# Bayesianism, Conditional Probability and Laplace Law of Succession
in Quantum Mechanics

Tsubasa Ichikawa  Center for Quantum Information and Quantum Biology, Osaka
University, Osaka 560-0043, Japan<EMAIL_ADDRESS>
###### Abstract

We present a comparative study between classical probability and quantum
probability from the Bayesian viewpoint, where probability is construed as our
rational degree of belief on whether a given statement is true. From this
viewpoint, including conditional probability, three issues are discussed: i)
Given a measure of the rational degree of belief, does it satisfy the axioms
of the probability? ii) Given the probability satisfying these axioms, is it
seen as the measure of the rational degree of belief? iii) Can the measure of
the rational degree of belief be evaluated in terms of the relative frequency
of events occurring? Here we show that as with the classical probability, all
these issues can be resolved affirmatively in the quantum probability,
provided that the relation to the relative frequency is slightly modified from
the Laplace law of succession in case of a small number of observations. This
implies that the relation between the Bayesian probability and the relative
frequency in quantum mechanics is the same as that in the classical
probability theory, including conditional probability.

##  I Introduction

Probability theory allows two mutually opposite viewpoints on its
interpretation [1, 2]. One is the Bayesian viewpoint [3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15], where the probability is understood as our degree of
belief on how a given statement is true under incomplete information. In other
words, the probability is an epistemic concept, measuring how our subjective
knowledge is likely. The other is the ontological viewpoint, where probability
represents a definite objective property, and it can be measured by the
relative frequency of the repeatable observations.

An attempt to a reconciliation of these two viewpoints was put forward by de
Finetti [4, 5] and extended in [9, 10, 11]. The strategy is as follows: i)
Assuming that the degree of belief is measured by bets, and defining a
condition called coherence, which ensures rationality of one’s degree of
belief. ii) Proving that if the degree of belief satisfies the coherence
condition [4, 5, 9], then it satisfies the axioms of probability theory. iii)
Proving the converse of ii), that is, showing that if the degree of belief
satisfies the axiom of probability, then it fulfills the coherence condition
[10, 11]. iv) Demonstrating that the degree of belief approaches the relative
frequency for infinitely repeatable and exchangeable events for which we have
no prior information. Steps ii) and iii), called Dutch Book Argument (DBA) and
the converse DBA, respectively, ensure the equivalence between the rational
degree of belief and probability.

In the last two decades, there is much literature to derive quantum mechanical
probability (the Born rule) from the Bayesian viewpoint [16]. Among them, one
of the influential works is [17], which derived the Born rule by resorting to
the quantum analogue of the DBA and discussed the relation to the relative
frequency by using a simple combinatorial argument to independent and
identically distributed (i.i.d.) arrays of the pure states. In comparison with
de Finetti’s strategy for the reconciliation, the following three analyses are
still requisites to extend the scope of the application of the Bayesian
viewpoint in quantum mechanics (QM): the quantum analogue of the DBA including
the conditional probability, the converse DBA, and the argument about the
relative frequency for the infinitely repeatable and exchangeable events on
the quantum systems.

This paper performs these mandatory analyses and shows a coherent picture of
the Bayesian viewpoint and relative frequency in QM. More precisely, we extend
the quantum analogue of the DBA utilized in [17] in such a way to include
conditional probability and consider its converse. Through these
considerations, we show that our degree of belief satisfying the coherence
condition takes the form of the Born rule and the quantum conditional
probability associated with the Lüders projection postulate [18, 19], and the
converse DBA holds even when the non-commutativity of projectors are taken
into account. These suggest that quantum probability is fully understandable
from the Bayesian point of view.

Moreover, our analyses uncover the relation between quantum probability and
relative frequency. In the Bayesian viewpoint, (classical) probability
approaches the relative frequency through the Laplace law of succession [1]
under no prior information. We will show that this is true even in QM, with
the slight modification to the Laplace law of succession. Thus, summing up,
the present work makes the counterpart in QM to the classical argument for the
Bayesian probability and relative frequency.

The rest of the paper is organized as follows: In Sec. II, we summarize our
motivation, research questions, and the results to be obtained in this paper.
In Sec. III, we review three issues raised in the Bayesian viewpoint in
classical probability, that is, the DBA, the converse of the DBA, and the
relation between the probability and relative frequency. In Sec. IV, we extend
the DBA and its converse to QM. We derive the Born rule and the concrete
expression of quantum conditional probability by using the DBA, and give a
proof of the converse of the DBA. In Sec. V, we show that quantum probability
approaches the relative frequency via the modified Laplace law of succession,
by using the quantum de Finetti representation theorem [20]. Section VI is
devoted for our conclusion and discussions. From Sec. III to Sec. V, we
present the whole picture of the Bayesian constructions of probability theory
and QM with technical details, and readers who are uninterested in the details
may skip these sections.

##  II Backgrounds and results

As announced, this paper provides a comparative study between classical and
quantum probability from the Bayesian viewpoint. In this section, we summarize
the necessary backgrounds, the research questions, and the results to be
obtained in the following sections.

###  II.1 Motivation

Quantum mechanics is a theory describing the outcomes of measurements on
microscopic systems. Its descriptions of experiments presuppose the measured
system (object) and the measurer (subject) [21]. Heisenberg was aware of the
subject-object distinction in the standard interpretation of QM and its
consequence:

> A real difficulty in the understanding of this [Copenghagen] interpretation
> arises, however, when one asks the famous question: But what happens
> ‘really’ in an atomic event? It has been said before that the mechanism and
> the results of an observation can always be stated in terms of the classical
> concepts. But what one deduces from an observation is a probability
> function, a mathematical expression that combines statements about
> possibilities or tendencies with statements about our knowledge of facts. So
> we cannot completely objectify the result of an observation, we cannot
> describe what ‘happens’ between this observation and the next. [22]

This statement suggests that QM is a formalism that inherently and implicitly
contains information acquiring processes, which are of the measurer’s
activities.

The coexistence of subject and object is not problematic for all practical
purposes. As Heisenberg pointed out, however, it triggers conceptual
difficulty if one wishes an ontological picture to the microscopic phenomena.
For the analysis of this difficulty, the resolution of the subjective and
objective parts in QM would be desirable, as stated by Jaynes:

> But our present QM formalism is not purely epistemological; it is a peculiar
> mixture describing in part realities of Nature, in part incomplete human
> information about Nature - all scrambled up by Heisenberg and Bohr into an
> omelette that nobody has seen how to unscramble. Yet we think that the
> unscrambling is a prerequisite for any further advance in basic physical
> theory. For, if we cannot separate the subjective and objective aspects of
> the formalism, we cannot know what we are talking about; it is just that
> simple. So we want to speculate on the proper tools to do this. [13]

The tool Jaynes chose was the Bayesian probability, on the ground of
resemblance between Bohr’s reply [23] to the Einstein-Podolsky-Rosen (EPR)
argument [24] and the Bayesian inference [12]. Here Bayesian probability means
the schools of interpretations of probability, where the probability is
understood as a measure of our degree of belief on how a given statement is
true under incomplete information.

We hereafter employ the Bayesian probability, as Jaynes and QBists [17, 20,
16, 25, 26] did so. In terms of the Bayesian probability, the resolution of
the objective and subjective parts in QM could be reduced to answering the
following question: what and how much of the formalism of QM is explained by
the Bayesian probability? Something unexplainable in terms of the Bayesian
probability, if exists, could be objective. To find it, all the probabilistic
aspects of QM should be investigated. We accomplish this programme in this
paper. Note that this programme means the expansion of the scope of
application of the Bayesian viewpoint in QM too.

In passing, we point out that the Bayesian probability has strong affinities
to pragmatism, which is a broad spectrum of schools of philosophy aiming at
“knowing the world as inseparable from agency within it” [27]. Indeed, it has
been known that several founders of the Bayesian probability were influenced
by pragmatism [28], and founders of QBism also express the influence,
especially from the pragmatist James [16]. Accordingly, the present work is,
by construction, also under the strong influence of pragmatism, because we
hereafter fill the gap between the Bayesian probability and QBism, both of
which share the influence of pragmatism.

###  II.2 Problem Setting

With the above motivation, we answer the questions by filling the gap between
the present achievements of QBism and the corresponding works in classical
probability theory given mainly by de Finetti. We chose these two schools
because QBism is progressing along the Jaynes’ programme with a more
subjective approach, which is partially based on the de Finetti’s works.

To identify the gap, we first make a review of the de Finetti’s arguments in
the classical probability theory. For details, see Sec. III. We next state the
outline of QBism associated with the de Finetti’s argument. We shall compare
these and find the gap.

In the de Finetti’s argument, we first suppose that our degree of belief can
be quantified using bets. More precisely, given a bet on whether a proposition
$a$ is true, the ratio between the bettor’s (maximal) wager and the stake that
the bettor would win measures the bettor’s degree of belief on how the
statement is certain. We call this ratio the betting quotient with respect to
the proposition $a$, hereafter denoted by $q(a)$, or $q$ for short. On the
assumption of measurability of the degree of belief, we further introduce a
consistency condition, called coherence, to the bettor’s set of the betting
quotients on several bets. Here the coherence is the requirement that the set
of the betting quotients should be constructed in such a way that the bettor
wins at least one combination of the outcomes of the bets under consideration.
Note that the coherence condition implies the rationality of the bettor:
without coherence, the bettor is allowed to make the bets in such a way that
he never wins and must have a sure loss. As such, we hereafter call the degree
of belief satisfying the coherent condition the coherent degree of belief or
the rational degree of belief.

