myctr

# Alexander S. Holevo’s Researches in Quantum Information Theory
in 20th Century

Masahito Hayashi  School of Data Science, The Chinese University of Hong Kong,
Shenzhen, Longgang District, Shenzhen, 518172, China
International Quantum Academy (SIQA), Futian District, Shenzhen 518048, China
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya
464-8602, Japan
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

This paper reviews Holevo’s contributions to quantum information theory during
the 20 century. At that time, he mainly studied three topics, classical-
quantum channel coding, quantum estimation with Craméro-Rao approach, and
quantum estimation with the group covariant approach. This paper addresses
these three topics.

###### keywords:

classical-quantum channel; quantum state estimation; group covariance.

##  1 Introduction

Alexander S. Holevo is undoubtedly one of the greatest researchers in quantum
information theory. In particular, before quantum information theory became a
popular scientific area, he led this research area as the leading person.
Nevertheless, this field could not attrach much attention at that time. He is
one of the greatest witnesses of the long winter experienced by the field of
quantum information theory. In 1994, the author started this area with reading
his textbook [1]:

A.S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory. North-
Holland, Amsterdam (1982). Originally published in Russian in 1980.

At that time, we were still in the long winter era/period of quantum
information theory. In the later part of the 1990s, the author researched many
of the findings that the author will eventually write about in the 20th
century. Therefore, the author is extremely grateful for Alexander S. Holevo’s
contributions to this area.

Looking back at his publication records, we see that his researches can be
mainly divided into three streams. His first stream is the first period on
quantum information, which starts in the beginning of 1970s and ends in the
middle of 1980s. The above textbook summarizes the results in this period. His
second stream is his researches in quantum dynamical system with quantum
stochastic process and semi-group, which starts in the middle of 1980s and
ends almost in the end of 20th century. The second stream is related to the
mathematical description of a quantum measurement process. His third stream is
the second period on quantum information, which starts in the following
international conference.

Third International Conference on Quantum Communication & Measurement, Mt.
Fuji-Hakone Land, Japan, September 25-30, (1996).

At the above conference, he knew the result for a pure-state channel coding by
Hausladen [2]. Inspired by this result, he obtained its generalization to the
case with mixed states when he stayed in Tamagawa University after this
conference [3]. This result was published as one of his most famous results
[4]:

A.S. Holevo, “The capacity of the quantum channel with general signal states,”
IEEE Transactions on Information Theory 44 (1), 269-273 (1998).

His third stream still continues until today. That is, he still actively
studies quantum information theory.

Unfortunately, the author is not familiar with his second stream. Since his
contributions in his third stream are well known, this paper focuses on his
first stream. We review his major results in his first stream, which is
composed of three topics. The first topic is classical-quantum channel coding.
The second topic is quantum state estimation with the Crámer-Rao approach. The
third topic is quantum state estimation with the group covariant approach.

The remaining of this paper is organized as follows: Section 2 reviews his
results for classical-quantum channel coding. Section 3 discusses his results
for quantum state estimation with the Crámer Rao approach. Section 4 explains
his results for quantum state estimation with the group covariant approach. In
addition, these sections discuss how his results influenced the subsequent
studies in the area of quantum information theory. Section 5 is the
conclusion.

##  2 Classical-quantum channel coding

Classical-quantum (cq-) channel coding is the subarea in quantum information
theory.

###  2.1 Information quantities

Information quantities play a central role in channel coding. Here, we
introduce classical and quantum information quantities. When a random variable
$X$ is generated according to a distribution $P_{X}=\\{P_{X}(x)\\}_{x\in{\cal
X}}$ on ${\cal X}$, its Shannon entropy $H(P_{X})$ is defined as

$\displaystyle H(P_{X}):=-\sum_{x\in{\cal X}}P_{X}(x)\log P_{X}(x).$ (1)

When two random variables $X,Y$ are subject to a joint distribution $P_{X,Y}$,
the mutual information is given as

$\displaystyle I(P_{X,Y}):=H(P_{X})-H(P_{Y})-H(P_{X,Y}),$ (2)

where $P_{X}$ and $P_{Y}$ are the marginal distribution of $P_{X,Y}$ with
respect to $X$ and $Y$, respectively. The mutual information has another form,
by using relative entropy. When two probability distributions $P_{X}$ and
$Q_{X}$ are given, the relative entropy is defined as

$\displaystyle D(P_{X}\|Q_{X}):=\sum_{x\in{\cal X}}P_{X}(x)(\log P_{X}(x)-\log
Q_{X}(x)).$ (3)

Then, the mutual information is rewritten as

$\displaystyle I(P_{X,Y}):=\sum_{x\in{\cal
X}}P_{X}(x)D\big{(}P_{Y|X=x}\big{\|}P_{Y}\big{)},$ (4)

where $P_{Y|X=x}$ is the conditional distribution $P_{XY}(x,y)/P_{X}(x)$.

