# G-dual teleparallel connections in Information Geometry

F. M. Ciaglia1,5 , F. Di Cosmo1,2,6 , A. Ibort1,2,7 , G. Marmo3,4,8

###### Abstract

Given a real, finite-dimensional, smooth parallelizable Riemannian manifold
$(\mathcal{N},G)$ endowed with a teleparallel connection $\nabla$ determined
by a choice of a global basis of vector fields on $\mathcal{N}$, we show that
the $G$-dual connection $\nabla^{*}$ of $\nabla$ in the sense of Information
Geometry must be the teleparallel connection determined by the basis of
$G$-gradient vector fields associated with a basis of differential one-forms
which is (almost) dual to the basis of vector fields determining $\nabla$. We
call any such pair $(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair. Then,
after defining a covariant $(0,3)$ tensor $T$ uniquely determined by
$(\mathcal{N},G,\nabla,\nabla^{*})$, we show that $T$ being symmetric in the
first two entries is equivalent to $\nabla$ being torsion-free, that $T$ being
symmetric in the first and third entry is equivalent to $\nabla^{*}$ being
torsion free, and that $T$ being symmetric in the second and third entries is
equivalent to the basis vectors determining $\nabla$ ($\nabla^{*}$) being
parallel-transported by $\nabla^{*}$ ($\nabla$). Therefore, $G$-dual
teleparallel pairs provide a generalization of the notion of Statistical
Manifolds usually employed in Information Geometry, and we present explicit
examples of $G$-dual teleparallel pairs arising both in the context of both
Classical and Quantum Information Geometry.

1 Department of Mathematics, University Carlos III de Madrid, Leganés, Madrid,
Spain
2 ICMAT, Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)
3 INFN-Sezione di Napoli, Naples, Italy
4 Department of Physics “E. Pancini”, University of Naples Federico II,
Naples, Italy

5fciaglia[at]math.uc3m.es, 6fcosmo[at]math.uc3m.es, 7albertoi[at]math.uc3m.es,
8marmo[at]na.infn.it

###### Contents

  1. 1 Introduction
  2. 2 G-dual teleparallel pairs
  3. 3 G-dual teleparallel pairs in Classical Information Geometry
  4. 4 G-dual teleparallel pairs in Quantum Information Geometry
  5. 5 Conclusions
  6. Funding

##  1 Introduction

From a purely mathematical point of view, the structure theory of Classical
Information Geometry is the theory of Riemannian smooth manifolds with an
additional structure. This additional structure may be described,
equivalently, using a pair of affine connections or a covariant $(0,3)$ tensor
(cf. [2, 4, 40]). Specifically, in the first case we have a Riemannian smooth
manifold $(\mathcal{N},G)$, whose points parametrize probability distributions
on a suitable outcome space, a torsion-free affine connection $\nabla$, and
another torsion-free affine connection $\nabla^{*}$ which is dual to $\nabla$
with respect to $G$ in the sense that

$X(G(Y,Z))=G\left(\nabla_{X}Y,Z\right)+G\left(Y,\nabla^{*}_{X}Z\right)$ (1)

for all vector fields $X,Y,Z$ on $\mathcal{N}$. In general,
$\nabla\neq\nabla^{*}$ unless $\nabla$ is the Levi-Civita connection
$\nabla^{G}$ of $G$. However, both $\nabla$ and $\nabla^{*}$ can be expressed
in terms of $\nabla^{G}$ and a covariant tensor essentially because the space
of affine connections is, quite appropriately, an affine space modelled on a
vector space of tensor fields. Specifically, it holds

$G(\nabla_{X}Y,Z)-G(\nabla^{G}_{X}Y,Z)=T(X,Y,Z)$ (2)

with $T$ a covariant $(0,3)$-tensor field on $\mathcal{N}$ which may also be
written according to

$G(Y,\nabla_{X}^{G}Z)-G(Y,\nabla^{*}_{X}Z)=T(X,Y,Z).$ (3)

because of equation (1) and because $\nabla^{G}$ is its own $g$-dual
connection. If both $\nabla$ and $\nabla^{*}$ are torsion-free, it can be
proved that $T$ is completely symmetric.

If instead of the pair $(\nabla,\nabla^{*})$ of torsion-free, $G$-dual affine
connections we endow the Riemannian smooth manifold $(\mathcal{N},G)$ with a
covariant $(0,3)$-tensor field $T$ which is totally symmetric, we can again
obtain a pair of torsion-free, $G$-dual affine connections simply reading
equation (2) and (3) from right to left. Therefore, we conclude that the two
approaches are equivalent. The triple $(\mathcal{N},G,T)$ is referred to as a
statistical manifold (cf. [40]), and the tensor $T$ is often called Amari-
Čencov tensor in honour of the two pioneers who thoroughly studied this
structure (cf. [2, 6, 56]).

It is worth noting how recently the theory of statistical manifolds has been
approached from the point of view of Lie groupoids (cf. [30, 31, 32]) where it
is shown how Information Geometry on statistical manifolds is but a particular
example of Information Geometry on Lie groupoids. This point of view on
Information Geometry may be particularly fruitful in connection with the
recent reformulation of Quantum Theories in the framework of groupoids (cf.
[13, 17, 18]), especially considering how this reformulation of Quantum
theories heavily relies on a categorical background which shares many
interesting similarities with the one exploited by Čencov in his pioneering
work (cf. [56]). However, we only plan to investigate these matters in future
works.

Coming back to statistical manifolds, of particular interest are those
statistical manifolds whose Amari-Čencov tensor $T$ gives rise to a pair
$(\nabla,\nabla^{*})$ of $G$-dual connections which are both flat in the sense
that, in addition to being torsion-free, their curvature vanishes. In this
case, the statistical manifold is called dually flat and some interesting
properties of both applied and purely mathematical flavour present themselves
(cf. [3, 4, 50]). Of course, the request of flatness for the connections
immediately imposes topological obstructions on the type of smooth manifold we
can consider, and these kind of considerations also apply to the case of
Quantum Information Geometry of finite-dimensional systems (cf. [7, 8, 27,
39]).

A typical example of a dually flat statistical manifold is given by the
interior of the n-simplex endowed with the Fisher-Rao metric tensor $G_{FR}$
and with the pair of $G_{FR}$-dual affine connections known as the mixture and
exponential connection. The mixture connection on the interior of the
n-simplex has a kind of privileged role because it is intimately connected
with the convex structure of the set of probability distributions. We refer to
section 3 for a thorough discussion of this example.

It is well-known that a statistical manifold $(\mathcal{N},G,T)$ can be built
suitably expanding up to third order a smooth function
$S\colon\mathcal{N}\times\mathcal{N}\rightarrow\mathbb{R}$ (also known as a
two-point function on $\mathcal{N}$) satisfying some specific conditions (cf.
[5, 14, 43]). It often happens that the function $S$ is a relative entropy
function, or a divergence function. The most relevant family of relative
entropies in the classical setting is that investigated by Csizar (cf. [21]),
and it turns out that every member in this family gives rise to the same
Riemannian metric tensor which is precisely the Fisher-Rao metric tensor.

In the (finite-dimensional) quantum setting, it turns out that the situation
is essentially different than in the classical setting. First of all, on the
manifold of faithful quantum states $\mathscr{S}(\mathcal{H})$ of a physical
system described by a finite-dimensional complex Hilbert space $\mathcal{H}$
(cf. section 4) there is an infinite number of Riemannian metric tensors whose
theoretical (and often also applied) relevance has been thoroughly
corroborated. These Riemannian metric tensors are known as quantum monotone
metric tensors and have been completely classified by Petz (cf. [48]). Every
quantum monotone metric tensor may be obtained suitably expanding a quantum
relative entropy which, in some sense, generalizes the classical relative
entropy of Csizar mentioned above (cf. [12, 41]). These instances are in stark
contrast with what happens in the classical case where the Riemannian
structure of essentially all the statistical manifolds considered are given by
the Fisher-Rao metric tensor111We are here deliberately ignoring all the
Wasserstein-type metric tensors not because we believe they are not relevant
but only because their very definition depend on additional structures (e.g.,
a metric distance on the outcome space in the classical case or a metric
distance on the space of pure states in the quantum case) that are not of
interest for us here., and where every relative entropy in the family
investigated by Csizar leads to the Fisher-Rao metric tensor.

Of course, the dual pairs of affine connections generated by the quantum
relative entropies associated with the quantum monotone metric tensors are
necessarily torsion-free. In particular, it is possible to build a mixture
connection on $\mathscr{S}(\mathcal{H})$ because, analogously to the space of
classical probability distributions, the space of quantum states is a convex
set (cf. section 4), and it turns out that there is only one monotone metric
tensor - the so-called Bogoliubov-Kubo-Mori metric tensor (cf. [45, 47, 49]) -
for which the mixture connection admits a dual connection which is the quantum
analogue of the exponential connection and which is torsion-free (cf. [34,
44]). In this case, the associated quantum relative entropy is the von
Neumann-Umegaki relative entropy (cf. [24, 55, 57]). Incidentally, the
resulting statistical manifold is dually flat.

However, it turns out that there are pairs of affine connections which are
dual with respect to the quantum monotone metric tensors and are not both
torsion-free (cf. [4, 26, 38]). In particular, examples of these dual pair
with non-vanishing torsion are found when generalizing Amari’s
$\alpha$-representation (cf. [2]) to the quantum case (cf. [38]). Moreover,
one of the results of this work is to show that the affine connection which is
dual to the mixture connection on $\mathscr{S}(\mathcal{H})$ with respect to a
specific quantum monotone metric tensor always has non-vanishing torsion
unless the gradient vector fields of the expectation value functions of
quantum mechanical observables commute. Since the mixture connection is
intimately connected with the convex structure of the space of quantum states
and thus reflects a structural feature of $\mathscr{S}(\mathcal{H})$ that goes
beyond the formalism of Information Geometry, the idea of discarding all the
cases in which the affine connection dual to the mixture one presents torsion
seems too restrictive. Indeed, already Amari and Nagaoka in their seminal work
[4, p. 19] noted how “the incorporation of torsion into the framework of
information geometry, which would relate it to such fields as quantum
mechanics (noncommutative probability theory) and systems theory, is an
interesting topic for the future.” Consequently, it seems that the notion of
statistical manifold is not enough to completely understand the mathematical
structures arising in Quantum Information Geometry.