####  II.2.1 Dutch Book Argument.

Despite its simplicity, the coherence condition derives all the axioms of the
probability theory. In other words, the set of the betting quotients
satisfying the coherence condition fulfills

$\displaystyle q\geq 0,$ (1a) $\displaystyle q(a\lor b)=q(a)+q(b)\quad{\rm
if}\quad a\land b=\emptyset,$ (1b) $\displaystyle q(\Omega)=1,$ (1c)
$\displaystyle q(a\land b)=q(a|b)q(b).$ (1d)

The argument deriving Eqs. (1) is called the DBA. According to our motivation,
the DBA shows that classical probability is subjective and it has no objective
portion, since the rational degree of belief explains the axioms of the
probability theory (1). Here, $\Omega$ and $\emptyset$ are the always true
proposition (tautology) and the always false proposition (contradiction),
respectively. Together with the NOT ($\lnot$) operation, AND operation
($\land$) and OR operation ($\lor$), the set of all the propositions forms the
Boolean algebra. The quantity $q(a|b)$ is the conditional betting quotient of
the proposition $a$ given $b$, which will be defined in Sec. III by
introducing conditional bettings.

####  II.2.2 Converse Dutch Book Argument.

Moreover, we can strengthen the above observation by proving the converse of
the DBA [11, 10], that is, any set of the betting quotients is coherent if it
satisfies Eqs. (1). Together with the DBA, the converse DBA shows that nothing
is inconsistent to assume that the probability is exactly the same as the
one’s rational degree of belief. Note that the converse DBA deals with any set
of betting quotients including those for conditional bettings. Thus, the DBA
with the conditional bettings and its coherence condition is a prerequisite to
proving the converse DBA.

####  II.2.3 Relative Frequency.

Since the probability is found to be subjective in our approach, the
probability assignment on a given event may depend on the one’s subjectivity,
and thereby it may become arbitrary, as far as one’s assignment obeys the
axioms of probability. In contrast, the relative frequency is uniquely
determined within the error margin and thereby considered as an objective
concept; and it has been employed in actual experiments. Bearing these in
mind, we may ask a further question that whether we can explain the relative
frequency using the Bayesian notions.

For this question, one can make an argument by considering the case that there
are independent repeatable events with constant but unknown probability $p$.
For example, consider a coin-flipping game, where the coin is biased and the
probability to find its head in a toss is given by an unknown constant
probability $p$. Then the probability to find $k$ heads in the $n$ tosses is
proportional to the binomial distribution $\tbinom{n}{k}p^{k}(1-p)^{n-k}$,
whose maximum is attained at $p=k/n$ and the probability of finding the
relative frequency $k/n$ close to $p$ approaches the unity in large $n$ limit
[7].

Although the above example properly derives the relative frequency, it is
difficult to justify the assumption of the constantness of $p$ from the
Bayesian viewpoint, since the probability assignment for $p$ may depend on
one’s subjectivity, and thereby arbitrary as stated before [4, 5]. Thus we
need to derive the relative frequency under more relaxed conditions.

Now, instead of the independent repeatable events with constant but unknown
probability, let us consider the infinitely exchangeable events for which we
have no prior information. We will give the precise definition of infinite
exchangeability in Sec. III. As to be shown later, the infinite
exchangeability assumption corresponds with introducing a probability
distribution that describes the observer’s subjective likelihood on how much
the coin is biased. On this setup, given the set ${\cal M}_{n,k}$ of the
outcomes that we find the $k$ heads in the $n$ tosses, the conditional
probability $q(k_{n+1}|{\cal M}_{n,k})$ to find the head at the $(n+1)$-th
toss is given as

$\displaystyle q(k_{n+1}|{\cal M}_{n,k})=\frac{k+1}{n+2},$ (2)

which is called the Laplace law of succession [1, 2, 4, 5, 7, 8], or Laplace
smoothing in machine learning community [29]. Clearly, the Laplace law of
succession shows that the conditional probability approaches the relative
frequency in the large $n$ limit.

####  II.2.4 Remaining issues as counterparts lacking in QBism.

QBism derived the Born rule and argued the relative frequency [17], through
the identification between the proposition $a$ in classical probability and
the projector $P$ in QM. The derivation of the Born rule using the DBA implies
that the density operator describes a subjective state of knowledge of an
observer, from which his rational degree of belief is evaluated. On the basis
of these observations, QBism provides subjective explanations of genuine
quantum phenomena such as the violation of the Bell inequality [25, 26].

The Born rule gives the concrete form of the probability with respect to the
quantum systems, but it does not give that of the conditional probability; it
is still unanswered whether the quantum conditional probability is explained
with the Bayesian notions. In comparison with the argument in probability
theory, conditional probability in QM should be also essential to develop the
converse DBA and the relative frequency derivation in QM. Moreover, the
rigorous treatment of the two-slit experiment requires the use of conditional
probability [30].

The derivation of the relative frequency in [17] assumes that an observer of
the repeatable events has the maximal knowledge of the quantum system. This
assumption allows the observer to uniquely assign a pure state to the system.
Performing single-system projective measurements on the i.i.d. array of the
assigned pure state, we obtain the multinomial distribution of the measurement
outcomes, whose maximal is attained when the probability of finding the
measurement outcome is given by the relative frequency.

The point of the above argument is the use of the maximal knowledge on a
quantum system, which ensures one’s unique pure state assignment and thereby
the probability of the measurement outcomes is inevitably constant, unlike the
classical probability. However, putting this unavoidability aside, the above
argument still formally corresponds to the classical case of infinitely
repeatable events with constant unknown probability. Thus, we may expect that
the relative frequency in QM could be derived on the ground of more relaxed
conditions.

Summing up, there are two remaining issues on the Bayesian explanation of QM:
One is the derivation of the concrete expression of the conditional
probability using DBA, followed by the converse DBA. The other is the
derivation of the relative frequency not employing the assumption of the
maximal knowledge of the quantum system. We shall perform these analyses in
Secs. IV and V.

###  II.3 Results

In this paper, we fill the gap between the classical probability and QM from
the Bayesian viewpoint by deriving three results, which are summarized as
follows. First, by extending the DBA to the conditional betting on the
projector $P$ given the outcome of the measurement of a projector $Q$, we
obtain the explicit expression of the quantum conditional probability

$\displaystyle q(P|Q)={\rm tr}(\rho_{Q}P),$ (3)

where

$\displaystyle\rho_{Q}=\frac{Q\rho Q}{{\rm tr}(\rho Q)}$ (4)

is the post-measurement state with respect to the projector $Q$ defined with
an appropriate density operator $\rho$.

The above expression was already found in connection with the Lüders
projection postulate [18, 19]. Our derivation gives an interesting
interpretation of the Lüders projection postulate from the Bayesian viewpoint:
Observer’s state of knowledge $\rho$ changes into $\rho_{Q}$ as he acquires
the knowledge of the measurement outcome with respect to $Q$. Given the
updated state of knowledge, one’s rational degree of belief is evaluated as
the conditional probability for the given quantum system. Although this
interpretation itself is of long-standing [26, 31], we gave its new derivation
in a way precisely consistent with the Bayesian viewpoint.

Moreover, we may further extend the above observation. Let us consider $n$
conditional bets on the projector $P$ conditioned by $Q_{i}$, where
$i=1,\dots,n$. Given the rational degrees of belief $q(P|Q_{i})$ and
$q(Q_{i})$, we can average these degrees of belief as

$\displaystyle\langle q(P)\rangle=g_{1}q(P|Q_{1})+\dots+g_{n}q(P|Q_{n}),$ (5)

where

$\displaystyle g_{i}=\frac{q(Q_{i})}{q(Q_{1})+\dots+q(Q_{n})}$ (6)

is the relative weight of the degree of belief that the conditioning $Q_{i}$
occurs. From the linearity of the trace, we may write

$\displaystyle\langle q(P)\rangle={\rm tr}(\rho_{\langle Q\rangle}P)$ (7)

with the use of the state of our knowledge averaged over the possible
conditionings

$\displaystyle\rho_{\langle Q\rangle}$ $\displaystyle=$ $\displaystyle
g_{1}\rho_{Q_{1}}+\dots+g_{m}\rho_{Q_{m}}=\frac{\sum_{i=1}^{m}Q_{i}\rho
Q_{i}}{\sum_{i=1}^{m}q(Q_{i})},$ (8)

which is the aggregated post-measurement state already postulated by Lüders
when the outcomes of the measurements of $\\{Q_{i}\\}_{i=1}^{n}$ are known
[18]. Thus, the coherent degree of belief can describe two post-measurement
states: Eq. (4) for the direct measurements, and Eq. (8) for the measurements
with the outcome aggregation.

Second, armed with Eqs. (3) and (4), we extend the converse DBA to quantum
systems. This is accomplished in a way parallel with the argument in classical
probability. Thus, we may conclude that even in QM, nothing is inconsistent to
assume that the probability is identical to the rational degree of belief.
Besides, since the (conditional) probability is evaluated from the density
operator, the converse DBA ensures the consistency of the interpretation that
the density operator is the state of knowledge; we can safely exclude the
“statements about possibilities or tendencies” of the system Heisenberg
addressed in the quotation.

(a) Flat measure

(b) Bures measure

Figure 1: (Color online) The correction term $I_{n+1,k+1}/I_{n,k}$ to the
Laplace law of succession for (a) the flat measure and (b) the Bures measure.
The correction term tends to unity for large $n$ in both the cases,
irrespective of the value of the relative frequency $k/n$. Note that we have
not plotted the correction term for the uniform measure of the pure state
space, since $I_{n+1,k+1}/I_{n,k}=1$ for any $n$ and $k$.

Third, we show that the Laplace law of succession holds in QM up to an overall
factor. More precisely, let us prepare an infinitely exchangeable state [20]
consisting of $n+1$ qubits and perform projective measurements
$\\{P,\openone-P\\}$ to each constituent qubit, where $\openone$ is the
identity operator of $\mathbb{C}^{2}$. Then, given the set ${\cal M}_{n,k}$ of
the outcomes that $P$ is true for $k$ times in the $n$ measurements, the
conditional probability of $P$ being true for the $(n+1)$-th measurement takes
the form of

$\displaystyle q(k_{n+1}|{\cal
M}_{n,k})=\frac{k+1}{n+2}\frac{I_{n+1,k+1}}{I_{n,k}},$ (9)

for three measures of quantum states we have investigated, i.e. the uniform
measure on the pure state space, the flat measure, and the Bure measure [32,
33], all of which may represent our ignorance on the system. Here, the overall
factor $I_{n+1,k+1}/I_{n,k}$, hereafter called the correction term, is a
polynomial of $n$ and $k$ determined by the measure we choose. For example,
the uniform measure on the pure state space yields $I_{n+1,k+1}/I_{n,k}=1$,
suggesting that the eigenvalue distribution of the density operator
contributes to the nontrivial correction term. Note that the infinitely
exchangeable state does not correspond with the maximal knowledge of the
system, since it is not an i.i.d. array of a pure state in general.

