# Law of Total Probability in Quantum Theory and Its Application in Wigner’s
Friend Scenario

Jianhao M. Yang<EMAIL_ADDRESS>Qualcomm, San Diego, CA
92121, USA

###### Abstract

It is well-known that the law of total probability does not hold in general in
quantum theory. However, the recent arguments on some of the fundamental
assumptions in quantum theory based on the extended Wigner’s Friend scenario
show a need to clarify how the law of total probability should be formulated
in quantum theory and under what conditions it still holds. In this work, the
definition of conditional probability in quantum theory is extended to POVM
measurements. Rule to assign two-time conditional probability is proposed for
incompatible POVM operators, which leads to a more general and precise
formulation of the law of total probability. Sufficient conditions under which
the law of total probability holds are identified. Applying the theory
developed here to analyze several quantum no-go theorems related to the
extended Wigner’s friend scenario reveals logical loopholes in these no-go
theorems. The loopholes exist as a consequence of taking for granted the
validity of the law of total probability without verifying the sufficient
conditions. Consequently, the contradictions in these no-go theorems only
reconfirm the invalidity of the law of total probability in quantum theory,
rather than invalidating the physical statements that the no-go theorems
attempt to refute.

Keywords

The Law of Total Probability, Extended Wigner’s Friend Scenario, PVOM
Measurement

###### pacs:

03.65.Ta, 03.65.-w

††preprint: APS/123-QED

##  I Introduction

In his seminal paper on the path integral formulation of quantum mechanics
Feynman , Feynman started the introduction of his new theory by pointing out
that the law of total probability in classical probability theory must be
replaced by a new form of rule. Specifically, in a slightly different
notation, the classical law of probability, $p(c|a)=\sum_{b}p(c|b)p(b|a)$
where $p(y|x)$ is the probability of obtaining measurement result $y$ given
measurement result $x$, is no longer true in quantum theory, and must be
replaced by $\varphi(c|a)=\sum_{b}\varphi(c|b)\varphi(b|a)$ where $\varphi$ is
a complex number called probability amplitude and related to the classical
probability by the Born’s rule $p(y|x)=|\varphi(y|x)|^{2}$. From this key
idea, Feynman continued to expand the theory that leads to the path integral
formulation of quantum mechanics. He also discussed when the new rule of
summation over probability amplitude can fall back to the classical law of
probability. This is when one “attempts to perform” intermediate measurements
that obtain results of all $b$. In modern terms, what Feynman means
“attempting to perform measurement” here can be understood as the decoherence
phenomenon Zurek03 .

The above example shows that it has been long known that the law of total
probability cannot be taken for granted in quantum theory. Indeed, many other
classical probability rules are only upheld in specific conditions. For
instance, a joint probability can be definitely assigned only when the two
measurement operators are commutative Nielsen ; Hayashi ; Fine ; Malley ;
There are many variants of definition of the conditional probability in
quantum theory (see a review in Bobo ). However, a family of no-go theorems
recently publishedBrukner ; Bong ; Guerin ; Rovelli21  appear to rely on the
total law of probability one way or another without considering the sufficient
conditions. These no-go theorems are related to the extensively discussed
Wigner’s Friend experiments. In quantum mechanics, the Wigner’s friend Wigner
; Wigner2  thought experiment has been widely discussed as it tests the
validity of many quantum interpretation theories. The significance of such
experiment is that Wigner and his friend give two different descriptions of
the same physical process happened inside the lab. Deutsch further extended
the thought experiment to be applicable to macroscopic system such as the lab
system Deutsch  itself. Based on that, a more sophisticate extended Wigner’s
friend experiment is put forwarded by Brukner Brukner0 ; Brukner . Such
experiment setup involves two remotely separated labs. Each lab contains half
of an entangled pair of spins and a local observer. Outside each lab there is
a super-observer who can choose to perform different types of measurements on
the lab as a whole. The intention of such experiment setup is to prove,
through a no-go theorem, that measured facts are observer-dependent in quantum
theory. Subsequent experiment Proietti19  has been carried out to confirm the
inequality developed in Brukner . A stronger version of no-go theorem is
further proposed for reaching the similar conclusion Bong . The statement that
measured facts are observer-dependent is considered of importance for the
quantum foundation and deserved rigorous theoretical proving and experimental
testing. However, proving the no-go theorems by taking the law of total
probability for granted casts doubt on the theoretical rigorousness.

The fact that there is still ambiguity to use the total law of probability in
quantum theory though it has been long recognized it is not upheld in quantum
mechanics shows the needs to provide a rigorous formulation of the law of
total probability in quantum theory, and to clarify under what conditions it
holds true. And this is indeed the motivation of the present work. The
formulation of the law of total probability depends on a clear definition of
conditional probability in quantum theory. There is already extensive research
literature on how the conditional probability is defined Bobo ; Bobo2 ; Luders
; Cassinelli ; Bub ; PW ; Dolby ; Lloyd ; Hohn ; Baumann . However, these
formulations are either based on projection measurements, or only considering
simultaneous measurements with commutative operators. In this work we will
extend a two-time conditional probability formulation from projection
measurement to more generic POVM measurements. Generalization for POVM
measurement is needed because some of the no-go theorems choose POVM operators
in their proofs. We then give several sufficient conditions for the law of
total probability to become true. The theory is applied to analyze several no-
go theorems related to the extended Wigner’s friend scenario. Logical
loopholes are shown in these no-go theorems because their proofs rely on the
law of total probability one way or another, but the conditions to validate
the law are not met. Thus, these no-go theorems do not really prove the
results they expect, such as “measured facts are observer-dependent”. Instead,
they just indirectly confirm that the law of total probability doesn’t not
hold in quantum theory.

It is worth to mention that other concerns on these no-go theorems have
already been pointed out Zukowski ; Relano . In particular, only when a
measurement is completed should a probability distribution be assigned.
Assigning probability distribution for pre-measurement without result leads to
contradiction Zukowski . The analysis in this work will go one step further by
showing that even assigning probability distribution for completed
measurement, there is still logical loopholes in the no-go theorem. This is
because the law of total probability that the proofs rely on does not hold
true with the specific measurement operators and initial quantum state being
chosen. Lastly, it is important to emphasize that we do not take a stand on
the assertions of the no-go theorems themselves. For instance, it could be
still a valid statement that “measured facts are observer-dependent”. What we
show here is that there are logical loopholes in the proof of the no-go
theorems.

In summary, this paper extends the formulation of conditional probability to
generic POVM measurements, and clarifies the conditions under which the law of
total probability can be valid in quantum theory. Applying the theory
developed in this work to the extended Wigner Friend scenario reveals logical
loopholes in several no-go theorems that take for granted on the validity of
the law of total probability. The contradictions in these no-go theorems only
reconfirm the invalidity of the law of total probability in quantum theorem,
rather than invalidating the physical statements that the no-go theorems are
intended to refute, such as “measured facts are independent of the observer”.
We hope the results presented here can inspire further researches to find more
convincing proof and experimental testing. This is important because the
implications of the extended Wigner’s friend scenario are conceptually
fundamental in quantum theory.

##  II The Law of Total Probability in Quantum Theory

First, we briefly review the classical probability theory. Suppose there are
two random variables X and Y. Without loss of generality, we assume X and Y
are discrete random variables. Measuring $X$ (or $Y$) will obtaining one of
values in $\\{a_{i}:i=1,2,3...\\}$ (or in $\\{b_{j}:j=1,2,3...\\}$) which is
finite or countable infinite. Denote the joint probability of measuring X with
result $X=a_{i}$ and measuring Y with result $Y=b_{j}$ as $p(a_{i},b_{j})$,
and the conditional probability of obtaining $X=a_{i}$ given that $Y=b_{j}$ as
$p(a_{i}|b_{j})$. They are related by the following axioms:

$\displaystyle p(a_{i},b_{j})$ $\displaystyle=p(b_{j}|a_{i})p(a_{i})$ (1)
$\displaystyle p(b_{j},a_{i})$ $\displaystyle=p(a_{i}|b_{j})p(b_{j})$ (2)
$\displaystyle p(a_{i},b_{j})$ $\displaystyle=p(b_{j},a_{i}),$ (3)

where $p(a_{i})$ is the marginal probability of measuring X with result
$X=a_{i}$, and similarly for $p(b_{j})$. Axiom (3) ensures the joint
probability is defined uniquely regardless it is defined by (1) or (2). We
explicitly call out (3) since it is not always true in the quantum theory.

The law of total probability can be derived111Axioms (1) to (3) give
$p(b_{j}|a_{i})p(a_{i})=p(a_{i}|b_{j})p(b_{j})$, which is the Bayes’ law.
Summing over $i$ in both sides and using identity
$\sum_{i}p(a_{i}|b_{j})p(b_{j})=p(b_{j})$, one obtains (4). from axioms (1) -
(3) and expressed as following,

$p(b_{j})=\sum_{i}p(b_{j}|a_{i})p(a_{i}).$ (4)

What we want to investigate here is how the equivalent version of (4) in
quantum theory can be formulated.

To start with, we need to examine how the conditional probability is
constructed in quantum theory. The subtlety of constructing the conditional
probability in quantum theory has been investigated long ago. G. Bobo gives an
extensive review and discussion Bobo . The generally accepted formulation of
conditional probability in quantum theory is provided by Lüders rule Luders
where the measurements are associated with projection operators. Lüders rule
is based on Gleason’s theorem which mathematically justifies Born’s rule. Here
we wish to follow the similar approach to generalize the formulation for
conditional probability when the measurements are associated with POVM
operators.

