Asymptotic estimation theory for a finite dimensional pure state model

Masahito Hayashi Department of Mathematics, Kyoto University Kyoto 606-8502, Japan e-mail address: masahito@kusm.kyoto-u.ac.jp

Abstract

The optimization of measurement for n samples of pure sates are studied. The error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample.

1 Introduction

Recently, there has been a rise in the necessity for studies about statistical estimation for the unknown state, related to the corresponding advance in measuring technologies in quantum optics. An investigation including both quantum theory and mathematical statistics is necessary for an essential understanding of quantum theory because it has statistical aspects [1, 2]. Therefore, it is indeed important to optimize the measuring process with respect to the estimation of the unknown state. Such research is known as quantum estimation, and was initiated by Helstrom in the late 1960s, originating in the optimization of the detecting process in optical communications [1]. In classical statistical estimation, one searches the most suitable estimator for which probability measure describes the objective probabilistic phenomenon. In quantum estimation, one searches the most suitable measurement for which density operator describes the objective quantum state.

Contained among important results are three estimation problems. The first is of the complex amplitude of coherent light in thermal noise and the second is of the expectation parameters of quantum Gaussian state. The former was studied by Yuen and Lax [3] and the latter by Holevo [2]. These studies discovered that heterodyning is the most suitable for the estimation of the complex amplitude of coherent light in thermal noise. The third is a formulation of the covariant measurement with respect to an action of a group. It was studied by Holevo [2, 4]. In the formulation, he established a quantum analogue of Hunt-Stein theorem.

Quantum estimation, was first used in the evaluation of the estimation error of a single sample of the unknown state as it had advanced in connection with the optimization of the measuring process in optical communications. Thus early studies were lacking in asymptotic aspects, i.e. there were few researches with respect to reducing the estimation error by quantum correlations between samples.

Recently, studies about the estimation of the unknown state are attracting many physicists [5, 6, 7, 8]. Some of them were drawn by the variation of the measuring precision with respect to the number of samples of the unknown state [9, 10].

Nagaoka [11] studied, for the first time, asymptotic aspects of quantum estimation. He paid particular attention to the quantum correlations between samples of the unknown state, and studied the relation between the asymptotic estimation and the local detection of a one-parameter family of quantum states.

In the early 1990s, Fujiwara and Nagaoka [12, 13, 14] studied the estimation problem for a multiparameter family consisting of pure states. They pioneered studies into the estimation problem of the complex amplitude of noiseless coherent light. The research found that heterodyning is the most suitable for the estimation of the complex amplitude of noiseless coherent light as for the one of coherent light in thermal noise. In 1996, Matsumoto [15] established a more general formulation of the estimation for a multi-parameter family consisting of pure states. Moreover in 1991, Nagaoka [16] treated the estimation problem for 2-parameter families of mixed states in spin 1/2 system, and in 1997 the author [17, 18] treated it for 3-parameter families of mixed states in spin 1/2 system. However, there are no asymptotic aspects in these works about multi-parameter families. There is more necessity of this type of investigation into one- and multi-parameter families.

Can quantum estimation reduce the estimation error by using the quantum correlations between samples, under the preparation of sufficient samples of the unknown state? To answer this question, in this paper, we treat a family, consisting of all of pure states on a Hilbert space H ^{1} under the preparation of n samples of the unknown state, with the estimation problem In §2, we use, as a tool, the composite system consisting of n samples as a single system. The quantum i.i.d. condition is introduced as the quantum counterpart of the independent and identical distributions condition (3). In §3, we review Holevo’s result concerning covariant measurements which will be used in the following sections. In §4, we apply Holevo’s result to the optimization of measurements on the composite system, which results in obtaining the most suitable measurement (Theorem 3). We asymptotically calculate the estimation error by the optimal measurement in the sense of both the mean square error (MSE) and large deviation. (see (9)(10)(11) (13).) The first term of the righthand side of (10) is consistent with the value conjectured from the results by Fujiwara, Nagaoka [14] and Matsumoto [15]. However, the optimal measurement may be too difficult for modern technology to realize when using more than one samples.

In §5, we use this estimation problem under the following guidelines. The samples are divided into pairs consisting of a maximum of m samples. By measuring each pair with the optimal measurement in section 4, we create some data. The estimated valued is given by manipulating these data. The restricted condition is called m-semiclassical (see (14)). We compare an m-semiclassical measurement with the optimal measurement of section 4 with respect to the estimation error under the preparation of a sufficient amount of samples. When we use the maximum likelihood estimator to manipulate the data, the MSE of both asymptotically coincide in the first order (see (10)(19)). However, when the radius of allowable errors is finite, the error of large deviation type in the latter is smaller than that in the former type (see (11)(20)).

Can we asymptotically realize a small estimation error as the optimal measurements in section 4 has? It is, physically, sufficient to construct the optimal measurement for one sample. In section 5, we show how to construct it (see (25)). And in this case, we can calculate the maximum likelihood estimator from data by using computer.

Most of the proofs of this paper are given in Appendices. In view of multiparameter families of mixed states in spin 1/2 system, Hayashi [19] has discussed the same problem by using Cramér-Rao type bound.

2 Pure state n-i.i.d. model

In this section, we use the mathematical formulation of the estimation for pure states. Let k be the dimension of the Hilbert space H, and P(H) be the set of pure states on H.

In quantum physics, the most general description of a quantum measurement is given by the mathematical concept of a positive operator valued measure (POVM) [1, 2] on the system of state space. Generally, if Ω is measurable space, a measurement M satisfies the following:

M(B) = M(B)^{∗}, M(B) ≥ 0, M(∅) = 0, M(Ω) = Id on H, for any B ⊂ Ω.
M(∪_{i}B_{i}) = ∑_{i} M(B_{i}), for B_{i} ∩ B_{j} = ∅(i ≠ j), {B_{i}} is countable subsets of Ω.

In this paper, M(Ω, H) denotes the set of POVMs on H whose measurable set is Ω. A measurement M ∈ M(Ω, H) is called simple if M(B) is a projection for any Borel B ⊂ Ω. A measurement M is random if it is described as a convex combination of simple measurements. A random measurement M = ∑_{i} a_{i}M_{i} (M_{i} is simple and a_{i} > 0.) can be realized when every measurement M_{i} is done with the probability a_{i}.

1Where H denotes a finite-dimensional Hilbert space which corresponds to the physical system of interest.

In this paper, we consider measurements whose measurable set is P(H) since it is known that the unknown state is included in P(H).

Next, we define two distances charactering the homogeneous space P(H).

Definition 1 the Fubini-Study distance d_{fs} (which is the geodesic distance of the Fubini-Study metric) is defined as:

cos d_{fs}(ρ, ρ̂) = \sqrt{tr ρρ̂} , 0 ≤ d_{fs}(ρ, ρ̂) ≤ \frac{π}{2} .  (1)

the Bures distance d_{b} is defined in the usual way:

d_{b}(ρ, ρ̂) := \sqrt{1 − tr ρρ̂}.  (2)

It is introduced by Bures [20] in a mathematical context.

Let W (ρ, ρ̂) be a measure of deviation of the measured value ρ̂ from the actual value ρ, then we have the following equivalent conditions:

∘ W (ρ, ρ̂) = W (gρg^{∗}, gρ̂g^{∗}) for g ∈ SU(k), ρ, ρ̂ ∈ P(H).
∘ There exists a function h on [0, 1] such that W (ρ, ρ̂) = h ∘ d_{fs}(ρ, ρ̂).

It is natural to assume that a deviation measure W (ρ, ρ̂) is monotone increasing with respect to the Fubini-Study distance d_{fs}.

If H_{1}, …, H_{n} are n Hilbert spaces which correspond to the physical systems, then their composite system is represented by the tensor Hilbert space:

H^{(n)} := H_{1} ⊗ ⋯ ⊗ H_{n} = ⊗_{i=1}^{n} H_{i}.

Thus, a state on the composite system is denoted by a density operator ρ on H^{(n)}. In particular if n element systems {H_{i}} of the composite system H^{(n)} are independent of each other, there exists a density ρ_{i} on H_{i} such that

ρ^{(n)} = ρ_{1} ⊗ ⋯ ⊗ ρ_{n} = ⊗_{i=1}^{n} ρ_{i}.

The condition:

H_{1} = ⋯ = H_{n} = H, ρ_{1} = ⋯ = ρ_{n} = ρ  (3)

corresponds to the independent and identically distributed condition (i.i.d. condition) in the classical case. In this paper, we treat with this estimation problem under this condition (3) called the quantum i.i.d. condition. This condition means that identical n samples are independently prepared. The model \{ρ^{(n)} = \underbrace{ρ ⊗ ⋯ ⊗ ρ}_{n} |ρ ∈ P(H)\} is called n-i.i.d. model. As ρ is a pure state,

H^{(n)} and ρ^{(n)} are simplified as follows. Letting ρ = |ϕ⟩⟨ϕ| ∈ P(H), we have

ρ^{(n)} = |ϕ^{(n)}⟩ ⟨ϕ^{(n)}| , ϕ^{(n)} := \overbrace{ϕ ⊗ ⋯ ⊗ ϕ}^{n} .

