shows these conditions (if any) under which the respective properties are
satisfied.

To address the second question, we now discuss the interpretation and the
limits of each definition of conditional probabilities presented in the last
section (see Table 5 for a summary of the definitions and their properties,
and Fig. 2 for a sketched graphical representation). As already noticed,
definition 1 has a clear interpretation in terms of “objective collapse”
probabilities, but does not give rise to a well-defined joint probability (see
Sec. 4). However, one could maintain that at the operational level conditional
probabilities come prior to joint probabilities, which are a posteriori
constructions.

On the other hand, definition 2b gives measurement statistics consistent with
standard quantum theory assuming $S$, $F$ and $W$ evolve unitarily and the
Born rule is applied after $W$’s measurement interaction has occurred. While
this definition also has a clear interpretation, it fails in representing
genuine two-time probabilities (only the final time $t_{2}$ appears in the
rule to assign conditional probabilities for the measurements of $F$ and $W$).

Definition 3 has the merit of being interpretable as a genuine two-time
probability, however it is a well-defined probability (i.e. real and non-
negative) only under the rather demanding “non-disturbance measurement”
conditions, that is, when Wigner measures in a basis that contains the joint
state of the system and the Friend (see Table 5). In this case, the
probabilities are the same as definition 2b. The condition required for
definition 3 to result in genuine probabilities can be interpreted as evidence
that only in the non-disturbance case there exists a record of the past
outcome $f$ of $F$’s measurement and the current outcome $w$ of $W$’s
measurement. Thus, only in this case is a joint probability distribution
associated with the measurement outcomes of $W$ and $F$ operationally
meaningful, since it corresponds to collectible statistics for the friend.
Remarkably, this limitation to the meaningful attribution of a joint
probability of the outcomes of $F$ and $W$ emerges naturally from our
formalism. This suggests a connection with a recent no-go theorem [5, 7]
stating that it is in general not possible to construct a joint probability
distribution associated with measurement outcomes of different observers in a
(more complicated) Wigner’s friend scenario. Moreover, we have shown that the
conditions for definition 3 to give well-defined probabilities ensure that the
measurement outcomes of Wigner and his Friend are elements of a family of
consistent quantum histories [22, 24] (see Appendix C).

Finally, definition 2a gives well-defined (i.e., normalized) probabilities
only under the conditions in Eq. (16). Note that the non-disturbance
measurement satisfies these conditions (see Fig. 2), meaning that if Eq. (20)
is satisfied, so is Eq. (16). In general, however, it is not clear to us how
to interpret the probabilities given by definition 2a operationally. It seems
that formally for any given initial state of the quantum system and fixed
measurement of the Friend, there always exists a measurement choice of Wigner
(besides the non-disturbance one) which ensures that definition 2a gives well-
defined probabilities. However, we do not have an intuition of what is special
about the measurement choices that satisfy Eqs. (16). Moreover, the
probabilities provided by definition 2a (see Table 2) do not have a
straightforward interpretation since they do not correspond to either
“collapse” or full unitary evolution up to the final measurement of Wigner and
unlike for definition 3 we do not find an operational principle distinguishing
them. Again, a systematic study of all possible probability rules that reduce
to standard quantum probabilities for non-Wigner’s friend scenarios might give
further insight. We leave it up for future work to do so.

Moreover, it will be fruitful to explicitly construct maps between the
perspectives of Wigner and his friend [25] using the timeless description of
the setup developed here.

In contrast to the standard quantum formalism –where there is a dichotomy
between the assignment of probabilities at the moment of measurements and
unitary evolution– the timeless formulation of quantum mechanics places the
rule to assign probabilities to measurement outcomes (Born rule) logically
prior to unitary dynamics. Thus, the timeless formulation is not directly
concerned with the “evolution of quantum states” (in particular the transition
from before to after a measurement), making it an appropriate tool to deal
with the ambiguity of the standard formalism in describing quantum
measurements.

As long as one considers typical quantum scenarios (i.e., a system measured
one or more times), there have been at least two different, yet equivalent,
proposal in the timeless formulation on how to assign probabilities to the
outcomes of measurements [19, 18]. However, we have shown that to deal with
Wigner’s friend scenarios these two-time probabilities rules ought to be
adapted. We presented two such adaptations and considered a third proposal,
all of which resolve the ambiguity of Wigner’s friend probability paradox in
different ways. Thus, in this framework, the ambiguity between the two
dynamical processes (i.e., unitary evolution versus “collapse”) is pushed back
to the choice of the probability-assignment rule: Finding compelling physical
arguments to rule out all but one of the proposed two-time probability-
assignments would significantly contribute to solving the Wigner’s friend
paradox.