Mathematical proofs for generalizing Gleason’s theorem to POVM measurements
are given by Busch ; Caves , which is our starting point. Suppose a quantum
system $S$ is prepared such that its state is described by density operator
$\rho$. $S$ could be a composite system, which we will discuss later. Let
$A=\\{A_{i}\\}$ be a POVM for $S$. The probability of measurement with element
$A_{i}$ resulting in value $a_{i}$ is Busch ; Caves

$p(a_{i}|\rho)=Tr(\rho A_{i}),$ (5)

and the post-measurement density operator $\rho_{i}$ are given by Hayashi

$\rho_{i}=\frac{\sqrt{A_{i}}\rho\sqrt{A_{i}}}{p(a_{i}|\rho)}.$ (6)

Let $B=\\{B_{j}\\}$ be another POVM for $S$. Given the post measurement state
$\rho_{i}$, the probability of measurement with element $B_{j}$ resulting in
value $b_{j}$ is, by applying recursively (5),
$p(b_{j}|\rho_{i})=Tr(\rho_{i}B_{j})$. Substituting the expression for
$\rho_{i}$ in (6), we obtain the conditional probability

$p(b_{j}|a_{i},\rho)=\frac{Tr(\sqrt{A_{i}}\rho\sqrt{A_{i}}B_{j})}{Tr(\rho
A_{i})}.$ (7)

There is an underlying assumption in this definition that the probability is
assigned only after the measurements are completed. In particular, the first
POVM measurement $A_{i}$ must be completed in order to be qualified as a
condition. We strictly follow this assumption as opposite to assigning a
probability with only “pre-measurement”. Pre-measurement refers to only the
unitary process that entangles the measured system and measuring apparatus
Neumann , but without the projection process to single out a particular
outcome.

Given the same initial state $\rho$, if we swap the order of measurements such
that $B_{j}$ goes first, following $A_{i}$, we obtain a conditional
probability

$p(a_{i}|b_{j},\rho)=\frac{Tr(\sqrt{B_{j}}\rho\sqrt{B_{j}}A_{i})}{Tr(\rho
B_{j})}.$ (8)

Note $p(a_{i}|\rho)=Tr(\rho A_{i})$ and $p(b_{i}|\rho)=Tr(\rho B_{j})$,
Eqs.(7) and (8) can be rewritten as

$\displaystyle p(b_{j}|a_{i},\rho)p(a_{i}|\rho)$
$\displaystyle=Tr(\sqrt{A_{i}}\rho\sqrt{A_{i}}B_{j})$ (9) $\displaystyle
p(a_{i}|b_{j},\rho)p(b_{j}|\rho)$
$\displaystyle=Tr(\sqrt{B_{j}}\rho\sqrt{B_{j}}A_{i}).$ (10)

Eqs.(9) and (10) are not necessarily equal, which shows that the quantum
version of Bayes’ theorem,

$p(a_{i}|b_{j},\rho)p(b_{j}|\rho)=p(b_{j}|a_{i},\rho)p(a_{i}|\rho)$ (11)

does not hold in general in quantum theory. This posts a difficulty to define
a joint probability as either
$p(a_{i},b_{j})=p(a_{i}|b_{j},\rho)p(b_{j}|\rho)$ or
$p(a_{i},b_{j})=p(b_{j}|a_{i},\rho)p(a_{i}|\rho)$, because it depends on the
order of measurement events. Another consequence is that the laws of total
probability, i.e., the quantum version of (4)

$\sum_{i}p(b_{j}|a_{i},\rho)p(a_{i}|\rho)=p(b_{j}|\rho)$ (12)

does not hold either in general. This is because from (9),
$\sum_{i}p(b_{j}|a_{i},\rho)p(a_{i}|\rho)=\sum_{i}Tr(\sqrt{A_{i}}\rho\sqrt{A_{i}}B_{j})$,
while $p(b_{j}|\rho)=Tr(\rho B_{j})$, and there are not equal in
general222Note that on the other hand, given (7) and the completeness of POVM
elements, $\sum_{i}A_{i}=I$ where $I$ is the identity operator, it is
straightforward to verify that
$\sum_{j}p(b_{j}|a_{i},\rho)p(a_{i}|\rho)=p(a_{i}|\rho)$.. We are interested
in finding the conditions under which (12) becomes true.

It is well-known that when $[A_{i},B_{j}]=0$, i.e., $A_{i}$ and $B_{j}$
commute, from (7) and (8), one gets
$p(b_{j}|a_{i},\rho)p(a_{i}|\rho)=p(a_{i}|b_{j},\rho)p(b_{j}|\rho)=Tr(\rho
A_{i}B_{j})$. Consequently, the law of total probability (12) becomes true and
a joint probability can be well-defined. However, the situation becomes much
complicated when $[A_{i},B_{j}]\neq 0$.

Strictly speaking, due to the uncertainty principle, when $A_{i}$ and $B_{j}$
are non-commutative, the two measurements cannot be performed to obtain
definite outcomes at the same time. The conditional probability defined in (7)
or (8) needs to be extended to a two-time formulation of conditional
probability in order to be applicable when $[A_{i},B_{j}]\neq 0$. There are
extensive research literature on how to construct two-time conditional
probability in quantum theory Bobo ; Bobo2 ; Luders ; Cassinelli ; Bub ; PW ;
Dolby ; Lloyd ; Hohn ; Baumann . One noticeable approach is based on the Page-
Wootters’ timeless formulation PW ; Dolby ; Lloyd ; Hohn ; Baumann . In this
work, we will continue to be based on the generalized Gleason theorem for POVM
Busch ; Caves  to derive the two-time conditional probability, and leave the
discussion of the Page-Wootters mechanism in Section IV.

For conceptual clarity, we start the analysis by considering that there is
finite non-zero duration for each measurement. After we construct the
conditional probability formulation, for practical purpose of calculation, we
can approximate the measurement duration to zero. Suppose the first
measurement starts at $t_{a}^{-}$ and completes at $t_{a}^{+}$. Here
$t_{a}^{+}-t_{a}^{-}$ covers the time duration for both the pre-measurement
unitary phase that entangles the measured system and the measuring apparatus,
and the projection phase. The measurement process333Theorem 5.2 of Hayashi
gives a detailed account on how this POVM measurement is physically realized
through indirect measurement. is represented by a POVM element $A_{i}$
associated with outcome $a_{i}$. Similarly, the second measurement starts at
$t_{b}^{-}$ and completes at $t_{b}^{+}$. Between $t_{a}^{+}$ and $t_{b}^{-}$
is a free time evolution for the measured system $S$, described by operator
$U(\Delta t)=e^{-iH\Delta t/\hbar}$ where $\Delta t=(t_{b}^{-}-t_{a}^{+})$.
Since it is only meaningful to assign a probability distribution after a
measurement is completed, the two-time conditional probability we want to
construct is “given the measurement outcome of $a_{i}$ at $t_{1}$ where
$t_{a}^{+}<t_{1}<t_{b}^{-}$, what is the probability of measurement outcome
$b$ at $t_{2}>t_{b}^{+}$. Mathematically, this two-time conditional
probability can be written as $p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho_{0})$, where $\rho_{0}\equiv\rho(t_{a}^{-})$ is the initial
density operator of $S$ when the first measurement starts. After the first
measurement with POVM element $A_{i}$, the post-measurement state is
$\rho_{i}(t_{a}^{+})=\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}}/Tr(\rho_{0}A_{i})$. The
quantum system $S$ then time evolves from $t_{a}^{+}$ to $t_{b}^{-}$ to a new
state $\rho_{i}(t_{b}^{-})=U(\Delta t)\rho_{i}(t_{a}^{+})U^{\dagger}(\Delta
t)$. At $t_{b}^{-}$, the second measurement occurs. This is represented by
applying POVM element $B_{j}$ on $\rho_{i}(t_{b}^{-})$ and obtain outcome
$b_{j}$ at $t_{b}^{+}$ with probability $Tr(B_{j}\rho_{i}(t_{b}^{-}))$.
Substituting $\rho_{i}(t_{b}^{-})$, the two-time conditional probability is

$\begin{split}&p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho_{0})=Tr(B_{j}\rho_{i}(t_{b}^{-}))\\\ &=\frac{Tr(B_{j}U(\Delta
t)\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}}U^{\dagger}(\Delta
t))}{Tr(\rho_{0}A_{i})}.\end{split}$ (13)

For practical purpose of calculation, we can assume the measurement duration
is very small compared to the free evolution time, i.e.,
$(t_{a}^{+}-t_{a}^{-})\ll\Delta t$ and $(t_{b}^{+}-t_{b}^{-})\ll\Delta t$.
Then we can denote $t_{a}^{-}\approx t_{a}^{+}$ as $t_{a}$, $t_{b}^{-}\approx
t_{b}^{+}$ as $t_{b}$, and $\Delta t=(t_{b}-t_{a})$.