Because all of the vectors ϕ^{(n)} is included in n-times symmetric tensor space, for any measurement M ∈ M(Ω, H^{(n)}) on the n-times tensor space H^{(n)}, the measurement M̃( dω) := P_{H_{s}^{(n)}}M( dω)P_{H_{s}^{(n)}} ∈ M(Ω, H_{s}^{(n)}) on the n-times symmetric tensor space H_{s}^{(n)} satisfies that:

tr M( dω)ρ^{(n)} = tr M̃( dω)ρ^{(n)} for any ρ ∈ H,

where H_{s}^{(n)} denotes the n-times symmetric tensor space on H. Therefore, all of possible measurements can be regarded as elements of M(P(H), H_{s}^{(n)}). The mean error of the measurement Π ∈ M(P(H), H_{s}^{(n)}) with respect to a deviation measure W (ρ, ρ̂), provided that the actual state is ρ, is equal to

D_{ρ}^{W,(n)}(Π) := ∫_{P(H)} W (ρ, ρ̂) tr(Π( dρ̂)ρ^{(n)}).

In minimax approach the maximum possible error with respect to a deviation measure W (ρ, ρ̂)

D^{W,(n)}(Π) := max_{ρ∈P(H)} D_{ρ}^{W,(n)}(Π)

is minimized.

3 Quantum Hunt-Stein theorem

In this section, the quantum Hunt-Stein theorem, established by Holevo [2, 4], is summarized. Let G be a compact transitive Lie group of all transformations on a compact parametric set Θ, and {V_{g}} a continuous unitary irreducible representation of G in a finite-dimensional Hilbert space H′ := C^{k′}, and μ a σ-finite invariant measure on group G such that μ(G) = 1. In this section, we consider the following condition for a measurement.

Definition 2 A measurement Π ∈ M(Θ, H′) is covariant with respect to {V_{g}} if

V_{g}^{∗}Π(B)V_{g} = Π(B_{g^{−1}})

for any g ∈ G and any Borel B ⊂ Θ, where

B_{g} := {gθ|θ ∈ B}.

M(Θ, V) denotes the set of covariant measurements with respect to {V_{g}}.

Covariant measurements are characterized by the following theorem.

Theorem 1 The map V^{θ} from the set S(H′) of densities on H′ to M(Θ, V) is surjective for any θ ∈ Θ, where V^{θ}(P) is defined as follows:

V^{θ}(P)(B) := k′ ∫_{{gθ∈B}} V_{g}P V_{g}^{∗}μ( dg) for B ∈ B(Θ)

for any P ∈ S(H′).

In this section, we treat with the following condition for a family of states.

Definition 3 The family is called covariant under the representation {V_{g}} of group G acting on Θ, if

S_{gθ} = V_{g}S_{θ}V_{g}^{∗}, ∀g ∈ G, ∀θ ∈ Θ.

Assuming that the object is prepared in one of the states {S_{θ}|θ ∈ Θ} but the actual value of θ is unknown, then the difficulty is estimating this value as close as possible to a measurement on the object. We shall solve this problem by means of the quantum statistical decision theory.

Let W (θ, θ̂) be a measure of deviation of the measured value θ̂ from the actual value θ. It is natural to assume that W (θ, θ̂) is invariant:

W (θ, θ̂) = W (gθ, gθ̂) for ∀g ∈ G, ∀θ, ∀θ̂ ∈ Θ.  (4)

The mean error of the measurement Π ∈ M(Θ, H′) with respect to a deviation measure W (θ, θ̂), provided that the actual state is S_{θ}, is equal to

D_{θ}^{W,S}(Π) := ∫_{Θ} W (θ, θ̂) tr(Π( dθ̂)S_{θ}).

Following the classical statistical decision theory, we can form two functionals of D_{θ}^{W} giving a total measure of precision of the measurement Π.

In Bayes’ approach we take the mean of D_{θ}^{W} with respect to a given prior distribution π( dθ). The measurement minimizing the resulting functional:

D_{π}^{W,S}(Π) := ∫_{Θ} D_{θ}^{W,S}(Π)π( dθ)

is called Bayesian. This quantity represents the mean error in the situation where θ is a random parameter with known distribution π( dθ). In particular, as Θ, G are compact and “nothing is known” about θ, it is natural to take for π( dθ) the “uniform” distribution, i.e. normalized invariant measure ν( dθ) defined as follows:

ν(B) := μ({gθ ∈ B}).

It is independent of the choice of θ ∈ Θ.

In minimax approach the maximum possible error with respect to a deviation measure W (θ, θ̂)

D^{W,S}(Π) := max_{θ∈Θ} D_{θ}^{W,S}(Π)

is minimized. The minimizing measurement is called minimax.

Because G is compact, we shall show that in the covariant case the minima of Bayes and minimax criteria coincide and are achieved on a covariant measurement. We obtain the following quantum Hunt-Stein theorem [2, 4]. It is easy to prove the theorem.

Theorem 2 For a covariant measurement Π ∈ M(Θ, V), we obtain the following equations:

D_{θ}^{W,S}(Π) = D_{ν}^{W,S}(Π) = D^{W,S}(Π).

For Π ∈ M(Θ, H′), denote

Π_{g}(B) := V_{g}Π(B_{g})V_{g}^{∗} for B ∈ B(Θ).

Introducing the “averaged” measurement

Π̄(B) := ∫_{G} Π_{g^{−1}}(B)μ( dg),

we have

D_{ν}^{W,S}(Π̄) = ∫_{G} D_{ν}^{W,S}(Π_{g^{−1}})μ( dg) = D_{ν}^{W,S}(Π).

Thus,

D^{W,S}(Π) ≥ D_{ν}^{W,S}(Π) = D_{ν}^{W,S}(Π̄).

In this case, minimax approach and Bayes’ approach with respect to ν( dθ) are equivalent. Therefore we minimize the following:

D_{θ}^{W,S} ∘ V^{θ}(P) = k′ ∫_{G} W (θ, gθ) tr S_{θ}V_{g}P V_{g}^{∗}μ( dg) = tr Ŵ (θ)P,

where

Ŵ (θ) := k′ ∫_{G} W (θ, gθ)V_{g}^{∗}S_{θ}V_{g}μ( dg)
= k′ ∫_{Θ} W (θ, θ̂)Sθ̂ν( dθ̂).

Thus, it is sufficient to consider the following minimization:

min_{P ∈S(H)} tr Ŵ (θ)P = min_{P ∈P(H′)} tr Ŵ (θ)P.

4 Optimal measurement in pure state n-i.i.d. model

In this section we apply the theory of §3 to the problem §2.

We let as follows:

Θ := P(H), H′ := H_{s}^{(n)}, G := SU(k), S_{ρ} := ρ^{(n)}.

Then, the invariant measure ν on P(H) is equivalent to the measure defined by the volume bundle induced by the Fubini-Study metric. We let the action {V_{g}} of G = SU(k) to H_{s}^{(n)} be the tensor representation of the natural representation. In this case, we have k′ = \binom{n+k−1}{k−1}.

Theorem 3 If a deviation measure W (ρ, ρ̂) is monotone increasing with respect to the Fubini-Study distance d_{fs}, then we get

min_{P_{0}∈P(H_{s}^{(n)})} tr Ŵ (ρ)P_{0} = tr Ŵ (ρ)ρ^{(n)}.

For a proof see Appendix A. Thus, V^{ρ}(ρ^{(n)}) is the optimal measurement with respect to a deviation measure W (ρ, ρ̂). The optimal measurement is independent of the choice of ρ and W since V^{ρ_{0}}(ρ_{0}^{(n)}) = V^{ρ}(ρ^{(n)}). This optimal measurement is denoted by Π_{n} and is described as follows:

Π_{n}( dρ̂) := \binom{n + k − 1}{k − 1}ρ̂^{(n)}ν( dρ̂).

Under the following chart (6), the optimal measurements are denoted as:

Π_{n}( dθ) = \binom{n + k − 1}{k − 1} |ϕ(θ)^{(n)}⟩ ⟨ϕ(θ)^{(n)}| ν( dθ)  (5)

for θ ∈ {θ ∈ R^{2k−2}|θ_{i} ∈ [0, 2π)1 ≤ j ≤ k − 1, θ_{j} ∈ [0, π/2]}, where we defined as follows:

ϕ(θ) := (  cos θ_{1} e^{iθ_{k}} sin θ_{1} cos θ_{2} e^{iθ_{k+1}} sin θ_{1} sin θ_{2} cos θ_{3} ⋮ e^{iθ_{2k−3}} sin θ_{1} sin θ_{2} sin θ_{3} ⋯ sin θ_{k−2} cos θ_{k−1} e^{iθ_{2k−2}} sin θ_{1} sin θ_{2} sin θ_{3} ⋯ sin θ_{k−2} sin θ_{k−1}) .  (6)

The invariant measures ν( dθ) described above is from [21,p.31].

