Autoparallelity of Quantum Statistical Manifolds in The Light of Quantum Estimation Theory

The autoparallelity is a multi-dimensional version (including the 1-dimensional case in particular) of the notion of geodesic for a manifold equipped with an affine connection. In the classical information geometry for manifolds of probability distributions, where the triple $(g,\nabla^{({\rm e})},\nabla^{({\rm m})})$ of the Fisher metric $g$, the e-connection $\nabla^{({\rm e})}$ and the m-connection $\nabla^{({\rm m})}$ plays the leading role, the autoparallelity w.r.t. (with respect to) the e-connection, which is called the e-autoparallelity, is particularly important. This is because the e-autoparallel submanifolds of the space ${\mathcal{P}}$ consisting of all strictly positive probability distributions on a finite set are the exponential families, which is one of the key concepts in probability theory and statistics. For quantum statistical manifolds, which are submanifolds of the space ${\mathcal{S}}$ consisting of all strictly positive density operators on a finite-dimensional Hilbert space, we may introduce an analogous notion of exponential families as autoparallel submanifolds w.r.t.  an affine connection on ${\mathcal{S}}$ analogous to the classical e-connection. However, the notion of quantum exponential families introduced in this way does not necessarily have statistical and/or physical importance. One of the main achievements of the present paper is that the e-autoparallelity for the SLD structure, which is among a family of information-geometric structures introduced on ${\mathcal{S}}$ in a unified way, has been shown to possess estimation-theoretical characterizations.

In order to clarify our motivation, we begin with an overview of a result in the classical estimation theory. Let ${\mathcal{P}}={\mathcal{P}}(\Omega)$ be the totality of strictly positive probability distributions (probability mass functions) on a finite set $\Omega$, and let ${\mathcal{M}}=\{p_{\xi}\,|\,\xi=(\xi^{1},\dots,\xi^{n})\in\Xi\}\subset{\mathcal{P}}$ be a statistical model whose elements $p_{\xi}$ are smoothly and injectively parametrized by an $n$-dimensional parameter $\xi$ ranging over an open subset $\Xi$ of $\mathbb{R}^{n}$. As is well known, the variance matrix $V_{\xi}\in\mathbb{R}^{n\times n}$ of an arbitrary unbiased estimator for the parameter $\xi$ satisfies the Cramér-Rao inequality $V_{\xi}\geq G_{\xi}^{-1}$, where $G_{\xi}\in\mathbb{R}^{n\times n}$ denotes the Fisher information matrix. An unbiased estimator achieving the equality $V_{\xi}=G_{\xi}^{-1}$ for every $\xi\in\Xi$ is called an efficient estimator for the parameter $\xi$, whose existence is known to impose strong restrictions on both the set ${\mathcal{M}}$ and the parametrization $\xi\mapsto p_{\xi}$. Namely, we have the following theorem (e.g. § ​5a.2 (p.324) of [1], Eq. (7.14) in §​ 7.2 of [2], Theorem 3.12 of [3]).

Condition (ii) in the above theorem means that the elements of ${\mathcal{M}}$ are represented as

$$p_{\xi}(\omega)=\exp[C(\omega)+\sum_{i=1}^{n}\theta_{i}(\xi)F^{i}(\omega)-\psi(\xi)]$$

(1.1)

and that the parameter $\xi$ satisfies

$$\xi^{i}=E_{\xi}(F^{i}),$$

(1.2)

where $E_{\xi}$ denotes the expectation w.r.t. the distribution $p_{\xi}$. This condition is expressed in the language of geometry as follows.

• (ii)’

  ${\mathcal{M}}$ is autoparallel in ${\mathcal{P}}$ w.r.t. the e-connection of ${\mathcal{P}}$, and $\xi$ forms an affine coordinate system w.r.t. the m-connection of ${\mathcal{M}}$.