Suppose the two POVM elements $A_{i}$ and $B_{j}$ are projection measurements,
$A_{i}=|\phi_{i}\rangle\langle\phi_{i}|$ and
$B_{j}=|\varphi_{j}\rangle\langle\varphi_{j}|$, one can verify that the
conditional probability defined in (13) gives the correct transition
probability in standard quantum mechanics:

$p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho_{0})=|\langle\phi_{i}|U(\Delta t)|\varphi_{j}|^{2}.$ (14)

However, Eq.(13) is more generic as it is defined with general POVM operators.
Note that the denominator in (13) $Tr(\rho_{0}A_{i})=p(a_{i}\text{ at
}t_{1}|\rho_{0})$, (13) can be rewritten as

$\begin{split}&p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho_{0})p(a_{i}\text{ at }t_{1}|\rho_{0})\\\ &=Tr(B_{j}U(\Delta
t)\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}}U^{\dagger}(\Delta t)).\end{split}$ (15)

To analyze the two-time version of total law of probability, which can be
expressed as

$\begin{split}p(b_{j}\text{ at }t_{2}|\rho_{0})&=\sum_{i}p(b_{j}\text{ at
}t_{2}|a_{i}\text{ at }t_{1},\rho_{0})p(a_{i}\text{ at
}t_{1}|\rho_{0}),\end{split}$ (16)

we consider a series of two-time measurements $\\{A_{i}\text{ at
}t_{a},B\text{ at }t_{b},i=1\ldots N\\}$ on $N$ copies of measured system $S$
with the same initial state $\rho_{0}$. Each two-time measurement consists a
first measurement from one possible POVM element from the complete set
$\\{A_{i},i=1\ldots N\\}$ at time $t_{a}$, and the same second measurement
$B_{j}$ at time $t_{b}$. For $t_{a}<t_{1}<t_{b}<t_{2}$, from (15) we have

$\begin{split}&\sum_{i}p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho_{0})p(a_{i}\text{ at }t_{1}|\rho_{0})\\\
&=\sum_{i}Tr(B_{j}U(\Delta
t)\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}}U^{\dagger}(\Delta t)).\end{split}$ (17)

But by definition, $p(b_{j}\text{ at
}t_{2}|\rho_{0})=Tr(B_{j}U\rho_{0}U^{\dagger})$. We can see (16) is not true
in general. The Theorem next attempts to address the question on under what
conditions (16) is valid.

###### Theorem 1

Let $\rho_{0}$ be the density operator for a quantum system $S$ before the
measurements. Let $A_{i}$ and $B_{j}$ be two POVM elements to measure $S$ at
time $t_{a}$ and $t_{b}$ respectively, and $U(t_{b},t_{a})$ is the unitary
time evolution operator from $t_{a}$ to $t_{b}$. Select $t_{1}$ and $t_{2}$
such that $t_{a}<t_{1}<t_{b}<t_{2}$. The law of total probability (16) is true
if one of the following conditions is met.

  1. C1.

$[A_{i},U^{{\dagger}}B_{j}U]=0$, $\forall\rho_{0}$,

  2. C2.

$\rho_{0}=\sum_{i}\lambda_{i}|\phi_{i}\rangle\langle\phi_{i}|$ and
$A_{i}=|\phi_{i}\rangle\langle\phi_{i}|$,

  3. C3.

$\rho_{0}$ is a pure state, given by $|\Psi\rangle\langle\Psi|$, $A_{i}$ is a
projection operator and
$\langle\Psi|[A_{i},U^{{\dagger}}B_{j}U]A_{i}|\Psi\rangle=0$.

The proof of Theorem 1 is left in Appendix A, but a few comments are in order
here. First, Condition $C1$ implies
$U^{{\dagger}}B_{j}UA_{i}=A_{i}U^{{\dagger}}B_{j}U$. The sequence of
operations for $U^{{\dagger}}B_{j}UA_{i}$ means performing measurement $A_{i}$
at $t_{a}$, time evolving the post-measurement state from $t_{a}$ to $t_{b}$,
performing measurement $B_{j}$ at $t_{b}$, and reversing time evolution of the
post-measurement state back to $t_{a}$. The sequence of operations
$A_{i}U^{{\dagger}}B_{j}U$ means time evolving the state from $t_{a}$ to
$t_{b}$, performing measurement $B_{j}$ at $t_{b}$, then reversing time
evolution of the state back to $t_{a}$, and performing measurement $A_{i}$ at
$t_{a}$. Condition $C1$ says that if these two sequences of operations are
equivalent, then the law of total probability (16) holds true.

Second, if the post-measurement state $\rho_{i}(t_{a})$ after the first
measurement does not change during free time evolution, such as the case of a
spin state in free space, we will have
$\rho_{i}(t_{b})=\rho_{i}(t_{a})=\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}}/Tr(\rho_{0}A_{i})$.
Then, (13) can be written as

$p(b_{j}\text{ at }t_{2}|a_{i}\text{ at
}t_{1},\rho)=\frac{Tr(\rho_{0}\sqrt{A_{i}}B_{j}\sqrt{A_{i}})}{Tr(\rho_{0}A_{i})}.$
(18)

Eq.(18) appears the same as (7), but the precisely meaning is different in
that the two measurements $A_{i}$ and $B_{j}$ in (18) are taken place at two
different times. With such special post-measurement quantum state, the
sufficient conditions in Theorem 1 become

  1. $\textit{C1}^{\prime}$.

$[A_{i},B_{j}]=0$, $\forall\rho_{0}$,

  2. $\textit{C2}^{\prime}$.

$\rho_{0}=\sum_{i}\lambda_{i}|\phi_{i}\rangle\langle\phi_{i}|$ and
$A_{i}=|\phi_{i}\rangle\langle\phi_{i}|$,

  3. $\textit{C3}^{\prime}$.

$\rho_{0}=|\Psi\rangle\langle\Psi|$, $A_{i}$ is a projection operator and
$\langle\Psi|[A_{i},B_{j}]A_{i}|\Psi\rangle=0$.

We close this section with two comments. First, when two measurement
operations are not commutative, the conditional probability needs to be
defined in the two-time formulation. Second, We can give an intuitive
explanation of why (16) does not hold in general in quantum theory. As shown
in (17), the right hand side of (16) refer to the summation of traces of
multiplication of operators from a series of experiments where two
measurements are carried out in a sequence. In the case of special post-
measurement state where (18) holds, this is
$\sum_{i}Tr(B_{j}\sqrt{A_{i}}\rho_{0}\sqrt{A_{i}})$. The measurement of
$A_{i}$ changes the initial quantum state such that it affects the probability
of outcome for a subsequent measurement $B_{j}$. However, the term on the left
hand side of (16) refers to the probability of an experiment where only
measurement $B_{j}$ is carried out with the same initial quantum state. There
is no reason to assume both sides are equal. Eq.(16) holds only in special
conditions such as those specified in Theorem 1.

The conclusion here is that one should not take for granted that the law of
total probability holds true in general. Instead, the sufficient conditions,
such as those provided in Theorem 1, need to be clearly called out. Failing to
do so may leave loophole in logical deduction.

Figure 1: Sketch of the extended Wigner’s Friend scenario described in Brukner
. Laboratory $L_{1}$ consists spin $s_{1}$ and Charlie, while Laboratory
$L_{2}$ consists spin $s_{2}$ and Debbie. The two laboratories are remotely
separated. The dot line between $s_{1}$ and $s_{2}$ symbolizes they are
entangled. Alice can measure $L_{1}$ as a whole, and Bob can measure $L_{2}$.

##  III Application to Composite Systems

In this subsection, we will apply the conditional probability definition to
composite quantum systems, and re-examine Theorem 1 when measuring composite
system. Suppose the measured system $S$ consists two subsystems $S_{1}$ and
$S_{2}$ that are space-like separated. Define $A_{i}=P_{i}\otimes I_{2}$ where
$P_{i}=|\phi_{i}\rangle\langle\phi_{i}|$ is a local POVM element on subsystem
$S_{1}$ and $I_{2}$ is an identity operator on subsystem $S_{2}$. Similarly,
define $B_{j}=I_{1}\otimes Q_{j}$ where $Q_{j}$ is a local POVM elements on
subsystem $S_{2}$. By the principle of locality, a local measurement on a
subsystem should not cause any impact on the other remote subsystem.
Therefore, $[A_{i},B_{j}]=0$. For measurement outcomes of such two local
measurements, Eqs.(7) and (8) are correct formulations for conditional
probability, the joint probability is well-defined. Consequently, Eqs.(11) and
(12) hold true. There is no need to use the two-time formulation of
conditional probability. This is the case for typical Bell tests and has been
used in deriving the Bell-CHSH inequalities444However, in the derivation of
Bell-CHSH inequalities, identity (1) is further expressed as
$p(a_{i},b_{j}|\lambda)=p(a_{i}|b_{j},\lambda)p(b_{j}|\lambda)=p(a_{i}|\lambda)p(b_{j}|\lambda),$
(19) where $\lambda$ is a hidden variable. This is known as the Outcome
Independence assumption Hall ; Hall2 ..

However, suppose $B_{j}=Q_{j}\otimes I_{2}$ where $Q_{j}$ is another local
POVM elements on subsystem $S_{1}$ and $[P_{i},Q_{j}]\neq 0$. In this case,
Eq.(7) is incorrect for conditional probability. The two-time conditional
probability formulation is needed and can be calculated as

$\begin{split}&p(b_{j}\text{ at }t_{2}|a_{i}\text{ at }t_{1},\rho_{0})\\\
&=\frac{Tr((Q_{j}\otimes I_{2})U(\Delta t)\sqrt{P_{i}\otimes
I_{2}}\rho_{0}\sqrt{P_{i}\otimes I_{2}}U^{\dagger}(\Delta
t))}{Tr(\rho_{0}P_{i}\otimes I_{2})},\end{split}$ (20)

where $U(\Delta t)=U_{S_{1}}(\Delta t)\otimes U_{S_{2}}(\Delta t)$.