Moreover, it is worth noting how the investigation of the appearance of
torsion also in Classical Information Geometry is recently gaining interest
(cf. [37, 59, 60, 61]). In particular, the present work may be thought of as
complementing some theoretical aspects already discussed in [59] (especially
in connection with the explicit construction of what is called the
g-biorthogonal frame), and providing a natural extension to the case of
Quantum Information Geometry.

The aim of this work is to introduce a variation on the theme of dually flat
connections that allows the appearance of torsion and to show that this type
of geometric structure naturally appears in Classical and Quantum Information
Geometry. The starting point is the idea that, at least locally, a flat
connection $\nabla$ on a smooth manifold $\mathcal{N}$ admits a basis of
vector fields $\\{X_{j}\\}_{j=1,...,n=\mathrm{dim}(\mathcal{N})}$ such that
$\nabla_{X_{j}}X_{k}=0$ for all $j,k$. If the vector fields are globally
defined so that $\\{X_{j}\\}_{j=1,...,n=\mathrm{dim}(\mathcal{N})}$ is a basis
for the module of vector fields on $\mathcal{N}$ (note that this requires
$\mathcal{N}$ to be parallelizable), then the condition
$\nabla_{X_{j}}X_{k}=0$ is what makes $\nabla$ be the so-called teleparallel
connection associated with
$\\{X_{j}\\}_{j=1,...,n=\mathrm{dim}(\mathcal{N})}$. Moreover, a teleparallel
connection associated with a basis
$\\{X_{j}\\}_{j=1,...,n=\mathrm{dim}(\mathcal{N})}$ of vector fields is
uniquely determined by the condition $\nabla_{X_{j}}X_{k}=0$ for all $j,k$. In
general, however, teleparallel connections need not be flat (i.e., curvature-
free), nor even torsion-free (indeed, they are torsion-free if and only if the
basis vector fields mutually commute as it can be easily seen from the very
definition of the torsion tensor). In almost all the cases arising from
Classical and Quantum Information Geometry, it turns out that there is one
preferred teleparallel connection which is basically the mixture connection
arising from the convex structure characteristic of both the space of
probability distributions and the space of quantum states.

Motivated by the previous discussion, in section 2, given a fixed teleparallel
connection on a smooth Riemannian manifold $(\mathcal{N},G)$, we investigate
the properties of the $G$-dual connection $\nabla^{*}$ and show that it must
necessarily be itself a teleparallel connection. We call any such pair
$(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair. Moreover, we also
investigate the symmetry properties of the tensor $T$ generalizing the Amari-
Čencov tensor to the case of $G$-dual teleparallel pairs. Then, in section 3
we review the case of the mixture and exponential connection on the interior
$\Delta_{n}$ of the n-simplex highlighting the teleparallel structure of both
connections, and showing how the teleparallel structure of the exponential
connection is connected with a particular homogeneous manifold structure for
$\Delta_{n}$. In section 4 we investigate the quantum case and characterize
the $G_{f}$-dual teleparallel pairs $(\nabla,\nabla^{*})$ on the manifold
$\mathscr{S}(\mathcal{H})$ of faithful quantum states in finite dimensions
endowed with a monotone quantum metric tensor $G_{f}$ in which $\nabla$ is the
mixture connection reflecting the convex structure of
$\mathscr{S}(\mathcal{H})$. Finally, in section 5 we gather some concluding
remarks and future perspectives.

##  2 G-dual teleparallel pairs

Let $\mathcal{N}$ be a parallelizable smooth manifold and denote with
$\mathfrak{X}(\mathcal{N})$ the module of smooth vector fields on
$\mathcal{N}$. Let $\\{X_{j}\\}_{j=1,...,n}$, with
$n=\mathrm{dim}(\mathcal{N})$, be a global basis of vector fields on
$\mathcal{N}$. Define the teleparallel connection222The term teleparallel my
be translated with ’distantly parallel’, and its use is mainly due to the fact
that the affine connection we are interested in is used in the context of the
so-called Teleparallel Gravity [1]. Note, however, that the teleparallel
connection is also known as the Weitzenböck connection of
$\\{X_{j}\\}_{j=1,...,n}$. $\nabla$ associated with $\\{X_{j}\\}_{j=1,...,n}$
by setting

$\nabla_{X_{j}}X_{k}:=0\quad\forall\;j,k=1,...,n.$ (4)

It is clear that every integral curve of $X_{j}$ is a geodesic of $\nabla$ for
every $j=1,...,n$. Moreover, since geodesics with fixed initial conditions are
unique, every geodesic of $\nabla$ starting at $x\in\mathcal{N}$ with initial
velocity $v_{x}\in T_{x}\mathcal{N}$ can be realized as an integral curve
starting at $x$ of the vector field $X=a^{j}X_{j}$ with $a^{j}\in\mathbb{R}$
for all $j=1,...,n$ such that $v_{x}=X(x)$.

The torsion tensor $T^{\nabla}$ of $\nabla$, in general, reads

$T^{\nabla}(Z,W):=\nabla_{Z}W-\nabla_{W}Z-[Z,W]$ (5)

for every $Z,W\in\mathfrak{X}(\mathcal{N})$. Therefore, equation (4) implies

$T^{\nabla}(X_{j},X_{k})=[X_{k},X_{j}]$ (6)

for every $j,k=1,...,n$, which means that, from the perspective of the basis
$\\{X_{j}\\}_{j=1,...,n}$, the torsion tensor is connected with the commutator
of the basis vector fields. Of course, from equation (6) it follows that
$\nabla$ is torsion-free if and only if the basis vector fields mutually
commute.

The curvature tensor $R^{\nabla}$ of $\nabla$ reads

$\left(R^{\nabla}(Z,W)\right)(V):=\nabla_{Z}\left(\nabla_{W}V\right)-\nabla_{W}\left(\nabla_{Z}V\right)-\nabla_{[Z,W]}V$
(7)

for every $Z,W,V\in\mathfrak{X}(\mathcal{N})$. Therefore, again equation (4)
implies

$\left(R^{\nabla}(X_{j},X_{k})\right)(X_{l}):=-\nabla_{[X_{j},X_{k}]}X_{l}=0$
(8)

for every $j,k,l=1,...,n$, because $\\{X_{j}\\}_{j=1,...,n}$ is a basis of
vector fields and thus $[X_{j},X_{k}]=F^{l}X_{l}$ with some smooth functions
$F^{l}$ globally defined on $\mathcal{N}$. We thus conclude that if $\nabla$
is torsion-free then it is necessarily flat (in the sense of being free of
curvature and torsion).

Let $G$ be a smooth Riemannian metric tensor on $\mathcal{N}$. Motivated by
the theory of Statistical Manifolds, we want to investigate the pairs
$(\nabla,\nabla^{*})$ of $G$-dual connections on $\mathcal{N}$ for which both
$\nabla$ and $\nabla^{*}$ are teleparallel connections on $\mathcal{N}$. We
call any such pair $(\nabla,\nabla^{*})$ a $G$-dual teleparallel pair.

At this purpose, it is instrumental that we understand how $\nabla$ behaves on
differential one-forms. First of all, we consider a basis
$\\{\theta^{j}\\}_{j=1,...,n}$ of globally-defined differential one-forms on
$\mathcal{N}$ such that

$\theta^{j}(X_{k})=C^{j}_{k}.$ (9)

where $C^{j}_{k}$ is a number for every $j,k=1,...,n$. We call any such basis
almost dual to $\\{X_{j}\\}_{j=1,...,n}$. In particular, when
$C^{j}_{k}=\delta^{j}_{k}$ we obtain precisely the dual basis of
$\\{X_{j}\\}_{j=1,...,n}$. We may think of $C_{jk}$ as a scaled version of
$\delta_{jk}$ where the value at $j=k$ actually depends on $j$ and it is not
necessarily always 1. The fact that all the constructions presented in the
work are valid for $C_{jk}$ instead of just $\delta_{jk}$ simply reflects the
fact that the choice of each single vector field in the basis is determined up
to a constant.

According to the general theory of affine connections (cf. [10]), the
behaviour of $\nabla$ on the almost dual basis $\\{\theta^{j}\\}_{j=1,...,n}$
is encoded in the equation

$\left(\nabla_{Z}\theta^{j}\right)(W)=\nabla_{Z}\left(\theta^{j}(W)\right)-\theta^{j}\left(\nabla_{Z}W\right)$
(10)

where $Z,W$ are arbitrary vector fields on $\mathcal{N}$. In particular, when
$Z=X_{k}$ and $W=X_{l}$, equation (10) becomes

$\left(\nabla_{X_{k}}\theta^{j}\right)(X_{l})=0\quad\forall\;j,k,l=1,...,n$
(11)

because of equation (4), and because the covariant derivatives of the
$C_{jk}$’s in equation (9) vanish being the $C_{jk}$’s constant.

Equation (11) is equivalent to

$\nabla_{Z}\theta^{j}=0\quad\forall\;j=1,...,n\quad\;\forall\;Z\in\mathfrak{X}(\mathcal{N})$
(12)

because $\\{X_{j}\\}_{j=1,...,n}$ is a basis for $\mathfrak{X}(\mathcal{N})$
and, for every smooth function $f$ on $\mathcal{N}$, it holds
$\nabla_{fZ}\alpha=f\nabla_{Z}\alpha$ for every differential one-form $\alpha$
on $\mathcal{N}$.