Next, we consider a quantum system ${\cal H}_{1}$. Given a density matrix
$\rho$ on ${\cal H}_{1}$, its von Neumann entropy is given as

$\displaystyle S(\rho):=-\mathop{\rm Tr}\nolimits\rho\log\rho.$ (5)

When a density matrix $\rho_{1,2}$ on a joint system ${\cal H}_{1}\otimes{\cal
H}_{2}$, the mutual information is given as

$\displaystyle I(\rho_{1,2}):=S(\rho_{1})+S(\rho_{2})+S(\rho_{1,2}),$ (6)

where $\rho_{1}:=\mathop{\rm Tr}\nolimits_{2}\rho_{1,2}$ and
$\rho_{2}:=\mathop{\rm Tr}\nolimits_{1}\rho_{1,2}$. The mutual information
also has another form, by using relative entropy. When two densities matrices
$\rho$ and $\sigma$ are given, the relative entropy is defined as

$\displaystyle D(\rho\|\sigma):=\mathop{\rm
Tr}\nolimits\rho(\log\rho-\log\sigma).$ (7)

Then, the mutual information is rewritten as

$\displaystyle I(\rho_{1,2})=D(\rho_{1,2}\|\rho_{1}\otimes\rho_{2}).$ (8)

Next, we assume that the state $\rho_{1,2}$ is a classical-quantum state,
i.e., it has the form $\rho_{1,2}=\sum_{x\in{\cal X}}P_{X}(x)|x\rangle\langle
x|\otimes\rho_{2|x}$, where $P_{X}$ is a distribution on ${\cal X}$. In
addition, ${\cal H}_{1}$ is assumed to be spanned by an orthogonal basis
$\\{|x\rangle\\}_{x\in{\cal X}}$, and $\rho_{2|x}$ is a density matrix on
${\cal H}_{2}$. In this case, the formula (6) of the mutual information is
rewritten as

$\displaystyle I(\rho_{1,2})=S\Big{(}\sum_{x\in{\cal
X}}P_{X}(x)\rho_{2|x}\Big{)}-\sum_{x\in{\cal X}}P_{X}(x)S(\rho_{2|x}).$ (9)

The formula (8) of the mutual information is rewritten as

$\displaystyle I(\rho_{1,2})=\sum_{x\in{\cal
X}}P_{X}(x)D\Big{(}\rho_{2|x}\Big{\|}\sum_{x\in{\cal
X}}P_{X}(x)\rho_{2|x}\Big{)}.$ (10)

###  2.2 Formulation and capacity theorem

The problem of cq-channel coding is formulated as follows: [9, 5, 6, 7, 8]
Consider the input alphabet ${\cal X}$ and the output quantum system ${\cal
H}$, which is represented as a Hilbert space. A cq-channel is given as
$\bm{W}:=\\{W_{x}\\}_{x\in{\cal X}}$, where $W_{x}$ expresses the density
matrix on ${\cal H}$ to express the output state with the input $x\in{\cal
X}$.

To discuss a cq-channel, we discuss the mutual information when a random
variable $X$ is generated according to distribution $P=\\{P(x)\\}_{x\in{\cal
X}}$ on ${\cal X}$. Then, the mutual information with the input distribution
$P$ is [11, 10]

$\displaystyle I(P,\bm{W})=I\Big{(}\sum_{x\in{\cal X}}P(x)|x\rangle\langle
x|\otimes W_{x}\Big{)}.$ (11)

The aim of cq-channel coding is the transmission of classical message via $n$
uses of the cq-channel $\bm{W}$. When we use the cq-channel $\bm{W}$ $n$
times, the channel is given as
$\bm{W}^{(n)}\\{W^{(n)}_{x^{n}}\\}_{x^{n}\in{\cal X}^{n}}$ [13, 12]. i.e., the
output state with the input $x^{n}=(x_{1},\ldots,x_{n})\in{\cal X}^{n}$ is
given as

$\displaystyle W^{(n)}_{x^{n}}:=W_{x_{1}}\otimes\cdots W_{x_{n}},$ (12)

which is a density matrix on the $n$-fold tensor product system ${\cal
H}^{\otimes n}$. We denote the set of messages by ${\cal
M}:=\\{1,\ldots,\mathsf{M}\\}$. An encoder is given as a map $\phi_{n}$ from
${\cal M}$ to ${\cal X}^{n}$. When $\phi(m)$ is given as
$(x_{1},\ldots,x_{n})$, the output state is

$\displaystyle W^{(n)}_{\phi(m)}=W_{x_{1}}\otimes\cdots W_{x_{n}}.$ (13)

A decoder is given as a positive operator-valued measure (POVM)
$\bm{\Pi}:=\\{\Pi_{m}\\}_{m\in{\cal M}}$ over the quantum system ${\cal
H}^{\otimes n}$, whose set of outcomes is given as ${\cal M}$. That is,
$\bm{\Pi}$ is a resolution of the identity on ${\cal H}^{\otimes n}$, i.e.,
$\Pi_{m}$ is a positive semi-definite matrix on ${\cal H}^{\otimes n}$, and
the relation $\sum_{m\in{\cal M}}\Pi_{m}=I$. The pair of an encoder $\phi$ and
a decoder $\bm{\Pi}$ is called a code and is written as $\Phi$. The
performance of a code $\Phi$ is evaluated by two parameters. One is the size
$|\Phi|$ of the code $\Phi$, which is given by $\mathsf{M}$. The other is the
decoding error probability $e(\Phi)$, which is given as