A quantum version of Theorem ​1.1 is known, which we state below. Let ${\mathcal{H}}$ be a finite-dimensional Hilbert space, ${\mathcal{S}}={\mathcal{S}}({\mathcal{H}})$ be the totality of strictly positive density operators on ${\mathcal{H}}$, and ${\mathcal{M}}=\{\rho_{\xi}\;|\;\xi\in\Xi\}$ be an arbitrary quantum statistical model consisting of states $\rho_{\xi}$ in ${\mathcal{S}}$. It is well known that the variance matrix $V_{\xi}$ of an arbitrary unbiased estimator for the parameter $\xi$ satisfies the SLD Cramér-Rao inequality $V_{\xi}\geq G_{\xi}^{-1}$ [6, 7], where $G_{\xi}$ is the SLD Fisher information matrix (see Section 4 for details.) When an unbiased estimator satisfies $V_{\xi}=G_{\xi}^{-1}$ for every $\xi\in\Xi$, we call it an efficient estimator for the parameter $\xi$ as in the classical case. Then we have the following theorem (Theorem 7.6 of [3]).

When a model ${\mathcal{M}}=\{\rho_{\xi}\}$ is represented as (1.3), we call it a quasi-classical exponential family. As is pointed out in [3] and will be verified in Section 4 of this paper, a quasi-classical exponential family is e-autoparallel in ${\mathcal{S}}$ w.r.t. the SLD structure. Note, however, that this is merely a special case of e-autoparallel submanifolds. Namely, the existence of efficient estimator is too strong as a characterization of the e-autoparallelity. Is it then possible to characterize the e-autoparallelity by an estimation-theoretic condition which is weaker than the existence of efficient estimator? We give an affirmative answer to this question in Section ​5. We also give another characterization of the e-autoparallelity in Section ​7 by considering estimation for scalar-valued functions instead of estimation for vector-valued parameters.

As mentioned above, the e-autoparallelity in the SLD structure has estimation-theoretical significance and is therefore a concept worth studying further. It should be noted here that the e-connection in the SLD structure is curvature-free but not torsion-free, so that the e-connection is not flat. This also means that ${\mathcal{S}}$ is not a dually flat space w.r.t. the SLD structure. In the case of a flat connection, an autoparallel submanifold corresponds to an affine subspace in the coordinate space of an affine coordinate system, so that the existence condition for autoparallel submanifolds is obvious. For a non-flat connection, on the other hand, we cannot see the whole picture of autoparallel submanifolds and are faced with the new problem of what kind of condition ensures the existence of autoparallel submanifold. Therefore, it is also important to study the autoparallelity from a purely geometrical point of view, away from estimation problems. This is another concern in this paper, along with the estimation-theoretical consideration.

The paper is organized as follows. In Section ​2, we explain the basic issues about the autoparallelity for an affine connection $\nabla$ on a general differential manifold, focusing in particular on the situation where the dual connection of $\nabla$ w.r.t. a Riemannian metric is flat. In Section ​3, we introduce a family of information-geometric structures on the space ${\mathcal{S}}({\mathcal{H}})$, and derive the basic issues concerning the e-autoparallelity by applying the results of Section ​2. These two sections are preliminaries for later sections. Although the results shown there are mostly known, we present them together with their derivations so that the descriptions are as self-contained as possible. Section ​4 also consists of basically known results, where Theorem ​1.3 is revisited, and it is clarified that the existence of efficient estimator only partially characterizes the e-autoparallelity for the SLD structure. This observation motivates Section ​5, where a sequence of estimators, which is called a filtration of estimators, is treated instead of a single estimator, and the existence of efficient filtration is shown to characterize the e-autoparallelity. In Section ​6, it is shown that a quantum Gaussian shift model has an efficient filtration and that the model in fact exhibits an analogous property to the e-autoparallelity in ${\mathcal{S}}({\mathcal{H}})$, although the Hilbert space ${\mathcal{H}}$ is infinite-dimensional in this case, so that we cannot fully develop a differential geometrical argument there. Section ​7 treats an estimation problem for scalar-valued functions, where it is shown that a quantum statistical manifold is e-autoparallel in ${\mathcal{S}}$ if and only if the linear space formed by functions having efficient estimators is of maximal dimension. In Section ​8, we move away from estimation theory and consider the condition for existence of e-autoparallel submanifolds from a purely geometrical point of view, where the involutivity of a parallel distribution of tangent spaces is studied in relation to the torsion tensor. In Section ​9, we treat the case when $\dim{\mathcal{H}}=2$ and study the SLD structure of the space ${\mathcal{S}}({\mathcal{H}})$ of qubit states. It is shown there that ${\mathcal{S}}({\mathcal{H}})$ in this case has a characteristic property that every e-parallel distribution is involutive. Section ​10 is devoted to concluding remarks. Some proofs and additional results are included in Appendix for the sake of readability of the main text.