## Acknowledgments

We want to thank Lorenzo Maccone for helpful discussions. This work was funded
by CoQuS, Vienna Doctoral School (VDS) and the QUOPROB project (no. I 2906-
G24) of the Austrian Science Fund (FWF) and supported in part by the Natural
Sciences and Engineering Research Council of Canada (NSERC) and the Dartmouth
College Society of Fellows. F.D.S. acknowledges the financial support through
a DOC Fellowship of the Austrian Academy of Sciences (OAW). E.C.-R. was
supported in part by the Program of Concerted Research Actions (ARC) of the
Université libre de Bruxelles. Research at Perimeter Institute is supported in
part by the Government of Canada through the Department of Innovation, Science
and Industry Canada and by the Province of Ontario through the Ministry of
Colleges and Universities. C.B., further, acknowledges the support of the
Austrian Science Fund (FWF) through the SFB project "BeyondC" and the project
I-2526-N27, a grant from the Foundational Questions Institute (FQXi) Fund and
a grant from the John Templeton Foundation (Project No. 60609). The opinions
expressed in this publication are those of the authors and do not necessarily
reflect the views of the John Templeton Foundation.

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##  Appendix A Reproducing standard quantum theory in non-Wigner’s friend
setups

For non-Wigner’s friend scenarios the two measurements are performed on the
same quantum system instead of one of them measuring both the system and an
observer. The constraint Hamiltonian then takes the form

$\displaystyle\hat{H}^{\prime}=\hat{P_{t}}+H_{S}+\delta(\hat{T}-t_{M})\hat{K}_{SM}+\delta(\hat{T}-t_{N})\hat{K}_{SN},$
(21)

where now $M$ and $N$ are apparatus or memories, in which the results of the
respective measurements are encoded. This gives the physical state

$\displaystyle\ket{\Psi^{\prime}}\rangle$
$\displaystyle=\int_{-\infty}^{t_{M}}dt\,\ket{t}U_{S}(t,t_{0})\ket{\psi_{S}(t_{0})}\ket{R_{M}}\ket{R_{N}}$
$\displaystyle\quad+\int_{t_{M}}^{t_{N}}dt\,\ket{t}\sum_{m}U_{S}(t,t_{M})\Pi_{m}\ket{\psi_{S}(t_{M})}\ket{m_{M}}\ket{R_{N}}$
$\displaystyle\quad+\int_{t_{N}}^{\infty}dt\,\ket{t}\sum_{m,n}\big{[}U_{S}(t,t_{N})\Pi_{n}U_{S}(t_{N},t_{M})\Pi_{m}\ket{\psi_{S}(t_{M})}\ket{m_{M}}\ket{n_{N}}\big{]},$
(22)

with both $\Pi_{m}$ and $\Pi_{n}$ acting on $\mathcal{H}_{S}$. Moreover, we
have

$\displaystyle\bra{t}P^{\rm
ph}\ket{t_{0}}\ket{\phi(t_{0})}=\mathcal{U}(t,t_{0})\ket{\phi(t_{0})}$ (23)

for arbitrary
$\ket{\phi(t_{0})}\in\mathcal{H}_{S}\operatorname*{\otimes}\mathcal{H}_{M}\operatorname*{\otimes}\mathcal{H}_{N}$,
where

$\displaystyle\mathcal{U}(t,t_{0})=\begin{cases}U_{S}(t,t_{0})&t_{0}<t\\\
U_{S}(t,t_{M})U_{M}U_{S}(t_{M},t_{0})&t_{0}<t_{M}<t\\\
U_{S}(t,t_{N})U_{N}U_{S}(t_{N},t_{0})&t_{0}<t_{N}<t\\\
U_{S}(t,t_{N})U_{N}U_{S}(t_{N},t_{M})U_{M}U_{S}(t_{M},t_{0})&t_{0}<t_{M}<t_{N}<t\\\
\end{cases}$ (24)

with $U_{M}=e^{-i\hat{K}_{SM}}$ and $U_{N}=e^{-i\hat{K}_{SN}}$ being the
measurement unitaries that entagle the measured system with the respective
memories:

$\displaystyle
U_{X}\ket{\psi_{S}}\ket{R_{X}}=\sum_{x}\Pi_{x}\ket{\psi_{S}}\ket{x_{X}},$

and $(x,X)\in\\{(m,M),(n,N)\\}$.