ν( dθ) = \frac{(k − 1)!}{π^{k−1}} sin^{2k−3} θ_{1} sin^{2k−5} θ_{2} ⋯ sin θ_{k−1} cos θ_{1} cos θ_{2} ⋯ cos θ_{k−1} dθ_{1} dθ_{2} ⋯ dθ_{2k−2}. (7)

Lemma 1 If the deviation measure W is characterized as W (ρ, ρ̂) = h ∘ d_{fs}(ρ, ρ̂), we can describe the maximum possible error of the optical measurement Π_{n} as:

D^{W,(n)}(Π_{n}) = 2(k − 1)\binom{n + k − 1}{k − 1} ∫0^{\frac{π}{2}} h(θ) cos^{2n+1} θ sin^{2k−3} θ dθ.

For a proof, see Appendix B.

Next, we asymptotically calculate the error of the optimal measurements Π_{n} in the third order.

Theorem 4 When the deviation measure W is described as W = d_{b}^{γ}, we can asymptotically calculate the maximum possible error of the optimal measurement as:

lim_{n→∞} D^{d_{b}^{γ},(n)}(Π_{n})n^{\frac{γ}{2}} = \frac{Γ(k − 1 + γ/2)}{Γ(k − 1)}.  (8)

Specially in the case of γ = 2, we have

D^{d_{b}^{2},(n)}(Π_{n})n = \frac{(k − 1)n}{n + k} = (k − 1) ∑_{i=0}^{∞} (−\frac{k}{n})^{i} → k − 1 as n → ∞.  (9)

When the deviation measure is defined by the square of the Fubini-Study distance, we can asymptotically calculate the maximum possible error of the optimal measurement as:

D^{d_{fs}^{2},(n)}(Π_{n})n ≅ (k − 1) − \frac{2}{3}k(k − 1)\frac{1}{n} + k(k − 1)\frac{23k − 7}{45} \frac{1}{n^{2}} as n → ∞.  (10)

The error of the sequence {Π_{n}}_{n=1}^{∞} of the optimal measurements can be calculated in the sense of large deviation as:

\frac{1}{n} log (Pr_{Π_{n}}^{ρ^{(n)}}\{ρ̂ ∈ P(H)|d_{fs}(ρ, ρ̂) ≥ ϵ\})
≅ log cos^{2} ϵ + (k − 2)\frac{log n}{n} + (− log(k − 2)! + 2(k − 2) log(sin ϵ) − 2 log(cos ϵ)) \frac{1}{n}
+ (\frac{k^{2} − k − 2}{2} + (k − 2) cot^{2} ϵ) \frac{1}{n^{2}} as n → ∞,  (11)

where Pr_{M}^{S} B denotes the probability of B with respect to the probability measure tr(M( dω)S) for a Borel B ⊂ Ω, a measurement M ∈ M(Ω, H′) and a state S ∈ S(H′).

For a proof, see Appendix C. The first term of the right hand of (11) coincide with the logarithm of the fidelity. About the fidelity, see Jozsa [22].

In this paper, ϵ in equations (11) is called admissible radius.

Since

lim_{ϵ→0} \frac{log cos^{2} ϵ}{ϵ^{2}} = −1,  (12)

we obtain the following large deviation approximation.

lim_{ϵ→0} lim_{n→∞} \frac{1}{ϵ^{2}n} log (Pr_{Π_{n}}^{ρ^{(n)}}\{ρ̂ ∈ P(H)|d_{fs}(ρ, ρ̂) ≥ ϵ\}) = −1.  (13)

5 Semiclassical measurement

In this section, we consider measurements allowed the quantum correlation between finite samples only. A measurement M on H^{(nm)} is called m-semiclassical if there exists a estimator T on the probability space \underbrace{P(H) × ⋯ × P(H)}_{n} whose range is P(H) such that

M(B) = ∫_{T^{−1}(B)} \underbrace{Π_{m}( dρ_{1}) ⊗ ⋯ ⊗ Π_{m}( dρ_{n})}_{n} ∀B ⊂ P(H).  (14)

We compare the error between m-semiclassical measurements and the optimal measurement Π_{nm} for nm samples of the unknown state as the equations (10),(11),(13).

In doing this comparison, we bear in mind asymptotic estimation theory in classical statics. In classical statics, it is assumed that the sequence of estimators satisfies the consistency.

Definition 4 A sequence {T^{(n)}}_{n=1}^{∞} of estimators on a probability space Ω is called consistent with respect to a family {p_{θ}|θ ∈ Θ} of probability distributions on Ω, if it satisfies the condition (15), where every T^{(n)} is a probability variable on the probability space \underbrace{Ω × ⋯ × Ω}_{n} whose range is Θ.

p_{θ}^{(n)}{d_{J}(T^{(n)}, θ) > ϵ} → 0 as n → ∞, ∀θ ∈ Θ, ∀ϵ > 0,  (15)

where d_{J} denotes the geodesic distance defined by the Fisher Information metric and p_{θ}^{(n)} denotes the probability measure \underbrace{p_{θ} × ⋯ × p_{θ}}_{n} on the probability space \underbrace{Ω × ⋯ × Ω}_{n}.

It is well known that the following theorem establishes under the preceding consistency [23, 24, 25].

Theorem 5 If a sequence {T^{(n)}}_{n=1}^{∞} of estimators is a consistent estimator with respect to a family {p_{θ}|θ ∈ Θ} of probability distributions on a probability space Ω which satisfies some regularity, then we have the following inequalities.

lim_{n→∞} n \underbrace{∫ ∫ ⋯ ∫}_{n} d_{J}^{2} (T^{(n)}(x_{1}, x_{2}, …, x_{n}), θ) p_{θ}^{(n)}( dx_{1}, …, dx_{n}) ≥ dim Θ  (16)

lim_{n→∞} \frac{1}{n} log (p_{θ}^{(n)}\{D(p_{T^{(n)}}∥p_{θ}) ≥ ϵ\}) ≥ −ϵ  (17)
lim_{ϵ→0} lim_{n→∞} \frac{1}{ϵ^{2}n} log (p_{θ}^{(n)}\{d_{J}(T^{(n)}, θ) ≥ ϵ\}) ≥ −\frac{1}{2},  (18)

where D(p∥q) denotes the information divergence of a probability distribution q with respect to another probability distribution p defined by:

D(p∥q) := ∫_{Ω} (log p(ω) − log q(ω)) p(ω) dω.

Under some regularity conditions, the lower bounds of (16),(18) can be attained by the maximum likelihood estimator(MLE). A regularity condition for attaining (16) is different from a one for (18). The lower bound of (17) can be attained by the MLE when the family {p_{θ}|θ ∈ Θ} is exponential, but generally cannot be attained.

For the comparison, we apply Theorem 5 to the family of distributions {tr Π_{m}( dρ̂)ρ^{(m)}|ρ ∈ P(H)} given by the measurement Π_{m} and the family of states {ρ^{(m)}|ρ ∈ P(H)}. Let T_{(n,m)} be the measurement on H^{(nm)} defined by the estimator T^{(n)} and n data given by the measurement \underbrace{Π_{m} ⊗ ⋯ ⊗ Π_{m}}_{n} and the state ρ^{(nm)}. We consider the sequence of measurements \{T_{(n,m)}\}_{n=1}^{∞}. From

the symmetry of P(H) and Π_{m}, the information divergence of a probability measure tr Π_{m}( dρ̂)ρ_{1}^{(m)} with respect to another a probability measure tr Π_{m}( dρ̂)ρ_{2}^{(m)} is determined by the the Fubini-Study distance ϵ between ρ_{1} and ρ_{2}. Thus, the divergence can be denoted by D_{m}(ϵ). From Lemma 2, the geodesic distance d_{Π_{m}} with respect to Fisher information metric in the family of distributions {tr Π_{m}( dρ̂)ρ^{(m)}|ρ ∈ P(H)} is given by:

d_{Π_{m}} = \sqrt{2m}d_{fs}.

Since dim P(H) = 2(k − 1), we have the following inequalities:

lim_{n→∞} nmD^{d_{fs}^{2},(nm)}(T_{(n,m)}) = lim_{n→∞} max_{ρ∈P(H)} nm ∫_{P(H)} d_{fs}^{2}(ρ, ρ̂) tr(T_{(n,m)}( dρ̂)ρ^{(nm)}) ≥ k − 1 (19)
lim_{n→∞} \frac{1}{nm} log Pr_{T_{(n,m)}}^{ρ^{(nm)}}\{ρ̂ ∈ P(H)|d_{fs}(ρ, ρ̂) ≥ ϵ\} ≥ −\frac{D_{m}(ϵ)}{m} (20)
lim_{ϵ→0} lim_{n→∞} \frac{1}{ϵ^{2}nm} log Pr_{T_{(n,m)}}^{ρ^{(nm)}}\{ρ̂ ∈ P(H)|d_{fs}(ρ, ρ̂) ≥ ϵ\} ≥ −1. (21)

The lower bound of (19) is consistent with the first term of the right hand of (10) and the lower bound of (21) is consistent with the right hand of (13). The family of distributions {tr Π_{m}( dρ̂)ρ^{(m)}|ρ ∈ P(H)} satisfies a regularity condition for (16) by MLE. But, we cannot show that it does a one for (18). We have the following lemma concerning the comparison of the lower bound −\frac{D_{m}(ϵ)}{m} of (20) and the first term 2 log cos ϵ of the right hand of (11).