Since the Riemannian metric tensor $G$ is non-degenerate, for every
$\theta^{j}$ we can define its associated $G$-gradient (or simply ’gradient’
if there is no risk of confusion) vector field $Y_{j}$ uniquely determined by

$\theta^{j}(Z)=G(Y_{j},Z)$ (13)

for every $Z\in\mathfrak{X}(\mathcal{N})$. Note that $\\{Y_{j}\\}_{j=1,...,n}$
is a basis of globally-defined vector fields on $\mathcal{N}$. Setting
$Z=Y_{k}$ and $W=X_{l}$ and exploiting equation (12), it follows that equation
(10) becomes

$0=\theta^{j}(\nabla_{Y_{k}}X_{l})$ (14)

for all $j,k,l=1,...,n,$ which is equivalent to

$\nabla_{Y_{k}}X_{l}=0$ (15)

for all $k,l=1,...,n$.

With these elements at our disposal, we now characterize the affine connection
$\nabla^{*}$ which is $G$-dual to the teleparallel connection $\nabla$. In
particular, it is proved that $\nabla^{*}$ is the teleparallel connection
associated with $\\{Y_{j}\\}_{j=1,...,n}$. It is worth noting that the content
of proposition 1 is essentially the same as that of Theorem 10 in [59].

######  Proposition 1.

Let $\mathcal{N}$ be a parallelizable smooth manifold of dimension
$n=\mathrm{dim}(\mathcal{N})$. Let $\\{X_{j}\\}_{j=1,...,n}$ be the global
basis of vector fields on $\mathcal{N}$, let $\nabla$ be its associated
teleparallel connection, and let $\\{\theta^{j}\\}_{j=1,...,n}$ be an almost
dual basis of differential one-forms for $\\{X_{j}\\}_{j=1,...,n}$ (cf.
equation (9)). For any smooth Riemannian metric tensor $G$ on $\mathcal{N}$,
let $\\{Y_{j}\\}_{j=1,...,n}$ be the global basis of vector fields on
$\mathcal{N}$ made up of $G$-gradient vector fields of the $\theta^{j}$’s (cf.
equation (13)). Then, the $G$-dual affine connection $\nabla^{*}$ of $\nabla$
(cf. equation (1)) is the teleparallel connection associated with
$\\{Y_{j}\\}_{j=1,...,n}$.

###### Proof.

The $G$-dual connection $\nabla^{*}$ is uniquely characterized by

$Z\left(G(W,V)\right)=G\left(\nabla_{Z}W,V\right)+G\left(W,\nabla_{Z}^{*}V\right)$
(16)

for every $Z,W,V\in\mathfrak{X}(\mathcal{N})$. Note that

$Y_{j}\left(G(Y_{k},X_{l})\right)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
gradient vector
field}}}}{{=}}Y_{j}\left(\theta^{k}(X_{l})\right)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
dual basis}}}}{{=}}Y_{j}(C^{k}_{l})=0.$ (17)

Therefore, setting $Z=Y_{j}$, $W=X_{k}$, and $V=Y_{l}$, equation (16) becomes

$0\stackrel{{\scriptstyle\mbox{\eqref{eqn: dually weitzenbock
3}}}}{{=}}Y_{j}\left(G(X_{k},Y_{l})\right)=G\left(\nabla_{Y_{j}}X_{k},Y_{l}\right)+G\left(X_{k},\nabla_{Y_{j}}^{*}Y_{l}\right)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
dually weitzenbock 2}}}}{{=}}G(X_{k},\nabla_{Y_{j}}^{*}Y_{l})$ (18)

for all $j,k,l=1,...,n$, which is equivalent to

$\nabla_{Y_{j}}^{*}Y_{l}=0$ (19)

for all $j,l=1,...,n$ because $\\{X_{j}\\}_{j=1,...,n}$ is a basis of vector
fields on $\mathcal{N}$. We thus conclude that the $G$-dual connection
$\nabla^{*}$ is precisely the teleparallel connection determined by the basis
$\\{Y_{j}\\}_{j=1,...,n}$ of gradient vector fields on $\mathcal{N}$
associated with the almost dual basis $\\{\theta^{j}\\}_{j=1,...,n}$ of
$\\{X_{j}\\}_{j=1,...,n}$ through $G$ as claimed. ∎∎

######  Remark 1.

We can introduce the basis $\\{\alpha^{j}\\}_{j=1,...,n}$ of differential one-
forms where each $\alpha^{j}$ is the $G$-dual of $X_{j}$ according to

$\alpha^{j}(Z)=G(X_{j},Z)$ (20)

for every $Z\in\mathfrak{X}(\mathcal{N})$. Then, it follows that
$\\{\alpha^{j}\\}_{j=1,...,n}$ is an almost dual basis of
$\\{Y_{j}\\}_{j=1,...,n}$ because

$\alpha^{j}(Y_{k})\stackrel{{\scriptstyle\mbox{\eqref{eqn: gradient vector
field 2}}}}{{=}}G(X_{j},Y_{k})\stackrel{{\scriptstyle\mbox{\eqref{eqn:
gradient vector
field}}}}{{=}}\theta^{k}(X_{j})\stackrel{{\scriptstyle\mbox{\eqref{eqn: dual
basis}}}}{{=}}C^{k}_{j}.$ (21)

Consequently, proceeding in perfect analogy with what is done to obtain
equation (15), we obtain

$\nabla_{X_{k}}^{*}Y_{l}=0$ (22)

for all $k,l=1,...,n$.

######  Remark 2.

Recalling the formula

$\mathrm{d}\beta(Z,W)=Z(\beta(W))-W(\beta(Z))-\beta([Z,W])$ (23)

valid for every differential one-form $\beta$ and every pair of vector fields
$Z,W$ on $\mathcal{N}$, we immediately see that

$\begin{split}\mathrm{d}\theta^{j}(X_{k},X_{l})&=X_{k}(\theta^{j}(X_{l}))-X_{k}(\theta^{j}(X_{l}))-\theta^{j}([X_{k},X_{l}])\stackrel{{\scriptstyle\mbox{\eqref{eqn:
dual basis}}}}{{=}}-\theta^{j}([X_{k},X_{l}]),\end{split}$ (24)

which means that $\nabla$ is torsion-free (cf. equation (6)) if and only if
$\\{\theta^{j}\\}_{j=1,...,n}$ is a basis of closed differential one-forms.
Similarly, we have

$\begin{split}\mathrm{d}\alpha^{j}(Y_{k},Y_{l})&=Y_{k}(\alpha^{j}(Y_{l}))-Y_{l}(\alpha^{j}(Y_{k}))-\alpha^{j}([Y_{k},Y_{l}])\stackrel{{\scriptstyle\mbox{\eqref{eqn:
dually dual basis}}}}{{=}}-\alpha^{j}([Y_{k},Y_{l}])\end{split}$ (25)

which means that $\nabla^{*}$ is torsion-free (cf. equation (6)) if and only
if $\\{\alpha^{j}\\}_{j=1,...,n}$ is a basis of closed differential one-forms.

If $(\nabla,\nabla^{*})$ is a $G$-dual pair on $(\mathcal{N},G)$ and
$\nabla^{G}$ is the Levi-Civita connection of $G$, we may introduce the
covariant $(0,3)$ tensor $T$ setting

$T(Z,W,V):=G(\nabla_{Z}W,V)-G(\nabla^{G}_{Z}W,V)$ (26)

for every $Z,W,V\in\mathfrak{X}(\mathcal{N})$. Since $\nabla^{*}$ is the
$G$-dual connection of $\nabla$, equation (26) can be re-written in terms of
$\nabla^{*}$ as

$T(Z,W,V):=G(W,\nabla^{G}_{Z}V)-G(W,\nabla_{Z}^{*}V).$ (27)

If both $\nabla$ and $\nabla^{*}$ are torsion-free, then $T$ is completely
symmetric and it is precisely the Amari-Čencov tensor of the Statistical
Manifold $(\mathcal{N},G,T)$. In general, however, $T$ is no longer completely
symmetric and we now investigate its properties when $(\nabla,\nabla^{*})$ is
a $G$-dual teleparallel pair.

######  Proposition 2.

Let $(\nabla,\nabla^{*})$ be a $G$-dual teleparallel pair on the real, finite-
dimensional, smooth, parallelizable Riemannian manifold $(\mathcal{N},G)$ of
dimension $\mathrm{dim}(\mathcal{N})=n$, and let $T$ be the covariant $(0,3)$
tensor defined in equation (26) (or equation (27)). Then:

  * •

$T$ is symmetric in the first two entries if and only if $\nabla$ is torsion-
free or, equivalently, if and only if $\\{\theta^{j}\\}_{j=1,...,n}$ is a
basis of closed differential one-forms;

  * •

$T$ is symmetric in the first and third entries if and only if $\nabla^{*}$ is
torsion-free or, equivalently, if and only if $\\{\alpha^{j}\\}_{j=1,...,n}$
is a basis of closed differential one-forms;

  * •

$T$ is symmetric in the second and third entries if and only if
$\nabla^{*}_{X_{j}}X_{k}=0$ for every $j,k=1,...,n$, or, equivalently, if and
only if $\nabla_{Y_{j}}Y_{k}=0$ $j,k=1,...,n$.

###### Proof.

In this case it holds

$T(X_{j},X_{k},Y_{l})\stackrel{{\scriptstyle\mbox{\eqref{eqn: 3-tensor
0}\eqref{eqn: weitzenbock
connection}}}}{{=}}-G(\nabla^{G}_{X_{j}}X_{k},Y_{l}).$ (28)

Consequently, since $\nabla^{G}$ is torsion-free, we have
$\nabla^{G}_{X_{j}}X_{k}-\nabla^{G}_{X_{k}}X_{j}=[X_{j},X_{k}]$ and thus

$T(X_{j},X_{k},Y_{l})-T(X_{k},X_{j},Y_{l})=G\left([X_{k},X_{j}],Y_{l}\right)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
gradient vector
field}}}}{{=}}-\theta^{l}([X_{j},X_{k}])\stackrel{{\scriptstyle\mbox{\eqref{eqn:
differential and commutators 1}}}}{{=}}\mathrm{d}\theta^{j}(X_{k},X_{l})$ (29)

showing that $T$ is symmetric in the first two entries if and only if $\nabla$
is torsion-free or, equivalently, if and only if
$\\{\theta^{j}\\}_{j=1,...,n}$ is a basis of closed differential one-forms
(cf. remark 2) as claimed. In particular, this is always true when we consider
the mixture connection in both the classical and quantum case (cf. sections 3
and 4).