According to definition 1 the conditional probability of result $n$ at time
$t_{2}\geq t_{N}$ given result $m$ at time $t_{1}\geq t_{M}$ is

$\displaystyle\frac{\left<\left<\Psi^{\prime}\right.\right|t_{1}\rangle\Pi^{m}\braket{t_{1}}{P^{\rm
ph}}{t_{2}}\Pi^{n}\braket{t_{2}}{P^{\rm
ph}}{t_{1}}\Pi^{m}\braket{t_{1}}{\Psi^{\prime}}\rangle}{\left<\left<\Psi^{\prime}\right.\right|t_{1}\rangle\Pi^{m}\braket{t_{1}}{\Psi^{\prime}}\rangle},$
(25)

with $\Pi^{m}$ and $\Pi^{n}$ acting on $\mathcal{H}_{M}$ and $\mathcal{H}_{N}$
respectively. From Eq. (22) we see that
$\ket{\phi(t_{1})}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\braket{t_{1}}{\Psi^{\prime}}\rangle=\sum_{m^{\prime}}U_{S}(t_{1},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}\ket{m^{\prime}_{M}}\ket{R_{N}}$,
and the denominator in Eq.(25) is

$\displaystyle\braket{\phi(t_{1})}{\Pi^{m}}{\phi(t_{1})}$
$\displaystyle=\sum_{m^{\prime},m^{\prime\prime}}\braket{m^{\prime\prime}_{M}}{\Pi^{m}}{m^{\prime}_{M}}\bra{\psi_{S}(t_{M})}\Pi_{m^{\prime\prime}}\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}$
$\displaystyle=|\braket{m}{\psi_{S}(t_{M})}|^{2}.$ (26)

Moreover, since

$\displaystyle\mathcal{U}(t_{2},t_{1})\Pi^{m}\ket{\phi(t_{1})}=\sum_{n^{\prime}}U_{S}(t_{2},t_{N})\Pi_{b^{\prime}}U_{S}(t_{N},t_{M})\Pi_{m}\ket{\psi_{S}(t_{M})}\ket{m_{M}}\ket{n^{\prime}_{N}},$
(27)

the numerator in Eq.(25) gives

$\displaystyle\braket{\phi(t_{1})}{\Pi^{m}\mathcal{U}^{\dagger}(t_{2},t_{1})\Pi^{n}\mathcal{U}(t_{2},t_{1})\Pi^{m}}{\phi(t_{1})}$
$\displaystyle\qquad=\sum_{n^{\prime},n^{\prime\prime}}\braket{n^{\prime\prime}_{N}}{\Pi^{n}}{n^{\prime}_{N}}\bra{\psi_{S}(t_{M})}\Pi_{m}U_{S}(t_{M},t_{N})\Pi_{n^{\prime\prime}}U_{S}(t_{N},t_{N})\Pi_{n^{\prime}}U_{S}(t_{N},t_{M})\Pi_{m}\ket{\psi_{S}(t_{M})}$
$\displaystyle\qquad=\braket{\psi_{S}(t_{M})}{\Pi_{m}U_{S}(t_{M},t_{N})\Pi_{n}U_{S}(t_{N},t_{M})\Pi_{m}}{\psi_{S}(t_{M})}$
$\displaystyle\qquad=|\braket{n}{U_{S}(t_{N},t_{M})}{m}|^{2}|\braket{m}{\psi_{S}(t_{M})}|^{2}$
(28)

and, hence,

$\displaystyle P_{1}\left(n\ {\rm when}\ t_{2}\,|\,m\ {\rm when}\
t_{1}\right)=|\braket{n}{U_{S}(t_{N},t_{M})}{m}|^{2},$ (29)

which is the standard quantum probabilities for two subsequent measurements on
a quantum system $S$.