In this section we summarize basic issues related to autoparallelity from the perspective of general differential geometry, which will be necessary for later discussions.

Let $S$ be an arbitrary manifold, and denote the totality of smooth functions and that of smooth vector fields on $S$ by ${\mathcal{F}}(S)$ and ${\mathscr{X}}(S)$, respectively. Suppose that $S$ is provided with an affine connection $\nabla$, which is a map ${\mathscr{X}}(S)\times{\mathscr{X}}(S)\rightarrow{\mathscr{X}}(S),\;(X,Y)\mapsto\nabla_{X}Y$. Given a submanifold $M$ of $S$, let ${\mathscr{X}}(S/M)$ denote the totality of smooth mappings which map each point $p\in M$ to a tangent vector in $T_{p}(S)$, i.e., sections of the vector bundle $\bigsqcup_{p\in M}T_{p}(S)$. Then $\nabla$ naturally induces a map ${\mathscr{X}}(M)\times{\mathscr{X}}(S/M)\rightarrow{\mathscr{X}}(S/M)$ so that for any $X\in{\mathscr{X}}(M)$ and any $Y\in{\mathscr{X}}(S/M)$, $\nabla_{X}Y$ is defined as an element of ${\mathscr{X}}(S/M)$. Since ${\mathscr{X}}(M)={\mathscr{X}}(M/M)\subset{\mathscr{X}}(S/M)$, $\nabla_{X}Y\in{\mathscr{X}}(S/M)$ is defined for any $X,Y\in{\mathscr{X}}(M)$, although it does not necessarily belong to ${\mathscr{X}}(M)$.

When $\nabla_{X}Y$ belongs to ${\mathscr{X}}(M)$ for every $X,Y\in{\mathscr{X}}(M)$, $M$ is said to be autoparallel w.r.t. ​$\nabla$ or $\nabla$-autoparallel in $S$ (e.g., Sec. ​8 in Chap. ​VII of [4]). In particular, $S$ itself is $\nabla$-autoparallel in $S$. An autoparallel curve is usually called a geodesic (or pregeodesic when we wish to clarify that our interest lies only in the image of the curve), so that the autoparallelity is a multi-dimensional extension of the notion of geodesic. When $M$ is $\nabla$-autoparallel in $S$, $\nabla$ defines an affine connection on $M$. We denote this connection by $\nabla|_{M}$ when we wish to distinguish it from the original connection $\nabla$ on $S$. The autoparallelity is transitive in the sense that if $M$ is $\nabla$-autoparallel in $S$ and $N$ is $\nabla|_{M}$-autoparallel in $M$, then $N$ is $\nabla$-autoparallel in $S$.

A vector field $X\in{\mathscr{X}}(S)$ is said to be parallel w.r.t. $\nabla$ or $\nabla$-parallel when $\forall Y\in{\mathscr{X}}(S),\,\nabla_{Y}X=0$. More generally, $X\in{\mathscr{X}}(S/M)$ (including the case $X\in{\mathscr{X}}(M)$) is said to be $\nabla$-parallel when $\forall Y\in{\mathscr{X}}(M),\,\nabla_{Y}X=0$.

When there exist on $S$ the same number of linearly independent $\nabla$-parallel vector fields as $\dim S$, we say that $(S,\nabla)$ is curvature-free. This condition is known to be equivalent to the curvature tensor of $\nabla$ vanishing on $S$. When $(S,\nabla)$ is curvature-free, the parallel transport $\Phi^{(\nabla)}_{p,q}:T_{p}(S)\rightarrow T_{q}(S)$ is defined for arbitrary two points $p,q\in S$ so that a vector field $X\in{\mathscr{X}}(S)$ is $\nabla$-parallel iff $\forall p,q\in S,\;\Phi^{(\nabla)}_{p,q}(X_{p})=X_{q}$.

The following two propositions are straightforward, where the curvature-freeness is essential.