Analogously, we have

$T(Y_{j},X_{k},Y_{l})\stackrel{{\scriptstyle\mbox{\eqref{eqn: 3-tensor
1}\eqref{eqn: weitzenbock
connection}}}}{{=}}G(X_{k},\nabla^{G}_{Y_{j}}Y_{l}),$ (30)

so that, again since $\nabla^{G}$ is torsion-free, we can use
$\nabla^{G}_{Y_{j}}Y_{l}-\nabla^{G}_{Y_{l}}Y_{j}=[Y_{j},Y_{l}]$ and obtain

$\begin{split}T(Y_{j},X_{k},Y_{l})-T(Y_{l},X_{k},Y_{j})&=G(X_{k},[Y_{j},Y_{l}])\stackrel{{\scriptstyle\mbox{\eqref{eqn:
gradient vector field
2}}}}{{=}}-\alpha^{k}([Y_{j},Y_{l}])\stackrel{{\scriptstyle\mbox{\eqref{eqn:
differential and commutators
2}}}}{{=}}\mathrm{d}\alpha^{k}(Y_{j},Y_{l}),\end{split}$ (31)

which means that $T$ is symmetric in the first and third entries if and only
if $\nabla^{*}$ is torsion-free or, equivalently, if and only if
$\\{\alpha^{j}\\}_{j=1,...,n}$ is a basis of closed differential one-forms
(cf. remark 2).

Finally, recalling that

$Z(G(W,V))=G(\nabla^{G}_{Z}W,V)+G(W,\nabla^{G}_{Z}V)$ (32)

for all $Z,W,V\in\mathfrak{X}(\mathcal{N})$ because $\nabla^{G}$ is its own
$G$-dual connection, we have

$\begin{split}T(X_{j},X_{k},X_{l})-T(X_{j},X_{l},X_{k})&\stackrel{{\scriptstyle\mbox{\eqref{eqn:
3-tensor
1}}}}{{=}}G\left(X_{k},\nabla^{G}_{X_{j}}X_{l}-\nabla^{*}_{X_{j}}X_{l}\right)-G\left(X_{l},\nabla^{G}_{X_{j}}X_{k}-\nabla^{*}_{X_{j}}X_{k}\right)=\\\
&\stackrel{{\scriptstyle\mbox{\eqref{eqn: dual connections 0}\eqref{eqn:
weitzenbock connection}\eqref{eqn: dual Levi-Civita
connection}}}}{{=}}G\left(X_{l},\nabla^{*}_{X_{j}}X_{k}\right),\end{split}$
(33)

which means that $T$ is symmetric in the second and third entries if and only
if $\nabla^{*}_{X_{j}}X_{k}=0$ for every $j,k=1,...,n$. Moreover, it also
holds

$\begin{split}T(Y_{j},Y_{k},Y_{l})-T(Y_{j},Y_{l},Y_{k})&\stackrel{{\scriptstyle\mbox{\eqref{eqn:
3-tensor
0}}}}{{=}}G\left(\nabla_{Y_{j}}Y_{k}-\nabla^{G}_{Y_{j}}Y_{k},Y_{l}\right)-G\left(\nabla_{Y_{j}}Y_{l}-\nabla^{G}_{Y_{j}}Y_{l},Y_{k}\right)=\\\
&\stackrel{{\scriptstyle\mbox{\eqref{eqn: dual connections 0}\eqref{eqn:
dually weitzenbock 5}\eqref{eqn: dual Levi-Civita
connection}}}}{{=}}-G\left(\nabla_{Y_{j}}Y_{l},Y_{k}\right)\end{split}$ (34)

which means that $T$ is symmetric in the second and third entries if and only
if $\nabla_{Y_{j}}Y_{k}=0$ for every $j,k=1,...,n$. We thus conclude that

$\nabla^{*}_{X_{j}}X_{k}=0\;\;\forall\;j,k=1,...,n\,\Longleftrightarrow\;\nabla_{Y_{j}}Y_{k}=0\;\;\forall\;j,k=1,...,n.$
(35)

Of course, if $\nabla$ and $\nabla^{*}$ are both torsion-free then it can
easily be checked that $T$ is completely symmetric. ∎

##  3 G-dual teleparallel pairs in Classical Information Geometry

Let $\overline{\Delta_{n}}$ be the n-simplex in $\mathbb{R}^{n}$:

$\overline{\Delta_{n}}:=\left\\{\vec{p}=(p^{1},...,p^{n})\in\mathbb{R}^{n}\,|\quad
p^{j}\geq 0\;\forall\;j=1,...,n,\mbox{ and }\sum_{j=1}^{n}p^{j}=1\right\\},$
(36)

and $\Delta_{n}$ its interior

$\Delta_{n}:=\left\\{\vec{p}=(p^{1},...,p^{n})\in\overline{\Delta_{n}}\,|\quad
p^{j}>0\;\forall\;j=1,...,n\right\\}.$ (37)

Note that both $\overline{\Delta_{n}}$ and $\Delta_{n}$ are convex subsets of
$\mathbb{R}^{n}$. If $\mathcal{X}_{n}$ denote a discrete set with $n$
elements, then $\overline{\Delta_{n}}$ can be interpreted as the set of
probability distributions on $\mathcal{X}_{n}$, and $\Delta_{n}$ as the subset
of nowhere-vanishing probability distributions on $\mathcal{X}_{n}$. The space
$\Delta_{n}$ is a smooth manifold which is also parallelizable because it is
essentially an open subset of the affine hyperplane determined by the
normalization condition $\sum_{j=1}^{n}p^{j}=1$.

A tangent vector at $\vec{p}\in\Delta_{n}$ can be identified with an element
$\vec{a}\in\mathbb{R}^{n}$ such that $\sum_{j=1}^{n}a^{j}=0$. If we choose
$(n-1)$ linearly independent vectors
$\vec{a}_{1},...,\vec{a}_{n-1}\in\mathbb{R}^{n}$ such that
$\sum_{j=1}^{n}a^{j}_{k}=0$ for all $k=1,...,(n-1)$, we can define a basis
$\\{L_{k}\\}_{k=1,...,(n-1)}$ of global vector fields on $\Delta_{n}$ setting

$L_{k}(\vec{p})=\vec{a}_{k}.$ (38)

The notational change from $X_{k}$ to $L_{k}$ is made to remark the relation
of these vector fields with the linear structure of $\mathbb{R}^{n}$, and thus
with the convex structures of probability vectors. Indeed, every $L_{k}$
generates a traslation on $\mathbb{R}^{n}$, and traslations are related with
the linear structure of $\mathbb{R}^{n}$.

The teleparallel connection associated with $\\{L_{k}\\}_{k=1,...,(n-1)}$ is
known as the mixture connection on $\Delta_{n}$ and it is often denoted by
$\nabla^{m}$. This affine connection is intimately connected with the convex
structure of $\Delta_{n}$. Indeed, as mentioned in section 2, every geodesic
of $\nabla^{m}$ starting at $\vec{p}$ and with intial velocity
$\vec{v}=v^{k}a_{k}=v^{k}L_{k}(\vec{p})$, where $v^{k}\in\mathbb{R}$ for all
$k=1,...,(n-1)$, can be realized as an integral curve of the vector field
$V=v^{k}L_{k}$, and has the simple expression

$\vec{p}_{\vec{v}}(t)=\vec{p}+t\,\vec{v}.$ (39)

Therefore, given two arbitrary points $\vec{p}_{1},\vec{p}_{2}\in\Delta_{n}$,
we can build a geodesic starting at $\vec{p}_{1}$ and ending at $\vec{p}_{2}$
at time $t=1$ setting $\vec{v}=\vec{p}_{2}-\vec{p}_{1}$ thus obtaining the
segment $(1-t)\vec{p}_{1}+t\vec{p}_{2}$ describing all the possible convex
combinations of $\vec{p}_{1}$ and $\vec{p}_{2}$.

It is perhaps seldom observed that, from the geometrical point of view,
$\Delta_{n}$ is actually an homogeneous space of the multiplicative Lie group
$\mathbb{R}^{n}_{+}$, i.e., the product of $n$ copies of $\mathbb{R}_{+}$.
Specifically, let $(q^{1},...,q^{n})=\vec{q}\in\mathbb{R}^{n}_{+}$ and set

$\gamma(\vec{q},\vec{p}):=\frac{1}{N_{\vec{q}}^{\vec{p}}}\,\left(q^{1}p^{1},...,q^{n}p^{n}\right)$
(40)

with $N_{\vec{q}}^{\vec{p}}=\sum_{j=1}^{n}q^{j}p^{j}$. It is a matter of
direct computation to check that $\gamma$ is indeed a smooth, transitive left
action of $\mathbb{R}^{n}_{+}$ on $\Delta_{n}$. Then, setting
$\vec{b}_{0}=(1,...,1)$ and selecting $(n-1)$ linearly independent vectors
$\vec{b}_{1},...,\vec{b}_{n-1}\in\mathbb{R}^{n}$ such that
$\sum_{j=1}^{n}b^{j}_{k}=0$ for all $k=1,...,(n-1)$, we obtain a basis
$\\{\vec{b}_{j}\\}_{j=0,...,(n-1)}$ of the Lie algebra $\mathbb{R}^{n}$ of
$\mathbb{R}^{n}$ such that the fundamental vector field of $\gamma$ associated
with $\vec{b}_{0}$ identically vanishes, while the fundamental vector field of
$\gamma$ associated with $\vec{b}_{j}$ reads

$Y_{j}(\vec{p})=\left(\left(b^{1}_{j}-N_{\vec{b}_{j}}^{\vec{p}}\right)p^{1},...\,,\left(b^{n}_{j}-N_{\vec{b}_{j}}^{\vec{p}}\right)p^{n}\right).$
(41)

It turns out that the teleparallel connection determined by the basis
$\\{Y_{j}\\}_{j=1,...,(n-1)}$ of vector fields on $\Delta_{n}$ is precisely
the exponential connection which is dual to $\nabla^{m}$ with respect to the
Fisher-Rao metric tensor.