$\displaystyle e(\Phi):=1-\frac{1}{\mathsf{M}}\sum_{m\in{\cal M}}\mathop{\rm
Tr}\nolimits W^{(n)}_{\phi(m)}\Pi_{m}.$ (14)

The channel capacity is defined as

$\displaystyle
C(\bm{W}):=\sup_{\\{\Phi_{n}\\}_{n}}\Big{\\{}\lim_{n\to\infty}\frac{1}{n}\log|\Phi_{n}|\Big{|}e(\Phi_{n})\to
0\Big{\\}}.$ (15)

The cq-channel coding theorem, i.e., the capacity theorem is formulated as

$\displaystyle C(\bm{W})=\max_{P\in{\cal P}({\cal X})}I(P,\bm{W}),$ (16)

where ${\cal P}({\cal X})$ expresses the set of probability distributions on
${\cal X}$. To show this result, we need to prove the following two
inequalities.

$\displaystyle C(\bm{W})\leq$ $\displaystyle\max_{P\in{\cal P}({\cal
X})}I(P,\bm{W})$ (17) $\displaystyle C(\bm{W})\geq$
$\displaystyle\max_{P\in{\cal P}({\cal X})}I(P,\bm{W}).$ (18)

To show (16), he tackled the inequality (17), which is called the converse
part in information theory. He proved the inequality (17) by dividing it into
two steps.

In his paper [10] in 1972, as the first step, Holevo showed

$\displaystyle I({\rm P}[P,\bm{W},\bm{\Pi}])\leq I(P,\bm{W})$ (19)

for any POVM $\bm{\Pi}$ on ${\cal Y}$, where ${\rm P}[P,\bm{W},\bm{\Pi}]$ is
the joint distribution on ${\cal X}\times{\cal Y}$ defined as

$\displaystyle{\rm P}[P,\bm{W},\bm{\Pi}](x,y):=P(x)\mathop{\rm Tr}\nolimits
W_{x}\Pi_{y}.$ (20)

Nowadays, the inequality (19) can be shown by using the monotonicity of the
quantum relative entropy [14, 15] for trace-preserving completely positive
maps. However, at that time, the above monotonicity was not known. We can say
that Holevo showed the monotonicity of the quantum relative entropy in the
case of the mutual information with cq-channel as (19).

In his paper [12] in 1979, as the second step, he showed that

$\displaystyle C(\bm{W})\leq\lim_{n\to\infty}\frac{1}{n}\sup_{P_{n}\in{\cal
P}({\cal X}^{n})}\sup_{\bm{\Pi}_{n}}I({\rm
P}[P,^{n},\bm{W}^{(n)},\bm{\Pi}_{n}]).$ (21)

In addition, he pointed out the relation

$\displaystyle n\max_{P\in{\cal P}({\cal X})}I(P,\bm{W})=\max_{P_{n}\in{\cal
P}({\cal X}^{n})}I(P_{n},\bm{W}^{(n)}).$ (22)

In fact, the above relation can be shown by using the chain rule. Combining
three relations (19), (21), and (22), he derived (17).

For the opposite direction (18), Hausladen et. al. [2] showed the inequality
(18) when all $W_{x}$ are pure states. Their key idea is typical subspaces
[16, 17]. Inspired by this result [2], using the idea of typical subspaces,
Holevo showed the inequality (18) with general mixed states $W_{x}$. Later,
independently, Schumacher and Westmoreland [18] showed the same result. In
addition, with Burnashev, Holevo derived an exponential decay date of the
decoding error probability when the transmission rate is smaller than the
capacity $C(\bm{W})$ and all $W_{x}$ are pure states [19].

Since the result (16) forms the foundation of quantum communication, it
motivated many studies in quantum information theory. For example, Ogawa and
Nagaoka [20] showed the strong converse theorem for cq-channel coding. To
prove (16) via information spectrum method [21], the reference [22] invented a
useful matrix inequality, which is often called Hayashi-Nagaoka inequality.
Bennet et al [23] considered cq-channel coding with entanglement-assistance,
which leads to reverse Shannon theorem [24], which is known as a new topic in
Shannon theory. Holevo [25] provided a more elegant proof. Also using the
result (16), Devetak [26] showed the coding theorem for sending a quantum
state via a noisy quantum channel. In this way, the result (16) yields various
results in quantum information theory.

##  3 Quantum estimation with Crámer Rao type bounds

Holevo made great contributions to quantum state estimation as well. In this
problem, we consider a parameterized family of density operators ${\cal
S}=\\{\rho_{\theta}\\}_{\theta\in\Theta}$ on a Hilbert space ${\cal H}$. When
$\Theta$ is a continuous subset of $\mathbb{R}^{d}$, an estimator is
formulated as a POVM $\bm{\Pi}$ on ${\cal H}$ whose data set is
$\mathbb{R}^{d}$. Since $\mathbb{R}^{d}$ is a continuous set, its rigorous
formulation requires measure theory. A POVM $\Pi$ is defined as a map from the
set ${\cal B}(\mathbb{R}^{d})$ of Borel sets to the set of positive semi-
definite operators on ${\cal H}$. The map $\Pi$ should satisfy the following
conditions.