In the following, we consider the case when $(S,\nabla)$ is additionally provided with a Riemannian metric $g$ for which the dual connection $\nabla^{*}$ of $\nabla$ is flat. Namely, the triple $(g,\nabla,\nabla^{*})$ satisfies the duality [5] [3]:

$$\forall X,Y,Z\in{\mathscr{X}}(S),\;Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}^{*}Z),$$

(2.1)

and $\nabla^{*}$ is flat in the sense that it is curvature-free and torsion free. The flatness is known to be equivalent to the existence of a coordinate system $\xi=(\xi^{i})$, which is called an affine coordinate system w.r.t.  $\nabla^{*}$, such that $\partial_{i}=\frac{\partial}{\partial\xi^{i}},i\in\{1,\ldots,\dim S\}$, are all $\nabla^{*}$-parallel. In this case, $\nabla$ turns out to be curvature-free, since the curvature-freeness is preserved by the duality of connections (Theorem ​3.3 of [3]), but is not necessarily torsion-free, and hence $(S,g,\nabla,\nabla^{*})$ is not necessarily dually flat.

Suppose that $M$ is $\nabla$-autoparallel in $S$. Then $M$ is a Riemannian submanifold of $(S,g)$ equipped with the affine connection $\nabla|_{M}$. Hence the dual connection of $\nabla|_{M}$ w.r.t. $g$ (more precisely, w.r.t. the induced metric $g|_{M}$ on $M$) is defined, which we denote by $\nabla^{*}_{M}:=(\nabla|_{M})^{*}$; i.e.,

$$\forall X,Y,Z\in{\mathscr{X}}(M),\;Xg(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,(\nabla^{*}_{M})_{X}Z),$$

(2.4)

where we have applied $(\nabla|_{M})_{X}Y=\nabla_{X}Y$. From (2.1) and (2.4), we have

$$\forall X,Y,Z\in{\mathscr{X}}(M),\;g(Y,(\nabla^{*}_{M})_{X}Z)=g(Y,\nabla^{*}_{X}Z),$$

(2.5)

which means that $(\nabla^{*}_{M})_{X}Z$ is the $g$-projection of $\nabla^{*}_{X}Z\in{\mathscr{X}}(S/M)$ onto ${\mathscr{X}}(M)$.

$$\begin{CD}\nabla @>{\text{restriction}}>{}>\nabla|_{M}\\ @V{\text{$g$-dual}}V{}V@V{}V{\text{$g|_{M}$-dual}}V\\ \nabla^{*}@>{\text{$g$-projection}}>{}>\nabla_{M}^{*}\end{CD}$$

Since the curvature-freeness and the torsion-freeness are respectively preserved by the duality and the projection, $\nabla^{*}_{M}$ is flat as in the case with $\nabla^{*}$, which ensures the existence of $\nabla^{*}_{M}$-affine coordinate system of $M$.

The following proposition will be of fundamental importance for later arguments.

We next show (ii) $\Rightarrow$ (i). Assume (ii). Then according to Prop. ​2.4, $M$ is $\nabla$-autoparallel in $S$. Noting that $g(X^{i},\partial_{j})$ is constant on $M$ and applying item (2) of Prop. ​2.5 to $(M,g|_{M},\nabla|_{M},\nabla^{*}_{M})$, we have that $\{\partial_{j}\}$ are $\nabla^{*}_{M}$-parallel, which means that $\xi$ is $\nabla^{*}_{M}$-affine. $\Box$

In this section, we introduce the information-geometric structures on quantum statistical manifolds and apply the results of the previous section to them. The geometric structure treated here is essentially the same as the one studied in § ​7.3 of [3].

Let ${\mathcal{L}}={\mathcal{L}}({\mathcal{H}})$, ${\mathcal{L}}_{{\rm h}}={\mathcal{L}}_{{\rm h}}({\mathcal{H}})$ and ${\mathcal{S}}={\mathcal{S}}({\mathcal{H}})$ be the totality of linear operators on ${\mathcal{H}}$, that of Hermitian operators on ${\mathcal{H}}$ and that of strictly positive density operators on ${\mathcal{H}}$, respectively. Then ${\mathcal{S}}$ is an open subset of the affine space ${{\mathcal{L}}_{{\rm h}}}_{,1}:=\{A\in{\mathcal{L}}_{{\rm h}}\,|\,{\rm Tr}A=1\}$, so that a flat affine connection is naturally introduced on ${\mathcal{S}}$, which we call the m-connection and denote by $\nabla^{({\rm m})}$. In order to express $\nabla^{({\rm m})}$ more explicitly, we introduce the embedding map $\iota:{\mathcal{S}}\rightarrow{{\mathcal{L}}_{{\rm h}}}_{,1}$ so that $\rho\in{\mathcal{S}}$ is denoted by $\iota(\rho)$ when treating it as an element of ${{\mathcal{L}}_{{\rm h}}}_{,1}$. Since $\iota$ is a smooth map, it has the differential at every point $\rho\in{\mathcal{S}}$, which we denote by $\iota_{*}=(d\iota)_{\rho}:T_{\rho}({\mathcal{S}})\rightarrow{{\mathcal{L}}_{{\rm h}}}_{,0}:=\{A\in{\mathcal{L}}_{{\rm h}}\,|\,{\rm Tr}A=0\}$. For a vector field $X\in{\mathscr{X}}({\mathcal{S}})$, the map $\iota_{*}(X):{\mathcal{S}}\rightarrow{{\mathcal{L}}_{{\rm h}}}_{,0}$ is defined to be $\rho\mapsto\iota_{*}(X_{\rho})=(d\iota)_{\rho}(X_{\rho})$. Then the definition of the m-connection is represented as follows:

$$\forall X,Y\in{\mathscr{X}}({\mathcal{S}}),\;\iota_{*}(\nabla^{({\rm m})}_{X}Y)=X\iota_{*}(Y),$$

(3.1)

where $X\iota_{*}(Y):{\mathcal{S}}\rightarrow{{\mathcal{L}}_{{\rm h}}}_{,0}$ is the derivative of $\iota_{*}(Y)$ w.r.t. ​$X$. When a coordinate system $\xi=(\xi^{i})$ is arbitrarily given and the elements of ${\mathcal{S}}$ is parametrized by it as $\rho_{\xi}$, we have

$$\iota_{*}\left(\partial_{i}\right)=\partial_{i}\rho_{\xi},$$

(3.2)

and

$$\iota_{*}\left(\nabla^{({\rm m})}_{\partial_{i}}\partial_{j}\right)=\partial_{i}\,\iota_{*}\left(\partial_{j}\right)=\partial_{i}\partial_{j}\rho_{\xi},$$

(3.3)

where $\partial_{i}:=\frac{\partial}{\partial\xi^{i}}$.

Suppose that we are given a family of inner products $\{\left\langle\cdot,\cdot\right\rangle_{\rho}\,|\,\rho\in{\mathcal{S}}({\mathcal{H}})\}$ on the $\mathbb{R}$-linear space ${\mathcal{L}}_{{\rm h}}$, where the correspondence $\rho\mapsto\left\langle\cdot,\cdot\right\rangle_{\rho}$ is smooth, and assume that

$$\forall\rho\in{\mathcal{S}},\forall A\in{\mathcal{L}}_{{\rm h}},\;\left\langle A,I\right\rangle_{\rho}=\left\langle A\right\rangle_{\rho}:={\rm Tr}(\rho A).$$

(3.4)

The inner products are represented as

$$\left\langle A,B\right\rangle_{\rho}=\left\langle A,\Omega_{\rho}(B)\right\rangle_{\rm HS}={\rm Tr}(A\,\Omega_{\rho}(B))$$

(3.5)

by a family of super-operators $\{\Omega_{\rho}:{\mathcal{L}}_{{\rm h}}\rightarrow{\mathcal{L}}_{{\rm h}}\}_{\rho\in{\mathcal{S}}}$, where $\left\langle\cdot,\cdot\right\rangle_{\rm HS}$ denotes the Hilbert-Schmidt inner product. Note that the assumption (3.4) is equivalent to

$$\forall\rho\in{\mathcal{S}},\;\Omega_{\rho}(I)=\rho.$$

(3.6)

For an arbitrary tangent vector $X_{\rho}\in T_{\rho}({\mathcal{S}})$, a Hermitian operator $L_{X_{\rho}}\in{\mathcal{L}}_{{\rm h}}$ is defined by the relation

$$\forall A\in{\mathcal{L}}_{{\rm h}},\;X_{\rho}\left\langle A\right\rangle=\left\langle L_{X_{\rho}},A\right\rangle_{\rho},$$

(3.7)

where the LHS denotes the derivative of the function $\left\langle A\right\rangle:{\mathcal{S}}\rightarrow\mathbb{R}$, $\rho\mapsto\left\langle A\right\rangle_{\rho}$ w.r.t. $X_{\rho}$. Noting that the LHS and the RHS are represented as ${\rm Tr}(\iota_{*}(X_{\rho})A)$ and ${\rm Tr}(\Omega_{\rho}(L_{X_{\rho}})\,A)$, respectively, we can rewrite (3.7) into

$$\iota_{*}(X_{\rho})=\Omega_{\rho}(L_{X_{\rho}}).$$

(3.8)