Lemma 2 We can calculate the divergence D_{m}(ϵ) and the distance d_{Π_{m}} as:

\frac{D_{m}(ϵ)}{m} = ∑_{i=1}^{m} \frac{sin^{2i} ϵ}{i} → − log (1 − sin^{2} ϵ) = − log cos^{2} ϵ as m → ∞  (22)
d_{Π_{m}} = \sqrt{2m}d_{fs}.  (23)

Therefore, \frac{D_{m}(ϵ)}{m} is monotone increasing with respect to m.

For a proof, see Appendix D. (22) derives that

0 < \frac{−m log cos^{2} ϵ − D_{m}(ϵ)}{mϵ^{2m}} → 0 as ϵ → 0  (24)

which means that the first term of (11) cannot be attained by a semi-classical measurement. However, it is an open problem as to whether the left-hand side of (13) can be asymptotically attained by a 1-semiclassical measurement. Concerning MSE, the first term of (10) can be asymptotically attained by it i.e. it can be asymptotically attained by measurements without using quantum correlations between samples. Thus, in order to attain it asymptotically, it is sufficient to physically realize the optimal measurement Π_{1} on a single sample. Indeed, Π_{1} is a random measurement as follows. To denote Π_{1} as a random measurement, we will define the simple measurement E_{g}(g ∈ SU(k)) whose measurable space P(H). For an element g ∈ SU(k), the vectors ϕ_{1}(g), ⋯, ϕ_{k}(g) in H are defined as:

(ϕ_{1}(g) ⋯ ϕ_{k}(g)) = g.

The measurement E_{g} is defined as:

E_{g} (|ϕ_{i}(g)⟩⟨ϕ_{i}(g)|) = |ϕ_{i}(g)⟩⟨ϕ_{i}(g)|.

Therefore, the optimal measurement Π_{1} for a single sample can be described as the following random measurement:

Π_{1} = ∫_{SU(k)} E_{g}μ( dg),  (25)

where μ is the invariant measure on SU(k) with μ(SU(k)) = 1. Therefore, in order to realize the optimal measurement Π_{1}, it is sufficient to realize the simple measurement E_{g} for any g ∈ SU(k).

6 Conclusion

We have compared two cases. One regards the system consisting of enough samples as the single system, the other regards it as separate systems. Under this comparison, the MSEs of both cases asymptotically coincide in the first order with respect to the Fubini-Study distance (see (10) and (19)). However we leave the question of whether they asymptotically coincide in the second order with respect to the Fubini-Study distance to a future study. On the other hand, in view of the evaluation of large deviation, if the allowable radius is finite, neither coincide (see (11) and (20)). However, in the case of the allowable radius goes to infinitesimal, it is an open problem as to whether both coincide (see (13) and (21)).

These results depend on the effect of a pure state. Therefore, it is an open question as to whether the MSEs of both cases asymptotically coincide in the first order in another family. In the case of large deviation, the same question is also open in the limit where the radius of allowing error goes to infinitesimal.

Acknowlegments

The author wishes to thank Prof. K. Ueno. Also, he wishes to thank Dr. A. Fujiwara, Dr. K. Matsumoto and Dr. H. Nagaoka.

Appendices

A Proof of Theorem 3

In this Appendix, assume that ρ = |ϕ(0)⟩⟨ϕ(0)|. Because H_{s}^{(n)} is irreducible with respect to the action of SU(k),

H_{s}^{(n)} = {∑_{i} a_{i}V_{g_{i}}ϕ(0)^{(n)}| a_{i} ∈ C, g_{i} ∈ SU(k)}

= {∑_{i} ϕ_{i}^{(n)}| ϕ_{i} ∈ H} .  (26)

We assume that W (ρ, ρ̂) = h(tr ρρ̂). As h is monotone decreasing, there exists a measure h′ on [0, 1] such that h(x) = h′([x, 1]).

The function h_{β} on [0, 1] and the deviation measure W_{β} are defined as follows:

h_{β}(x) := \begin{cases}1 & for & x ≤ β \\ 0 & for & x > β \end{cases}
W_{β}(ρ, ρ̂) := h_{β}(tr ρρ̂).

From Lemma 3, for any measurement Π we have

D_{ρ}^{W,(n)}(Π) = ∫_{[0,1]} D_{ρ}^{W_{β},(n)}(Π)h′( dβ).

From (26), it is sufficient to show the following for {ϕ_{i}} ⊂ H in the case of W = W_{β}.

\frac{tr Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ ⟨∑_{i} ϕ_{i}^{(n)}|}{⟨∑_{i} ϕ_{i}^{(n)} |∑_{i} ϕ_{i}^{(n)} ⟩} ≥ tr Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ ⟨ϕ(0)^{(n)}| .  (27)

From Lemma 4 it is sufficient for (27) to prove the following:

⟨∑_{i} ϕ_{i}^{(n)}| Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ · ⟨ϕ(0)^{(n)}| Id −Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩

≥ ⟨ϕ(0)^{(n)}| Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ · ⟨∑_{i} ϕ_{i}^{(n)}| Id −Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ .  (28)

Remark that |⟨ϕ(θ)|ϕ(0)⟩|^{2} = cos^{2} θ_{1}. From Lemma 5, we get

⟨∑_{i} ϕ_{i}^{(n)}| Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫α^{\frac{π}{2}} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}

⟨∑_{i} ϕ_{i}^{(n)}| Id −Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫_{0}^{α} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}

⟨ϕ(0)^{(n)}| Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = C ∫α^{\frac{π}{2}} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨ϕ(0)^{(n)}| Id −Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = C ∫_{0}^{α} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1},

where

β := cos^{2} α
f_{1}(θ_{1}) := \underbrace{∫0^{\frac{π}{2}} ⋯ ∫0^{\frac{π}{2}}}_{k−2} f_{2}(θ_{1}, …, θ_{k−1})λ( dθ_{2} ⋯ dθ_{k−1})

f_{2}(θ_{1}, …, θ_{k−1}) := \underbrace{∫_{0}^{2π} ⋯ ∫_{0}^{2π}}_{k−1} ∑_{i,j}⟨ϕ_{i}|ϕ(θ)⟩^{n}⟨ϕ(θ)|ϕ_{j}⟩^{n} dθ_{k} ⋯ dθ_{2k−2}

C := \frac{k′ · (k − 1)!}{π^{(k−1)}} \underbrace{∫_{0}^{2π} ⋯ ∫_{0}^{2π}}_{k−1} \underbrace{∫0^{\frac{π}{2}} ⋯ ∫0^{\frac{π}{2}}}_{k−2} λ( dθ_{2} dθ_{3} ⋯ dθ_{k−1}) dθ_{k} ⋯ dθ_{2k−2}

λ( dθ_{2}, …, dθ_{k−1}) := sin^{2k−5} θ_{2} ⋯ sin θ_{k−1} cos θ_{2} ⋯ cos θ_{k−1} dθ_{2} ⋯ dθ_{k−1}.

Therefore, it is sufficient for the equation (28) to show that for π/2 ≥ θ_{1} > θ′_{1} ≥ 0

f_{1}(θ_{1}) sin^{2k−3} θ_{1} cos^{2n+1} θ′_{1} sin^{2k−3} θ′_{1} ≥ f_{1}(θ′_{1}) sin^{2k−3} θ′_{1} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1}.

It suffices to verify that for θ_{i} ∈ [0, \frac{π}{2} ], 2 ≤ i ≤ k − 1, π/2 ≥ θ_{1} > θ′_{1} ≥ 0

\frac{f_{2}(θ_{1}, θ_{2}, …, θ_{n−1})}{cos^{2n} θ_{1}} ≥ \frac{f_{2}(θ′_{1}, θ_{2}, …, θ_{n−1})}{cos^{2n} θ′_{1}}.