In case of non-Wigner’s friend scenarios definition 2a and definition 2b are
equal since

$\displaystyle\left<\left<\Psi^{\prime}\right.\right|t_{2}\rangle\Pi^{m}\braket{t_{2}}{\Psi^{\prime}}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}m^{\prime},n\\\
m^{\prime\prime},n^{\prime}\end{subarray}}\Big{[}\braket{n^{\prime}_{N}}{n_{N}}\braket{m^{\prime\prime}_{M}}{\Pi^{m}}{m^{\prime}_{M}}\bra{\psi_{S}(t_{M})}\Pi_{m^{\prime\prime}}U_{S}(t_{M},t_{N})\Pi_{n^{\prime}}$
$\displaystyle\qquad\qquad\cdot
U_{S}(t_{N},t_{2})U_{S}(t_{2},t_{N})\Pi_{n}U_{S}(t_{N},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}\Big{]}$
$\displaystyle=\bra{\psi_{S}(t_{M})}\Pi_{m}U_{S}(t_{M},t_{N})\left(\sum_{n}\Pi_{n}\right)U_{S}(t_{N},t_{M})\Pi_{m}\ket{\psi_{S}(t_{M})}$
$\displaystyle=|\braket{m}{\psi_{S}(t_{M})}|^{2}$
$\displaystyle=\left<\left<\Psi^{\prime}\right.\right|t_{1}\rangle\Pi^{m}\braket{t_{1}}{\Psi^{\prime}}\rangle.$

From Eq. (22) we get that
$\ket{\phi(t_{2})}\mathrel{\mathchoice{\vbox{\hbox{$\displaystyle:$}}}{\vbox{\hbox{$\textstyle:$}}}{\vbox{\hbox{$\scriptstyle:$}}}{\vbox{\hbox{$\scriptscriptstyle:$}}}{=}}\braket{t_{2}}{\Psi^{\prime}}\rangle=\sum_{m^{\prime},n^{\prime}}U_{S}(t,t_{N})\Pi_{n^{\prime}}U_{S}(t_{N},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}\ket{m^{\prime}_{M}}\ket{n^{\prime}_{N}}$,
and the numerator in Defs. 2 gives

$\displaystyle\langle\langle{\Psi^{\prime}}\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{n}\operatorname*{\otimes}\Pi^{m}\left|\left.\Psi^{\prime}\right>\right>$
$\displaystyle=\braket{\phi(t_{2})}{\Pi^{n}\operatorname*{\otimes}\Pi^{m}}{\phi(t_{2})}$
$\displaystyle=\sum_{\begin{subarray}{c}m^{\prime},n^{\prime}\\\
m^{\prime\prime},n^{\prime\prime}\end{subarray}}\Big{[}\braket{n^{\prime\prime}_{N}}{\Pi^{n}}{n^{\prime}_{N}}\braket{m^{\prime\prime}_{M}}{\Pi^{m}}{m^{\prime}_{M}}\bra{\psi_{S}(t_{M})}$
$\displaystyle\qquad\qquad\cdot\Pi_{m^{\prime\prime}}U_{S}(t_{M},t_{N})\Pi_{n^{\prime\prime}}\Pi_{n^{\prime}}U_{S}(t_{N},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}\Big{]}$
$\displaystyle=|\braket{b}{U_{S}(t_{N},t_{M})}{m}|^{2}|\braket{m}{\psi_{S}(t_{M})}|^{2}$
(30)

Therefore, both Defs. 2 give the standard quantum probabilities

$\displaystyle P_{2a}\left(n\ {\rm when}\ t_{2}\,|\,m\ {\rm when}\
t_{1}\right)=|\braket{n}{U_{S}(t_{N},t_{M})}{m}|^{2}=P_{2b}\left(n\ {\rm
when}\ t_{2}\,|\,m\ {\rm when}\ t_{2}\right).$

According to definition 3 the conditional probability of result $n$ at time
$t_{2}\geq t_{N}$ given result $m$ at time $t_{1}\geq t_{M}$ is

$\displaystyle\frac{\left<\left<\Psi^{\prime}\right.\right|t_{1}\rangle\Pi^{m}\braket{t_{1}}{P^{\rm
ph}}{t_{2}}\Pi^{n}\braket{t_{2}}{\Psi^{\prime}}\rangle}{\left<\left<\Psi^{\prime}\right.\right|t_{1}\rangle\Pi^{m}\braket{t_{1}}{\Psi^{\prime}}\rangle}.$
(31)

The denominator is the same as in definition 1 and definition 2a and given by
Eq. (26). The numerator of Eq. (31) gives