According to Čencov’s pioneering work (cf. [56]), the Riemannian manifold
structure which is relevant in the context of Classical Information Geometry
is the so-called Fisher-Rao metric tensor $G_{FR}$ given by

$\left(G_{FR}(X,Y)\right)(\vec{p}):=\langle
X(\vec{p}),Y(\vec{p})\rangle_{\vec{p}}$ (42)

where $\langle\cdot,\cdot\rangle_{\vec{p}}$ is the weighted scalar product on
$\mathbb{R}^{n}$ given by

$\langle\vec{a},\vec{b}\rangle_{\vec{p}}:=\sum_{j=1}^{n}\frac{a^{j}b^{j}}{p^{j}}.$
(43)

It is well-known that the $G_{FR}$-dual connection of $\nabla^{m}$ is the so-
called exponential connection $\nabla^{e}$ (cf. [4, 56]). Moreover, from
proposition 1 we know that $\nabla^{e}$ is the teleparallel connection
associated with the gradient vector fields of an almost dual basis of
$\\{L_{k}\\}_{k=1,...,(n-1)}$. At this purpose, an almost dual basis
$\\{\theta^{k}\\}_{k=1,...,(n-1)}$ of $\\{L_{k}\\}_{k=1,...,(n-1)}$ is
associated with a choice of $(n-1)$ linearly independent vectors
$\\{\vec{b}_{k}\\}_{k=1,...,(n-1)}$ such that $\sum_{j=1}^{n}b^{j}_{k}=0$ for
all $k=1,...,(n-1)$. Indeed, setting $\theta^{k}=\mathrm{d}l^{k}$ with

$l^{k}(\vec{p}):=\vec{p}\cdot\vec{b}_{k},$ (44)

it immediately follows that

$\theta^{k}(L_{j})=\frac{\mathrm{d}}{\mathrm{d}t}\left((\vec{p}+t\vec{a}_{j})\cdot\vec{b}_{k}\right)_{t=0}=\vec{a}_{j}\cdot\vec{b}_{k},$
(45)

and the right hand side clearly does not depend on the point $\vec{p}$, thus
showing that $\\{\theta^{k}\\}_{k=1,...,(n-1)}$ is indeed an almost dual basis
of $\\{L_{k}\\}_{k=1,...,(n-1)}$. Then, a direct computation using the very
definition of gradient vector field (cf. equation (13)), and equations (42)
and (45) shows that the gradient vector field associated with $\theta^{k}$ by
the Fisher-Rao metric tensor $G_{FR}$ is precisely the vector field $Y_{k}$
given in equation (41) as claimed. Since the exponential connection
$\nabla^{e}$ is the teleparallel connection determined by the basis
$\\{Y_{j}\\}_{j=1,...,(n-1)}$, every geodesic of $\nabla^{e}$ starting at
$\vec{p}$ and with intial velocity $\vec{v}=v^{j}a_{j}=v^{j}Y_{j}(\vec{p})$,
where $v^{j}\in\mathbb{R}$ for all $j=1,...,(n-1)$, can be realized as an
integral curve of the vector field $V=v^{j}Y_{j}$. Then, since the $Y_{j}$’s
are the fundamental vector fields of the action of $\mathbb{R}^{n}_{+}$ on
$\Delta_{n}$ described in equation (40), it immediately follows that every
integral curve of $V$ starting at $\vec{p}$ reads

$\vec{p}_{\vec{v}}(t)=\frac{1}{\sum_{j=1}^{n}\mathrm{e}^{tv^{j}}p^{j}}\,\left(\mathrm{e}^{tv^{1}}\,p^{1},...,\mathrm{e}^{tv^{n}}\,p^{n}\right),$
(46)

from which it is evident that $\nabla^{e}$ is complete.

##  4 G-dual teleparallel pairs in Quantum Information Geometry

The purpose of this section is to discuss the $G$-dual teleparallel pairs that
naturally arise in the context of finite-dimensional Quantum Information
Geometry when the reference teleparallel connection comes from the natural
convex structure of the space of quantum states, and the metric tensors are
the so-called monotone metric tensors replacing the Fisher-Rao metric tensor
in the quantum context and classified by Petz [48].

Let $\mathcal{H}$ be a complex Hilbert space of dimension $n<\infty$. Let
$\langle\cdot,\cdot\rangle$ denote the Hilbert space inner product on
$\mathcal{H}$, $\mathcal{B}(\mathcal{H})$ denote the space of bounded linear
operators on $\mathcal{H}$, and $\mathcal{B}_{sa}(\mathcal{H})$ the space of
self-adjoint elements in $\mathcal{B}(\mathcal{H})$. The vector space
$\mathcal{B}_{sa}(\mathcal{H})$ is a real Hilbert space with respect to the
so-called Hilbert-Schmidt product

$\langle\mathbf{a},\mathbf{b}\rangle_{HS}:=\textit{Tr}(\mathbf{a}\mathbf{b}),$
(47)

where Tr is the Hilbert space trace. We can always select a basis
$\\{\sigma_{j}\\}_{j=0,...,n^{2}-1}$ for $\mathcal{B}_{sa}(\mathcal{H})$ in
such a way that $\sigma_{0}=\mathbb{I}$, the identity operator on
$\mathcal{H}$, and $\textit{Tr}(\sigma_{j})=0$ for all $j=1,...,n^{2}-1$. When
$\mathcal{H}\cong\mathbb{C}^{2}$, the typical example of such a basis is the
one in which $\sigma_{j}$ is the j-th Pauli matrix for $j=1,2,3$.

Every element $\mathbf{a}\in\mathcal{B}_{sa}(\mathcal{H})$ can then be
expressed as

$\mathbf{a}=x^{0}\sigma_{0}+\vec{x}\cdot\vec{\sigma}$ (48)

where $\vec{x}=(x^{1},...,x^{n^{2}-1})\in\mathbb{R}^{n^{2}-1}$,
$\vec{\sigma}=(\sigma^{1},...,\sigma^{n^{2}-1})$, and $\cdot$ is the Euclidean
scalar product. The real numbers $(x^{0},\vec{x})$ determine a Cartesian
coordinate system on $\mathcal{B}_{sa}(\mathcal{H})$.

An element in $\mathcal{B}_{sa}(\mathcal{H})$ is called positive if its
spectrum lies on the non-negative half-line, and strictly positive when its
spectrum lies on the positive half-line. In particular, a strictly positive
operator is invertible. It is customary to write $\rho\geq 0$ to denote a
positive operator, and $\rho>0$ to denote a strictly positive one. The space
$\mathscr{P}(\mathcal{H})$ of strictly positive operators on $\mathcal{H}$ is
an open convex cone in $\mathcal{B}_{sa}(\mathcal{H})$, which means it is an
open submanifold of $\mathcal{B}_{sa}(\mathcal{H})$, and thus a parallelizable
smooth manifold since $\mathcal{B}_{sa}(\mathcal{H})$ is a vector space.

The space of faithful quantum states of a finite-level quantum systems
associated with $\mathcal{H}$ is identified with the space of invertible
density operators on $\mathcal{H}$ given by

$\mathscr{S}(\mathcal{H}):=\left\\{\rho\in\mathscr{P}(\mathcal{H})|\;\textit{Tr}(\rho)=1\right\\}.$
(49)

Accordingly, $\mathscr{S}(\mathcal{H})$ is a codimension-one submanifold of
$\mathscr{P}(\mathcal{H})$ which is also parallelizable because it is
essentially an open subset of the affine hyperplane in
$\mathcal{B}_{sa}(\mathcal{H})$ determined by the trace-normalization
condition. Moreover, it is also a convex set.

The space of all quantum states $\overline{\mathscr{S}(\mathcal{H})}$ is the
norm-closure of $\mathscr{S}(\mathcal{H})$ inside $\mathcal{B}(\mathcal{H})$
(or $\mathcal{B}_{sa}(\mathcal{H})$), and it is a compact, convex subset of
$\mathcal{B}_{sa}(\mathcal{H})$ which is also a stratified manifold [22, 33].

Recalling equation (48) and the normalization condition in equation (49), it
is clear that every $\rho\in\overline{\mathscr{S}(\mathcal{H})}$ is such that
$x^{0}=\frac{1}{n}$. However, it is not in general possible to explicitly find
all the constraints the vector $\vec{x}$ in equation equation (48) must
satisfy in order to ensure the positivity condition $\rho\geq 0$. The two-
dimensional case is the only case in which a direct computation of the
eigenvalues shows that $\rho\geq 0$ is equivalent to
$\vec{x}\cdot\vec{x}\leq\frac{1}{4}$.

Because of the normalization condition, and because of dimensional reasons
related with the fact that $\mathscr{S}(\mathcal{H})$ is a codimension-one
submanifold of the open submanifold
$\mathscr{P}(\mathcal{H})\subset\mathcal{B}_{sa}(\mathcal{H})$, the tangent
space at $\rho$ can be expressed as the linear span

$T_{\rho}\mathscr{S}(\mathcal{H})\cong\mathrm{span}\left(\sigma_{1},...,\sigma_{n^{2}-1}\right),$
(50)

and we can define the global basis $\\{L_{k}\\}_{k=1,...,n^{2}-1}$ of vector
fields on $\mathscr{S}(\mathcal{H})$ setting

$L_{k}(\rho)=\sigma_{k}.$ (51)

Again, the notational change from $X_{k}$ to $L_{k}$ is made to emphasize the
relation of these vector fields with the linear structure of
$\mathcal{B}_{sa}(\mathcal{H})$ (cf. the comment after equation (38)).