$\displaystyle\Pi(\emptyset)=0,\quad\Pi(\mathbb{R}^{d})=I,\quad\Pi(\cup_{j}B_{j})=\sum_{j}\bm{\Pi}(B_{j}),$
(23)

where $B_{j}\in{\cal B}(\mathbb{R}^{d})$ are disjoint sets. We often consider
the unbiased condition, which is formulated as

$\displaystyle\int_{\mathbb{R}^{d}}\hat{\theta}\mathop{\rm
Tr}\nolimits\rho_{\theta}\Pi(d\hat{\theta})=\theta$ (24)

for $\theta\in\Theta$. Since the unbiased condition is too restrictive, taking
the Taylor expansion in (24) at $\theta_{0}\in\Theta$, we often consider its
relaxed version, the locally unbiased condition at $\theta_{0}\in\Theta$,
which is formulated as

$\displaystyle\int_{\mathbb{R}^{d}}\hat{\theta}\mathop{\rm
Tr}\nolimits\rho_{\theta_{0}}\Pi(d\hat{\theta})$ $\displaystyle=\theta_{0}$
(25)
$\displaystyle\frac{\partial}{\partial\theta^{k}}\int_{\mathbb{R}^{d}}\hat{\theta}^{l}\mathop{\rm
Tr}\nolimits\rho_{\theta_{0}}\Pi(d\hat{\theta})$ $\displaystyle=\delta_{k,l}.$
(26)

To measure the precision of an estimator $\Pi$, we focus on the mean square
matrix $V(\Pi)$ as

$\displaystyle
V^{k,l}_{\theta}(\Pi):=\int_{\mathbb{R}^{d}}(\hat{\theta}^{k}-\theta^{k})(\hat{\theta}^{l}-\theta^{l})\mathop{\rm
Tr}\nolimits\rho_{\theta}\Pi(d\hat{\theta}).$ (27)

When $\Pi$ is an unbiased estimator, the mean square matrix $V_{\theta}(\Pi)$
coincides with the covariance matrix. For $d=1$, we simply call
$V_{\theta}(\Pi)$ the mean square.

Helstrom defined the SLD $L_{\theta,k}$ as a Hermitian (self-adjoint) operator
satisfying [27, 8]

$\displaystyle
L_{\theta,k}\circ\rho_{\theta}=\frac{\partial\rho_{\theta}}{\partial\theta^{k}}.$
(28)

where $X\circ Y:=\frac{1}{2}(XY+YX)$ Then, Helstrom defined the SLD Fisher
information matrix $J_{\theta,k,l}$ as

$\displaystyle J_{\theta,k,l}:=\mathop{\rm Tr}\nolimits
L_{\theta,k}(L_{\theta,l}\circ\rho_{\theta}).$ (29)

For a locally unbiased estimator $\Pi$ at $\theta$, using Schwarz inequality,
we obtain the matrix inequality based on positive semi-definite matrices.

$\displaystyle V_{\theta}(\Pi)\geq J_{\theta}^{-1}.$ (30)

In the following, we use matrix inequalities in the above sense. For $d=1$,
the following locally unbiased estimator achieves the equality in (30). When
$\Pi$ is the spectral decomposition of the operator
$\frac{1}{J_{\theta}}(L_{\theta}+\theta)$, the equality in (30) holds.

However, the equality in (30) does not hold in the multiple-parameter case.
This difficulty is caused by the non-commutativity of SLDs $L_{\theta,k}$. The
first attempt to tackle this problem was done by Yuen and Lax [28]. They
considered complex parameters to identify the state, i.e., they assumed
$\Theta\subset\mathbb{C}^{d}$. They focused on the unbiased condition (24) by
replacing $\mathbb{R}^{d}$ by $\mathbb{C}^{d}$, and considered right
logarithmic derivatives based on complex parameters. Their method works well
for the quantum Gaussian family, whose mathematical formulation is given in
the references [31, 30, 29][1, Chapter 5], which has crucial non-commutativity
related to the canonical observables $Q$ and $P$. But, it cannot be applied to
a general real-multiple-parameter model. At that time, several Russian
researchers [32, 34] focused this topic.