$\displaystyle\braket{\phi(t_{1})}{\Pi^{m}\mathcal{U}(t_{1},t_{2})\Pi^{n}}{\phi(t_{2})}$
$\displaystyle=\bra{R_{N}}\bra{m_{M}}\bra{\psi_{S}(t_{M})}\Pi_{m}U_{S}(t_{M},t_{N})U^{\dagger}_{N}U_{S}(t_{N},t_{2})$
$\displaystyle\qquad\cdot\sum_{m^{\prime}}U_{S}(t_{2},t_{N})\Pi_{n}U_{S}(t_{N},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}\ket{m^{\prime}_{M}}\ket{n_{N}}$
$\displaystyle=\sum_{m^{\prime},n^{\prime}}\braket{n^{\prime}_{N}}{n_{N}}\braket{m_{M}}{m^{\prime}_{M}}\bra{\psi_{S}(t_{M})}\Pi_{m}U_{S}(t_{M},t_{N})\Pi_{n^{\prime}}$
$\displaystyle\qquad\qquad\cdot\Pi_{n}U_{S}(t_{N},t_{M})\Pi_{m^{\prime}}\ket{\psi_{S}(t_{M})}$
$\displaystyle=|\braket{n}{U_{S}(t_{N},t_{M})}{m}|^{2}|\braket{m}{\psi_{S}(t_{M})}|^{2}.$
(32)

Hence, also definition 3 gives the standard quantum probabilities for non-
Wigner’s friend scenarios.

##  Appendix B Conditions for Definition 2a

$\bm{\frac{\langle\langle{\Psi}\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{w}\otimes\Pi^{f}\left|\left.\Psi\right>\right>}{\langle\langle{\Psi}\ket{t_{1}}\bra{t_{1}}\operatorname*{\otimes}\Pi^{f}\left|\left.\Psi\right>\right>}}$
---
| $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\frac{a^{2}\alpha^{4}+2ab\alpha^{3}\beta\cos(\phi)+b^{2}\alpha^{2}\beta^{2}}{a^{2}}$ | $\frac{a^{2}\beta^{4}-2ab\alpha\beta^{3}\cos(\phi)+b^{2}\alpha^{2}\beta^{2}}{a^{2}}$
$\downarrow$ | $\frac{b^{2}\beta^{4}+2ab\alpha\beta^{3}\cos(\phi)+a^{2}\alpha^{2}\beta^{2}}{b^{2}}$ | $\frac{b^{2}\alpha^{4}-2ab\alpha^{3}\beta\cos(\phi)+a^{2}\alpha^{2}\beta^{2}}{b^{2}}$
Table 6: Definition 2a evaluated for arbitrary $a,b,\alpha,\beta$ and
$\phi=\phi_{S}-\phi_{SF}$. These expressions constitute possible conditional
probabilities of Wigner seeing result $w\in\\{\rm{yes,no}\\}$ at time $t_{2}$
given that the Friend saw result $f\in\\{\uparrow,\downarrow\\}$ at time
$t_{1}$ only if conditions (33) and (34) are both satisfied.

Evaluating definition 2a for the setup in Fig. 1 gives the terms listed in
Table 6. Normalization requires that $\forall
f:\sum_{w}\frac{\langle\langle{\Psi}\ket{t_{2}}\bra{t_{2}}\operatorname*{\otimes}\Pi^{w}\otimes\Pi^{f}\left|\left.\Psi\right>\right>\rangle}{\langle\langle{\Psi}\ket{t_{1}}\bra{t_{1}}\operatorname*{\otimes}\Pi^{f}\left|\left.\Psi\right>\right>}=1$,
which gives the following conditions

$\displaystyle\alpha^{4}+\beta^{4}+2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{b}{a}+2\alpha^{2}\beta^{2}\left(\frac{b}{a}\right)^{2}=1$
(33)
$\displaystyle\alpha^{4}+\beta^{4}-2\cos(\phi)(\alpha^{3}\beta-\alpha\beta^{3})\frac{a}{b}+2\alpha^{2}\beta^{2}\left(\frac{a}{b}\right)^{2}=1.$
(34)

These quadratic equations in $\frac{b}{a}$ and $\frac{a}{b}$ have solutions

$\displaystyle{\left(\frac{b}{a}\right)}_{\pm}=\frac{-\cos\phi(\alpha^{2}-\beta^{2})\pm\sqrt{1-\sin^{2}\phi(\alpha^{2}-\beta^{2})^{2}}}{2\alpha\beta}$

and

$\displaystyle{\left(\frac{a}{b}\right)}_{\pm}=\frac{\cos\phi(\alpha^{2}-\beta^{2})\pm\sqrt{1-\sin^{2}\phi(\alpha^{2}-\beta^{2})^{2}}}{2\alpha\beta},$

which exist for any $\alpha,\beta$ and $\phi$. However, requiring that
${\left(\frac{b}{a}\right)}_{\pm}=1/{\left(\frac{a}{b}\right)}_{\pm}$ has
solutions only when combining either the two “$+$” or the two “$-$” solutions.
These combinations are the common solutions to both Eqs. (33) and (34) and,
hence, give normalized probabilities for definition 2a.