Figure 1 shows that the correction terms tend to unity as the number of
measurements $n$ increases with relative frequency $k/n$ fixed. Here we have
used the flat measure and Bures measure. Quantum conditional probability
clearly approaches the Laplace law of succession in the large $n$ limit. This
implies that our degree of belief in terms of conditional probability obeys
the correspondence principle, and can be evaluated with the relative frequency
in a large number of the measurements.

As a technical remark, we point out that the correction term for the Bures
measure yields a smaller deviation from the Laplace law of succession than
that for the flat measure. This is understood by the fact that the Bures
measure is localized on the states of high purity [32], whence the behavior of
the conditional probability should resemble that of the uniform measure on the
pure state space, yielding the Laplace law of succession.

These results, as a whole, formally show that the probabilistic aspects of QM
are subjective as much as probability theory, even taking the conditional
probability and the Lüders projection postulate into account. Nonetheless, we
may employ the relative frequency even without invoking the maximal
information of the system. We hereafter give the technical details of the
derivations, with a review of the arguments in classical probability theory.

##  III Arguments in Probability Theory

In the Bayesian viewpoint, probability is construed as a measure of our degree
of belief on how a given statement is true under incomplete information. To
develop this idea precisely, the DBA was proposed by de Finetti [4, 5]. Since
then, the DBA has given rise to a myriad of succeeding research not only for
classical probability [9, 10, 11, 2] but also for QM [16, 17].

In the DBA, our degree of belief is supposed to be measured using bets. We
consider a bet played by a bettor and a bookmaker on whether a given
proposition $a$ is true. If the proposition is true, then the bettor obtains
the stake $S$, which is a real number. Otherwise, he obtains nothing. To
participate in the bet, the bettor pays the wager $qS$, where $q$ is the real
number called the betting quotient. The betting quotient $q$ plays a role as a
measure of bettor’s degree of belief on how much the proposition is certain.
Depending on the betting outcome, the stake is paid to the bettor. The payoff
$G$ is given by $G=S-qS$ when the proposition is true, whereas $G=-qS$ when it
is false. The betting quotient, stake, and payoff with respect to the
proposition $a$ are denoted by $q(a)$, $S(a)$ and $G(a)$, or $q_{a}$, $S_{a}$
and $G_{a}$ for short, respectively.

Let us consider the case that the bettor and the bookmaker make bets on
several propositions $a_{1},a_{2},\dots,a_{n}$. As before, the bettor assigns
the betting quotient $q_{i}$ for the proposition $a_{i}$, pays the wager
$q_{i}S_{i}$, and obtains the stake $S_{i}$ when the proposition $a_{i}$ is
true. Then, the set of the betting quotients $q_{1},q_{2},\dots,q_{n}$ is
called coherent [5, 9] if the bettor wins at least one combination of the
truth values of $a_{i}$. In other words, the payoff is non-negative with at
least one combination of the truth values of $a_{i}$ for any sets of the
stakes $S_{i}$. In contrast, the set of the bets is called Dutch book if the
bettor loses in any combinations of the betting outcomes.

We hereafter deal with the betting for the composite propositions. Given the
propositions, say $a_{1}$ and $a_{2}$, by using the Boolean operations
AND($\land$), OR($\lor$), NOT($\lnot$), we can make the composite propositions
$a_{1}\land a_{2}$, $a_{1}\lor a_{2}$, $\lnot a_{i}$, respectively. Here the
Boolean operations are defined by the following properties [34]:

  1. 1.

Commutativity

$\displaystyle a_{1}\land a_{2}=a_{2}\land a_{1},\qquad a_{1}\lor
a_{2}=a_{2}\lor a_{1}.$ (10)

  2. 2.

Distributivity

$\displaystyle a_{1}\land(a_{2}\lor a_{3})$ $\displaystyle=$
$\displaystyle(a_{1}\land a_{2})\lor(a_{1}\land a_{3}),$ $\displaystyle
a_{1}\lor(a_{2}\land a_{3})$ $\displaystyle=$ $\displaystyle(a_{1}\lor
a_{2})\land(a_{1}\lor a_{3}).$ (11)

  3. 3.

Identity

$\displaystyle a_{1}\land\Omega=a_{1},\qquad a_{1}\lor\emptyset=a_{1}.$ (12)

  4. 4.

Complements

$\displaystyle a_{1}\land\lnot a_{1}=\emptyset,\qquad a_{1}\lor\lnot
a_{1}=\Omega.$ (13)

Equations (12) show that $\emptyset$ and $\Omega$ are the identities with
respect to the OR operation and AND operation, respectively. We regard these
identities as also propositions: $\Omega$ is the always true proposition
(tautology), whereas $\emptyset$ is the always false proposition
(contradiction). Note that $\lnot\Omega=\lnot\Omega\land\Omega=\emptyset,$ and
$\lnot\emptyset=\lnot\emptyset\lor\emptyset=\emptyset\lor\lnot\emptyset=\Omega$,
from Eqs. (10), (12), (13).

Let the propositions $a_{1},a_{2},\dots,a_{n}$ be mutually exclusive and
exhaustive: $a_{i}\land a_{j}=\emptyset$ for $i\neq j$, and $a_{1}\lor
a_{2}\lor\dots\lor a_{n}=\Omega$. Let $A$ be the set whose elements are
$\emptyset,\Omega,a_{1},a_{2},\dots,a_{n}$, and all the composite propositions
thereof. It has been shown in [4, 5, 9, 10, 11, 2, 17] that if the set of the
betting quotients with respect to the elements of $A$ are coherent, then it
satisfies the axioms of the probability: as a function of the propositions,
the betting quotient $q(\cdot)$ is non-negative (1a), additive with respect to
the mutually exclusive propositions (1b), and yields unity on the tautology
$\Omega$ (1c). To make the description self-contained, we hereafter show
these.

We first show the non-negativity (1a) of $q$, if the betting quotient $q$ is
coherent. This is shown as follows: Assume $q<0$. Then by setting $S<0$, the
payoff is $G=S-qS<0$ for the proposition being true, and $G=-qS<0$ for that
being false. Thus, $q<0$ leads to the Dutch book situation. By taking the
contraposition of this observation, we can find Ineq. (1a) if the betting
quotient is coherent.

Second, we prove the addition law of probability (1b), for mutually exclusive
(composite) propositions $a,b$. To show this, we consider the bettings on the
propositions $a,b$, and $c=a\lor b$. Depending on which combination of the
propositions is found to be true, the payoff is given as follows:

$\displaystyle G_{a}$ $\displaystyle=$
$\displaystyle(1-q_{a})S_{a}-q_{b}S_{b}+(1-q_{c})S_{c}\quad\,\,\,\,\text{if
$a$ is true},$ $\displaystyle G_{b}$ $\displaystyle=$ $\displaystyle-
q_{a}S_{a}+(1-q_{b})S_{b}+(1-q_{c})S_{c}\quad\text{if $b$ is true},$
$\displaystyle G_{c}$ $\displaystyle=$ $\displaystyle-
q_{a}S_{b}-q_{b}S_{b}-q_{c}S_{c}\qquad\qquad\quad\text{if $c$ is false}.$ (14)

Two remarks are in order: First, $a\land b$ is always false, since the
propositions $a$ and $b$ are mutually exclusive. Second, Eqs. (14) comprise a
system of linear equations from the set of stakes to that of the possible
payoffs.

Now we come to the point. If Eqs. (14) have the solution as the linear system,
the bookmaker is allowed to adjust his stakes in such a way that all the
possible payoffs $G_{a},G_{b},G_{c}$ are negative, resulting in the Dutch book
situation. Taking the contraposition of this observation, we observe that the
following must hold if the set of the betting quotients is coherent:

$\displaystyle\left|\begin{array}[]{ccc}1-q_{a}&-q_{b}&1-q_{c}\\\
-q_{a}&1-q_{b}&1-q_{c}\\\
-q_{a}&-q_{b}&-q_{c}\end{array}\right|=q_{a}+q_{b}-q_{c}=0,$ (18)

showing that Eq. (1b) holds if the set of the betting quotient
$\\{q_{a},q_{b},q_{c}\\}$ is coherent.

Third, we show that for the mutually exclusive and exhaustive propositions
$a_{1},a_{2},\dots,a_{n}$,

$\displaystyle q(a_{1})+q(a_{2})+\dots+q(a_{n})=1$ (19)

holds if the betting quotients are coherent. To prove this, we consider the
bettings on the propositions $a_{1},a_{2},\dots,a_{n}$. Since these
propositions are mutually exclusive, the possible payoffs are given by

$\displaystyle G_{i}=(1-q_{i})S_{i}-\sum_{j\neq i}q_{j}S_{j}\quad\text{if
$a_{i}$ is true},$ (20)

for all $i$. Here we have introduced abbreviated notation $q_{i}=q(a_{i})$ and
$S_{i}=S(a_{i})$. Similarly to the derivation of Eq. (1b), the bookmaker can
realize the Dutch book situation unless

$\displaystyle\left|\begin{array}[]{cccc}1-q_{1}&-q_{2}&\cdots&-q_{n}\\\
-q_{1}&1-q_{2}&\cdots&-q_{n}\\\ \vdots&\vdots&\ddots&\vdots\\\
-q_{1}&-q_{2}&\cdots&1-q_{n}\end{array}\right|=1-\sum_{i=1}^{n}q_{i}=0,$ (25)

which leads to Eq. (19). Furthermore, by using Eq. (1b) and Eq. (19), we find

$\displaystyle q(\Omega)$ $\displaystyle=$ $\displaystyle q(a_{1}\lor
a_{2}\lor\dots\lor a_{n})$ (26) $\displaystyle=$ $\displaystyle
q(a_{1})+q(a_{2})+\dots+q(a_{n})$ $\displaystyle=$ $\displaystyle 1,$

which is one of the axioms of the probability (1c) announced before. Moreover,
since the betting quotients are non-negative, Eq. (26) implies that $0\leq
q\leq 1$ for all the betting quotients $q$, including those for the composite
propositions. Together with Eqs. (1b) and (26), we can conclude that the
betting quotient obeys the axioms of probability.