To resolve this problem, for a general real-multiple-parameter model, Holevo
[35, 1] defined the right logarithmic derivative $L_{\theta,k}^{R}$ as

$\displaystyle\rho_{\theta}L_{\theta,k}^{R}=\frac{\partial\rho_{\theta}}{\partial\theta^{k}}.$
(31)

Here, $L_{\theta,k}^{R}$ is not necessarily self-adjoint operator. Holevo then
defined the RLD Fisher information matrix $J^{R}_{\theta}$ as

$\displaystyle J^{R}_{\theta,k,l}:=\mathop{\rm Tr}\nolimits\rho
L_{\theta,k}^{R}(L_{\theta,l}^{R})^{\dagger}.$ (32)

For a locally unbiased estimator $\Pi$ at $\theta$, using Schwarz inequality,
we obtain the matrix inequality

$\displaystyle V_{\theta}(\Pi)\geq(J_{\theta}^{R})^{-1}.$ (33)

In the one-parameter case, the inequality $J_{\theta}^{R}\geq J_{\theta}$
holds so that (33) does not give a better bound than (30). However, in the
multiple parameter case, due to the effect of the off-diagonal part, there are
several cases where (33) gives a better bound than (30). To clarify such a
case, he introduced the D operator ${\cal D}_{\theta}$ on the set of self-
adjoint operators as

$\displaystyle\rho_{\theta}\circ{\cal D}_{\theta}(X)=i[X,\rho_{\theta}].$ (34)

When the space spanned by $\\{L_{\theta,k}\\}_{k}$ is invariant with respect
to the D operator ${\cal D}_{\theta}$, which is called the D invariant
property, Holevo [35, 1] showed that

$\displaystyle(J_{\theta}^{R})^{-1}=J_{\theta}^{-1}+\frac{i}{2}J_{\theta}^{-1}D_{\theta}J_{\theta}^{-1}$
(35)

where the antisymmetric matrix $D_{\theta}$ is defined as

$\displaystyle D_{\theta,k,l}:=\mathop{\rm Tr}\nolimits{\cal
D}_{\theta}(L_{\theta,k})(L_{\theta,l}\circ\rho_{\theta}).$ (36)

This fact means that (33) is a stricter matrix inequality than (30) in the
above case.

Furthermore, to utilize the imaginary part of the inequality (33), Holevo [35,
1] considered the weighted sum of the components of the mean square error
matrix $V_{\theta}(\Pi)$. We choose a $d\times d$ positive semi-definite
matrix $G$, and focus on

$\displaystyle\mathop{\rm Tr}\nolimits GV_{\theta}(\Pi).$ (37)

In fact, since it is impossible to simultaneously minimize all diagonal
components in $V_{\theta}(\Pi)$, we need to handle their trade-off. The above
strategy enables us to discuss their trade-off. To discuss this problem,
Holevo focused on the following lemma.

######  Lemma 3.1.

[1, Lemma 6.6.1],[33, (2.9)],[34, (8.1 1)] Given a Hermitian matrix $R$, we
have

$\displaystyle\min_{V:\hbox{Hermitian}}\\{\mathop{\rm Tr}\nolimits GV|V\geq\pm
R\\}=\mathop{\rm Tr}\nolimits|\sqrt{G}R\sqrt{G}|.$ (38)

The minimum holds when $V=\sqrt{G}^{-1}|\sqrt{G}R\sqrt{G}|\sqrt{G}^{-1}$ .

This lemma can be easily shown by diagonalizing the Hermitian matrix
$\sqrt{G}R\sqrt{G}$.

Combining the matrix inequality (33) and Lemma 3.1, Holevo [35, 1] showed that

$\displaystyle\mathop{\rm Tr}\nolimits GV_{\theta}(\Pi)\geq\mathop{\rm
Tr}\nolimits G{\rm\bf Re}(J_{\theta}^{R})^{-1}+\mathop{\rm
Tr}\nolimits|\sqrt{G}{\rm\bf Im}(J_{\theta}^{R})^{-1}\sqrt{G}|$ (39)

for a locally unbiased estimator $\Pi$. Hence, the right hand side of (39) is
called the RLD bound. When the D invariant property holds, the relation (35)
simplifies (39) to

$\displaystyle\mathop{\rm Tr}\nolimits GV_{\theta}(\Pi)\geq\mathop{\rm
Tr}\nolimits GJ_{\theta}^{-1}+\frac{1}{2}\mathop{\rm
Tr}\nolimits|\sqrt{G}J_{\theta}^{-1}D_{\theta}J_{\theta}^{-1}\sqrt{G}|.$ (40)

When we employ (30) instead of (33), the lower bound is composed only of the
first term in (40). Also, Holevo [36] also showed the equality in (40).

However, the above approach works only when the D invariant property holds. To
resolve this problem, we consider a more general approach. When $\Pi$ is a
locally unbiased estimator, we choose $d$ self-adjoint operators
$\bm{X}(\Pi)=(X^{1}(\Pi),\ldots,X^{d}(\Pi))$ as

$\displaystyle
X^{k}(\Pi):=\int_{\mathbb{R}^{d}}(\hat{\theta}^{k}-\theta^{k})\Pi(d\hat{\theta}).$
(41)

Then, we have the condition

$\displaystyle\mathop{\rm Tr}\nolimits
X^{k}(\Pi)\frac{\partial\rho_{\theta}}{\partial\theta^{l}}=\delta_{k,l}.$ (42)

Also, Holevo [1, (6.7.73)] showed the matrix inequality

$\displaystyle V_{\theta}(\Pi)\geq Z(\bm{X}(\Pi)),$ (43)

where the Hermitian matrix $Z(\bm{X})$ is defined as

$\displaystyle Z^{k,l}(\bm{X}):=$ $\displaystyle\mathop{\rm Tr}\nolimits
X^{k}X^{l}\rho_{\theta}.$ (44)