The teleparallel connection associated with $\\{L_{k}\\}_{k=1,...,n^{2}-1}$ is
known as the mixture connection on $\mathscr{S}(\mathcal{H})$ and it is
denoted as $\nabla^{m}$. Very much like it happens in the classical setting
for the mixture connection on the interior of the n-simplex, the mixture
connection on $\mathscr{S}(\mathcal{H})$ turns out to have a natural geometric
interpretation in terms of the convex structure of $\mathscr{S}(\mathcal{H})$.
Indeed, every geodesic of $\nabla^{m}$ starting at $\rho$ with initial tangent
vector $\mathbf{v}=v^{k}\sigma_{k}=v^{k}L_{k}(\rho)$, where
$v^{k}\in\mathbb{R}$ for every $k=1,...,(n^{2}-1)$, can be realized as the
integral curve of the vector field $V=v^{k}L_{k}$ (cf. section 2) which reads

$\rho_{\mathbf{v}}(t)=\rho+t\mathbf{v}.$ (52)

Therefore, given two arbitrary points
$\rho_{1},\rho_{2}\in\mathscr{S}(\mathcal{H})$, we can build a geodesic
starting at $\rho_{1}$ and ending at $\rho_{2}$ at time $t=1$ by setting
$\mathbf{v}=\rho_{2}-\rho_{1}$, thus obtaining the segment
$(1-t)\rho_{1}+t\rho_{2}$ describing all the possible convex combinations of
$\rho_{1}$ and $\rho_{2}$. Moreover, it follows from equation (52) that
$\nabla^{m}$ is not complete on $\mathscr{S}(\mathcal{H})$.

In the context of Quantum Information Geometry, and, contrary to the situation
in Classical Information Geometry, the Riemannian structure on
$\mathscr{S}(\mathcal{H})$ is not uniquely determined, and there is an
infinite family of so-called quantum monotone metric tensors (cf. [48]). In
particular, given an operator monotone function
$f\colon(0,\infty)\rightarrow(0,\infty)$ such that

$f(x)=xf(x^{-1}),\quad f(1)=1,$ (53)

Petz’s classification of quantum monotone metric tensor assures us that every
such metric tensor $G_{f}$ is associated with $f$ by means of

$\left(G_{f}(V,W)\right)(\rho)=\textit{Tr}\left(V_{\rho}\,T_{\rho}^{f}(W_{\rho})\right),$
(54)

where $V,W$ are vector fields on $\mathscr{S}(\mathcal{H})$, and where
$T_{\rho}^{f}=(K^{f}_{\rho})^{-1}$ with

$K^{f}_{\rho}=f(L_{\rho}\,R_{\rho^{-1}})\,R_{\rho},$ (55)

where $L_{\rho}(\mathbf{a})=\rho\mathbf{a}$ and
$R_{\rho}(\mathbf{a})=\mathbf{a}\rho$. Note that both $L_{\rho}$ and
$R_{\rho}$ are (invertible) superoperators on $\mathcal{B}(\mathcal{H})$.

Being self-adjoint, every $\rho$ can be diagonalized using an orthonormal
basis $\\{|j\rangle\\}_{j=1,...,\mathrm{dim}(\mathcal{H})}$ of $\mathcal{H}$.
Writing $\mathbf{e}^{\rho}_{lm}=|l\rangle\langle m|$, we have

$\rho=\sum_{j=1}^{n}\,p^{j}_{\rho}\,\mathbf{e}_{jj}^{\rho}.$ (56)

Note that the collection of all $\mathbf{e}^{\rho}_{lm}$’s forms a basis of
$\mathcal{B}(\mathcal{H})$.

We can also introduce the super-operators $E_{kj}^{\rho}$ acting on
$\mathcal{B}(\mathcal{H})$ according to

$E_{kj}^{\rho}\left(\mathbf{e}^{\rho}_{lm}\right)\,=\,\delta_{jl}\,\delta_{km}\mathbf{e}_{jk}^{\rho},$
(57)

and it is then a matter of straightforward computation to check that

$K^{f}_{\rho}=\sum_{j,k=1}^{n}\,p^{k}_{\rho}\,f\left(\frac{p^{j}_{\rho}}{p^{k}_{\rho}}\right)\,E_{kj}^{\rho},$
(58)

and

$T^{f}_{\rho}=\sum_{j,k=1}^{n}\,\left(p^{k}_{\rho}\,f\left(\frac{p^{j}_{\rho}}{p^{k}_{\rho}}\right)\right)^{-1}\,E_{kj}^{\rho}.$
(59)

As a side remark, note that, for every vector field $W$ such that its
associated tangent vector $W_{\rho}$ at $\rho$ commutes with $\rho$, and for
every vector field $V$, using equations (59) and (42) it follows that

$\left(G_{f}(V,W)\right)(\rho)=\left(G_{FR}(\vec{V}_{d},\vec{W}_{d})\right)(\vec{p}_{\rho})$
(60)

with $\vec{p}_{\rho}=(p^{1}_{\rho},...,p^{n}_{\rho})$, and
$\vec{V}_{d}(\vec{p}_{\rho}),\vec{W}_{d}(\vec{p}_{\rho})$ the vectors built
out of the diagonal elements of $V_{\rho}$ and $W_{\rho}$ with respect to the
basis of eigenvectors of $\rho$. Note that equation (60) holds for every
choice of the operator monotone function $f$.

From proposition 1 we know that the $G_{f}$-dual connection $\nabla^{f}$ of
the mixture connection $\nabla^{m}$ is the teleparallel connection associated
with the gradient vector fields of an almost dual basis of
$\\{L_{k}\\}_{k=1,...,n}$. An almost dual basis
$\\{\theta^{k}\\}_{k=1,...,n^{2}-1}$ of $\\{L_{k}\\}_{k=1,...,n^{2}-1}$ is
then easily seen to be associated with a choice of basis
$\\{\omega_{k}\\}_{k=0,...,n^{2}-1}$ for $\mathcal{B}_{sa}(\mathcal{H})$ in
such a way that $\omega_{0}=\mathbb{I}$ and $\textit{Tr}(\omega_{k})=0$.
Indeed, setting $\theta^{k}:=\mathrm{d}l^{k}$ with

$l^{k}(\rho):=\textit{Tr}(\rho\omega_{k})$ (61)

the expectation-value function associated with $\sigma_{k}$, a direct
computation shows that

$\theta^{k}(L_{j})=\frac{\mathrm{d}}{\mathrm{d}t}\left(l^{k}(\rho_{j}(t))\right)_{t=0}\stackrel{{\scriptstyle\mbox{\eqref{eqn:
geodesic of mixture connection quantum
case}}}}{{=}}\textit{Tr}(\sigma_{j}\omega_{k}),$ (62)

and the right-hand-side does not depend on the point $\rho$. Then, we have

$\textit{Tr}(\sigma_{j}\omega_{k})\stackrel{{\scriptstyle\mbox{\eqref{eqn:
almost dual basis
quantum}}}}{{=}}\left(\theta^{k}(L_{j})\right)(\rho)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
gradient vector
field}}}}{{=}}\left(G_{f}(L_{j},Y^{f}_{k})\right)(\rho)\stackrel{{\scriptstyle\mbox{\eqref{eqn:
Petz metric 1}\eqref{eqn: linear vector field
quantum}}}}{{=}}\textit{Tr}\left(\sigma_{j}\,T_{\rho}^{f}(Y^{f}_{k}(\rho))\right)$
(63)

for all $k=1,...,(n^{2}-1)$, from which it follows that the gradient vector
field $Y^{f}_{k}$ associated with $\theta^{k}$ by means of $G_{f}$ must be
such that

$\omega_{j}-T_{\rho}^{f}(Y^{f}_{k}(\rho))=c_{\rho}\mathbb{I}\equiv
c_{\rho}\omega_{0}.$ (64)

Therefore, since $T_{\rho}^{f}=(K^{f}_{\rho})^{-1}$, we have

$Y_{k}^{f}(\rho)=K^{f}_{\rho}(\omega_{k}-c_{\rho}\mathbb{I})\stackrel{{\scriptstyle\mbox{\eqref{eqn:
K superoperator on
eigenprojectors}}}}{{=}}K^{f}_{\rho}(\omega_{k})-c_{\rho}\rho,$ (65)

with $c_{\rho}=\textit{Tr}(K^{f}_{\rho}(\omega_{k}))$ because it must be
$\textit{Tr}(Y_{k}^{f}(\rho))=0$ since $Y_{k}^{f}(\rho)\in
T_{\rho}\mathscr{S}(\mathcal{H})$.

From equation (6) we know that the torsion tensor of a teleparallel connection
is connected with the commutator of the basis vector fields defining the
teleparallel connection. It is then relevant to review the results presented
in [11, 15, 16] in relation with the teleparallel connection $\nabla^{f}$
determined by $\\{Y^{f}_{k}\\}_{k=1,...,n}$. In the above-mentioned works,
even if their role with respect to $G_{f}$-dual pairs was not known, the
gradient vector fields associated with the expectation value functions by
means of monotone quantum metric tensors have been studied in relation with
the actions of Lie groups suitably extending the unitary group
$\mathcal{U}(\mathcal{H})$ and its canonical action

$\rho\mapsto\alpha(\mathbf{U},\rho)=\mathbf{U}\rho\mathbf{U}^{\dagger}$ (66)

on $\mathscr{S}(\mathcal{H})$, and a great deal of attention has been given to
the commutators between these vector fields.