From (3.4) and (3.7), we have

$$\left\langle L_{X_{\rho}}\right\rangle_{\rho}=\left\langle L_{X_{\rho}},I\right\rangle_{\rho}=X_{\rho}\left\langle I\right\rangle=X_{\rho}1=0.$$

(3.9)

Since $X_{\rho}\leftrightarrow\iota_{*}(X_{\rho})\leftrightarrow L_{X_{\rho}}$ are one-to-one correspondences, we obtain the following identity:

$$\{L_{X_{\rho}}\,|\,X_{\rho}\in T_{\rho}({\mathcal{S}})\}=\{A\in{\mathcal{L}}_{{\rm h}}\,|\,\left\langle A\right\rangle_{\rho}=0\}.$$

(3.10)

In the following, we often express (3.4) as

$$\forall A\in{\mathcal{L}}_{{\rm h}},\;\left\langle A,I\right\rangle=\left\langle A\right\rangle$$

(3.11)

as an identity for functions on ${\mathcal{S}}$. Similarly, (3.7) is expressed as

$$\forall A\in{\mathcal{L}}_{{\rm h}},\;X\left\langle A\right\rangle=\left\langle L_{X},A\right\rangle,$$

(3.12)

where $X$ is a vector field on ${\mathcal{S}}$, $L_{X}$ denotes the map ${\mathcal{S}}\rightarrow{\mathcal{L}}_{{\rm h}}$, $\rho\mapsto L_{X_{\rho}}$, and $\left\langle L_{X},A\right\rangle$ denotes the function $\rho\mapsto\left\langle L_{X_{\rho}},A\right\rangle_{\rho}$.

For a submanifold ${\mathcal{M}}$ of ${\mathcal{S}}$ and a vector field $X\in{\mathscr{X}}({\mathcal{M}})$ on it, the map $L_{X}:{\mathcal{M}}\rightarrow{\mathcal{L}}_{{\rm h}},\rho\mapsto L_{X_{\rho}}$ is defined, for which (3.12) holds as an identity for functions on ${\mathcal{M}}$. We may write $X\left\langle A\right\rangle$ as $X\left\langle A\right\rangle\!|_{{\mathcal{M}}}$ in this case. In particular, given a coordinate system $\xi=(\xi^{i})$ of ${\mathcal{M}}$, we have

$$\forall A\in{\mathcal{L}}_{{\rm h}},\;\partial_{i}\left\langle A\right\rangle=\partial_{i}\left\langle A\right\rangle\!|_{{\mathcal{M}}}=\left\langle L_{i},A\right\rangle,$$

(3.13)

where $\partial_{i}:=\frac{\partial}{\partial\xi^{i}}$ and $L_{i}:=L_{\partial_{i}}$.

Now, we define a Riemannian metric $g$ on ${\mathcal{S}}$ by

$$\forall\rho\in{\mathcal{S}},\forall X_{\rho},Y_{\rho}\in T_{\rho}({\mathcal{S}}),\;g_{\rho}(X_{\rho},Y_{\rho}):=\left\langle L_{X_{\rho}},L_{Y_{\rho}}\right\rangle_{\rho}={\rm Tr}\left(\iota_{*}(X_{\rho})L_{Y_{\rho}}\right),$$

(3.17)

which is equivalently written as

$$\forall X,Y\in{\mathscr{X}}({\mathcal{S}}),\;g(X,Y):=\left\langle L_{X},L_{Y}\right\rangle={\rm Tr}\left(\iota_{*}(X)L_{Y}\right).$$

(3.18)

We also have the expression

$$\forall X,Y\in{\mathscr{X}}({\mathcal{S}}),\;g(X,Y)=-\left\langle XL_{Y}\right\rangle=-\left\langle YL_{X}\right\rangle,$$

(3.19)

where $XL_{Y}:{\mathcal{S}}\rightarrow{\mathcal{L}}_{{\rm h}}$, $\rho\mapsto X_{\rho}L_{Y}$, denotes the derivative of the map $L_{Y}:{\mathcal{S}}\rightarrow{\mathcal{L}}_{{\rm h}}$, $\rho\mapsto L_{Y_{\rho}}$, w.r.t. $X$, and $\left\langle XL_{Y}\right\rangle$ denotes the function $\rho\mapsto\left\langle X_{\rho}L_{Y}\right\rangle_{\rho}$. This expression is derived as