Then, we define

$\displaystyle C_{\theta}(G,\bm{X}):=\inf_{V:{\rm Symmetric}}\\{\mathop{\rm
Tr}\nolimits GV|V\geq Z(\bm{X})\\}.$ (45)

Since the relation (43) guarantees that the matrix $V_{\theta}(\Pi)$ satisfies
the condition in (45), we find the inequality

$\displaystyle\mathop{\rm Tr}\nolimits GV_{\theta}(\Pi)\geq
C_{\theta}(G,\bm{X}(\Pi)).$ (46)

Therefore, for a locally unbiased estimator $\Pi$, we obtain

$\displaystyle\mathop{\rm Tr}\nolimits GV_{\theta}(\Pi)\geq
C_{\theta}^{HN}(G):=\inf_{\bm{X}}C_{\theta}(G,\bm{X}),$ (47)

where the above infimum is taken for $d$ self-adjoint operators
$\bm{X}=(X^{1},\ldots,X^{d})$ satisfying the condition (42).

When the D invariant property holds, Holevo [1, Section 6.7] showed that the
lower bound in (47) equals the right hand side of (40). He also showed that
the infimum in (47) is attained when the $d$ self-adjoint operators $\bm{X}$
is given as

$\displaystyle X^{k}=\sum_{l=1}^{d}(J_{\theta}^{-1})^{k,l}L_{\theta,l}.$ (48)

Using this fact, Holevo [36, 1] constructed an estimator to attain the lower
bound (47) under the quantum Gaussian family.

Later, using Lemma 3.1, Nagaoka solved the minimization (45) as

$\displaystyle C_{\theta}(G,\bm{X})=\mathop{\rm Tr}\nolimits\sqrt{G}{\rm\bf
Re}Z(\bm{X})\sqrt{G}+\mathop{\rm Tr}\nolimits|\sqrt{G}{\rm\bf
Im}Z(\bm{X})\sqrt{G}|.$ (49)

Nagaoka [37] wrote that Holevo introduced the lower bound
$C_{\theta}^{HN}(G)$. In fact, the aim of the paper [37] was to present
Nagaoka’s new bound, which is the two-parameter case of the bound called
Nagaoka-Hayashi bound in the recent paper [38], as well as to present its
advantage over the lower bound $C_{\theta}^{HN}(G)$. Also, since the
inequality (47) can be obtained with a few steps and a combination of the
matrix inequality (42) and Lemma 3.1, Nagaoka considered that the inequality
(47) should be attributed to the contribution by Holevo. Following to
Nagaoka’s idea, many subsequent studies [39, 40, 41, 42, 43, 44, 45, 38, 46]
called it the Holevo bound. In these studies, one of the authors was directly
suggested by Nagaoka or his collaborators about the use of the terminology
“the Holevo bound”. Recently, due to these studies, other studies [49, 47, 48]
followed this terminology without direct suggestion from Nagaoka or his
collaborator.

However, Holevo [1, Section 6.7] considered the infimum (47) only when the D
invariant property holds. Nagaoka [37] wrote down the infimum (47) for the
general case at the first time. While Nagaoka’s paper [37] was written as a
technical report, the reference [39] provided the full derivation of (47) with
the form (49) as a journal publication. In the reference [39, Remark 2], the
author mentioned that this bound was essentially introduced in [1, Sec. VI-7]
while the notation of [39] is based on the reference [37]. At that time, the
author was overconfident about the explanation in the reference [37] for the
reference [1]. The main issue of the reference [1, Sec. VI-7] is a statement
different from (47). Since Nagaoka wrote down the infimum
$C_{\theta}(G,\bm{X})$ as (49) and the expression (49) has been widely used
for the calculation of this bound, it is suitable to call it the Holevo-
Nagaoka bound as has been done in the reference [50]. Indeed, some of the
recent papers cited only the reference [1] to discuss this bound whereas it is
hard to derive the bound (47) with the form (49) only from the reference [1].
When a paper uses this bound, it might be better to cite paper [37] (and
reference [39]) in addition to reference [1].

At a later time, the importance of the lower bound $C_{\theta}^{HN}(G)$ was
revealed. In the following, we discuss its importance in subsequent studies.
Nagaoka [51] introduced the concept of the $n$-fold independent and identical
distributed (i.i.d.) extension of a given state family ${\cal
S}=\\{\rho_{\theta}\\}$ as ${\cal S}_{n}=\\{\rho_{\theta}^{\otimes n}\\}$. At
that time, many strong objections to this extension were presented from the
viewpoint of physics due to the no-cloning theorem. Therefore, this research
direction was not well accepted at that time. The paper [39] showed the
asymptotic attainability of the lower bound $C_{\theta}^{HN}(G)$ under the
i.i.d. extension of any submodel of the qubit system including the full model.
To show this fact, the paper [39] showed the asymptotic approximation of the
i.i.d. extension by the qubit system by the Gaussian state family, which is
called the local asymptotic normality. The paper [52] showed the local
asymptotic normality of full parameter model in the qudit system, which
implies the asymptotic attainability of the RLD bound under the i.i.d.
extension of the full parameter model of the qubit system. Finally, the papers
[40, 43] showed the asymptotic attainability of the lower bound
$C_{\theta}^{HN}(G)$ under the i.i.d. extension of any submodel of the qubit
system in various settings. In this way, the lower bound $C_{\theta}^{HN}(G)$
plays the key role among subsequent works. Nowadays, it is widely considered
that the lower bound $C_{\theta}^{HN}(G)$ gives the ultimate bound in state
estimation.