Thus, it is sufficient to prove that the following is monotone decreasing about θ_{1} for any θ_{2}, …, θ_{k−1}:

\frac{1}{cos^{2n} θ_{1}} \underbrace{∫_{0}^{2π} ⋯ ∫_{0}^{2π}}_{k−1} ∑_{i,j}⟨ϕ_{i}|ϕ(θ)⟩^{n}⟨ϕ(θ)|ϕ_{j}⟩^{n} dθ_{k} ⋯ dθ_{2k−2}.  (29)

Letting

ϕ_{i} := (e^{iψ_{i}^{1}}ϕ_{i}^{1} ; e^{iψ_{i}^{2}}ϕ_{i}^{2} ; ⋮ ; e^{iψ_{i}^{k}}ϕ_{i}^{k}) ,

we get

\frac{⟨ϕ_{i}|ϕ(θ)⟩^{n}}{cos^{n} θ_{1}}

= (e^{iψ_{i}^{1}}ϕ_{i}^{1} + ∑_{j=2}^{k−1} e^{i(θ_{k−2+j}−ψ_{i}^{j})} tan θ_{1} sin θ_{2} ⋯ sin θ_{j} cos θ_{j+1}ϕ_{i}^{j}

+e^{i(θ_{2k−2}−ψ_{i}^{k−1})} tan θ_{1} sin θ_{2} ⋯ sin θ_{k−1}ϕ_{i}^{k})^{n}.

Letting x := tan θ_{1}, Lemma 6 induce that (29) is monotone decreasing about θ_{1}. The proof is complete.

Lemma 3 If the deviation measure W (ρ, ρ̂) = h′([tr ρρ̂, 1]), then

D_{ρ}^{W,(n)}(Π) = ∫_{[0,1]} D_{ρ}^{W_{β},(n)}(Π)h′( dβ).  (30)

Proof For the probability measure π on P(H), we have

∫_{P(H)} W (ρ, ρ̂)π( dρ̂) = ∫_{P(H)} h(tr ρρ̂)π( dρ̂)
= ∫_{P(H)} ∫_{[0,1]} h_{β}(tr ρρ̂)h′( dβ)π( dρ̂)
= ∫_{[0,1]}(∫_{P(H)} h_{β}(tr ρρ̂)π( dρ̂))h′( dβ)
= ∫_{[0,1]}(∫_{P(H)} W_{β}(ρ, ρ̂)π( dρ̂))h′( dβ).

Substituting π( dρ̂) for tr(Π( dρ̂)ρ^{(n)}), then we obtain (30). □ Lemma 4 Let H be any finite dimensional Hilbert space. For any elements ϕ, ψ ∈ H and any selfadjoint operator A on H, the following are equivalent.

∘ \frac{⟨ϕ|A|ϕ⟩}{⟨ϕ|ϕ⟩} ≥ \frac{⟨ψ|A|ψ⟩}{⟨ψ|ψ⟩}.
∘ ⟨ϕ|A|ϕ⟩⟨ψ| Id −A|ψ⟩ ≥ ⟨ψ|A|ψ⟩⟨ϕ| Id −A|ϕ⟩.

Lemma 5 we have

⟨∑_{i} ϕ_{i}^{(n)}| Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫α^{\frac{π}{2}} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨∑_{i} ϕ_{i}^{(n)}| Id −Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫_{0}^{α} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨ϕ(0)^{(n)}| Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = C ∫α^{\frac{π}{2}} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨ϕ(0)^{(n)}| Id −Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = C ∫_{0}^{α} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}.

Proof Ŵ_{β}(ρ) is denoted as follows:

Ŵ_{β}(ρ) = k′ ∫_{P(H)} W_{β}(ρ, ρ̂)ρ̂^{(n)}ν( dρ̂)
= k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ≤β}} ρ̂^{(n)}ν( dρ̂).

We obtain

⟨∑_{i} ϕ_{i}^{(n)}| Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = ⟨∑_{i} ϕ_{i}^{(n)}| k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ≤β}} ρ̂^{(n)}ν( dρ̂) |∑_{i} ϕ_{i}^{(n)}⟩
= ∑_{i,j} k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ≤β}} ⟨ϕ_{i}^{(n)}| ρ̂^{(n)} |ϕ_{j}^{(n)}⟩ ν( dρ̂)
= ∑_{i,j} k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ≤β}}⟨ϕ_{i}|ρ̂|ϕ_{j}⟩^{n}ν( dρ̂)
= \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫α^{\frac{π}{2}} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}.

Similarly,

⟨∑_{i} ϕ_{i}^{(n)}| Id −Ŵ_{β}(ρ) |∑_{i} ϕ_{i}^{(n)}⟩ = ∑_{i,j} k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ >β}}⟨ϕ_{i}|ρ̂|ϕ_{j}⟩^{n}ν( dρ̂)
= \frac{k′ · (k − 1)!}{π^{(k−1)}} ∫_{0}^{α} f_{1}(θ_{1}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨ϕ(0)^{(n)}| Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ≤β}}⟨ϕ(0)|ρ̂|ϕ(0)⟩^{n}ν( dρ̂)
= C ∫α^{\frac{π}{2}} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}
⟨ϕ(0)^{(n)}| Id −Ŵ_{β}(ρ) |ϕ(0)^{(n)}⟩ = k′ ∫_{{ρ̂∈P(H)| tr ρ̂ρ >β}}⟨ϕ(0)|ρ̂|ϕ(0)⟩^{n}ν( dρ̂)
= C ∫_{0}^{α} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}.
□

Lemma 6 The following function f(x) is monotone decreasing on [0, ∞):

f(x) := ∑_{a=1}^{m} ∑_{b=1}^{m} \underbrace{∫_{0}^{2π} … ∫_{0}^{2π}}_{k}(c_{a}^{0}e^{id_{a}^{0}} + x ∑_{j=1}^{k} e^{i(θ_{j}+d_{a}^{j})}c_{a}^{j})^{n}(c_{b}^{0}e^{id_{b}^{0}} + x ∑_{j=1}^{k} e^{−i(θ_{j}+d_{b}^{j})}c_{b}^{j})^{n} dθ_{1} ⋯ dθ_{k}.

where c_{n}^{j}, d_{n}^{j} are any real numbers. Proof The set K_{n}^{m} is defined as follows:

K_{n}^{m} := {I = (I_{1}, ⋯, I_{m}) ∈ (N^{+,0})^{m} |∑_{j=1}^{m} I_{j} = n} .

The number C(I) is defined for I ∈ K_{n}^{m} as sufficing the following condition:

(∑_{j=1}^{m} x_{j})^{n} = ∑_{I∈K_{n}^{m}} C(I)x_{1}^{I_{0}} … x_{m}^{I_{m}}.

Therefore,

(c_{a}^{0} + x ∑_{j=1}^{k} e^{i(θ_{j}+d_{a}^{j})}c_{a}^{j})^{n} = ∑_{I∈K_{n}^{k+1}} C(I)e^{id_{a}^{0}}(c_{a}^{0})^{I_{0}}e^{iI_{1}(θ_{1}+d_{a}^{1})}(c_{a}^{1})^{I_{1}} … e^{iI_{d}(θ_{k}+d_{a}^{k})}(c_{a}^{k})^{I_{k}}x^{n−I_{0}}.

Thus,

f(x)
= ∑_{a=1}^{m} ∑_{b=1}^{m}(2π)^{k} ∑_{I} C(I)e^{iI_{0}(d_{a}^{0}−d_{b}^{0})}(c_{a}^{0}c_{b}^{0})^{I_{0}}e^{iI_{1}(d_{a}^{1}−d_{b}^{1})}(c_{a}^{1}c_{b}^{1})^{I_{1}} … e^{iI_{k}(d_{a}^{k}−d_{b}^{k})}(c_{a}^{k}c_{b}^{k})^{I_{k}}x^{2n−2I_{0}}
= (2π)^{k} ∑_{I} C(I) ∑_{a=1}^{m} ∑_{b=1}^{m} e^{i(∑_{j=0}^{k} I_{i}d_{a}^{i}−∑_{j=0}^{k} I_{i}d_{b}^{i})}(c_{a}^{0})^{I_{0}} … (c_{a}^{k})^{I_{k}}(c_{b}^{0})^{I_{0}} … (c_{b}^{k})^{I_{k}}x^{2n−2I_{0}}
= (2π)^{k} ∑_{I} C(I)D(I)x^{2n−2I_{0}},

D(I) := ∑_{a=1}^{m} ∑_{b=1}^{m} e^{i(∑_{j=0}^{k} I_{i}d_{a}^{i}−∑_{j=0}^{k} I_{i}d_{b}^{i})}(c_{a}^{0})^{I_{0}} … (c_{a}^{k})^{I_{k}}(c_{b}^{0})^{I_{0}} … (c_{b}^{k})^{I_{k}}.

It is sufficient to show D(I) ≥ 0. Letting

v_{a} := (c_{a}^{0})^{I_{0}} … (c_{a}^{k})^{I_{k}}
y_{a} := ∑_{j=0}^{k} I_{i}d_{a}^{i}
w_{a,b} := cos(y_{a} − y_{b}),

we have

D(I) = ∑_{a=1}^{m} ∑_{b=1}^{m} v_{a}w_{a,b}v_{b}.

Then

w_{a,b} = cos(y_{a} − y_{b}) = cos y_{a} cos y_{b} + sin y_{a} sin y_{b}.