	$\displaystyle 0=X_{\rho}\left\langle L_{Y}\right\rangle=X_{\rho}\left\langle L_{Y_{\bm{\cdot}}}\right\rangle_{\bm{\cdot}}$	$\displaystyle=X_{\rho}\left\langle L_{Y_{\rho}}\right\rangle_{\bm{\cdot}}+X_{\rho}\left\langle L_{Y_{\bm{\cdot}}}\right\rangle_{\rho}$
		$\displaystyle=\left\langle L_{X_{\rho}},L_{Y_{\rho}}\right\rangle_{\rho}+\left\langle X_{\rho}L_{Y}\right\rangle_{\rho},$		(3.20)

where the first equality follows from (3.9) and the last from (3.7), with dots ${\bm{\cdot}}$ added to clarify the positions of variables of maps. An important class of such Riemannian metrics is that of monotone metrics [9] for which $\Omega_{\rho}$ is represented as

$$\Omega_{\rho}(A)=f(\Delta_{\rho})(A\rho)=f(\Delta_{\rho})(A)\rho,$$

(3.21)

where $f:(0,\infty)\rightarrow(0,\infty)$ is an operator monotone function satisfying $\forall x>0,\,xf(1/x)=f(x)$ and $f(1)=1$, and $\Delta_{\rho}:{\mathcal{L}}\rightarrow{\mathcal{L}}$ is the modular operator defined by $\Delta_{\rho}(A)=\rho A\rho^{-1}$. The class contains the SLD metric, which plays the main role in this paper, defined by $f(x)=(x+1)/2$ and

$$\left\langle A,B\right\rangle_{\rho}={\rm Re}\,{\rm Tr}(\rho AB)={\rm Tr}(\rho(A\circ B))={\rm Tr}((\rho\circ A)B),\;\;A,B\in{\mathcal{L}}_{{\rm h}},$$

(3.22)

where $\circ$ denotes the symmetrized product: $A\circ B=\frac{1}{2}(AB+BA)$. In this case, $L_{X_{\rho}}$ ($L_{X}$, resp.) is called the SLD (symmetric logarithmic derivative) of the tangent vector $X_{\rho}$ (the vector field $X$, resp.). In particular, $L_{i}:=L_{\partial_{i}}$ obeys the equation

$$\partial_{i}\rho_{\xi}=\rho_{\xi}\circ L_{i,\xi},$$

(3.23)

which is a popular expression for the SLD.

Given a family of inner products $\{\left\langle\cdot,\cdot\right\rangle_{\rho}\,|\,\rho\in{\mathcal{S}}\}$ which determines a Riemannian metric $g$, let the e-connection $\nabla^{({\rm e})}$ be defined as the dual connection of $\nabla^{({\rm m})}$ w.r.t. ​$g$; i.e.,

$$\forall X,Y,Z\in{\mathscr{X}}({\mathcal{S}}),\;Xg(Y,Z)=g(\nabla^{({\rm e})}_{X}Y,Z)+g(Y,\nabla^{({\rm m})}_{X}Z).$$

(3.24)

We have thus obtained $({\mathcal{S}},g,\nabla^{({\rm e})},\nabla^{({\rm m})})$ as an example of $(S,g,\nabla,\nabla^{*})$ treated in the previous section, where $\nabla^{({\rm e})}$ and $\nabla^{({\rm m})}$ are dual w.r.t. $g$, $\nabla^{({\rm e})}$ is curvature-free and $\nabla^{({\rm m})}$ is flat. The triple $({\mathcal{S}},g,\nabla^{({\rm e})},\nabla^{({\rm m})})$ is called the information-geometric structure on ${\mathcal{S}}$ induced from a family of inner products $\{\left\langle\cdot,\cdot\right\rangle_{\rho}\,|\,\rho\in{\mathcal{S}}\}$. In particular, the information-geometric structure induced from the symmetrized inner product (3.22) is called the SLD structure. It is the SLD structure that will play a leading role in subsequent sections in relation to estimation theory, but this section will continue the discussion on general information-geometric structures.

Henceforth, we use the prefixes e- and m- for notions concerning the e-connection and m-connection; e.g., e-parallel, e-autoparallel, m-affine, etc.