##  4 Quantum estimation with group covariant estimator

Quantum system has useful group symmetry, which is an essential difference
from classical system. It is not impossible to consider a group symmetry in a
classical system, but the irreducible decomposition does not have a simple
form in a classical system. In contrast, Since a quantum system often has a
very elegant irreducible decomposition, we can expect that group symmetry
greatly simplifies the problem for quantum state estimation. Based on this
idea, Holevo formulated quantum state estimation theory when the state family
${\cal S}=\\{\rho_{\theta}\\}_{\theta\in\Theta}$ has the symmetry with respect
to a representation $f$ of a group $G$ on a Hilbert space ${\cal H}$ [53, 1].
When the parameter space $\Theta$ is closed with respect to an action of the
group $G$ and the state family ${\cal S}$ satisfies the following condition,
the state family is called a covariant family with respect to the
representation $f$.

$\displaystyle f(g)\rho_{\theta}f(g)^{\dagger}=\rho_{g\theta}$ (50)

for $g\in G$ and $\theta\in\Theta$. In this section, we assume the above
condition for our state family ${\cal S}$. Next, we consider a POVM $\Pi$
whose data set is $\Theta$. We say that a POVM $\Pi$ is covariant with respect
to the representation $f$ when the following condition holds:

$\displaystyle f(g)M(B)f(g)^{\dagger}=M(gB)$ (51)

for any Borel set $B$ of $\Theta$, where $gB:=\\{g\theta|\theta\in B\\}$. The
concept for the group covariant measurement can be attributed back to Mackey
[54].

To discuss the precision of our estimate, we need to employ an error function
$R(\theta,\hat{\theta})$. Then, the average error is given as

$\displaystyle R_{\theta}(\Pi):=\int_{\Theta}R(\theta,\hat{\theta})\mathop{\rm
Tr}\nolimits\Pi(d\hat{\theta})\rho_{\theta}.$ (52)

When we focus on the average for $\theta$, using a prior distribution $\nu$ on
$\Theta$, we discuss the Bayesian risk

$\displaystyle R_{\nu}(\Pi):=\int_{\Theta}R_{\theta}(\Pi)\nu(d\theta).$ (53)

When $G$ is a compact group, the invariant probability measure $\mu$ on $G$
can be defined. We often employ $\mu$ as our prior distribution. When we are
interested in the worst case $\theta$, we discuss the minimax risk

$\displaystyle R(\Pi):=\max_{\theta\in\Theta}R_{\theta}(\Pi).$ (54)

In this problem setting, it is natural to impose the group invariance to the
error function $R(\theta,\hat{\theta})$, which is formulated as

$\displaystyle R(g\theta,g\hat{\theta})=R(\theta,\hat{\theta}).$ (55)

When the error function $R$ satisfies (55) and the estimator $\Pi$ is
covariant, we have

$\displaystyle R_{\theta}(\Pi)=R_{\theta^{\prime}}(\Pi)=R(\Pi)=R_{\nu}(\Pi)$
(56)

for $\theta^{\prime},\theta\in\Theta$.

When the error function $R$ satisfies (55) and $G$ is a compact group, as a
quantum version of Hunt-Stein theorem [55], Holevo [53, 1] showed the
following relations.

$\displaystyle\min_{\Pi\in{\cal M}}R(\Pi)=\min_{\Pi\in{\cal
M}}R_{\mu}(\Pi)=\min_{\Pi\in{\cal M}_{cov}}R(\Pi)=\min_{\Pi\in{\cal
M}_{cov}}R_{\mu}(\Pi)=\min_{\Pi\in{\cal M}_{cov}}R_{\theta}(\Pi),$ (57)

where ${\cal M}$ expresses the set of POVM whose data set is $\Theta$ and
${\cal M}_{cov}$ expresses the set of covariant POVM whose data set is
$\Theta$.