As {cos y_{a} cos y_{b}} and {sin y_{a} sin y_{b}} are nonnegative, {w_{a,b}} is nonnegative matrix. Therefore, we obtain D(I) ≥ 0. □

B Proof of Lemma 1

D^{W,(n)}(Π_{n}) = ∫_{P(H)} h(d_{fs}(ρ, ρ̂) tr(Π_{n}( dρ̂)ρ^{(n)})
= ∫_{P(H)} h(θ_{1})\binom{n + k − 1}{k − 1}|⟨ϕ(θ)^{(n)}|ϕ(0)^{(n)}⟩|^{2}ν( dθ)

= ∫0^{\frac{π}{2}} h(θ_{1})\binom{n + k − 1}{k − 1}\frac{(k − 1)!}{π^{k−1}} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}

× \underbrace{∫_{0}^{2π} ⋯ ∫_{0}^{2π}}_{k−1} \underbrace{∫0^{\frac{π}{2}} ⋯ ∫0^{\frac{π}{2}}}_{k−2} sin^{2k−5} θ_{2} ⋯ sin θ_{k−1} cos θ_{2} ⋯ cos θ_{k−1} dθ_{2} ⋯ dθ_{2k−2}

= ∫0^{\frac{π}{2}} h(θ_{1})\binom{n + k − 1}{k − 1}\frac{(k − 1)!}{π^{k−1}} cos^{2n+1} θ_{1} sin^{2k−3} θ_{1} dθ_{1}

× \underbrace{∫_{0}^{1} x^{2k−5} dx ⋯ ∫_{0}^{1} x dx}_{k−2} ·(2π)^{k−1}

= ∫0^{\frac{π}{2}} h(θ)\binom{n + k − 1}{k − 1}\frac{(k − 1)!}{π^{k−1}} cos^{2n+1} θ sin^{2k−3} θ dθ \frac{(2π)^{k−1}}{2^{k−2}(k − 2)!}
= 2(k − 1)\binom{n + k − 1}{k − 1} ∫0^{\frac{π}{2}} h(θ) cos^{2n+1} θ sin^{2k−3} θ dθ.

The proof is complete.

C Proof of Thereon 4

Definition 1 and Lemma 1 means that:

D^{d_{b}^{γ},(n)}(Π_{n}) = 2(k − 1)\binom{n + k − 1}{k − 1} ∫0^{\frac{π}{2}} cos^{2n+1} θ sin^{2k−3+γ} θ dθ.  (31)

Since

∫0^{\frac{π}{2}} cos^{x} θ sin^{y} θ dθ = \frac{Γ(\frac{x+1}{2})Γ(\frac{y+1}{2})}{2Γ(\frac{x+y}{2} + 1)} ∀x, y ∈ R,

we have

D^{d_{b}^{γ},(n)}(Π_{n}) = 2(k − 1)\binom{n + k − 1}{k − 1}\frac{Γ(n + 1)Γ(k − 1 + γ/2)}{Γ(n + k + γ/2)}
= \frac{Γ(n + 1)Γ(k − 1 + γ/2)Γ(n + k)}{Γ(n + k + γ/2)Γ(n + 1)Γ(k − 1)}
= \frac{Γ(n + k)}{Γ(n + k + γ/2)} \frac{Γ(k − 1 + γ/2)}{Γ(k − 1)}.  (32)

Therefore we obtain (8) from the following formula of Γ function:

lim_{n→∞} \frac{Γ(n + x)}{Γ(n)n^{x}} = 1.

Letting γ := 2, we obtain

D^{d_{b}^{2},(n)}(Π_{n}) = \frac{Γ(n + k)}{Γ(n + k + 1)} \frac{Γ(k − 1 + 1)}{Γ(k − 1)} = \frac{k − 1}{n + k} .

Thus, we get (9).

Next, we will prove (10). θ^{2} can be expanded as:

θ^{2} = ∑_{i=0}^{∞} \frac{(2i − 2)!!}{(2i − 1)!!} \frac{sin^{2i} θ}{i} ,

where we put (2n)!! = 2n(2n − 2) ⋯ 4 · 2, (2n − 1)!! = (2n − 1)(2n − 3) ⋯ 3 · 1, 0!! = (−1)!! = 1. From (32) we have

D^{d_{fs}^{2},(n)}(Π_{n}) = ∑_{i=0}^{∞} \frac{(2i − 2)!!}{(2i − 1)!!i}D^{d_{b}^{2i},(n)}(Π_{n})

= ∑_{i=0}^{∞} \frac{(2i − 2)!!}{(2i − 1)!!i} ∏_{j=0}^{i−1} \frac{k − 1 + j}{n + k + j}

≅ (k − 1)\frac{1}{n} − \frac{2}{3}k(k − 1)\frac{1}{n^{2}} + k(k − 1)\frac{23k − 7}{45} \frac{1}{n^{3}} .

Thus we obtain (10). Lemma 1 derives that

log Pr_{Π_{n}}^{ρ^{(n)}}{ρ̂ ∈ P(H)|d_{fs}(ρ, ρ̂) ≥ ϵ}
= log ((k − 1)\binom{n + k − 1}{k − 1} ∫_{0}^{cos^{2} ϵ} x(1 − x)^{k−2} dx) .  (33)

Therefore, it is sufficient for (13) to show that

log \binom{n + k − 1}{k − 1} ≅ (k − 1) log n − log(k − 1)! + \frac{1}{n} \frac{(k − 1)k}{2}  (34)
log (∫_{0}^{cos^{2} ϵ} x^{n}(1 − x)^{k−2} dx) ≅ 2n log cos ϵ − log n + 2(k − 2) log sin ϵ − 2 log cos ϵ
−\frac{1}{n}(1 + (k − 2) cot^{2} ϵ).  (35)

The left hand of (34) is calculated as:

log \binom{n + k − 1}{k − 1} = ∑_{i=0}^{k−1} log \frac{n + i}{i}
= (k − 1) log n − log(k − 1)! + ∑_{i=1}^{k−1} log(1 + \frac{i}{n})
≅ (k − 1) log n − log(k − 1)! + ∑_{i=1}^{k−1} \frac{i}{n}
= (k − 1) log n − log(k − 1)! + \frac{1}{n} \frac{(k − 1)k}{2} .

Therefore, we have (34). The left hand of (35) is calculated as:

log (∫_{0}^{cos^{2} ϵ} x^{n}(1 − x)^{k−2} dx − 2n log cos ϵ)
= log (∫_{0}^{cos^{2} ϵ} (\frac{x}{cos^{2} ϵ})^{n} (1 − x)^{k−2} dx)
= log (∫_{0}^{1} x^{n} (1 − cos^{2} x)^{k−2} \frac{1}{cos^{2} ϵ} dx)
= −2 log cos ϵ + log (∑_{i=1}^{k−2} \binom{k − 2}{i} (− cos^{2} ϵ)^{i} \frac{1}{n + i + 1})
= −2 log cos ϵ − log n + log (∑_{i=1}^{k−2} \binom{k − 2}{i} (− cos^{2} ϵ)^{i} \frac{1}{1 + \frac{i+1}{n}} )
≅ −2 log cos ϵ − log n + log (∑_{i=1}^{k−2} \binom{k − 2}{i} (− cos^{2} ϵ)^{i} (1 − \frac{i + 1}{n}))
= −2 log cos ϵ − log n + log ((1 − cos^{2} ϵ)^{k−2} − \frac{1}{n} (1 − cos^{2} ϵ)^{k−3} (1 − (k − 1) cos^{2} ϵ))
= −2 log cos ϵ − log n + log (1 − cos^{2} ϵ)^{k−2} + log (1 − \frac{1}{n} \frac{1 − (k − 1) cos^{2} ϵ}{1 − cos^{2} ϵ})
≅ −2 log cos ϵ − log n + (k − 2) log (1 − cos^{2} ϵ) − \frac{1}{n} \frac{1 − (k − 1) cos^{2} ϵ}{1 − cos^{2} ϵ}.

We obtain (35).

D Proof of Lemma 2

From the symmetry of P(H) and Π_{m}, we may assume that ρ_{1} = |ϕ_{0}⟩⟨ϕ_{0}|, ρ_{2} = |ϕ_{ϵ}⟩⟨ϕ_{ϵ}|. First, we consider the case of k = 2. For the following calculation, we prepare the following equations:

|⟨ϕ_{ϵ}|ϕ(θ)⟩|^{2n}

= (cos^{2} ϵ cos^{2} θ_{1} + sin^{2} θ_{1} sin^{2} ϵ + 2 cos ϵ sin ϵ cos θ_{1} sin θ_{1} cos θ_{2})^{n}  (36)
∫_{0}^{2π} log(1 + 2a cos θ + a^{2}) dθ = 4πψ(|a|) log |a|,  (37)

where the function ψ is defined as:

ψ(x) = \{ \begin{matrix}1 & x ≥ 1 \\ 0 & x < 0 \end{matrix} .