##  Appendix C Consistent histories for Wigner’s-friend setups

In the consistent histories framework sequences of physical properties are
assigned to a closed quantum system. These sequences are represented by tensor
products of orthogonal projectors (i.e. a _quantum history_)

$Y^{\rm i}=\rho_{0}\otimes P_{1}^{\rm{i}_{1}}\dots\otimes P_{f}^{\rm{i}_{f}},$
(35)

where $\rho_{0}$ is the initial state and each $P_{k}^{\rm{i}_{k}}$
corresponds to some physical property at a certain time $k$. A consistent
family of histories is a complete set of histories {$Y^{i}$} which satisfy the
consistency condition

$\operatorname{tr}\Big{(}K^{\dagger}(Y^{\rm i})\rho_{0}K(Y^{\rm
i^{\prime}})\Big{)}=0\quad\text{for }\rm{i}\neq\rm{i}^{\prime}$ (36)

where $\rm{i}=(\rm{i}_{1}\dots\rm{i}_{f})$ and $K$ is the so called chain
operator defined by

$K(Y^{\rm i})=P_{0}^{\rm{i}_{1}}\cdot P_{0}^{\rm{i}_{2}}\dots\cdot
P_{0}^{\rm{i}_{f}},$ (37)

with $P_{0}^{\rm{i}_{k}}=U(t_{0},t_{k})P_{k}^{\rm{i}_{k}}U(t_{k},t_{0})$. Only
within a consistent family the dynamics of quantum theory describe the
respective properties over time.

For the simple Wigner’s-friend setup in Fig. 1, these properties are the
results observed by Wigner and his Friend, $\rm{i}=(f,w)$ and
$\rho_{0}=\ket{R_{W}}\ket{R_{F}}|{\psi_{S}}\vphantom{\psi_{S}}\rangle\langle{\psi_{S}}\vphantom{\psi_{S}}|\bra{R_{F}}\bra{R_{W}}$.
Condition (19) –under which definition 3 gives proper probabilities– implies
that

$\displaystyle\operatorname{tr}\Big{(}K^{\dagger}(Y^{(f,w)})\rho_{0}K(Y^{(f^{\prime},w^{\prime})})\Big{)}=\delta_{ff^{\prime}}\delta_{ww^{\prime}}\operatorname{tr}\big{(}\rho(t_{2})\Pi^{w}U(t_{2},t_{1})\Pi^{f}U(t_{1},t_{2})\big{)},$
(38)

where $\rho(t_{2})=\mathcal{U}(t_{2},t_{0})\rho_{0}\mathcal{U}(t_{0},t_{2})$.
Hence, in this case, the consistency condition is satisfied.

The solutions to the conditions on definition 2a, however, in general do _not_
satisfy Eq. (36). Consider the concrete counterexample of
$\ket{\psi_{S}}=\sqrt{\frac{1}{2}}(\ket{\uparrow}+\ket{\downarrow})$ and
$\ket{\rm
yes}=\alpha\ket{\uparrow,\uparrow}+i\beta\ket{\downarrow,\downarrow}$. In this
case one obtains

$\operatorname{tr}\Big{(}K^{\dagger}(Y^{i})\rho_{0}K(Y^{i^{\prime}})\Big{)}=\pm\delta_{ww^{\prime}}\frac{i}{2}\alpha\beta\quad\text{for
}f\neq f^{\prime},$ (39)

which still satisfies the so called weak consistency condition ${\rm
Re}\left[\operatorname{tr}\Big{(}K^{\dagger}(Y^{i})\rho_{0}K(Y^{i^{\prime}})\Big{)}\right]=0$.
In contrast to the consistency condition of (36), however, this has been shown
to be highly problematic concerning trivial combination of independent
subsystems as well as dynamical stability, see Ref. [26].