When $G$ is not compact, the invariant prior $\mu$ does not exist. Instead of
(57), we have [56, 57]

$\displaystyle\min_{\Pi\in{\cal M}}R(\Pi)=\min_{\Pi\in{\cal
M}_{cov}}R(\Pi)=\min_{\Pi\in{\cal M}_{cov}}R_{\theta}(\Pi).$ (58)

Now, we assume that there exists an invariant measure $\mu$ on $\Theta$, which
is a weaker condition than the compactness of $\Theta$. For a covariant POVM
$\Pi$, there exists a positive semi-definite operator $T_{0}$ such that

$\displaystyle
M(d\theta)=f(g_{\theta})T_{0}f(g_{\theta})^{\dagger}\mu(d\theta),$ (59)

where $g_{\theta}$ is chosen as $g_{\theta}\theta_{0}=\theta$, and

$\displaystyle\int_{\Theta}f(g_{\theta})T_{0}f(g_{\theta})^{\dagger}\mu(d\theta)=I.$
(60)

Conversely, given a positive semi-definite operator $T_{0}$ satisfying (60),
we can construct a covariant POVM $\Pi$ with (59). Therefore, our optimization
in (57) and (58) is simplified to the optimization for the choice of a
positive semi-definite operator $T_{0}$ satisfying (60).

Furthermore, condition (60) can be simplified into the case with irreducible
representation and the case of a commutative group $G$ without multiplicity.
In the case with irreducible representation, the left hand side of (60) is a
constant multiplication of $I$. Hence, the condition (60) is replaced by the
normalization of the trace of $T_{0}$. In the case of a commutative compact
group $G$ without multiplicity, each irreducible representation is one-
dimensional. The set of irreducible representations is denoted by $\hat{G}$.
We denote the vector in the irreducible representation space identified by
$\lambda\in\hat{G}$ by $|e_{\lambda}\rangle$. The condition “without
multiplicity” means that the uniqueness of the vector $|e_{\lambda}\rangle$.
We denote the set of irreducible representations appearing in ${\cal H}$ by
$\hat{S}$. Then, any vector in ${\cal H}$ is written as
$\sum_{\lambda\in\hat{S}}a_{\lambda}|e_{\lambda}\rangle$. The condition (60)
is simplified as $\|a_{\lambda}\|^{2}=1$ for $\lambda\in\hat{S}$.

Holevo [53, 1] applied this approach to several covariant models. The first
example is the model with the group U(1), in which Helstrom [58] studied the
case when the error function is the delta function. In this model, a state
$\rho$ and a representation of the group U(1) are fixed. This model can be
considered as the estimation of unknown phase operation. Holevo derived the
optimal estimator under proper error functions. The second example is a model
with Weyl-Heinsenberg group. He discusses the estimation in quantum Gaussian
family [1, Chapter 5]. The third example is a model with group SU(2) [53].

Later, this approach diverged into various directions. The first one is the
asymptotic estimation on the full pure state family [59]. The i.i.d. extension
of the full pure state family is written as a state family on the symmetric
tensor product space, which is an irreducible space. Hence, this approach
exactly derives the optimal estimator with a finite size. In contrast, the
Cramér-Rao approach gives only an asymptotically approximately optimal
estimator. In this way, while the group covariant approach can be applied
limited cases, this approach can drive a stronger conclusion in this way.

The second one is the estimation of unitary action. When we estimate the
unknown applied unitary and the set of possible unitaries forms a group, the
group covariant approach works well. In this case, we can optimize the input
state as well as the measurement. To consider this problem, many researchers
[62, 63, 60, 66, 67, 64, 65, 61] used covariant measurements. Thanks to the
covariant state estimation theory, the problem is simplified to the
optimization of the input state. As results, they could derive Heisenberg
scaling in this problem setting. In this approach, some of them exactly
derived the optimal error with the finite size. Indeed, many researchers [68,
69, 70, 71, 72] claimed the achievement of Heisenberg scaling by using the
Cramér-Rao approach. However, the Cramér-Rao approach has serious drawback in
the estimation of unitary action. Basically, Cramér-Rao approach realizes the
local optimal estimator. Using the two-step adaptive method [73], this
approach realizes the global optimal estimator in the case of state
estimation. However, the two-step adaptive method works only with the usual
scaling [74]. In addition, the bound obtained by the Cramér-Rao approach is
smaller than the limit of the minimum error with a finite size [74]. This
relation shows that it is impossible to show the achievability of the bound
obtained by the Cramér-Rao approach. That is, if it is possible, we can derive
an asymptotic bound that contradicts with the limit of the minimum error with
a finite size. In this way, Holevo’s covariant state estimation theory greatly
supports many researches for Heisenberg scaling.

The key idea of Holevo’s group symmetric approach is reducing the number of
free parameters by using the symmetry. Nowadays, this idea is widely accepted
in quantum information theory, and his result is the first successful example
in this direction. For example, approximate cloning employs the group symmetry
[75]. That is, there exists this philosophy behind many recent results of
quantum information theory with group covariance [76].

##  5 Conclusion

We have reviewed Holevo’s researches in quantum information theory in the 20th
century. Although majority of the research were done in the 1970s, they have
strongly influenced our current studies in quantum information theory. They
form the foundations of many topics in quantum information theory. That is to
say, he contributed many tools for these areas.

Interestingly, he contributed quantum estimation via two different approaches,
the Craméro-Rao approach and the group covariant approach - he addressed. That
is, he addressed quantum estimation without sticking to one approach. This
fact shows that he has the ability to tackle key problem without being
constrained to a specific strategy. The author would like to conclude this
article by expressing his sincere respect for Holevo’s significant
contributions to quantum information theory.

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