When the operator monotone function is $f(x)=\kappa(x-1)/\ln(x)$ with
$\kappa>0$, the Riemannian metric tensor $G_{f}$ reduces to the Bogoliubov-
Kubo-Mori metric tensor (cf. [48]) and the gradient vector fields mutually
commute. Consequently, the teleparallel connection $\nabla^{f}$ is torsion-
free and flat. This is not surprising because it is well-known that the
connection which is dual to $\nabla^{m}$ with respect to the Bogoliubov-Kubo-
Mori metric tensor is the quantum version of the exponential connection which
is torsion-free, flat, and also complete (cf. [24, 34, 44]). What is
surprising is the following fact (we refer to [11, 15, 16] for more details on
the construction we are about to resume). Together with the fundamental vector
fields of the standard action of $\mathcal{U}(\mathcal{H})$ on
$\mathscr{S}(\mathcal{H})$ (cf. equation (66)), the gradient vector fields
giving rise to connection $\nabla^{f}$ can be used to build an anti-
representation of the Lie algebra of the cotangent group
$T^{*}\mathcal{U}(\mathcal{H})$. This anti-representation integrates to a
transitive and smooth left action on $\mathscr{S}(\mathcal{H})$ which may be
thought of as the quantum counterpart of the action $\gamma$ (cf. equation
(40)) of $\mathbb{R}^{n}_{+}$ on $\Delta_{n}$. In particular, every geodesic
of $\nabla^{f}$ starting at $\rho$ with initial velocity $\mathbf{v}$ is of
the form

$\rho_{\mathbf{v}}(t)=\frac{\mathrm{e}^{\ln\rho+t\mathbf{v}}}{\textit{Tr}\left(\mathrm{e}^{\ln\rho+t\mathbf{v}}\right)}$
(67)

from which it clearly follows that $\nabla^{f}$ is complete on
$\mathscr{S}(\mathcal{H})$. Moreover, when $[\mathbf{v},\rho]=\mathbf{0}$, the
geodesic in equation (67), if diagonalized, has basically the form of the
geodesic of the classical exponential connection given in equation (46). Since
both $\nabla^{m}$ and $\nabla^{f}$ are torsion-free in this case, the
covariant tensor $T$ is completely symmetric and
$(\mathscr{S}(\mathcal{H}),G_{f},T)$ is a Statistical Manifold in the sense of
[40].

It is also worth noting that, when
$f(x)=\frac{\kappa(x-1)(x^{\kappa}+1)}{2(x^{\kappa}-1)}$ with $0<\kappa<1$, a
similar instance manifests itself. Indeed, again together with the fundamental
vector fields of the standard action of $\mathcal{U}(\mathcal{H})$ on
$\mathscr{S}(\mathcal{H})$ (cf. equation (66)), the gradient vector fields
giving rise to connection $\nabla^{f}$ provide a Lie algebra anti-
representation. However, this time the anti-representation is of the Lie
algebra $\mathfrak{gl}(\mathcal{H})$ of the general linear group
$\mathcal{GL}(\mathcal{H})$. This representation integrates to a transitive
and smooth left action of $\mathcal{GL}(\mathcal{H})$ on
$\mathscr{S}(\mathcal{H})$. In particular, every geodesic of $\nabla^{f}$
starting at $\rho$ and with initial tangent vector
$\mathbf{v}=v^{j}\sigma_{j}$, where $v^{j}\in\mathbb{R}$ for every
$j=1,...,(n^{2}-1)$, is of the form

$\rho_{\mathbf{v}}(t)=\frac{\left(\mathrm{e}^{t\mathbf{v}}\rho^{\sqrt{\kappa}}\mathrm{e}^{t\mathbf{v}}\right)^{\frac{1}{\sqrt{\kappa}}}}{\textit{Tr}\left(\left(\mathrm{e}^{t\mathbf{v}}\rho^{\sqrt{\kappa}}\mathrm{e}^{t\mathbf{v}}\right)^{\frac{1}{\sqrt{\kappa}}}\right)},$
(68)

from which it is clear that $\nabla^{f}$ is always complete.

When $\kappa=1$, the monotone quantum metric tensor $G_{f}$ coincides with the
Bures-Helstrom metric tensor (cf. [9, 20, 23, 36, 54, 58]) ubiquitous in
quantum estimation theory (cf. [19, 25, 42, 46, 51, 52, 53]), while when
$\kappa=\frac{1}{4}$, the monotone quantum metric tensor $G_{f}$ coincides
with the Wigner-Yanase metric tensor [28, 29, 35, 39]. In all these cases, the
commutator between gradient vector fields is a fundamental vector field of
$\alpha$ and, in general, it does not vanish so that the connection
$\nabla^{f}$ presents torsion. Therefore, the tensor $T$ determined by
$\nabla^{m},\nabla^{f}$, and $G_{f}$ will not be symmetric in the first and
second entries so that $(\mathscr{S}(\mathcal{H}),G_{f},T)$ is not a
Statistical Manifold in the sense of [40].

######  Remark 3.

Incidentally, for a two-level quantum system, the cases
$f(x)=\kappa(x-1)/\ln(x)$ with $\kappa>0$ and
$f(x)=\frac{\kappa(x-1)(x^{\kappa}+1)}{2(x^{\kappa}-1)}$ with $0<\kappa<1$ are
the only cases of monotone quantum metric tensors for which the gradient
vector fields and the fundamental vector fields of the standard action of
$\mathcal{U}(\mathcal{H})$ on $\mathscr{S}(\mathcal{H})$ give rise to a
transitive group action on $\mathscr{S}(\mathcal{H})$ (cf. [15]).

##  5 Conclusions

In this work we investigated the notion of $G$-dual teleparallel pairs
$(\nabla,\nabla^{*})$ of affine connections on a real, finite-dimensional,
smooth, parallelizable Riemannian manifold $(\mathcal{N},G)$. We showed how
every such pair gives rise to a covariant $(0,3)$ tensor $T$ generalizing the
Amari-Čencov tensor of Statistical Manifolds to the case where connections
present torsion. Then, we reviewed some examples of $G$-dual teleparallel
pairs in the context of Classical and Quantum Information Geometry, where the
existence of a suitable (torsion-free and flat) mixture connection
$\nabla^{m}$ (associated with a suitable convex structure) gifts us with a
preferred teleparallel connection.

In the classical case, it turns out that the dual connection is also torsion-
free and flat, and it is known as the exponential connection, a fact which is
well-known. Following the theory developed in section 2, we find the explicit
form of a basis of vector fields determining the exponential connection as the
associated teleparallel connection, and show how these vector fields provide a
homogeneous manifold structure for the interior of the n-simplex $\Delta_{n}$
such that the geodesics of the exponential connection coincides with the
images of one-parameter subgroups. As far as we know, these geometric
considerations regarding the exponential connection are new.

In the quantum case, we investigate $G_{f}$-dual pairs $(\nabla,\nabla^{*})$
where $G_{f}$ is a monotone quantum metric tensor, and $\nabla$ is the mixture
connection $\nabla^{m}$ on the manifold $\mathscr{S}(\mathcal{H})$ of faithful
quantum states of a quantum system with finite-dimensional Hilbert space
$\mathcal{H}$. The theory developed in section 2 allows us to conclude that
$\nabla^{*}\equiv\nabla^{f}$ is the teleparallel connection associated with
the gradient vector fields of the expectation value functions. These gradient
vector fields already appeared in the literature in connection with transitive
actions on $\mathscr{S}(\mathcal{H})$ of suitable extensions of the unitary
group $\mathcal{U}(\mathcal{H})$, and we are here able to make a point of
contact between these group actions and the properties of $\nabla^{f}$. In
particular, whenever these groups appear, we are able to explicitly write down
the geodesics of $\nabla^{f}$ again as the images of one-parameter subgroups,
from which it follows that $\nabla^{f}$ is a complete connection. However, it
is in general true that $\nabla^{f}$ presents torsion unless $G_{f}$ is the
so-called Bogoliubov-Kubo-Mori metric tensor, in which case $\nabla^{f}$ is
the quantum counterpart of the exponential connection.

A natural evolution of the ideas presented here would be the investigation of
how $G$-dual teleparallel pairs “projects down” on homogeneous manifolds.
Indeed, it is clear that $G$-dual teleparallel pairs only exist on
parallelizable manifolds, but not all manifolds of interest in Classical and
Quantum Information Geometry are parallelizable. A very relevant example is
given by the manifold of pure states of a finite-level quantum system, which
is diffeomorphic with the complex projective space $\mathbb{P}(\mathcal{H})$.
In this and other cases, however, the manifold of interest $\mathcal{N}$ is
actually a homogeneous manifold for some Lie group $\mathscr{G}$ (e.g., the
unitary group $\mathcal{U}(\mathcal{H})$ for the manifold of pure quantum
states), and Lie groups are obviously parallelizable manifolds. Then, we may
pull-back the Riemannian metric tensor on $\mathcal{N}$ “coming from”
Information Geometry to the Lie group $\mathscr{G}$ by means of the canonical
projection $\pi\colon\mathscr{G}\rightarrow\mathcal{N}$, and complement it to
a Riemannian metric tensor $G$ on $\mathscr{G}$ for which $G$-dual
teleparallel pairs exist. At this point, $\pi$ becomes a Riemannian submersion
and the idea would be to understand if and how it is possible to “project
down” a $G$-dual teleparallel pair on the homogeneous manifold in a suitable
way, perhaps exploiting a suitable projector to describe the module of vector
fields on $\mathcal{N}$ in terms of the module of vector fields on
$\mathscr{G}$. We are currently investigating this problem, and we plan to
discuss it in a future publication.

## Funding

This work has been supported by the Madrid Government (Comunidad de Madrid-
Spain) under the Multiannual Agreement with UC3M in the line of “Research
Funds for Beatriz Galindo Fellowships” (C&QIG-BG-CM-UC3M), and in the context
of the V PRICIT (Regional Programme of Research and Technological Innovation).
The authors acknowledge financial support from the Spanish Ministry of Economy
and Competitiveness, through the Severo Ochoa Programme for Centres of
Excellence in RD (SEV-2015/0554), the MINECO research project
PID2020-117477GB-I00, and Comunidad de Madrid project QUITEMAD++,
S2018/TCS-A4342. G. M. is also a member of the Gruppo Nazionale di Fisica
Matematica (INDAM), Italy. F. D. C. acknowledges support from the CONEX-Plus
programme funded by Universidad Carlos III de Madrid and the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No. 801538.