So far, we have used the DBA to show that the coherent betting quotient is
probability. Not only this, we hereafter apply the DBA to argue the
conditional probability. For this, let us introduce conditional betting
comprised of two propositions $a$ and $b$. In conditional betting, the
bookmaker first reveals whether $b$ is true. If $b$ is true, the bookmaker
proceeds to reveal whether $a$ is true. If $b$ is false, the bet is called off
and all the stakes and wagers associated with the conditioning by $b$ are
returned.

On this regulation, suppose that we are given three bets: $b$, $a\land b$, and
$a$ conditioned by whether $b$ is true. Let $q(a|b)$ and $S(a|b)$ be the
betting quotient and stake associated with $a$ conditioned by $b$,
respectively. Further we introduce the shorthand notations $q=q(a\land b)$,
$q^{\prime}=q(a|b)$, $q^{\prime\prime}=q(b)$ and $S=S(a\land b)$,
$S^{\prime}=S(a|b)$, $S^{\prime\prime}=S(b)$. The possible payoffs are written
as follows:

$\displaystyle G_{1}$ $\displaystyle=$
$\displaystyle(1-q)S+(1-q^{\prime})S^{\prime}+(1-q^{\prime\prime})S^{\prime\prime}\,\text{if
$a\land b$ is true},$ $\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle-
qS-q^{\prime}S^{\prime}+(1-q^{\prime\prime})S^{\prime\prime}\qquad\qquad\,\,\text{if
$\lnot a\land b$ is true},$ $\displaystyle G_{3}$ $\displaystyle=$
$\displaystyle-
qS-q^{\prime\prime}S^{\prime\prime}\qquad\qquad\qquad\qquad\quad\,\,\,\,\,\text{if
$b$ is false}.$ (27)

As before, the bets become Dutch book unless

$\displaystyle\left|\begin{array}[]{ccc}1-q&1-q^{\prime}&1-q^{\prime\prime}\\\
-q&-q^{\prime}&1-q^{\prime\prime}\\\
-q&0&-q^{\prime\prime}\end{array}\right|=q^{\prime}q^{\prime\prime}-q=0,$ (31)

which results in the multiplication law of probability (1d), showing that the
conditional betting quotient $q(a|b)$ is the conditional probability. Note
that

$\displaystyle q(a)=q(a|\Omega),$ (32)

by setting $b=\Omega$ in Eq. (1d).

We now turn to the converse DBA [10, 11, 2]: In other words, we show that if
the set of the betting quotients satisfies the axiom of the probability, then
the betting quotients are coherent. This ensures that our degree of belief is
identical to (conditional) probability if it is coherent.

To show this, let us give a general form of the payoff of the $n$ conditional
bets. Note that it suffices to consider the set of the conditional bets, since
the outright bets are thought of as the conditional bets with tautology
$\Omega$, due to Eq. (32). Each conditional bet consists of the proposition
$a_{i}$ conditioned by $b_{i}$, whose betting quotient and stake are denoted
by $q(a_{i}|b_{i})$ and $S_{i}$, respectively.

Now consider $2^{2n}$ mutually exclusive composite propositions

$\displaystyle\omega$ $\displaystyle=$ $\displaystyle c_{1}\land\dots\land
c_{n}\land d_{1}\land\dots\land d_{n},$ (33)

where $c_{i}\in\\{a_{i},\lnot a_{i}\\}$ and $d_{i}\in\\{b_{i},\lnot b_{i}\\}$.
The proposition $\omega$ should be understood as a collective expression of
the outcomes of the $n$ conditional bets. Given the proposition $\omega$,
then, the payoff of the conditional bets is written as

$\displaystyle G(\omega)=\sum_{i\in
T(\omega)}\left[1-q(a_{i}|b_{i})\right]S_{i}-\sum_{i\in
F(\omega)}q(a_{i}|b_{i})S_{i},$ (34)

where we have introduced the sets of indices $i$ such that

$\displaystyle T(\omega)$ $\displaystyle=$
$\displaystyle\\{i\,|\,c_{i}=a_{i}\,\,{\rm and}\,\,d_{i}=b_{i}\\},$
$\displaystyle F(\omega)$ $\displaystyle=$ $\displaystyle\\{i\,|\,c_{i}=\lnot
a_{i}\,\,{\rm and}\,\,d_{i}=b_{i}\\}.$ (35)

While $a_{i}\land b_{i}$ is found to be true in the conditional bet $i\in
T(\omega)$, $\lnot a_{i}\land b_{i}$ is found to be true in the conditional
bet $i\in F(\omega)$.

Following the standard procedure to show the converse DBA, we introduce the
average payoff over the composite propositions $\omega$:

$\displaystyle\langle G\rangle=\sum_{\omega}q(\omega)G(\omega).$ (36)

Rewriting the average payoff as the function of the stakes, we obtain

$\displaystyle\langle G\rangle=x_{1}S_{1}+\dots+x_{n}S_{n},$ (37)

where

$\displaystyle x_{i}=\left[1-q(a_{i}|b_{i})\right]\sum_{\omega:i\in
T(\omega)}q(\omega)-q(a_{i}|b_{i})\sum_{\omega:i\in F(\omega)}q(\omega).$ (38)

Here the first sum is taken over $\omega$ on which the given proposition
$a_{i}\land b_{i}$ is true, whereas the second sum is done over $\omega$ on
which $\lnot a_{i}\land b_{i}$ is true. The first sum reduces to $q(a_{i}\land
b_{i})$ since the elements of the set $\\{\omega\\}$ of the $2^{2n}$
propositions are mutually exclusive. Similarly, the second sum reduces to
$q(\lnot a_{i}\land b_{i})$. It follows from Eqs. (11), (1b), and (1d) that

$\displaystyle x_{i}$ $\displaystyle=$
$\displaystyle\left[1-q(a_{i}|b_{i})\right]q(a_{i}\land
b_{i})-q(a_{i}|b_{i})q(\lnot a_{i}\land b_{i})$ (39) $\displaystyle=$
$\displaystyle q(a_{i}\land b_{i})-q(a_{i}|b_{i})\left[q(a_{i}\land
b_{i})+q(\lnot a_{i}\land b_{i})\right]$ $\displaystyle=$ $\displaystyle
q(a_{i}\land b_{i})-q(a_{i}|b_{i})q(b_{i})$ $\displaystyle=$ $\displaystyle 0$

for all $i$, which leads to $\langle G\rangle=0$. Since $q(\omega)\geq 0$ and
$\sum_{\omega}q(\omega)=1$, not all payoffs $G(\omega)$ are negative, showing
coherence. This completes the proof of the converse DBA.

Two remarks are in order. First, the proposition $a_{i}$ may be the same
proposition as $a_{j}$ for some $j\neq i$, and $b_{i}$ is also the same as
$b_{j}$. This implies that the outcomes of the bets on a given proposition may
change in the $n$ bets. In other words, the outcome $c_{i}$ of the $i$-th bet
may differ from $c_{j}$ even though $a_{i}=a_{j}$, and $d_{i}$ also may differ
from $d_{j}$ even when $b_{i}=b_{j}$. Second, we can complete the proof of the
converse DBA even using

$\displaystyle
q^{\prime}(\omega)=\prod_{i=1}^{n}q(c_{i}|d_{i})q(d_{i})=\prod_{i=1}^{n}q(c_{i}\land
d_{i}),$ (40)

instead of $q(\omega)$. The possible correlation among the outcomes of the
mutually distinct conditional bettings in the joint probability $q(\omega)$ is
irrelevant to show the converse DBA. This alternative proof of the converse
DBA shall be used in Sec. IV.

The arguments so far have shown that our rational degree of belief measured by
the coherent betting quotient is equivalent to the probability. Now we move to
discuss how the rational degree of belief is related to the relative
frequency, which provides another relevant interpretation of the probability:
the frequency interpretation [2]. To argue this, let us consider a coin-
flipping game for simplicity and introduce infinite exchangeability of the
trials [5, 2, 20], which is the prerequisite of the de Finetti representation
theorem [5, 2, 20].

Let $k_{i}=0,1$ be the outcome of the $i$-th coin flipping. The degree of
belief $q(k_{1},\dots,k_{n})$ to obtain the outcomes $k_{1},\dots,k_{n}$ is
infinitely exchangeable if $q(k_{1},\dots,k_{n})$ is symmetric under all the
permutations, that is,

$\displaystyle q(k_{1},\dots,k_{n})=q(k_{\sigma(1)},\dots,k_{\sigma(n)})$ (41)

for any elements $\sigma\in\mathfrak{S}_{n}$ of the symmetric group
$\mathfrak{S}_{n}$ and there exists a symmetric distribution
$q(k_{1},\dots,k_{n+m})$ such that

$\displaystyle q(k_{1},\dots,k_{n})$
$\displaystyle=\sum_{k_{n+1},\dots,k_{n+m}}q(k_{1},\dots,k_{n},k_{n+1},\dots,k_{n+m})$
(42)

for all positive integer $m$. Then the de Finetti representation theorem
states that there exists a probability measure $d\mu(p)$ such that

$\displaystyle q(k_{1},\dots,k_{n})=\int_{0}^{1}d\mu(p)p^{k}(1-p)^{n-k},$ (43)

where $p$ is the probability for the outcome being $1$, and $k$ is the number
of the events whose outcomes are $1$. See [20] for the proof. The probability
measure $d\mu(p)$ is a prior probability distribution representing our initial
knowledge on the outcome distribution of the coin-flipping games.