##  Appendix D Conditions for Definition 3

$\bm{\frac{\langle\langle{\Psi}(\ket{t_{2}}\bra{t_{2}}\otimes\Pi^{w})P^{\rm
ph}(\ket{t_{1}}\bra{t_{1}}\otimes\Pi^{f})\left|\left.\Psi\right>\right>}{\langle\langle\Psi\ket{t_{1}}\bra{t_{1}}\otimes\Pi^{f}\left|\left.\Psi\right>\right>\rangle}}$
---
| $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\alpha^{2}+\frac{b}{a}\alpha\beta e^{-i(\phi)}$ | $\beta^{2}-\frac{b}{a}\alpha\beta e^{-i(\phi)}$
$\downarrow$ | $\beta^{2}+\frac{a}{b}\alpha\beta e^{-i(\phi)}$ | $\alpha^{2}-\frac{a}{b}\alpha\beta e^{-i(\phi)}$
$\bm{\frac{\langle\langle{\Psi}(\ket{t_{2}}\bra{t_{2}}\otimes\Pi^{w})P^{\rm
ph}(\ket{t_{1}}\bra{t_{1}}\otimes\Pi^{f})\left|\left.\Psi\right>\right>}{\langle\langle{\Psi}\ket{t_{2}}\bra{t_{2}}\otimes\Pi^{w}\left|\left.\Psi\right>\right>}}$
| $\rm{yes}$ | $\rm{no}$
$\uparrow$ | $\frac{a\alpha}{a\alpha+{b\beta}e^{-i(\phi)}}$ | $\frac{b\beta}{b\beta+a\alpha e^{i(\phi)}}$
$\downarrow$ | $\frac{a^{2}\beta^{2}-ab\alpha\beta e^{i(\phi)}}{a^{2}\beta^{2}+b^{2}\alpha^{2}-2ab\alpha\beta\cos\left(\phi\right)}$ | $\frac{b^{2}\alpha^{2}-ab\alpha\beta e^{-i(\phi)}}{a^{2}\beta^{2}+b^{2}\alpha^{2}-2ab\alpha\beta\cos\left(\phi\right)}$
Table 7: Definition 3 evaluated for arbitrary $a,b,\alpha,\beta$ and $\phi$
possibly constituting the conditional probabilities of Wigner seeing result
$w\in\\{\rm{yes,no}\\}$ at time $t_{2}$ given that the Friend saw result
$f\in\\{\uparrow,\downarrow\\}$ at time $t_{1}$ and that of the Friend seeing
$f$ at $t_{1}$ given that Wigner will see $w$ at $t_{2}$. These expressions
are real and positive and, hence, probabilities only if conditions (D) and
either (45) or (46) are satisfied.

Evaluating definition 3 for the setup in Fig. 1 gives the terms listed in
Table 7. Those expressions are in general not probabilities, since they are
neither necessarily real nor positive although they do add up to one. For them
to be real numbers we require that

$\displaystyle\phi=n\pi.$ (40)

For all the terms in Table 7 to be positive, of the following conditions one
each must hold. If either

$\displaystyle\alpha^{2}-\frac{a}{b}\alpha\beta\geq 0$ $\displaystyle\ \text{
and }\ \beta^{2}-\frac{b}{a}\alpha\beta\geq 0$ (41) or
$\displaystyle\alpha^{2}-\frac{b}{a}\alpha\beta\geq 0$ $\displaystyle\ \text{
and }\ \beta^{2}-\frac{a}{b}\alpha\beta\geq 0,$ (42)

the conditional probabilities for result $w$ at $t_{2}$ given result $f$ at
$t_{1}$ are well-defined. And if, either

$\displaystyle 1-\frac{b\alpha}{a\beta}\geq 0$ $\displaystyle\ \text{ and }\
1-\frac{a\beta}{b\alpha}\geq 0$ (43) or $\displaystyle
1-\frac{b\beta}{a\alpha}\geq 0$ $\displaystyle\ \text{ and }\
1-\frac{a\alpha}{b\beta}\geq 0,$ (44)

the conditional probabilities for result $f$ at $t_{1}$ given result $w$ at
$t_{2}$ are well-defined. The solutions to both cases are

$\displaystyle a=\alpha\quad$ $\displaystyle,\quad b=\beta$ (45) or
$\displaystyle a=\beta\quad$ $\displaystyle,\quad b=-\alpha,$ (46)

which means Wigner’s measurement is aligned with the initial state. The joint
system of $F$ and $S$ is in an eigenstate of $W$’s observable, i.e., the
measurement is non disturbing.