Next we wish to apply the two-time conditional probability to the extended
Wigner’s friend (EWF) scenario introduced in Brukner . As shown in Figure 1,
the EWF scenario consists two space-like separated laboratories $L_{1}$ and
$L_{2}$. Each laboratory contains half of an entangled pair of systems $s_{1}$
and $s_{2}$. $L_{1}$ also contains a friend Charlie who can perform
measurements on $s_{1}$. Outside $L_{1}$ there is a super-observer Alice who
can perform different types of measurements on $L_{1}$ as a whole. Similarly,
there is a friend Debbie in $L_{2}$, and a super-observer Bob outside $L_{2}$.
Here, we need to consider four POVM measurements, represented by POVM elements
$A,B,C,D$. Here operators $A$ and $C$ act on Hilbert space
$\mathcal{H}_{L_{1}}$, and $B$ and $D$ act on Hilbert space
$\mathcal{H}_{L_{2}}$. We drop the subscripts of the operators and $\rho_{0}$
for simplifying notations. In a typical EWF experiment, the chosen operators
are not all commutative with one another. Specifically, $[A,C]\neq 0$ and
$[B,D]\neq 0$, while $[C\otimes I_{L_{2}},I_{L_{1}}\otimes D]=0$ and
$[A\otimes I_{L_{2}},I_{L_{1}}\otimes B]=0$. The two-time probability
formulation to compute the conditional probability is needed because the
measurements $C$ and $D$ are taken place before the measurements of $A$ and
$B$. Since $[C\otimes I_{L_{2}},I_{L_{1}}\otimes D]=0$ and $[A\otimes
I_{L_{2}},I_{L_{1}}\otimes B]=0$, we can assume measurement $C$ and $D$ are
taken place at the same time, $t_{a}$, as $C\otimes D$ while measurements of
$A$ and $B$ at a same later time $t_{b}$ as $A\otimes B$. Without loss of
clarity, we drop the symbol $\otimes$ hereafter. Then the conditional
probability for $t_{a}<t_{1}<t_{b}<t_{2}$ is given by

$\begin{split}&p(ab\text{ at }t_{2}|cd\text{ at }t_{1},\rho,xy)\\\
&=\frac{Tr(\rho\sqrt{C}\sqrt{D}U^{{\dagger}}ABU\sqrt{C}\sqrt{D})}{Tr(\rho
CD)},\end{split}$ (21)

where $U=U_{L_{1}}\otimes U_{L_{2}}$ is the time evolution operator from
$t_{a}$ to $t_{b}$. The law of total probability we are interested is

$\begin{split}&p(ab\text{ at }t_{2}|\rho,xy)=\\\ &\sum_{cd}p(ab\text{ at
}t_{2}|cd\text{ at }t_{1},\rho,xy)p(cd\text{ at }t_{1}|\rho,xy)\\\
\end{split}$ (22)

From (21), the R.H.S. of (22) becomes

$\begin{split}&\sum_{cd}p(ab\text{ at }t_{2}|cd\text{ at
}t_{1},\rho,xy)p(cd\text{ at }t_{1}|\rho,xy)\\\
&=\sum_{CD}Tr(\rho\sqrt{C}\sqrt{D}U^{{\dagger}}ABU\sqrt{C}\sqrt{D}).\end{split}$
(23)

The summation is over POVM element sets for $\\{C\\}$ and $\\{D\\}$. Since
$[A\otimes I_{L_{2}},I_{L_{1}}\otimes B]=0$, the L.H.S. of (22) is $p(ab\text{
at }t_{2}|\rho,xy)=Tr(\rho U^{{\dagger}}ABU)$. Both sides are not equal in
general.

In the case that the post-measurement state after the first measurement is
unchanged during free time evolution555This is indeed the assumption in the
no-go theorems we will analyze in next section., Eq.(21) becomes

$p(ab\text{ at }t_{2}|cd\text{ at
}t_{1},\rho,xy)=\frac{Tr(\rho\sqrt{C}\sqrt{D}AB\sqrt{C}\sqrt{D})}{Tr(\rho
CD)}.$ (24)

Eq.(23) is simplified to

$\begin{split}&\sum_{cd}p(ab\text{ at }t_{2}|cd\text{ at
}t_{1},\rho,xy)p(cd\text{ at }t_{1}|\rho,xy)\\\
&=\sum_{CD}Tr(\rho\sqrt{C}\sqrt{D}AB\sqrt{C}\sqrt{D}),\end{split}$ (25)

and $p(ab\text{ at }t_{2}|\rho,xy)=Tr(\rho AB)$. In this case, one can derive
the following corollary based on Theorem 1.

###### Corollary 1.1

In the Extended Wigner’s Friend scenario setup, suppose the post-measurement
state is unchanged during free time evolution from $t_{a}$ to $t_{b}$. Select
$t_{1}$ and $t_{2}$ such that $t_{a}<t_{1}<t_{b}<t_{2}$. The law of total
probability (22) is true if one of the following conditions is met.

  1. C4.

$[A,C]=0$ and $[B,D]=0$, $\forall\rho_{0}$,

  2. C5.

$\rho_{0}=|\Psi\rangle\langle\Psi|$, $C$ and $D$ are projection operators, and

$\langle\Psi|[CD,AB]CD|\Psi\rangle=0.$ (26)

Condition $C4$ is quite obvious. Proof of condition (26) is given in Appendix
B.

##  IV Logical Loopholes in No-go Theorems Related to the Wigner’s Friend
Scenario

A no-go theorem usually starts from the conventional probability theory, which
is widely regarded as the true representation of logical deduction, and
assumes certain additional plausible physical premises: realism, locality, no
superdeterminism, observer independence, etc. One then shows that such a model
leads to prediction which is contradicted by quantum mechanics. Hence, one
concludes that, at least one of the assumptions, or the rules of conventional
probability must be violated by quantum mechanics. Let’s denote the physical
assumption that a no-go theorem tries to prove to be violated by quantum
theory as $\mathcal{W}$. For instance, $\mathcal{W}$ could be “measured facts
are observer independent”. The no-go theorem may be constructed independent of
the underlying physical theory. But if the logical deduction in the proof of
theorem utilizes the law of total probability in one of the forms of (12),
(16), or (22) without calling out the appropriate sufficient condition $C$,
then we know the resulting statement (could be in a form of inequality) will
not hold in quantum theory. This leaves a loophole in the logical deduction.
Because the contradiction shown in the no-go theorem could be just due to the
fact that $C$ is not met in the experiment setup, instead of the intended
conclusion that $\mathcal{W}$ is violated by quantum theory. Thus, the no-go
theorem does not reach the conclusion as desired. We will examine several such
no-go theorems in this section666We do not include the no-go theorem Renner
widely discussed in the literature since its proof does not invoke the law of
total probability..

###  IV.1 A strong no-go theorem on the Wigner’s friend paradox

Bong et al. introduce a no-go theorem that assuming quantum mechanics is
applicable to the scale of an observer, then one of the three assumptions,
‘Locality’, ‘No-superdeterminism’, or ‘Absoluteness of Observed Events (AOE)’
must be false Bong . Here AOE means that “every observed event exists
absolutely, not relative to anything or anyone”. The no-go theorem is supposed
to be independent of underlying physical theory and is proved in the context
of extended Wigner’s friend (EWF) scenario Brukner , as shown in Figure 1. The
measurement results from the friend in the lab can be correlated with the
super-observer’s subsequent measurement results. Suppose the measurement
outcomes from Alice, Bob, Charlie, and Debbie are $a,b,c,d$ respectively.
Alice can have three different measurement settings, labeled by parameter
$x\in\\{1,2,3\\}$. When $x=1$, Alice opens the $L_{1}$ and asks Charlie’s
measurement outcome, while when $x=2,3$, Alice performs a different
measurement on $L_{1}$. Similar measurement settings for Bob are labeled as
$y\in\\{1,2,3\\}$.

Among the three assumptions, it is more well accepted that ‘Locality’ and ‘No-
superdeterminism’ cannot be violated by any physical theory. The focus is on
the assumption of AOE, which is defined mathematically as following Bong .
There exists a joint probability distribution $p(abcd|xy)$ such that

  1. i

$p(ab|xy)=\sum_{cd}p(abcd|xy)\forall a,b,x,y$

  2. ii

$p(a|cd,x=1,y)=\delta_{a,c}\forall a,c,d,y$

  3. iii

$p(b|cd,x,y=1)=\delta_{b,d}\forall b,c,d,x$.

With the three assumptions, Bong  derives a number of inequalities and
experimentally confirms that quantum theory violates these inequalities when
proper measurement settings $x$ and $y$ and initial quantum state are chosen.
Therefore, the AOE assumption should be refuted.

However, closer examination of the derivation shows that the no-go theorem
assumes the law of total probability. The definition of AOE states that
$p(ab|xy)=\sum_{cd}p(abcd|xy)$. Then, in Eq.(3) of Ref. Bong , it implicitly
assumes $p(abcd|xy)=p(ab|cdxy)p(cd|xy)$. Together, they imply

$p(ab|xy)=\sum_{cd}p(ab|cdxy)p(cd|xy).$ (27)

But as discussed in Section II, the law of total probability does not hold
true in quantum theory unless certain condition is met.

We can apply Corollary 1.1 to analyze the validity of (27). In Appendix C, we
show that the operators chosen in Ref. Bong  are not all commutative with each
other. Specifically, $[A,C]\neq 0$ and $[B,D]\neq 0$, while $[C\otimes
I_{L_{2}},I_{L_{1}}\otimes D]=0$ and $[A\otimes I_{L_{2}},I_{L_{1}}\otimes
B]=0$. With these choices of operators, the corresponding two-time version of
the law of total probability is given by (22) and (25). However, we already
see condition $C4$ is not satisfied. The choice of initial quantum state,
i.e., Eq.(1) in Ref. Bong  and the forms of operator $A,B,C,D$ do not satisfy
conditions $C5$ in Corollary 1.1 either.