It follows from the de Finetti representation theorem and Eq. (1d) that, given
the set ${\cal M}_{n,k}$ of the outcomes that the $k$ heads are found in the
$n$ tosses, the conditional probability distribution of $k_{n+1}=1$ takes the
form

$\displaystyle q(k_{n+1}|{\cal
M}_{n,k})=\frac{\int_{0}^{1}d\mu(p)p^{k+1}(1-p)^{n-k}}{\int_{0}^{1}d\mu(p)p^{k}(1-p)^{n-k}}.$
(44)

Assuming that we have no prior knowledge of the outcome distribution of the
coin-flipping game, we take the uniform distribution $d\mu(p)=dp$ as the
prior. We readily find

$\displaystyle q(k_{1},\dots,k_{n})=B(n-k+1,k+1),$ (45)

where $B(n,k)$ is the Beta function

$\displaystyle B(n,k)=\int_{0}^{1}dpp^{n-1}(1-p)^{k-1}.$ (46)

Since $B(n,k)=(n-1)!(k-1)!/(n+k-1)!$, we then arrive at Eq. (2), which is the
Laplace law of succession [1, 2, 4, 5, 7, 8]. Thus, in large $n$ and $k$, the
degree of belief $q(k_{n+1}|{\cal M}_{n,k})$ approaches the relative frequency
$k/n$ under no prior knowledge situation.

##  IV Quantum Dutch Book Argument and its Converse

In this section, we extend the argument given in Sec. III to QM. We shall find
that the quantum probability, as well as conditional probability, are derived
from the DBA, and the converse DBA holds. These suggest that our coherent
degree of belief on the quantum system is equivalent to quantum (conditional)
probability.

Let ${\cal H}$ be a $d$-dimensional Hilbert space ($d<\infty$), and $P$ be a
projector acting on ${\cal H}$. ${\cal L}({\cal H})$ denotes the set of all
the projectors acting on ${\cal H}$. Every projector can be treated as a
proposition because the eigenvalues of the projectors are either 1 or 0, seen
as the truth values. In the same way as Sec. III, we introduce the betting
quotient $q(P)$, which measures our degree of belief for the proposition $P$
being true on a quantum system in ${\cal H}$ under consideration.

Suppose that we are given the propositions $P$ and $Q$. Then the composite
projectors $P\land Q$ and $P\lor Q$ are defined as the projectors to the meet
and join of the range of $P$ and that of $Q$, respectively. The negation
$\lnot P$ is the projector on the orthogonal complement of the range of $P$.
In particular, if $P$ and $Q$ commute, then

$\displaystyle P\land Q=PQ,\quad P\lor Q=P+Q-PQ,\quad\lnot P=\mathbbm{1}-P,$
(47)

where $\mathbbm{1}$ is the identity operator on ${\cal H}$. For the commuting
projectors $P,Q$ such that $[P,Q]=0$, we define $P\leq Q$, which means
$PQ=QP=P$.

To proceed, we hereafter assume that the betting quotient is coherent for the
bettings comprised of commuting projectors. The DBA in Sec. III is then
readily applied to the bets made of the commuting projectors. Indeed, the Born
rule was derived in [17] as follows: We derive

$\displaystyle q(P)\geq 0,\qquad q(\openone)=1,$ $\displaystyle q(P\lor
Q)=q(P)+q(Q)$ (48)

for the mutually exclusive commuting projectors such that $P\land Q=PQ=0$, by
repeating the DBA with formal replacement of the propositions in the previous
section by the projectors. Since Eqs. (48) are sufficient conditions of the
Gleason theorem [35], we obtain the Born rule

$\displaystyle q(P)={\rm tr}(\rho P),$ (49)

where $\rho$ is a density operator acting on ${\cal H}$.

In parallel with the Born rule, we can derive a concrete expression of the
degree of belief for conditional betting on the projectors. We begin with
defining the conditional betting of $P$ conditioned by $Q$ in a way similar to
the classical one, even though $P$ and $Q$ do not commute: The bookmaker first
measures $Q$ on a quantum system and reveals whether $Q$ is true. If $Q$ is
true, the bookmaker further measures $P$ on the system after the measurement
of $Q$ and reveals whether $P$ is true. If $Q$ is false, the bet is called off
as in the case of classical conditional betting. We denote by $S(P|Q)$ the
stake of the bet on $P$ conditioned by $Q$.

Let $q(P|Q)$ be the degree of belief for the projector $P$, given that $Q$ is
true. We now make two assumptions: First, $q(P|Q)$ is well-defined even for
not necessarily commuting projectors $[P,Q]\neq 0$. Second, similarly to
$q(\cdot)$ for the outright bet, $q(\cdot|Q)$ is coherent for the conditional
bettings comprised of commuting projectors conditioned by another projector
$Q$. Note that the projector $Q$ used for the conditioning is not necessarily
commuting with $P$. For example, given two conditional bettings, $P_{1}$
conditioned by $Q$ and $P_{2}$ conditioned by $Q$, we require coherence only
when $P_{1}$ and $P_{2}$ commute.

On the basis of these assumptions, it is straightforward to show that the
degree of belief $q(P|Q)$ is the probability distribution when $Q$ is fixed.
More precisely, $q(P|Q)$ satisfies i) $q(P|Q)\geq 0,$ ii) $q(\openone|Q)=1$
for any pairs of $P,Q$, and iii) the additivity

$\displaystyle q(P\lor Q|R)=q(P|R)+q(Q|R),$ (50)

for any projectors $P,Q,R$ such that $P\land Q=PQ=0$.

Similarly to the derivation of Ineq. (1a), let us show $q(P|Q)\geq 0$:
Consider the conditional bet on $P$ conditioned by $Q$ and suppose that the
proposition $Q$ is found to be true. If the proposition $P$ is also found to
be true and $q(P|Q)<0$, the payoff is given as $S(P|Q)-q(P|Q)S(P|Q)$, which is
negative when the bookmaker chooses $S(P|Q)<0$. On the other hand, if the
proposition $P$ is also found to be false and $q(P|Q)<0$, the payoff is given
as $-q(P|Q)S(P|Q)$, which is also negative when the bookmaker chooses
$S(P|Q)<0$. Thus, the payoff is always negative regardless of whether $P$ is
true or false, suggesting the Dutch book.

Next, we show the additivity $q(P\lor Q|R)=q(P|R)+q(Q|R)$ for the projectors
such that $PQ=0$: Consider the conditional bets $P$, $Q$ and $P\lor Q$ given
the condition $R$. Since all the conditional bets are called off if $R$ is
false, it suffices to consider the case that $R$ is found to be true, where
the payoffs take the form of Eqs. (14) with suitable replacements of the
indices, and thereby the coherence condition yields the law of the additivity
for the conditional bets. Note that $q(\openone|Q)=1$ is also derived in the
same manner. These properties of the conditional betting quotient show that
$q(P|Q)$ is the probability distribution when $Q$ is fixed, as in the case of
the outright betting quotient $q(P)$.

Having found that $q(\cdot|Q)$ is the probability distribution, we consider
three bets consisting of the commuting projectors $P$ and $Q$: $Q$, $P\land
Q=PQ$, and $P$ conditioned by whether $Q$ is true. Since $P$ and $Q$ commute,
the argument in the previous section can be applied to the present bets, and
thereby we obtain

$\displaystyle q(P\land Q)=q(P|Q)q(Q)$ (51)

for any pairs of commuting projectors. Note that

$\displaystyle q(P|Q)=\frac{q(P)}{q(Q)},\quad{\rm if}\,\,P\leq Q\,\,{\rm
and}\,\,q(Q)\neq 0,$ (52)

since $P\leq Q$ implies $P\land Q=PQ=P$.

With Eq. (52) for the degree of belief $q(P|Q)$, we employ the following
lemma:

######  Lemma 1 (Beltrametti and Cassinelli [30, 36, 37])

For Hilbert space ${\cal H}$, let $q(\cdot)$ be a probability measure on
${\cal L}({\cal H})$, and let $Q$ be any projector such that $q(Q)\neq 0$.
Then there exists a unique probability measure on ${\cal L}({\cal H})$, which
we denote by $q(\cdot|Q)$, such that for all projectors $P\leq Q$ it is the
case that $q(P|Q)=q(P)/q(Q)$.

This lemma ensures the uniqueness of the degree of belief $q(P|Q)$ as the
conditional probability distribution. Due to the additivity $q(P\lor
Q|R)=q(P|R)+q(Q|R)$ for the projectors such that $PQ=0$, we apply the Gleason
theorem to $q(\cdot|Q)$, and find that there exists an appropriate density
operator $\rho_{Q}$ with which the coherent degree of belief takes the form of
Eq. (3). Since $q(Q|Q)=1$ from Eq. (52), we find that ${\rm tr}(\rho_{Q}Q)=1$,
which suggests that the density operator $\rho_{Q}$ has non-vanishing support
only on the range of $Q$. Hence, there exists a density operator $\rho$ such
that $\rho_{Q}$ takes the form (4), from which we obtain the concrete
expression of the coherent degree of belief

$\displaystyle q(P|Q)=\frac{{\rm tr}(Q\rho QP)}{{\rm tr}(\rho Q)}.$ (53)

As stated in Sec. II, our derivation of quantum conditional probability leads
to the Bayesian interpretation of the Lüders projection postulate.

The argument developed so far is the quantum analogue of the DBA, showing that
our degree of belief on what occurs in quantum systems takes the forms of the
Born rule as well as the quantum conditional probability (3) on the assumption
of the coherence of the (conditional) betting quotients of commuting
projectors. We can further show the converse DBA in QM: the Born rule and the
quantum conditional probability (3) lead to the coherence of the betting
quotients.

The proof is completely parallel to the argument used for classical
probability: As in Sec. III, we begin with a general form of the payoff of the
$n$ conditional bets on projectors. Since $q(P|\openone)=q(P)$, it is
sufficient to consider the set of conditional bets. Each conditional bet
consists of the proposition $P_{i}$ conditioned by $Q_{i}$, whose betting
quotient and stake are denoted by $q(P_{i}|Q_{i})$ and $S_{i}$, respectively.