Paying attention to (5) and (7), we have

\frac{−D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)}) − m log cos^{2} ϵ}{m}

= −\frac{1}{m} ((m + 1) ∫_{P(H)} log \frac{|⟨ϕ_{0}|ϕ(θ)⟩|^{2m}}{|⟨ϕ_{ϵ}|ϕ(θ)⟩|^{2m}} |⟨ϕ_{0}|ϕ(θ)⟩|^{2m}ν( dθ) + m log cos ϵ)
= −\frac{2(m + 1)}{π}

· ∫_{0}^{2π} ∫0^{\frac{π}{2}} log (\frac{cos^{2} θ_{1} cos^{2} ϵ}{(cos θ_{1} cos ϵ + sin θ_{1} cos θ_{2} sin ϵ)^{2} + (sin θ_{1} sin θ_{2} sin ϵ)^{2}} )

· cos^{2m+1} θ_{1} sin θ_{1} dθ_{1} dθ_{2}

= \frac{(m + 1)}{π}

· ∫0^{\frac{π}{2}} ∫_{0}^{2π} log (1 + 2 tan θ_{1} tan ϵ cos θ_{2} + (tan θ_{1} tan ϵ)^{2}) dθ_{2} cos^{2m+1} θ_{1} sin θ_{1} dθ_{1}

= \frac{(m + 1)}{π} ∫0^{\frac{π}{2}} 4π log(tan θ_{1} tan ϵ)ψ(tan θ_{1} tan ϵ) cos^{2m+1} θ_{1} sin θ_{1} dθ_{1}
= \frac{(m + 1)}{π} ∫\frac{π}{2} −ϵ^{\frac{π}{2}} 4π log(tan θ_{1} tan ϵ) cos^{2m+1} θ_{1} sin θ_{1} dθ_{1}
= 2(m + 1) ∫\frac{π}{2} −ϵ^{\frac{π}{2}} log (tan^{2} θ_{1} tan^{2} ϵ) cos^{2m+1} θ_{1} sin θ_{1} dθ_{1}
= (m + 1) ∫_{0}^{sin^{2} ϵ} log (\frac{1 − x}{x} tan^{2} ϵ) x^{m} dx.  (38)

Substitute a = tan^{2} ϵ in (51) of Lemma 8 , then

(m + 1) ∫_{0}^{sin^{2} ϵ} x^{m} log (\frac{1 − x}{x} tan^{2} ϵ) dx = − log cos^{2} ϵ − ∑_{i=1}^{m} \frac{sin^{2i} ϵ}{i}.  (39)

From (38) and (39), we have

\frac{D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)})}{m} = ∑_{i=1}^{m} \frac{sin^{2i} ϵ}{i}.  (40)

Therefore, we can prove (22).

Next, we consider the case of k ≥ 3. In this case, we have:

|⟨ϕ_{ϵ}|ϕ(θ)⟩|^{2n}

= (cos^{2} ϵ cos^{2} θ_{1} + sin^{2} θ_{1} cos^{2} θ_{2} sin^{2} ϵ + 2 cos ϵ sin ϵ cos θ_{1} sin θ_{1} cos θ_{2} cos θ_{k})^{n}. (41)

Paying attention to (5),(37) and Lemma 7, we can calculate as:

\frac{−D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)}) − m log cos^{2} ϵ}{m}

= −\frac{1}{m} (\binom{m + k − 1}{k − 1} ∫_{P(H)} log (\frac{|⟨ϕ_{0}|ϕ(θ)⟩|^{2m}}{|⟨ϕ_{ϵ}|ϕ(θ)⟩|^{2m}} ) |⟨ϕ_{0}|ϕ(θ)⟩|^{2m}ν( dθ) + 2m log cos ϵ)
= −\frac{2(k − 1)(k − 2)}{π}\binom{m + k − 1}{k − 1}

· ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} log (\frac{cos^{2} θ_{1} cos^{2} ϵ}{(cos^{2} ϵ cos^{2} θ_{1} + sin^{2} θ_{1} cos^{2} θ_{2} sin^{2} ϵ + 2 cos ϵ sin ϵ cos θ_{1} sin θ_{1} cos θ_{2} cos θ_{k})})

· cos^{2m+1} θ_{1} sin^{2k−3} θ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{1} dθ_{2} dθ_{k}

= \frac{2(k − 1)(k − 2)}{π}\binom{m + k − 1}{k − 1} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}}

· ∫_{0}^{2π} log (1 + 2 tan θ_{1} cos θ_{2} tan ϵ cos θ_{k} + (tan θ_{1} cos θ_{2} tan ϵ)^{2}) dθ_{k}

· cos^{2m+1} θ_{1} sin^{2k−3} θ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{1} dθ_{2}

= \frac{2(k − 1)(k − 2)}{π}\binom{m + k − 1}{k − 1} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} 4πψ(tan θ_{1} cos θ_{2} tan ϵ)
· log (tan θ_{1} cos θ_{2} tan ϵ) cos^{2m+1} θ_{1} sin^{2k−3} θ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{1} dθ_{2}.  (42)

Substitute that a := tan^{2} ϵ, s := sin^{2} θ_{2}, y := cos^{2} θ_{1}, then the condition

tan θ_{1} cos θ_{2} tan ϵ ≥ 1 turns into the following conditions:

1 − \frac{y}{a(1 − y)} ≥ x ≥ 0, \frac{a}{1 + a} ≥ y ≥ 0

Using (42) and (50), we have

\frac{−D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)}) − m log cos^{2} ϵ}{m}

= (k − 1)(k − 2)\binom{m + k − 1}{k − 1} ∫0^{\frac{a}{1+a}} (∫_{0}^{1− \frac{y}{a(1−y)}} x^{k−3} log ((1 − x)\frac{a(1 − y)}{y}) dx) y^{m}(1 − y)^{k−2} dy

= (k − 1)\binom{m + k − 1}{k − 1} ∫0^{\frac{a}{1+a}} (− log (\frac{y}{a(1 − y)}) − ∑_{i=1}^{k−2} \frac{1}{i} (\frac{a − (1 + a)y}{a(1 − y)} )^{i}) y^{m}(1 − y)^{k−2} dy

= −(k − 1)\binom{m + k − 1}{k − 1} ∫0^{\frac{a}{1+a}} log (\frac{y}{a(1 − y)}) y^{m}(1 − y)^{k−2} dy − (k − 1)\binom{m + k − 1}{k − 1}f (\frac{a}{1 + a}) , (43)

where f(x) is defined as:

f(x) := ∫_{0}^{x} ∑_{i=1}^{k−2} \frac{1}{i} (1 − \frac{y}{x})^{i} y^{m}(1 − y)^{k−2−i} dy.

From Lemma 9, the derivative of f(x) can be calculated as:

f′(x) = ∫_{0}^{x} \frac{y}{x^{2}} (∑_{i=1}^{k−2} (\frac{1 − \frac{y}{x}}{1 − y} )^{i−1}) y^{m}(1 − y)^{k−3} dy

= ∫_{0}^{x} \frac{y}{x^{2}} (\frac{1 − (\frac{1− \frac{y}{x}}{1−y} )^{k−2}}{1 − \frac{1− \frac{y}{x}}{1−y}} ) y^{m}(1 − y)^{k−3} dy

= \frac{1}{x(1 − x)} ∫_{0}^{x} (1 − (\frac{1 − \frac{y}{x}}{1 − y} )^{k−2}) y^{m}(1 − y)^{k−2} dy

= \frac{1}{x(1 − x)} ∫_{0}^{x} y^{m}(1 − y)^{k−2} dy − \frac{1}{x(1 − x)} ∫_{0}^{x} (1 − \frac{y}{x})^{k−2} y^{m} dy

= \frac{1}{x(1 − x)} ∑_{i=0}^{k−2} \binom{k − 2}{i} ∫_{0}^{x}(−1)^{i}y^{m+i} dy − \frac{x^{m+1}}{x(1 − x)} ∫_{0}^{1} (1 − t)^{k−2} t^{m} dt

= \frac{x^{m}}{x(1 − x)} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−x)^{i}}{m + i + 1} − \frac{x^{m+1}}{x(1 − x)}\binom{m + k − 1}{k − 2}^{−1} \frac{1}{m + 1}.  (44)

By (51) and Lemma 9, the first term of (43) is calculated as:

−(k − 1)\binom{m + k − 1}{k − 1} ∫0^{\frac{a}{1+a}} y^{m}(1 − y)^{k−2} log (\frac{y}{a(1 − y)}) dy

= −(k − 1)\binom{m + k − 1}{k − 1} ∑_{i=0}^{k−2} \binom{k − 2}{i}(−1)^{i} ∫0^{\frac{a}{1+a}} y^{m+i} log (\frac{y}{a(1 − y)}) dy

= −(k − 1)\binom{m + k − 1}{k − 1} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{m + i + 1} (− log(1 + a) + ∑_{j=1}^{m+i} \frac{1}{j} (\frac{a}{1 + a})^{j})

= −(k − 1)\binom{m + k − 1}{k − 1} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{m + i + 1}

× (− log(1 + a) + ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j} + ∑_{j=m+1}^{m+i} \frac{1}{j} (\frac{a}{1 + a})^{j})

= −(k − 1)\binom{m + k − 1}{k − 1} (∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{m + i + 1}) (− log(1 + a) + ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j})

−(k − 1)\binom{m + k − 1}{k − 1} ∑_{i=0}^{k−2} ∑_{j=1}^{i} \binom{k − 2}{i}\frac{(−1)^{i}}{(m + i + 1)(j + m)} (\frac{a}{1 + a})^{j+m}

= −(k − 1)\binom{m + k − 1}{k − 1} (\binom{m + k − 1}{k − 2}^{−1} \frac{1}{m + 1}) (− log(1 + a) + ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j})

+(k − 1)\binom{m + k − 1}{k − 1}g (\frac{a}{1 + a})

= log(1 + a) − ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j} − (k − 1)\binom{m + k − 1}{k − 1}g (\frac{a}{1 + a}) ,  (45)

where g(x) is defined as:

g(x) := ∑_{i=0}^{k−2} ∑_{j=1}^{i} \binom{k − 2}{i}\frac{(−1)^{i}}{(m + i + 1)(j + m)}x^{j+m}.