Therefore, in general (27) does not hold with the conditions specified in Bong
. Inequalities derived based on (27) will be violated by quantum theory with
the choice of initial quantum state and measurement operators described in
Bong . But this raises the question of exactly what the no-go theorem refutes.
We agree with the authors that the violation of the inequalities by quantum
theory points to the validity of AOE in quantum theory. But the definition of
AOE and the derivation of the theorem implicitly assume the validity of the
law of total probability. The root cause of the violation of the inequalities
is due to the fact that the experimental setup does not satisfy the conditions
to make the law of total probability hold true, not because of the AOE
statement that “an observed event is not relative to anything or anyone”. One
may argue that the AOE statement is equivalent to the invalidity of the law of
total probability. But as discussed earlier, the invalidity of the law of
total probability is due to the fact that measurement $C$ (or $D$) alters the
initial quantum state that impacts the probability of outcome for measurement
$A$ (or $B$) since $[A,C]\neq 0$ (or $[B,D]\neq 0$). There is a logical gap to
equate this reason with the statement “an observed event is not relative to
anything or anyone”. Therefore, it appears that the violation of inequalities
in Bong  just reconfirms the consequence of non-commutative measurements and
therefore the invalidity of the law of total probability in quantum theory,
instead of the AOE statement.

###  IV.2 A no-go theorem for observer-independent facts

The no-go theorem for observer-independent facts by Brukner Brukner  was
actually introduced earlier than Bong  in a similar effort to prove that
measured facts are observer-dependent. The experimental setup is shown in
Figure 1. In Ref. Brukner , there are only two different measurement setups
either Alice or Bob will perform, compared to three different measurement
setups in Ref. Bong . Furthermore, the way the no-go theorem in is proved is
different. The proof in Brukner  leverages the well-known Bell-CHSH inequality
thus inherits all the assumptions associated with the Bell-CHSH theorem, while
the no-go theorem in Bong  is proved independent of Bell-CHSH theorem. As we
will explain next, the proof in Brukner  is more subtle, as it carefully
chooses a quantum state and a set of measurement operators such that the law
of total probability holds true if only considering Alice’s (or Bob’s)
measurements.

The initial wave function is chosen such that after Charlie and Debbie each
perform a measurement of the respective half of entangled spin along $z$ axis,
from Alice’s or Bob’s perspective, it becomes (see Eqs.(5)-(9) in Brukner ):

$\begin{split}|\Psi\rangle&=-\frac{1}{\sqrt{2}}\sin\frac{\theta}{2}(|00\rangle_{L_{1}}|00\rangle_{L_{2}}+|11\rangle_{L_{1}}|11\rangle_{L_{2}})\\\
&+\frac{1}{\sqrt{2}}\cos\frac{\theta}{2}(|00\rangle_{L_{1}}|11\rangle_{L_{2}}-|11\rangle_{L_{1}}|00\rangle_{L_{2}}),\end{split}$
(28)

where $|00\rangle_{L_{2}}$ represents that $s_{1}$ is in the spin up state and
Charlie’s pointer variable associated with the up state, and
$|11\rangle_{L_{1}}$ corresponds to the spin down state for $s_{1}$ and
Charlie’s pointer variable. Similar meanings for $|00\rangle_{L_{2}}$ and
$|11\rangle_{L_{2}}$ for $s_{2}$ and Debbie’s pointer variable. The key point
of Brukner  is to assume there exists a joint probability
$p(a_{1}a_{2}b_{1}b_{2})$, where $a_{1},a_{2}\in\\{0,1\\}$ are the measurement
results corresponding to Alice’s choice of two types of measurement operations
and $b_{1},b_{2}\in\\{0,1\\}$ are the measurement results of Bob. Alice can
choose to either measure $L_{1}$ with projection operator $A_{1}$, or with
projection operator $A_{2}$. Here $A_{1}$ is represented777In Brukner  the two
types of operations for Alice are defined as
$\mathcal{A}_{1}=|00\rangle\langle 00|_{L_{1}}-|11\rangle\langle
11|_{L_{1}},i\in\\{0,1\\}$, and $\mathcal{A}_{2}=|00\rangle\langle
11|_{L_{1}}-|00\rangle\langle 11|_{L_{1}}$. Here we use the spectral
decomposition theorem to decompose $\mathcal{A}_{1}$ into projection
operators, and represent it by $A_{1}$. Similar approach for the definition of
$A_{2}$. There is an important difference here compared to the setup in Bong .
Here $a_{1},a_{2}$ are results for Alice from completed measurement
$A_{1},A_{2}$ respectively, whereas in Bong  $a,c$ are measurement results for
Alice and Charlie respectively. The issue of $c$ as result of Charlie’s “pre-
measurement” in Bong  does not exist here in Brukner . by
$|\phi_{i}\rangle\langle\phi_{i}|_{L_{1}}$, and $|\phi_{0}\rangle=|00\rangle$
for $a_{1}=0$ or $|\phi_{1}\rangle=|11\rangle$ for $a_{1}=1$. $A_{2}$ is
chosen to be $A_{2}=|\chi_{0}\rangle\langle\chi_{0}|$ for $a_{2}=0$ or
$A_{2}=|\chi_{1}\rangle\langle\chi_{1}|$ for $a_{2}=1$, where

$|\chi_{i}\rangle=\frac{1}{\sqrt{2}}((-1)^{i}|00\rangle_{L_{1}}+|11\rangle_{L_{1}}).$
(29)

From these definition of $A_{1}$ and $A_{2}$, one can verify that

$[A_{1},A_{2}]=(-1)^{a_{1}+a_{2}}(|00\rangle\langle
11|_{L_{1}}-|11\rangle\langle 00|_{L_{1}}).$ (30)

Similar definitions for operators $B_{1}$ and $B_{2}$. The problem in Ref.
Brukner  is that it assumes the law of marginal probability holds true, for
instance, $p(a_{2}b_{2})=\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{1}a_{2}b_{1}b_{2})$.
Ref. Brukner  does not provide details on how the joint probability is
defined. As discussed earlier, the joint probability cannot be well defined
unless the measurement operators are commutative. If we further assume the
validity of the classical probability axiom in Eq. (1) and apply it
recursively, we have

$\begin{split}p(a_{1}a_{2}b_{1}b_{2})&=p(a_{2}b_{1}b_{2}|a_{1})p(a_{1})\\\
&=p(a_{2}b_{2}|a_{1}b_{1})p(b_{1}|a_{1})p(a_{1})\\\
&=p(a_{2}b_{2}|a_{1}b_{1})p(a_{1}b_{1}),\end{split}$ (31)

then the law of marginal probability is equivalent to the law of total
probability such that

$p(a_{2}b_{2})=\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{2}b_{2}|a_{1}b_{1},\rho)p(a_{1}b_{1}).$
(32)

Let’s analyze if Eqs. (31) and (32) hold true with the chosen operators
$\\{A_{1},B_{1},A_{2},B_{2}\\}$ and the quantum state in (28). To do this, we
apply Corollary 1.1 by replacing operators $\\{A,B,C,D\\}$ with operators
$\\{A_{2},B_{2},A_{1},B_{1}\\}$, respectively, and setting
$\rho_{0}=|\Psi\rangle\langle\Psi|$ where $|\Psi\rangle$ is defined in (28).
The conditional probability is similar to (24),

$\begin{split}&p(a_{2}b_{2}\text{ at }t_{2}|a_{1}b_{1}\text{ at
}t_{1},\rho,xy)\\\
&=\frac{Tr(\rho\sqrt{A_{1}}\sqrt{B_{1}}A_{2}B_{2}\sqrt{A_{1}}\sqrt{B_{1}})}{Tr(\rho
A_{1}B_{1})},\end{split}$ (33)

where we drop the subscript of $\rho_{0}$. The desired law of total
probability is

$\begin{split}&p(a_{2}b_{2}\text{ at }t_{2}|\rho)=\\\
&\sum_{a_{1},b_{1}}p(a_{2}b_{2}\text{ at }t_{2}|a_{1}b_{1}\text{ at
}t_{1},\rho)p(a_{1}b_{1}\text{ at }t_{1}|\rho).\end{split}$ (34)

In Appendix D, we show that for the choices of the set of operators
$\\{A_{1},A_{2},B_{1},B_{2}\\}$ prescribed earlier, $[A_{1},A_{2}]\neq 0$ and
$[B_{1},B_{2}]\neq 0$. With the quantum state (28), Condition
$\textit{C3}^{\prime}$ is satisfied such that
$p(a_{2}|\rho)=\sum_{a_{1}}p(a_{2}|a_{1},\rho)p(a_{1}|\rho)$ and
$p(b_{2}|\rho)=\sum_{b_{1}}p(b_{2}|b_{1},\rho)p(b_{1}|\rho)$. But
unfortunately, we also show that (34) and the law of marginal probability
$p(a_{2}b_{2})=\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{1}a_{2}b_{1}b_{2})$ are still
not valid.

Since the prove of the no-go theorem in Brukner  depends on the law of
marginal probability, and the law of marginal probability does not hold true
by the choice of quantum state and measurement operators, there is a logical
loophole in the no-go theorem. The violation of the inequality in Brukner  in
quantum theory does not necessarily imply that measured facts are observer
dependent. Instead, the violation just reconfirms that the law of marginal
probability does not hold for the choice of quantum state and the measurement
operators. The logical gap to equate the statement that “measured facts are
observer dependent” to the invalidity of the law of marginal probability in
quantum theory is similar to what we discussed in the last paragraph in
Section IV.1.