Let $w$ be a combination of the outcomes of the $n$ conditional bets such that

$\displaystyle\\!\\!w=(v_{1}(P_{1}),\dots,v_{n}(P_{n}),v_{n+1}(Q_{1}),\dots,v_{2n}(Q_{n})),$
(54)

where we have introduced the outcome

$\displaystyle v_{i}(P)\in\\{0,1\\},$ (55)

for $i=1,\dots,2n$. Associated with this, we define

$\displaystyle\tau_{i}(P)=\begin{cases}P&\text{if\,\,\,}v_{i}(P)=1,\\\ \lnot
P&\text{if\,\,\,}v_{i}(P)=0.\end{cases}$ (56)

Given the combination $w$, then, the payoff of the conditional bets is written
as

$\displaystyle G(w)=\sum_{i\in
T(w)}\left[1-q(P_{i}|Q_{i})\right]S_{i}-\sum_{i\in F(w)}q(P_{i}|Q_{i})S_{i},$

where we have introduced the sets of indices $i$ such that

$\displaystyle T(w)$ $\displaystyle=$
$\displaystyle\\{i\,|\,v_{i}(P_{i})=1\,\,{\rm and}\,\,v_{n+i}(Q_{i})=1\\},$
$\displaystyle F(w)$ $\displaystyle=$
$\displaystyle\\{i\,|\,v_{i}(P_{i})=0\,\,{\rm and}\,\,v_{n+i}(Q_{i})=1\\},$
(58)

which are analogues of $T(\omega)$ and $F(\omega)$ in Sec. III.

We take the average of the payoff $G(w)$ over the combinations of the outcomes
of the conditional bets $w$:

$\displaystyle\langle G\rangle_{\rm QM}=\sum_{w}q(w)G(w),$ (59)

where

$\displaystyle
q(w)=\prod_{i=1}^{n}q(\tau_{i}(P_{i})|\tau_{n+i}(Q_{i}))\,q(\tau_{n+i}(Q_{i}))$
(60)

is the probability to obtain the outcomes $w$ in the $n$ conditional bets.
Indeed, we observe

$\displaystyle q(w)\geq 0,\qquad\sum_{w}q(w)=1$ (61)

from

$\displaystyle q(P|Q)+q(\lnot P|Q)=1,\qquad q(P)+q(\lnot P)=1.$ (62)

We rewrite the average payoff as the function of the stakes, and obtain

$\displaystyle\langle G\rangle_{\rm QM}=x_{1}S_{1}+\dots+x_{n}S_{n},$ (63)

where

$\displaystyle x_{i}=\left[1-q(P_{i}|Q_{i})\right]\sum_{w:i\in
T(w)}q(w)-q(P_{i}|Q_{i})\sum_{w:i\in F(w)}q(w).$ (64)

Here the first sum is over $w$ with $v_{i}(P_{i})=v_{n+i}(Q_{i})=1$, whereas
the second sum is over $w$ with $v_{i}(P_{i})=0$ and $v_{n+i}(Q_{i})=1$. Since

$\displaystyle\sum_{w:i\in T(w)}q(w)$ $\displaystyle=$ $\displaystyle
q(P_{i}|Q_{i})q(Q_{i}),$ $\displaystyle\sum_{w:i\in F(w)}q(w)$
$\displaystyle=$ $\displaystyle q(\lnot P_{i}|Q_{i})q(Q_{i}),$ (65)

from Eqs. (62), by plugging Eqs. (65) into Eq. (64), we observe

$\displaystyle x_{i}$ $\displaystyle=$
$\displaystyle\left[1-q(P_{i}|Q_{i})\right]q(P_{i}|Q_{i})q(Q_{i})$ (66)
$\displaystyle-q(P_{i}|Q_{i})q(\lnot P_{i}|Q_{i})q(Q_{i})$ $\displaystyle=$
$\displaystyle\left[1-q(P_{i}|Q_{i})-q(\lnot
P_{i}|Q_{i})\right]q(P_{i}|Q_{i})q(Q_{i})$ $\displaystyle=$ $\displaystyle 0$

for all $i$, from Eqs. (62). This leads to $\langle G\rangle_{\rm QM}=0$.
Equations (61) then imply that not all payoffs $G(w)$ are negative, showing
coherence. This shows that the converse DBA holds in QM.

##  V Connection to Relative Frequency

We turn to discuss the relation between the conditional degree of belief
$q(P|Q)$ and relative frequency. Similarly to classical probability, we argue
this with the measurements of two-level systems, by introducing the quantum de
Finetti representation theorem [20], which requires exchange symmetry on the
density operators.

Consider a tensor product Hilbert space ${\cal K}^{n}={\cal H}^{\otimes n}$
with ${\cal H}=\mathbbm{C}^{N}$. Given a permutation $\sigma$ of the symmetric
group $\mathfrak{S}_{n}$, let us suppose that there exists a self-adjoint
operator $\pi_{\sigma}$ such that

$\displaystyle\pi_{\sigma}\bigotimes_{i=1}^{n}|e_{i}\rangle_{i}=\bigotimes_{i=1}^{n}|e_{i}\rangle_{\sigma(i)},$
(67)

where $\\{|e_{i}\rangle_{i}\\}$ is a complete orthonormal basis of the $i$-th
constituent Hilbert space. We define the totally symmetric subspace of the
tensor product Hilbert space ${\cal K}^{n}$ as

$\displaystyle{\cal K}_{\cal S}^{n}=\\{|\psi\rangle\in{\cal
K}^{n}\,|\,\pi_{\sigma}|\psi\rangle=|\psi\rangle,\,\forall\sigma\in\mathfrak{S}_{n}\\}.$
(68)

Let $\rho_{n}$ be a density operator whose non-vanishing support is only on
${\cal K}_{\cal S}^{n}$. We call such density operator a symmetric density
operator. The symmetric density operator $\rho_{n}$ is infinitely exchangeable
if there exists a symmetric density operator $\rho_{n+m}$ acting on ${\cal
K}^{n+m}$ such that

$\displaystyle\rho_{n}={\rm tr}_{m}(\rho_{n+m}),$ (69)

for all positive integer $m$. Here, ${\rm tr}_{m}(\cdot)$ denotes the partial
trace operation with respect to an $m$-partite subsystem. Note that the choice
of the subsystem to be traced out is irrelevant, due to the exchange symmetry
of $\rho_{n+m}$.

Quantum de Finetti representation theorem states that given an infinitely
exchangeable density operator $\rho_{n}$, there exists a density operator
$\rho$ acting on ${\cal H}$ such that

$\displaystyle\rho_{n}=\int d\mu(\rho)\rho^{\otimes n},$ (70)

where $\mu(\rho)$ is a probability measure of the density operator $\rho$. Now
we introduce the projector

$\displaystyle P_{n,k}=P^{\otimes k}\otimes(\lnot P)^{\otimes(n-k)}$ (71)

and define the probability distribution of observing the projector $P$ being
true for the first $k$ times in $n$ repeated measurements by

$\displaystyle q(P_{n,k})={\rm tr}(\rho_{n}P_{n,k})=\int
d\mu(\rho)q(P)^{k}(1-q(P))^{n-k}.$ (72)

Furthermore, by following the notation in Sec. III, we denote by
$q(k_{n+1}|{\cal M}_{n,k})$ the conditional probability of the $(n+1)$-th
measurement of $P$ being true, given the set ${\cal M}_{n,k}$ of the outcomes
that $P$ is true for $k$ times in the $n$ measurements. Since the ordering of
the outcomes occurring is immaterial in this case, the projection operator
associated with ${\cal M}_{n,k}$ is given by

$\displaystyle\Pi_{n,k}$ $\displaystyle=$ $\displaystyle P_{n,k}+(\lnot
P)\otimes P_{n-1,k}+P_{2,1}\otimes P_{n-2,k-1}+\cdots$ (73)
$\displaystyle+(\lnot P)^{\otimes(n-k)}\otimes P^{\otimes k},$

consisting of the $\binom{n}{k}$ mutually orthogonal projectors. With the
projector $\Pi_{n,k}$, the conditional probability $q(k_{n+1}|{\cal M}_{n,k})$
takes the form of

$\displaystyle q(k_{n+1}|{\cal M}_{n,k})$ $\displaystyle=$ $\displaystyle
q(P\otimes\openone^{\otimes n}|\openone\otimes\Pi_{n,k})$ (74)
$\displaystyle=$ $\displaystyle\frac{q(P_{n+1,k+1})}{q(P_{n,k})},$

where we have employed the density operator $\rho_{n+1}$. For the derivation
of Eq. (74), see Appendix A.

The classical uniform probability measure leads to the Laplace law of
succession (2). For comparison, we wish to employ concrete expressions of the
uniform measures on the density operators. Notably enough, it has been known
that the metric of the space of the density operators is not uniquely
determined in general [32]. For example, there exists a family of infinite
numbers of monotone metrics with respect to the completely positive and trace-
preserving (CPTP) operations. Taking into account this non-uniqueness feature,
we shall consider the three measures as possible substitutes for the classical
uniform measure: the uniform measure on the pure state space, flat measure,
and Bures measure. Note that the flat measure and Bures measure are those of
the space of the density operators.

To deal with the three measures in a unified manner, let us follow [33] and
introduce a family of the measures called product measures by paying attention
to the spectral decomposition of the density operator

$\displaystyle\rho=U\Lambda U^{\dagger},\qquad\Lambda={\rm
diag}(\lambda_{1},\dots,\lambda_{N}),$ (75)

where $U$ is a unitary operator of $SU(N)$ and $\lambda_{i}$ are the
eigenvalues of $\rho$ such that

$\displaystyle\lambda_{1}+\dots+\lambda_{N}=1,\qquad\lambda_{i}\geq 0,$ (76)

for all $i=1,\dots,N$. This decomposition enables us to see the set of the
density operators as a direct product of $SU(N)$ and the set of the
eigenvalues $\\{\lambda_{i}\\}$ satisfying Eq. (76). Accordingly, we may
construct the measures of the density operators as the product of the measure
$dV$ of $SU(N)$ elements and that of $\\{\lambda_{i}\\}$ denoted by $dF$,
i.e.,

$\displaystyle d\mu(\rho)=dVdF,$ (77)

which is the product measure announced before. As we shall see, all the
measures aforementioned are of the family of the product measures.