By Lemma 9, the derivative of g(x) is calculated as:

g′(x) = ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{(m + i + 1)} ∑_{j=1}^{i} x^{j+m−1}

= ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{(m + i + 1)}x^{m} \frac{1 − x^{i}}{1 − x}

= \frac{x^{m}}{1 − x} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−1)^{i}}{(m + i + 1)} − \frac{x^{m}}{1 − x} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−x)^{i}}{(m + i + 1)}

= \frac{x^{m}}{1 − x}\binom{m + k − 1}{k − 2}^{−1} \frac{1}{m + 1} − \frac{x^{m}}{1 − x} ∑_{i=0}^{k−2} \binom{k − 2}{i}\frac{(−x)^{i}}{(m + i + 1)}.  (46)

From (44) and (46), we have f′(x) = −g′(x). The definitions of f(x) and g(x) means that f(0) = g(0) = 0. Then we obtain f(x) = −g(x). By (43) and (45), we have

\frac{−D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)}) − 2m log cos ϵ}{m}

= (k − 1)(k − 2)\binom{m + k − 1}{k − 1} ∫0^{\frac{a}{1+a}} (∫_{0}^{1− \frac{y}{a(1−y)}} x^{k−3} log ((1 − x)\frac{a(1 − y)}{y}) dx) y^{m}(1 − y)^{k−2} dy

= log(1 + a) − ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j}

−(k − 1)\binom{m + k − 1}{k − 1} (g (\frac{a}{1 + a}) + f (\frac{a}{1 + a}))

= log(1 + a) − ∑_{j=1}^{m} \frac{1}{j} (\frac{a}{1 + a})^{j}

= − log cos^{2} ϵ − ∑_{j=1}^{m} \frac{1}{j} sin^{2j} ϵ.

Then, we obtain:

\frac{D_{Π_{m}}(ρ_{1}^{(m)}∥ρ_{2}^{(m)})}{m} = ∑_{j=1}^{m} \frac{sin^{2i} ϵ}{i}.

We proved (22).

Next we will prove (23). We consider the tangent space T_{ρ}P(H) at ρ := |ϕ(0)⟩⟨ϕ(0)|. If c(t) is a curve on P(H) such that c(0) = ρ, ċ denotes the element of T_{ρ}P(H) defined by c(t). the Fubini-Study metric g_{fs} is defined as:

g_{fs}(ċ, ċ) := (lim_{t→0} \frac{d_{fs}(c(0), c(t))}{t})^{2}

Therefore, it is sufficient to show that

J_{Π_{n}}^{ρ} = 2ng_{fs}.

Let c(t) := |ϕ_{t}⟩⟨ϕ_{t}|, ϕ_{t} := ϕ(t, 0, …, 0). (See the equation (6).) Because g_{fs}(ċ, ċ) = 1, it is sufficient to prove that

J_{Π_{n}}^{ρ}(ċ, ċ) = 2n.

We assume that k ≥ 3. From (41), we have

( \frac{d}{dt} log (|⟨ϕ_{t}|ϕ(θ)⟩|^{2n})|_{t=0})^{2} |⟨ϕ_{0}|ϕ(θ)⟩|^{2n} = 4n^{2} cos^{2n−2} θ_{1} sin^{2} θ_{1} cos^{2} θ_{2} cos^{2} θ_{k}.  (47)

By (47) and (49), we have:

\binom{m + k − 1}{k − 1} ∫_{P(H)} ( \frac{d}{dt} log (|⟨ϕ_{t}|ϕ(θ)⟩|^{2m})|_{t=0})^{2} |⟨ϕ_{0}|ϕ(θ)⟩|^{2m}ν( dθ)

= \frac{2(k − 1)(k − 2)}{π}\binom{m + k − 1}{k − 1}4m^{2} ∫0^{\frac{π}{2}} cos^{2n−1} θ_{1} sin^{2k−1} θ_{1} dθ_{1}

· ∫0^{\frac{π}{2}} cos^{3} θ_{2} sin^{2k−5} θ_{2} dθ_{2} ∫_{0}^{2π} cos^{2} θ_{k} dθ_{k}

= \frac{2(k − 1)(k − 2)}{π}\binom{m + k − 1}{k − 1}4m^{2} \frac{(m − 1)!(k − 1)!}{2(m + k − 1)!} \frac{1!(k − 3)!}{2(k − 1)!}π
= 2m.  (48)

We get (23). In the case of k = 2, similarly we can prove (23).

Lemma 7 If k ≥ 3, then we have

∫_{P(H)} f(θ_{1}, θ_{2}, θ_{k})ν( dθ)

= \frac{2(k − 1)(k − 2)}{π} ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} f(θ_{1}, θ_{2}, θ_{k}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{2} dθ_{k}.(49)

Proof From (7) the left hand of (49) is calculated as:

∫_{P(H)} f(θ_{1}, θ_{2}, θ_{k})ν( dθ)

= \frac{(k − 1)!}{π^{k−1}} ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} f(θ_{1}, θ_{2}, θ_{k}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{2} dθ_{k}

× \underbrace{∫0^{\frac{π}{2}} ⋯ ∫0^{\frac{π}{2}}}_{k−3} sin^{2k−7} θ_{3} ⋯ sin θ_{k−1} cos θ_{3} ⋯ cos θ_{k−1} dθ_{2} ⋯ dθ_{k−1} \underbrace{∫_{0}^{2π} ⋯ ∫_{0}^{2π}}_{k−2} dθ_{k+1} ⋯ dθ_{2k−2}

= ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} f(θ_{1}, θ_{2}, θ_{k}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{2} dθ_{k}

×\frac{(k − 1)!}{π^{k−1}} \underbrace{∫_{0}^{1} x^{2k−7} dx ⋯ ∫_{0}^{1} x dx}_{k−3} ·(2π)^{k−2}

= ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} f(θ_{1}, θ_{2}, θ_{k}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{2} dθ_{k}

×\frac{(k − 1)!}{π^{k−1}} \frac{(2π)^{k−2}}{2^{k−3}(k − 3)!}

= \frac{2(k − 1)(k − 2)}{π} ∫_{0}^{2π} ∫0^{\frac{π}{2}} ∫0^{\frac{π}{2}} f(θ_{1}, θ_{2}, θ_{k}) cos θ_{1} sin^{2k−3} θ_{1} dθ_{1} cos θ_{2} sin^{2k−5} θ_{2} dθ_{2} dθ_{k}.

Then we obtain (49). □

Lemma 8 The following integral can be calculated as:

∫_{0}^{1−a} x^{m} log (\frac{1 − x}{a}) dx = \frac{1}{m + 1} (− log a − ∑_{i=1}^{m+1} \frac{(1 − a)^{i}}{i} )  (50)
∫0^{\frac{a}{1+a}} x^{m} log (\frac{x}{a(1 − x)}) dx = \frac{1}{m + 1} (− log(1 + a) + ∑_{i=1}^{m} \frac{1}{i} (\frac{a}{1 + a})^{i}) .  (51)

Proof The equation (50) is derived by the following:

∫_{0}^{α} x^{m} log(1 − x) dx = \frac{1}{m + 1} ((α^{m+1} − 1) log(1 − α) − ∑_{i=1}^{m+1} \frac{α^{i}}{i} ) .  (52)

Also, the equation (51) is derived by (52) and the following:

∫_{0}^{α} x^{m} log x dx = \frac{1}{m + 1} (α^{m+1} (log α − \frac{1}{m + 1})) .  (53)
□

Lemma 9 We have the following equations:

∑_{i=0}^{n} \binom{n}{i}\frac{(−1)^{i}}{m + i} = ∫_{0}^{1} x^{m−1}(1 − x)^{n} dx = \binom{m + n}{n}^{−1} \frac{1}{m}.  (54)

It is easily derived.

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