Note that besides depending on the law of marginal probability, the proof of
no-go theorem in Brukner  also inherits the assumptions for the proof of the
Bell-CHSH inequality Hall ; Hall2 , particularly the dependency on the outcome
independence assumption (19). The no-go theorem in Bong , on the other hand,
does not depend on the outcome independence assumption.

###  IV.3 A no-go theorem for the persistent reality of Wigner’s friend’s
perception

In Ref. Guerin , another no-go theorem is introduced to show that in the
extended Wigner’s friend scenario, Wigner’s friend cannot “treat her perceived
measurement outcome as having reality across multiple times” without
contradicting one of the following assumptions in quantum mechanics Guerin .

  1. P1

Let $f_{1}$ and $f_{2}$ be perceived measurement records of the friend at time
$t_{1}$ and $t_{2}$, respectively. A joint probability distribution
$p(f_{1},f_{2})$ can be assigned that also satisfies the law of total
probability $p(f_{1})=\sum_{f_{2}}p(f_{1},f_{2})$ and
$p(f_{2})=\sum_{f_{1}}p(f_{1},f_{2})$;

  2. P2

One time probability is assigned according to
$p(f_{i})=Tr(|f_{i}\rangle\langle f_{i}|\rho)$ using unitary quantum theory
where no state collapse is considered occurred;

  3. P3

The joint probability of the friend’s perceived outcomes $p(f_{1},f_{2})$ has
a convex linear dependence on the initial state $\rho$.

In traditional quantum measurement theory Neumann , the unitary process is
considered as a process to entangle the measured system with the measuring
apparatus before the projection process. The projection process gives definite
final outcome. P2 essentially assumes the unitary process itself can have
measurement result and can be assigned a (one-time) probability. Zukowski and
Markiewicz have already pointed out that such assumption leads to a
contradiction. But there is another problem with P2 Zukowski . The derivation
in Ref. Guerin  assumes that the joint probability $p(f_{1},f_{2})$ is derived
through the standard probability axiom
$p(f_{1},f_{2})=p(f_{2}|f_{1})p(f_{1})$, but does not give details on how the
conditional probability is calculated in quantum theory. It is not clear how
the unitary formulation presented in Guerin  can be applied to derive the
conditional probability $p(f_{2}|f_{1})$, because P2 assumes there is no
“collapse” after the first measurement. It is not a problem to compute the
one-time probability $p(f_{1})$ and $p(f_{2})$. But in order to be able to
calculate a two-time probability such as $p(f_{2}|f_{1})$, one will have to
apply the state update rule after the first measurement at time $t_{1}$, as
shown in (13) for the two-time conditional probability.

More crucially, even if we are able to calculate the conditional probability,
there is still problem with P1 as P1 assumes the law of total probability,
$p(f_{1})=\sum_{f_{2}}p(f_{1},f_{2})$ is always true. We have shown in Theorem
1 and subsequent corollaries that the law of total probability is true in
quantum theory only with certain conditions. The two POVM elements chosen in
Guerin  are non-commutative, as shown in Eq.(17) in Guerin . Thus,
$p(f_{1})=\sum_{f_{2}}p(f_{1},f_{2})$ does not necessarily hold. The proof in
Guerin  assumes that $p(f_{1})=\sum_{f_{2}}p(f_{1},f_{2})$ always holds based
on P1, then deduces eventually that the two POVM elements should be
commutative, and claims there is a contradiction. But such contradiction is
due to the invalid assumption of $p(f_{1})=\sum_{f_{2}}p(f_{1},f_{2})$ in P1,
which in turn due to the fact that the two POVM elements are non-commutative.
Since P1 is invalid, the contradiction does not lead to the desired conclusion
that Wigner’s friend cannot “treat her perceived measurement outcome as having
reality across multiple times”.

###  IV.4 Relative facts, stable facts

In relational interpretation of quantum mechanics (RQM) Rovelli96 ; Rovelli07
; Transs2018 ; Rovelli18 ; Yang , measurement result is considered meaningful
only relative to the system that interacts with the measured system. A
definite measurement result is referred as a fact. Quantum theory is about
conditional probability for facts, given other facts. Recently, Biagio and
Rovelli introduce the concept of stable fact in the following sense Rovelli21
. If, given the probability $p(a_{i})$ for $N$ mutual exclusive facts
$a_{i}(i=1\ldots N)$ and the conditional probability of another fact $b$,
$p(b|a_{i})$, the probability $p(b)$ (dropping index $j$ for $b_{j}$) is given
by (4), then facts $a_{i}$ are considered stable.

RQM states that fact is relative. Formally, if two systems $S$ and $F$
interact such that a variable $L_{F}$ of $F$ depends on the value of a
variable $L_{S}$ of $S$, then the value of $L_{S}$ is said a fact relative to
$F$ Rovelli21 . However, not all relative facts are stable. The main thesis of
Ref. Rovelli21  comprises two claims. First, The law of total probability (4)
is satisfied only if $b$ and $a_{i}$ are facts relative to the same system,
say relative to system $F$; If $b$ is relative to another system $W\neq F$,
(4) is not true in general. Mathematically, these can be expressed as

$\displaystyle p(b^{(F)})$
$\displaystyle=\sum_{i}p(b^{(F)}|a_{i}^{(F)})p(a_{i}^{(F)})$ (35)
$\displaystyle p(b^{(W)})$
$\displaystyle\neq\sum_{i}p(b^{(W)}|a_{i}^{(F)})p(a_{i}^{(F)}).$ (36)

Here it is important to label the reference system the fact is relative to.
Second, if system $F$ goes through a decoherence process by interacting with
an environmental system $E$, the resulting density matrix for $F$ is
approximately given by $\rho=\sum_{i}\lambda_{i}|Fa_{i}\rangle\langle
Fa_{i}|$, where $Fa_{i}$ is eigenvalues of $L_{F}$. Then, Eq.(36) can be
rewritten as

$p(b^{(W)})=\sum_{i}p(b^{(W)}|Fa_{i}^{(E)})p(Fa_{i}^{(E)}).$ (37)

In such a case, facts $Fa_{i}$ relative to $E$ are stable for $W$.

Now let’s examine the two claims more carefully. For the first claim, from
Theorem 1, Eq.(35) is not necessarily true even if both $b$ and $a_{i}$ are
facts relative to a same system. That facts $b$ and $a_{i}$ are both relative
to a same system means both facts are obtained through interactions with the
same system, and the interactions can be represented by measurement operators
$B$ and $A_{i}$, respectively. If $[B,A_{i}]=0$, Eq.(35) is true. But there is
no reason that $B$ and $A_{i}$ have to be commutative. If $[B,A_{i}]\neq 0$,
Eq.(35) is not true in general, unless other conditions such as condition
$\textit{C2}^{\prime}$ or $\textit{C3}^{\prime}$ in Theorem 1 is satisfied.
Indeed, the second claim (37) is precisely the case where condition
$\textit{C2}^{\prime}$ is met. Note that the reasoning from (35) to (37) is
also applicable when facts $b$ and $a_{i}$ are relative to the same system but
the corresponding measurement operators are non-commutative, $[B,A_{i}]\neq
0$.

Therefore, it is not clear that one can use the validity of the law of total
probability (35) and (37) to distinguish stable facts from non-stable facts.
Again, we are not opposite to the idea that facts are relative. What we are
questioning here are the validity of (35) without specifying the conditions,
and the rigorousness of reasoning from (36) to (37). It appears that more
careful investigation is needed in order to search for the criteria to define
a “stable” fact.

##  V Discussion and Conclusion

###  V.1 The Page-Wootters Timeless Formulation

In the timeless formulation of quantum theory developed by Page and Wootters
PW , time evolution is naturally emerged from quantum correlation between a
clock and a system whose dynamics is tracked by the clock. Baumann  proposed
several two-time formulations of conditional probability based on the Page-
Wootters timeless mechanism. The advantage of such formulation is that from a
timeless quantum state one can derive probability of a measurement event
conditional on another event regardless of the temporary order of the two
events.

Although the formulation in the present work is based on the regular time
evolution dynamics in the Schrodinger picture, our definition of two-time
conditional probability (13) is consistent with the definitions in Baumann .
For instance, for the case of two projection measurements $A_{i}$ and $B_{j}$
at $t_{a}$ and $t_{b}$, respectively, (13) gives the same transition
probability (14), as that in Eq. (29) of Baumann .

However, the timeless formulations of conditional probability in Baumann  are
applicable only to projection measurements, while the theory developed here is
more general in the sense that it is applicable to POVM measurements. A two-
time conditional probability formulation for projection measurements is
insufficient to analyze the no-go theorems in Bong . Moreover, our focus here
is the validity of the law of total probability that is built on the
definition of two-time conditional probability, which is missing in Baumann
as its focus in only on the rules for two-time conditional probability.

It will be interesting to generalize the timeless Page-Wootters formulation of
two-time conditional probability in Baumann  to be able to handle POVM
measurements. Although we expect such generalization should produce similar
results as what are presented in this work.

###  V.2 Limitations

One limitation of the present works is that in Theorem 1, we are only able to
derive three sufficient conditions for the law of total probability to hold
true. In theory, there can be many other sufficient conditions. It is
desirable to find the sufficient and necessary condition for the law of total
probability to hold true in quantum theory. This remains a future
investigation topic. Nevertheless, for the purpose of analyzing the EWF
scenario and identifying the loopholes of the relevant no-go theorems, the
conditions specified in Theorem 1 and subsequent Corollaries are sufficient.

###  V.3 Conclusions

In this paper, the standard rule to assign conditional probability in quantum
theory, i.e., Lüders rule, is extended to include two-time POVM measurements.
The extension is strictly based on the recursive application of the POVM
measurement theory as shown in (5) and (6), and the assumption that
probability distribution can be assigned only for completed quantum
measurement. The resulting definition (13) is consistent with other works
based on Page-Wootters formulation Baumann  but with advantage of being able
to apply to POVM measurements instead of just projection measurements.