We hereafter work with a system comprised of the spins ${\cal
H}=\mathbbm{C}^{2}$ and consider the measurements of the projector

$\displaystyle P=|\uparrow\rangle\langle\uparrow|,$ (78)

where $|\uparrow\rangle=(\begin{smallmatrix}1\\\ 0\end{smallmatrix})$. We
employ the spectral decomposition (75) and use the Euler angle representation
of $SU(2)$, that is, for $U\in SU(2)$,

$\displaystyle
U=e^{i\alpha\sigma_{3}}e^{i\beta\sigma_{2}}e^{i\gamma\sigma_{3}},$ (79)

with $\sigma_{2}=(\begin{smallmatrix}0&-i\\\ i&0\end{smallmatrix})$ and
$\sigma_{3}=(\begin{smallmatrix}1&0\\\ 0&-1\end{smallmatrix})$. Here, the
ranges of the parameters $\alpha,\beta,\gamma$ are
$0\leq\alpha,\gamma\leq\pi$, and $0\leq\beta\leq\pi/2$, respetively. In the
Euler angle representation, we immediately find

$\displaystyle q(P)=\lambda_{1}\cos^{2}\beta+\lambda_{2}\sin^{2}\beta.$ (80)

In addition to this, we employ the Haar measure of $SU(2)$ group as $dV$,
which takes the form of

$\displaystyle dV=\frac{\sin 2\beta}{\pi^{2}}d\alpha d\beta d\gamma,$ (81)

and we may write

$\displaystyle dF=F(\lambda_{1},\lambda_{2})d\lambda_{1}d\lambda_{2},$ (82)

in view of the decomposition (75) [38, 39]. The choice of the measure
$F(\lambda_{1},\lambda_{2})$ leads to three distinct measures mentioned above.

###  V.1 Uniform measure on pure state space

We begin with the product measure with

$\displaystyle
F(\lambda_{1},\lambda_{2})=\delta(\lambda_{1}-1)\delta(\lambda_{1}+\lambda_{2}-1),$
(83)

where $\delta(x)$ is the Dirac delta function. The measure
$F(\lambda_{1},\lambda_{2})$ has non-vanishing support only on pure states.
Since the Haar measure $dV$ is invariant under the action of $SU(2)$,
$d\mu(\rho)$ with Eq. (83) is the uniform measure on the space of the pure
states. By performing the integration in Eq. (72) as shown in Appendix B, we
readily find

$\displaystyle q(P_{n,k})=B(n-k+1,k+1),$ (84)

which leads to the Laplace law of succession (2).

###  V.2 Flat measure

We next turn to the flat measure, which is the product measure composed of the
Haar measure (81) and

$\displaystyle F(\lambda_{1},\lambda_{2})=\delta(\lambda_{1}+\lambda_{2}-1),$
(85)

indicating uniformity with respect to the eigenvalue distribution. The
probability associated with the flat measure takes the form of

$\displaystyle q(P_{n,k})=B(n-k+1,k+1)I_{n,k},$ (86)

with

$\displaystyle I_{n,k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\sum_{l=0}^{n-k}\binom{j+l}{j}\binom{n-j-l}{k-j}B(n-r+1,r+1),$

where we have introduced

$\displaystyle r=k-j+l.$ (88)

For the derivation of Eq. (LABEL:ink_flat), see Appendix C. The conditional
probability is, accordingly, given by Eq. (9).

###  V.3 Bures measure

The Bures measure [32, 33] is also the product measure consisting of the Haar
measure and

$\displaystyle
F(\lambda_{1},\lambda_{2})=\frac{2}{\pi}\frac{(\lambda_{1}-\lambda_{2})^{2}}{\sqrt{\lambda_{1}\lambda_{2}}}\delta(\lambda_{1}+\lambda_{2}-1).$
(89)

The Bures measure is derived from the Bures metric, which is a natural metric
equipped in the space of the density operators [32]. The distance with respect
to the Bures metric is called the Bures distance, which is the function of
fidelity. Since fidelity is a measure of the distinguishability of the density
operators, the Bures measure has a clear operational meaning as the measure of
state discrimination.

The probability distribution $q(P_{n,k})$ and conditional probability
pertaining to the Bures measure take the forms of Eq. (86) and Eq. (9),
respectively, with

$\displaystyle I_{n,k}$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\sum_{j=0}^{k}\sum_{l=0}^{n-k}\binom{j+l}{j}\binom{n-j-l}{k-j}$
(90)
$\displaystyle\times\frac{(n-2r)^{2}+n+1}{(r+\frac{1}{2})(n-r+\frac{1}{2})}B(n-r+\frac{3}{2},r+\frac{3}{2}),$

which is derived in Appendix D.

##  VI Conclusions and Discussion

We have examined whether the quantum probability, as well as conditional
probability, can be construed as our degree of belief on the quantum systems
satisfying the coherence condition. Our finding is the following: exactly the
same as classical probability, the quantum probability is identical to the
degree of belief with coherence. More precisely, our degree of belief with
coherence can take the form of the Born rule as well as the quantum
conditional probability (74), and conversely, they fulfill the coherence
condition to any combinations of the (conditional) bettings. Hence, we can
understand any quantum mechanical descriptions of the quantum systems in terms
of Bayesian notions.

The coincidence between quantum probability and the coherent degree of belief
should be distinguished from other major interpretations of classical
probability. For example, in the logical interpretation of classical
probability, (conditional) probability is equivalent to the measure of the
plausibility of a logical statement under incomplete information [3, 6, 7, 8,
14, 15], which obeys natural inference rules. In contrast, quantum conditional
probability does not obey the inference rules in the case of non-commuting
projectors [37]. Bearing the above case in mind, we may say that this work is
the first derivation of quantum conditional probability and the post-
measurement state (4) in a way consistent with the epistemic viewpoint of the
probability theory.

Our derivation of the post-measurement state (4) associated with the quantum
conditional probability is complementary to the derivation of the
(unnormalized) post-measurement state (8) in [40] for the case of the outcome
aggregation. Whereas we have derived the state (4) by using the quantum
analogue of the DBA, [40] has derived the state (8) by considering the
diachronic DBA [41] on the assumption of the Lüders rule
$\rho\rightarrow\rho_{Q}$ without the outcome aggregation. Here the diachronic
DBA is a variant of the DBA in such a way as to take into account the time
development of the degree of belief. Therefore our derivation may support the
assumption employed in [40].

In addition to the investigation of the above foundational issues, we ensured
that quantum probability approaches the relative frequency in large $n$ limit,
through the Laplace law of succession with the correction terms. Our argument
is perfectly in parallel with that developed by de Finetti for probability
theory, and thereby directly comparable with it. Note that the i.i.d. pure
states are the special cases of infinitely exchangeable states. Thus, our
result supports the result in [17] on the ground of the weaker assumption.

Here we point out the resemblance between the Laplace law of succession (in
QM) and the concept of the convergence to the truth, proposed by the
pragmatist Peirce, which is summarized in the following quote:

> Different minds may set out with the most antagonistic views, but the
> progress of investigation carries them by a force outside of themselves to
> one and the same conclusion [42].

In parallel with this convergence, the Laplace law of succession ensures that
the degrees of belief of all the rational agents converge approximately to the
relative frequency, through the repetition of the observations. This
resemblance may show the affinity between QBism and the pragmatism not only of
James and Dewey [43] but also of Peirce.

It is interesting to mention the relation between our work presented here and
an ontological viewpoint on QM: Pusey, Barrett, Rudolph (PBR) theorem [44, 45]
is a no-go theorem that any epistemic descriptions properly formulated in
terms of some hidden variables contradict with QM, on the assumption of
independent preparation of the system. Moreover, it is pointed out in [46, 45]
that without the independent preparation assumption, we construct epistemic
models consistent with QM. Bearing these works in mind, it is notable that our
analysis is based on two assumptions: our degree of belief is measurable using
bets in principle, and the propositions on the quantum systems are described
by the projection operators. Since the Bayesian probability is an epistemic
notion and the assumptions we have used seem weaker than those of the PBR
theorem, our result may support the counterexample to the PBR theorem without
invoking the hidden variable description.

In closing, we turn to the subject-object distinction in QM, which is one of
the roots of our investigation. On the basis of our results, the objective
aspects of QM lie not on the Born rule and the Lüders projection postulate,
but on the other axioms unexplored here. In particular, to use Jaynes’ phrase,
it could be of interest to ask whether we can “separate the subjective and
objective aspects” of our assumption that the projectors represent the
propositions on the quantum system, since the contextuality [47, 48], one of
the genuine quantum phenomena, is the direct consequence of the non-Boolean
lattice structure of the projectors. We close this paper by quoting
Heisenberg’s phrase again, which could imply the subject-object coexistence in
the algebraic structure of the projectors:

> […, and] we have to remember that what we observe is not nature in itself
> but nature exposed to our method of questioning. Our scientific work in
> physics consists in asking questions about nature in the language that we
> possess and trying to get an answer from experiment by the means that are at
> our disposal. [22]

## Acknowledgment

We thank S. Tanimura, G. Kimura, and H. Hakoshima for fruitful discussions.
This work was supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP)
Grant Number JPMXS0120319794.

##  Appendix A Proof of Eq. (74)

To derive Eq. (74), we first note that

$\displaystyle\Pi_{n,k}^{2}=\Pi_{n,k},$ (91)

from which we find

$\displaystyle q(P\otimes\openone^{\otimes
n}|\openone\otimes\Pi_{n,k})=\frac{{\rm tr}(\rho_{n+1}P\otimes\Pi_{n,k})}{{\rm
tr}(\rho_{n+1}\openone\otimes\Pi_{n,k})}.$ (92)

For the numerator, we find

$\displaystyle\\!\\!{\rm tr}(\rho_{n+1}P\otimes\Pi_{n,k})$ $\displaystyle=$
$\displaystyle\int d\mu(\rho){\rm tr}[(\rho P)\otimes(\rho^{\otimes
n}\Pi_{n,k})]$ (93) $\displaystyle=$ $\displaystyle\int d\mu(\rho){\rm
tr}(\rho P){\rm tr}(\rho^{\otimes n}\Pi_{n,k}).$

It follows from the exchange symmetry of the density operator $\rho^{\otimes
n}$ that

$\displaystyle{\rm tr}(\rho^{\otimes n}\Pi_{n,k})=\binom{n}{k}{\rm
tr}(\rho^{\otimes n}P_{n,k}).$ (94)

Plugging this expression to Eq. (93), we obtain

$\displaystyle{\rm tr}(\rho_{n+1}P\otimes\Pi_{n,k})$ $\displaystyle=$
$\displaystyle\binom{n}{k}\int d\mu(\rho){\rm tr}(\rho P){\rm
tr}(\rho^{\otimes n}P_{n,k})$ (95) $\displaystyle=$
$\displaystyle\binom{n}{k}\int d\mu(\rho){\rm tr}[(\rho
P)\otimes(\rho^{\otimes n}P_{n,k})]$ $\displaystyle=$
$\displaystyle\binom{n}{k}{\rm tr}(\rho_{n+1}P_{n+1,k+1})$ $\displaystyle=$
$\displaystyle\binom{n}{k}q(P_{n+1,k+1}).$

Similarly to the numerator, we enumerate the denominator as follows.

$\displaystyle{\rm tr}(\rho_{n+1}\openone\otimes\Pi_{n,k})$ $\displaystyle=$
$\displaystyle\int d\mu(\rho){\rm tr}[\rho\otimes(\rho^{\otimes n}\Pi_{n,k})]$
(96) $\displaystyle=$ $\displaystyle\int d\mu(\rho){\rm tr}(\rho){\rm
tr}(\rho^{\otimes n}\Pi_{n,k})$ $\displaystyle=$ $\displaystyle\int
d\mu(\rho){\rm tr}(\rho^{\otimes n}\Pi_{n,k}).$

By using Eq. (94), we observe

$\displaystyle{\rm tr}(\rho_{n+1}\openone\otimes\Pi_{n,k})$ $\displaystyle=$
$\displaystyle\binom{n}{k}\int d\mu(\rho){\rm tr}(\rho^{\otimes n}P_{n,k})$
(97) $\displaystyle=$ $\displaystyle\binom{n}{k}{\rm tr}(\rho_{n}P_{n,k})$
$\displaystyle=$ $\displaystyle\binom{n}{k}q(P_{n,k}),$

which completes the proof.

Note that Eq. (94) reads an instance of the principle of indifference [1, 14];
when the ordering of the outcomes is irrelevant, it is reasonable to assign
the probability for a possible ordering of the outcomes by evenly distributing
the value of the original probability.

##  Appendix B Derivation of Eq. (84)

To compute $q(P_{n,k})$ with the measure (83), we first perform the
integration with respect to $\lambda_{1},\lambda_{2}$, which results in

$\displaystyle q(P_{n,k})$ $\displaystyle=\iiint_{\mathcal{R}}d\alpha d\beta
d\gamma\frac{\sin 2\beta}{\pi^{2}}(\cos\beta)^{2k}(\sin\beta)^{2(n-k)},$

where
$\mathcal{R}=\\{(\alpha,\beta,\gamma)\,|\,0\leq\alpha,\gamma\leq\pi,\,\,0\leq\beta\leq\pi/2\\}$.
We further integrate $\alpha$ and $\gamma$ to obtain

$\displaystyle
q(P_{n,k})=2\int_{0}^{\frac{\pi}{2}}d\beta(\sin\beta)^{2(n-k)+1}(\cos\beta)^{2k+1}.$
(99)

By comparing this expression to the definition of the Beta function

$\displaystyle
B(n,k)=2\int_{0}^{\frac{\pi}{2}}d\beta(\sin\beta)^{2n-1}(\cos\beta)^{2k-1},$
(100)

we obtain Eq. (84).

##  Appendix C Derivation of Eqs. (LABEL:ink_flat)

We first perform the integration with respect to $\alpha$ and $\gamma$, which
leads to

$\displaystyle
q(P_{n,k})=\int_{0}^{1}d\lambda_{1}\int_{0}^{1}d\lambda_{2}F(\lambda_{1},\lambda_{2})J_{n,k},$
(101)

where

$\displaystyle J_{n,k}=\int_{0}^{\frac{\pi}{2}}d\beta\sin 2\beta
q(P)^{k}(1-q(P))^{n-k}.$ (102)

Let us next evaluate $J_{n,k}$. The binomial theorem and Eq. (80) lead to

$\displaystyle
q(P)^{k}=\sum_{j=0}^{k}\binom{k}{j}(\lambda_{1}\cos^{2}\beta)^{j}(\lambda_{2}\sin^{2}\beta)^{k-j},$
(103)

and

$\displaystyle(1-q(P))^{n-k}$
$\displaystyle=\sum_{l=0}^{n-k}\binom{n-k}{l}(\lambda_{1}\sin^{2}\beta)^{n-k-l}(\lambda_{2}\cos^{2}\beta)^{l}.$
(104)

Substituting Eq. (103) and Eq. (104) to Eq. (102), we obtain

$\displaystyle J_{n,k}$ $\displaystyle=$ $\displaystyle
2\sum_{j=0}^{k}\sum_{l=0}^{n-k}\binom{k}{j}\binom{n-k}{l}\lambda_{1}^{n-k+j-l}\lambda_{2}^{k-j+l}$
(105)
$\displaystyle\times\int_{0}^{\frac{\pi}{2}}d\beta(\sin\beta)^{2(n-j-l)+1}(\cos\beta)^{2(j+l)+1}.$

From Eq. (100), we find

$\displaystyle
2\int_{0}^{\frac{\pi}{2}}d\beta(\sin\beta)^{2(n-j-l)+1}(\cos\beta)^{2(j+l)+1}$
$\displaystyle=B(n-j-l+1,j+l+1).$ (106)

Furthermore, direct calculation shows

$\displaystyle\binom{k}{j}\binom{n-k}{l}B(n-j-l+1,j+l+1)$
$\displaystyle=\binom{j+l}{j}\binom{n-j-l}{k-j}B(n-k+1,k+1).$ (107)

Therefore, we may write

$\displaystyle J_{n,k}$ $\displaystyle=$ $\displaystyle B(n-k+1,k+1)$ (108)
$\displaystyle\times\sum_{j=0}^{k}\sum_{l=0}^{n-k}\binom{j+l}{j}\binom{n-j-l}{k-j}\lambda_{1}^{n-r}\lambda_{2}^{r},$

where $r=k-j+l.$ Plugging Eq. (108) into Eq. (101), and comparing the obtained
expression with Eq. (86), we find

$\displaystyle I_{n,k}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{k}\sum_{l=0}^{n-k}\binom{j+l}{j}\binom{n-j-l}{k-j}K,$
(109)

where

$\displaystyle
K=\int_{0}^{1}d\lambda_{1}\int_{0}^{1}d\lambda_{2}F(\lambda_{1},\lambda_{2})\lambda_{1}^{n-r}\lambda_{2}^{r}.$
(110)

Substituting Eq. (85) to the RHS of Eq. (110) and performing the integral, we
find

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!K$
$\displaystyle=\int_{0}^{1}d\lambda_{1}\lambda_{1}^{n-r}(1-\lambda_{1})^{r}=B(n-r+1,r+1).$
(111)

Plugging Eq. (111) into Eq. (109), we find Eq. (LABEL:ink_flat).

##  Appendix D Derivation of Eq. (90)

Similarly to Appendix C, we first integrate $\alpha,\beta,\gamma$ and obtain
Eq. (109). By using Eq. (46), the integration of $\lambda_{1}$ and
$\lambda_{2}$ with the measure (89) leads to a sum of the Beta functions:

$\displaystyle K$
$\displaystyle=\frac{2}{\pi}\int_{0}^{1}d\lambda_{1}\Big{\\{}\left[\lambda_{1}^{2}-2\lambda_{1}(1-\lambda_{1})+(1-\lambda_{1})^{2}\right]\lambda_{1}^{n-r-\frac{1}{2}}(1-\lambda_{1})^{r-\frac{1}{2}}\Big{\\}}$
$\displaystyle=\frac{2}{\pi}\left[\int_{0}^{1}d\lambda_{1}\lambda_{1}^{n-r+\frac{3}{2}}(1-\lambda_{1})^{r-\frac{1}{2}}-2\int_{0}^{1}d\lambda_{1}\lambda_{1}^{n-r+\frac{1}{2}}(1-\lambda_{1})^{r+\frac{1}{2}}+\int_{0}^{1}d\lambda_{1}\lambda_{1}^{n-r-\frac{1}{2}}(1-\lambda_{1})^{r+\frac{3}{2}}\right]$
$\displaystyle=\frac{2}{\pi}\left[B(n-r+\frac{5}{2},r+\frac{1}{2})-2B(n-r+\frac{3}{2},r+\frac{3}{2})+B(n-r+\frac{1}{2},r+\frac{5}{2})\right].$
(112)

Using $yB(x+1,y)=xB(x,y+1)$, we further obtain

$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\left(\frac{n-r+\frac{3}{2}}{r+\frac{1}{2}}-2+\frac{r+\frac{3}{2}}{n-r+\frac{1}{2}}\right)B(n-r+\frac{3}{2},r+\frac{3}{2})$
(113) $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{(n-2r)^{2}+n+1}{(r+\frac{1}{2})(n-r+\frac{1}{2})}B(n-r+\frac{3}{2},r+\frac{3}{2}).$

Substitution of this expression into Eq. (109) leads to Eq. (90).

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