More importantly, with the generalized two-time conditional probability
formulation, we analyze the validity of the law of total probability. It is
shown that the quantum version of the law of total probability does not hold
true in general. Certain conditions related to the choice of measurement
operators and the initial quantum state must be met in order for the law of
total probability to hold. Specifically, such sufficient conditions are
derived in Theorem 1 and Corollary 1.1.

Application of the theory developed here to the Extended Wigner’s Friend
scenario reveals logical loopholes in several no-go theorems. These no-go
theorems take for granted on the validity of the law of total probability (or,
the law of marginal probability) in quantum theory. However, this is not the
case as shown in Theorem 1 and Corollary 1.1. Thus, the no-go theorems do not
lead to the desired conclusions. For instance, the violation of the
inequalities developed in Brukner  and Bong  in quantum theory does not
necessarily lead to to desired statement that “measured facts are observed-
dependent”. Instead, it just reconfirms the invalidity of the law of total
probability or the law of marginal probability in quantum theory. We do not
take a stand on the assertions themselves of the no-go theorems. It could be
still a valid statement that “measured facts are observed-dependent”. What we
show here is that there are logical loopholes to reach such statement. It is
desirable to find more convincing proof and experimental testing because the
implications of the extended Wigner’s friend scenario are conceptually
fundamental in quantum theory.

###### Acknowledgements.

The author sincerely thanks the anonymous reviewers of this paper for their
careful reviews. The valuable comments provided help to improve the clarity of
the presentations and discussions.

## Data Availability Statement

The data that support the findings of this study are available within the
article.

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##  Appendix A Proof of Theorem 1

With Condition $C1$, $[A_{i},U^{{\dagger}}B_{j}U]=0$. Thus,
$[\sqrt{A_{i}},U^{{\dagger}}B_{j}U]=0$. The right hand side of (15) becomes
$Tr(U^{{\dagger}}B_{j}UA_{i}\rho)$. Given the completeness of POVM operators,
$\sum_{i}A_{i}=I$, we have
$\sum_{i}Tr(U^{{\dagger}}B_{j}UA_{i}\rho)=Tr(U^{{\dagger}}B_{j}U\rho)=Tr(B_{j}\rho(t_{2}))=p(b_{j}\text{
at }t_{2}|\rho)$.

Given Condition $C2$,
$\rho=\sum_{k}\lambda_{k}|\phi_{k}\rangle\langle\phi_{k}|$ and
$A_{i}=|\phi_{i}\rangle\langle\phi_{i}|$, we get
$\sqrt{A_{i}}\rho\sqrt{A_{i}}=\lambda_{i}|\phi_{i}\rangle\langle\phi_{i}|$.
Then the right hand side of (15) becomes
$Tr(U^{{\dagger}}B_{j}U\lambda_{i}|\phi_{i}\rangle\langle\phi_{i}|)=\lambda_{i}\langle\phi_{i}|U^{{\dagger}}B_{j}U|\phi_{i}\rangle$.
The right hand side of (16) becomes
$\sum_{i}\lambda_{i}\langle\phi_{i}|U^{{\dagger}}B_{j}U|\phi_{i}\rangle$. On
the other hand, $p(b_{j}\text{ at
}t_{2}|\rho)=Tr(U^{{\dagger}}B_{j}U\rho)=\sum_{k}Tr(U^{{\dagger}}B_{j}U\lambda_{k}|\phi_{k}\rangle\langle\phi_{k}|)=\sum_{i}\lambda_{i}\langle\phi_{i}|U^{{\dagger}}B_{j}U|\phi_{i}\rangle$,
same as the right hand side of (16).

Given $\rho=|\Psi\rangle\langle\Psi|$, and $A_{i}$ is a projection operator,
in Condition $C3$, we have $A_{i}^{2}=A_{i}$, and the right hand side of (16)
becomes $\sum_{i}\langle\Psi|A_{i}U^{{\dagger}}B_{j}UA_{i}|\Psi\rangle$. The
left hand side $p(b_{j}\text{ at
}t_{2}|\rho)=\langle\Psi|U^{{\dagger}}B_{j}U|\Psi\rangle$. Again, by the
completeness of POVM operators, $\sum_{i}A_{i}=I$, and we get $p(b_{j}\text{
at
}t_{2}|\rho)=\sum_{i}\langle\Psi|U^{{\dagger}}B_{j}UA_{i}A_{i}|\Psi\rangle$.
To have both sides of (16) equal, we need
$\sum_{i}\langle\Psi|A_{i}U^{{\dagger}}B_{j}UA_{i}|\Psi\rangle-\sum_{i}\langle\Psi|U^{{\dagger}}B_{j}UA_{i}A_{i}|\Psi\rangle=0$.
This can be rearranged to
$\sum_{i}\langle\Psi|[A_{i},U^{{\dagger}}B_{j}U]A_{i}|\Psi\rangle=0$, and
Condition $C3$ ensures this is the case.

##  Appendix B Proof of (26)

To avoid confusion, we need to restore the subscripts of operators as $C_{i}$
and $D_{j}$. Since $C_{i}$ and $D_{j}$ are projection operators, and
$\rho=|\Psi\rangle\langle\Psi|$, the R.H.S. of (23) becomes
$\sum_{ij}\langle\Psi|C_{i}D_{j}ABC_{i}D_{j}|\Psi\rangle$. Given the
completeness of $\\{C_{i}\\}$ and $\\{D_{j}\\}$, we have
$\sum_{ij}C_{i}D_{j}=I$. Since $C_{i}^{2}=C_{i}$ and $D_{j}^{2}=D_{j}$, we
further obtain $\sum_{ij}C^{2}_{i}D^{2}_{j}=I$. Then $p(ab\text{ at
}t_{2}|\rho,xy)=Tr(\rho
AB)=\sum_{ij}\langle\Psi|ABC^{2}_{i}D^{2}_{j}|\Psi\rangle$. To make this equal
to the R.H.S. of (23), one condition is to have
$\langle\Psi|C_{i}D_{j}ABC_{i}D_{j}|\Psi\rangle=\langle\Psi|ABC^{2}_{i}D^{2}_{j}|\Psi\rangle$.
This is equivalent to
$\langle\Psi|(C_{i}D_{j}ABC_{i}D_{j}-ABC^{2}_{i}D^{2}_{j})|\Psi\rangle=0$. But
$(C_{i}D_{j}ABC_{i}D_{j}-ABC^{2}_{i}D^{2}_{j})=[C_{i}D_{j},AB]C_{i}D_{j}$.
Thus we have $\langle\Psi|[C_{i}D_{j},AB]C_{i}D_{j}|\Psi\rangle=0$. Omitting
the subscripts of $C_{i}$ and $D_{j}$ again gives (26).

##  Appendix C Non-commutation of Operators in Bong

The key characteristic of the EWF experiment is that the super-observer Alice
(or Bob) performs measurements on the laboratory $L_{1}$ (or $L_{2}$) as a
whole. Thus the measurement operator acts on both the observed system and the
friend in the lab. Bong  carefully chooses the operators as following. When
$x=1$, Alice’s measurement is represented as $A(x=1)=|c\rangle\langle
c|_{F_{1}}\otimes I_{s_{1}}$ where $c$ is the outcome Charlie obtains from his
measurement on $s_{1}$ and $F_{1}$ refers to Charlie himself. For
$x\in\\{2,3\\}$, Alice’s measurement operator is
$A(x)=U_{L_{1}}(I_{F_{1}}\otimes E^{x}_{s_{1}})U_{L_{1}}^{-1}$, where
$U_{L_{1}}^{-1}$ is a unitary evolution that reverses the entanglement between
$F_{1}$ and $s_{1}$, and $E^{x}_{s_{1}}$ is a positive operator on $s_{1}$
associated with outcome $a$ for measurement setting $x$. The operator
associated with Charlie’s measurement on $s_{1}$, according to Bong , is
described by an unitary operator $C(x)=U_{L_{1}}$ from Alice’s perspective.
$U_{L_{1}}$ acts on the same Hilbert space
${\cal{H}}_{F_{1}}\otimes{\cal{H}}_{s_{1}}$ and entangles $s_{1}$ and $F_{1}$.
For $x=1$, $[A(x),C(x)]=|c\rangle\langle c|_{F_{1}}\otimes
I_{s_{1}}U_{L_{1}}-U_{L_{1}}|c\rangle\langle c|_{F_{1}}\otimes I_{s_{1}}\neq
0$, and for $x\in\\{2,3\\}$, $[A(x),C(x)]=U_{L_{1}}(I_{F_{1}}\otimes
E^{x}_{s_{1}})-U^{2}_{L_{1}}(I_{F_{1}}\otimes E^{x}_{s_{1}})U_{L_{1}}^{-1}\neq
0$.

As already pointed out Zukowski , defining $C(x)$ as $U_{L_{1}}$ implies pre-
measurement only with no measurement result, and leads to contradictions. An
alternative choice of operation is that Charles performs a projection
operation after the pre-measurement. This refines the definition of $C(x)$ to
include both $U_{L_{1}}$ and a projection operation on $s_{1}$, i.e.,
$C(x)=U^{{\dagger}}_{L_{1}}(I_{F_{1}}\otimes|c\rangle\langle c|)U_{L_{1}}$.
With this refined definition, one can verify that $[A(x),C(x)]\neq 0$ still
true. Choosing $C(x)=U^{{\dagger}}_{L_{1}}(I_{F_{1}}\otimes|c\rangle\langle
c|)U_{L_{1}}$ implies operator $C(x)$ is from Charlie’s point of view. This
may not be the original intention in Bong . However, the key point here is
that either choice of $C(x)$, $[A(x),C(x)]\neq 0$. The same analysis goes to
operators $B(y)$ and $D(y)$, and the conclusion is that $[B(y),D(y)]\neq 0$
for $y\in\\{1,2,3\\}$.

##  Appendix D Proof That (34) Does Not Hold

First we consider a simpler case that only Alice performs the two type of
measurements and Bob does nothing. The law of total probability in this case
can take the form of
$p(a_{2}|\rho)=\sum_{a_{1}=\\{0,1\\}}p(a_{2}|a_{1},\rho)p(a_{1}|\rho)$. We
will show this is true due to the fact that the selected operators $A_{1}$ and
$A_{2}$, and wave function (28) together meet condition
$\textit{C3}^{\prime}$. To see this, substitute
$A_{1}=|\phi_{i}\rangle\langle\phi_{i}|_{L_{1}}$ into condition
$\textit{C3}^{\prime}$, $\textit{C3}^{\prime}$ becomes

$\langle\varphi_{i}|[A_{1},A_{2}]|\phi_{i}\rangle=0,\text{ where
}|\varphi_{i}\rangle=\langle\Psi|\phi_{i}\rangle|\Psi\rangle,$ (38)

Now consider the case $a_{1}=0$, where $|\phi_{0}\rangle=|00\rangle_{L_{1}}$.
From (28) one can calculate

$\displaystyle\langle\phi_{0}|\Psi\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}(-\sin\frac{\theta}{2}|00\rangle_{L_{2}}+\cos\frac{\theta}{2}|11\rangle_{L_{2}})$
(39) $\displaystyle|\varphi_{0}\rangle$
$\displaystyle=\langle\Psi|\phi_{0}\rangle|\Psi\rangle=\frac{1}{2}|00\rangle_{L_{1}}$
(40)

Then from (30) and dropping the unimportant factor of $1/2$ for
$|\varphi_{0}\rangle$, we have for the case of $a_{1}=0$

$\begin{split}\langle\varphi_{0}|[A_{1},A_{2}]|\phi_{0}\rangle&=\langle
00|(-1)^{a_{2}}(|00\rangle\langle 11|-|11\rangle\langle 00|)|00\rangle\\\
&=0.\end{split}$

For the case of $a_{2}=1$, we can verify that
$|\varphi_{1}\rangle=|11\rangle_{L_{1}}$ and

$\begin{split}\langle\varphi_{1}|[A_{1},A_{2}]|\phi_{1}\rangle&=\langle
11|(-1)^{1+a_{2}}(|00\rangle\langle 11|-|11\rangle\langle 11|)|00\rangle\\\
&=0.\end{split}$

Therefore, condition $\textit{C3}^{\prime}$ is met with the choices of
wavefunction and Alice’s measurement operation. Similarly, if only Bob
performs the two types of measurements and Alice does not perform any
measurement, and let
$B_{1}=|\phi^{\prime}_{i}\rangle\langle\phi^{\prime}_{i}|_{L_{2}}$, we can
verify that

$\langle\varphi^{\prime}_{i}|[B_{1},B_{2}]|\phi^{\prime}_{i}\rangle=0,\text{
where
}|\varphi^{\prime}_{i}\rangle=\langle\Psi|\phi^{\prime}_{i}\rangle|\Psi\rangle.$
(41)

Thus, $p(b_{2}|\rho)=\sum_{b_{1}=\\{0,1\\}}p(b_{2}|b_{1},\rho)p(b_{1}|\rho)$
holds true per Theorem 1.

However, when we consider both Alice and Bob perform the measurements
$\\{A_{1},A_{2}\\}$ and $\\{B_{1},B_{2}\\}$, respectively, the situation is
different. By replacing operators $A,B,C,D$ in (26) with operators
$A_{2},B_{2},A_{1},B_{1}$, (26) reads

$\langle\Psi|[A_{1}B_{1},A_{2}B_{2}]A_{1}B_{1}|\Psi\rangle=0.$ (42)

(38) and (41) together are not sufficient to ensure (42) is valid.
Consequently, the law of total probability such as
$p(a_{2}b_{2}|\rho)=\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{2}b_{2}|a_{1}b_{1},\rho)p(a_{2}b_{2}|\rho)$,
is not valid. Let’s confirm this by direct calculation for the case $a_{2}=0$
and $b_{2}=0$, where the corresponding projection operators are

$\displaystyle A_{1}$
$\displaystyle=|\phi_{i}\rangle\langle\phi_{i}|_{L_{1}},B_{1}=|\phi_{i}\rangle\langle\phi_{i}|_{L_{2}}$
$\displaystyle A_{2}$
$\displaystyle=\frac{1}{2}(|00\rangle_{L_{1}}+|11\rangle_{L_{1}})(\langle
00|_{L_{1}}+\langle 11|_{L_{1}})$ $\displaystyle B_{2}$
$\displaystyle=\frac{1}{2}(|00\rangle_{L_{2}}+|11\rangle_{L_{2}})(\langle
00|_{L_{2}}+\langle 11|_{L_{2}}).$

where $|\phi_{0}\rangle=|00\rangle$ and $|\phi_{1}\rangle=|11\rangle$. From
(23), one can calculate that

$\displaystyle\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{2}b_{2}|a_{1}b_{1},\rho)p(a_{2}b_{2}|\rho)$
(43) $\displaystyle=$
$\displaystyle\sum_{a_{1},b_{1}=\\{0,1\\}}\langle\Psi|A_{1}B_{1}A_{2}B_{2}A_{1}B_{1}|\Psi\rangle$
(44)
$\displaystyle=\sum_{i,j=\\{0,1\\}}|\langle\Psi|\phi_{i}\phi_{j}\rangle|^{2}\langle\phi_{i}|A_{2}|\phi_{i}\rangle\langle\phi_{j}|B_{2}|\phi_{j}\rangle$
(45) $\displaystyle=\frac{1}{4}.$ (46)

Meanwhile, given $[A_{2}\otimes I_{L_{2}},I_{L_{1}}\otimes B_{2}]=0$, the
joint probability $p(a_{2}b_{2}|\rho)$ is well-defined as

$\displaystyle
p(a_{2}=0,b_{2}=0|\rho)=Tr(A_{2}B_{2}\rho)=\langle\Psi|A_{2}B_{2}|\Psi\rangle$
$\displaystyle=\frac{1}{16}|\langle\Psi|(|00\rangle_{s_{1}C}+|11\rangle_{s_{1}C})(|00\rangle_{s_{2}D}+|11\rangle_{s_{2}D})|^{2}$
$\displaystyle=\frac{1}{8}\sin^{2}\frac{\theta}{2}.$

Thus,
$p(a_{2}b_{2}|\rho)\neq\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{2}b_{2}|a_{1}b_{1},\rho)p(a_{2}b_{2}|\rho)$
for the case of $a_{2}=0$ and $b_{2}=0$. For other values of
$a_{2},b_{2}\in\\{0,1\\}$, similar results can be calculated. Consequently,
the law of marginal probability
$p(a_{2}b_{2}|\rho)=\sum_{a_{1},b_{1}=\\{0,1\\}}p(a_{1}b_{1},a_{2}b_{2}|\rho)$
does not hold, if we define the joint probability
$p(a_{1}b_{1},a_{2}b_{2}|\rho)=p(a_{2}b_{2}|a_{1}b_{1},\rho)p(a_{2}b_{2}|\rho)$.

If we add another condition that the quantum state $|\Psi\rangle$ is chosen as
a product state of Hilbert space $\mathcal{H}_{L_{1}}$ and
$\mathcal{H}_{L_{2}}$, then together with (38) and (41), (34) becomes true. To
see this, let
$|\Psi^{\prime}\rangle=|\xi\rangle_{L_{1}}\otimes|\zeta\rangle_{L_{2}}$, (45)
becomes

$\displaystyle\sum_{i,j=\\{0,1\\}}|\langle\xi|\phi_{i}\rangle\langle\zeta|\phi_{j}\rangle|^{2}\langle\phi_{i}|A_{2}|\phi_{i}\rangle\langle\phi_{j}|B_{2}|\phi_{j}\rangle$
(47)
$\displaystyle=\sum_{i}|\langle\xi|\phi_{i}\rangle|^{2}\langle\phi_{i}|A_{2}|\phi_{i}\rangle\sum_{j}|\langle\zeta|\phi_{j}\rangle|^{2}\langle\phi_{j}|B_{2}|\phi_{j}\rangle.$
(48)

But (38) implies
$\sum_{i}|\langle\xi|\phi_{i}\rangle|^{2}\langle\phi_{i}|A_{2}|\phi_{i}\rangle=\langle\xi|A_{2}|\xi\rangle$,
and (41) implies
$\sum_{j}|\langle\zeta|\phi_{j}\rangle|^{2}\langle\phi_{j}|B_{2}|\phi_{j}\rangle=\langle\zeta|B_{2}|\zeta\rangle$.
Thus, (48) becomes
$\langle\xi|A_{2}|\xi\rangle\langle\zeta|B_{2}|\zeta\rangle=\langle\Psi^{\prime}|A_{2}B_{2}|\Psi^{\prime}\rangle=p(a_{2}b_{2}|\rho)$.
This confirms (34) is valid.

However, $|\Psi\rangle$ in (28) is an entangled state between Hilbert space
$\mathcal{H}_{L_{1}}$ and $\mathcal{H}_{L_{2}}$, so that (34) does not hold.