## References

  * [1] R. Aldrovandi and J. G. Pereira.  Teleparallel Gravity.  Springer Netherlands, 2013.  DOI: 10.1007/978-94-007-5143-9.
  * [2] S. I. Amari.  Differential-Geometrical Methods in Statistics, volume 28 of Lecture Notes in Statistics.  Springer-Verlag, Berlin Heidelberg, 1985.  DOI: 10.1007/978-1-4612-5056-2.
  * [3] S. I. Amari.  Information Geometry and its Application.  Springer, Japan, 2016.  DOI: 10.1007/978-4-431-55978-8.
  * [4] S. I. Amari and H. Nagaoka.  Methods of Information Geometry.  American Mathematical Society, Providence, RI, 2000.  DOI: 10.1090/mmono/191.
  * [5] N. Ay and S. I. Amari.  A novel approach to canonical divergences within information geometry.  Entropy, 17(12):8111 – 8129, 2015.  DOI: 10.3390/e17127866, MPI-MIS Preprint.
  * [6] N. Ay, J. Jost, H. V. Le, and L. Schwachhöfer.  Information Geometry.  Springer International Publishing, 2017.  DOI: 10.1007/978-3-319-56478-4.
  * [7] N. Ay and W. Tuschmann.  Dually flat manifolds and global information geometry.  Open Systems & Information Dynamics, 9(2):195–200, 2002.  DOI: 10.1023/a:1015604927654, MPI-MIS Preprint.
  * [8] N. Ay and W. Tuschmann.  Duality versus dual flatness in quantum information geometry.  Journal of Mathematical Physics, 44(4):1512 – 1518, 2003.  DOI: 10.1063/1.1556192, MPI-MIS Preprint.
  * [9] I. Bengtsson and K. Życzkowski.  Geometry of Quantum States: An Introduction to Quantum Entanglement.  Cambridge University Press, New York, 2006.  DOI: 10.1017/cbo9780511535048.
  * [10] A. L. Besse.  Einstein Manifolds.  Springer, Berlin, Heidelberg, 1987.  DOI: 10.1007/978-3-540-74311-8.
  * [11] F. M. Ciaglia.  Quantum states, groups and monotone metric tensors.  European Physical Journal Plus, 135:530–16pp, 2020.  DOI: 10.1140/epjp/s13360-020-00537-y, arXiv:2006.10595 [math-ph].
  * [12] F. M. Ciaglia, F. Di Cosmo, F. Di Nocera, and P. Vitale.  Monotone metric tensors in Quantum Information Geometry.  arXiv, 2022.  arXiv:2203.10857 [quant-ph].
  * [13] F. M. Ciaglia, F. Di Cosmo, A. Ibort, and G. Marmo.  Schwinger’s Picture of Quantum Mechanics.  International Journal of Geometric Methods in Modern Physics, 17(04):2050054–14, 2020.  DOI: 10.1142/S0219887820500541, arXiv:2002.09326 [math-ph].
  * [14] F. M. Ciaglia, F. Di Cosmo, M. Laudato, G. Marmo, G. Mele, F. Ventriglia, and P. Vitale.  A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the q-z family.  Annals of Physics, 395:238–274, 2018.  DOI: 10.1016/j.aop.2018.05.015, arXiv:1711.09769 [quant-ph].
  * [15] F. M. Ciaglia and F. Di Nocera.  Group Actions and Monotone Metric Tensors: The Qubit Case.  In Frank Nielsen and Frédéric Barbaresco, editors, Geometric Science of Information 2021, volume 12829 of Lecture Notes in Computer Science, pages 145–153. Springer International Publishing, 2021.  DOI: 10.1007/978-3-030-80209-717, arXiv: 2105.15067 [quant-ph].
  * [16] F. M. Ciaglia and F. Di Nocera.  Group Actions and Monotone Quantum Metric Tensors.  Mathematics, 10(15):2613, 2022.  DOI: 10.3390/math10152613, arXiv:2206.10394 [math-ph].
  * [17] F. M. Ciaglia, A. Ibort, and G. Marmo.  A gentle introduction to Schwinger’s formulation of quantum mechanics: the groupoid picture.  Modern Physics Letters A, 33(20):1850122–8, 2018.  DOI: 10.1142/s0217732318501225, arXiv:1807.00519 [quant-ph].
  * [18] F. M. Ciaglia, A. Ibort, and G. Marmo.  Schwinger’s Picture of Quantum Mechanics I: Groupoids.  International Journal of Geometric Methods in Modern Physics, 16(08):1950119 (31), 2019.  DOI: 10.1142/S0219887819501196, arXiv:1905.12274 [math-ph].
  * [19] F. M. Ciaglia, J. Jost, and L. Schwachhöfer.  Differential geometric aspects of parametric estimation theory for states on finite-dimensional C*-algebras.  Entropy, 22(11):1332, 2020.  DOI: 10.3390/e22111332, arXiv: 2010.14394 [math-ph].
  * [20] F. M. Ciaglia, J. Jost, and L. Schwachhöfer.  From the Jordan product to Riemannian geometries on classical and quantum states.  Entropy, 22(06):637–27, 2020.  DOI: 10.3390/e22060637, arXiv:2005.02023 [math-ph].
  * [21] I. Csizár.  Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten.  A Magyar Tudományos Akadémia Matematikaiés Fizikai Tudományok Osztályának Közleményei, 8:85–108, 1963.
  * [22] F. D’Andrea and D. Franco.  On the pseudo-manifold of quantum states.  Differential Geometry and its Applications, 78:101800, 2021.  DOI: 10.1016/j.difgeo.2021.101800, arXiv:2011.04488 [math.OA].
  * [23] J. Dittmann.  On the Riemannian metric on the space of density matrices.  Reports on Mathematical Physics, 36(3):309–315, 1995.  DOI: 10.1016/0034-4877(96)83627-5.
  * [24] D. Felice, S. Mancini, and N. Ay.  Canonical Divergence for Measuring Classical and Quantum Complexity.  Entropy, 21(4):435, 2019.  DOI: 10.3390/e21040435, arXiv: 1903.09797 [math-ph].
  * [25] C. A. Fuchs.  Distinguishability and Accessible Information in Quantum Theory.  PhD thesis, Departement IRO, Universite de Montreal, 1996.  arXiv:9601020 [quant-ph].
  * [26] A. Fujiwara.  Geometry of Quantum Information Systems.  In O. E. Barndorff-Nielsen and E. B. V. Jensen, editors, Geometry in Present Day Science, pages 35–48, 1999.  DOI: 10.1142/3958.
  * [27] Fujiwara, A.  Dually flat structures induced from monotone metrics on a two-level quantum state space.  European Physical Journal Plus, 135(10), 2020.  DOI: 10.1140/epjp/s13360-020-00877-9, arXiv: [].
  * [28] P. Gibilisco and T. Isola.  A characterization of Wigner-Yanase skew information among statistically monotone metrics.  Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4(4):553–557, 2001.  DOI: 10.1142/s0219025701000644, UniRoma Preprint.
  * [29] P. Gibilisco and T. Isola.  Wigner-Yanase information on quantum state space: the geometric approach.  Journal of Mathematical Physics, 44(9):3752–3762, 2003.  DOI: 10.1063/1.1598279, arXiv:0304170 [math.PR].
  * [30] K. Grabowska, J. Grabowski, M. Kuś, and G. Marmo.  Lie groupoids in information geometry.  Journal of Physics A: Mathematical and General, 52(50):505202, 2019\.  DOI: 10.1088/1751-8121/ab542e.
  * [31] K. Grabowska, J. Grabowski, M. Kuś, and G. Marmo.  Information Geometry on Groupoids: The Case of Singular Metrics.  Open Systems & Information Dynamics, 27(03):2050015, 2020.  DOI: 10.1142/S1230161220500158.
  * [32] K. Grabowska, J. Grabowski, M. Kuś, and G. Marmo.  Lifting statistical structures.  Reviews in Mathematical Physics, 35(02), 2023.  DOI: 10.1142/S0129055X22500428, arXiv:2203.02938 [math.DG].
  * [33] J. Grabowski, M. Kuś, and G. Marmo.  Geometry of quantum systems: density states and entanglement.  Journal of Physics A: Mathematical and General, 38(47):10217–10244, 2005.  DOI: 10.1088/0305-4470/38/47/011.
  * [34] M. R. Grasselli and R. F. Streater.  On the Uniqueness of the Chentsov Metric in Quantum Information Geometry.  Infinite Dimensional Analysis, Quantum Probability and Related Topics, 4(2):173–182, 2001.  DOI: 10.1142/S0219025701000462.
  * [35] H. Hasegawa.  Dual geometry of the Wigner-Yanase-Dyson information content.  Infinite Dimensional Analysis, Quantum Probability and Related Topics, 6(3):413–430, 2003.  DOI: 10.1142/S021902570300133X.
  * [36] C. W. Helstrom.  Minimum mean-squared error of estimates in quantum statistics.  Physics Letters A, 25(2):101–102, 1967.  DOI: 10.1016/0375-9601(67)90366-0.
  * [37] M. Henmi and H. Matsuzoe.  Statistical Manifolds Admitting Torsion and Partially Flat Spaces, pages 37–50.  Springer International Publishing, 2019.  DOI: 10.1007/978-3-030-02520-5_3.
  * [38] A. Jenčová.  Geometry of Quantum States: Dual Connections and Divergence Functions.  Reports in Mathematical Physics, 47(1):121 – 138, 2001.
  * [39] A. Jenčová.  Flat connections and Wigner-Yanase-Dyson metrics.  Reports on Mathematical Physics, 52(3):331–351, 2003.  DOI: 10.1016/S0034-4877(03)80033-2.
  * [40] S. Lauritzen.  Statistical Manifolds.  In S. I. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao, editors, Differential geometry in statistical inference, volume 10, pages 163–216. Institute of Mathematical Statistics, 1987\.  DOI: 10.1214/lnms/1215467061, Project Euclid.
  * [41] A. Lesniewski and M. B. Ruskai.  Monotone riemannian metrics and relative entropy on noncommutative probability spaces.  Journal of Mathematical Physics, 40(11):5702–5724, 1999.  DOI: 10.1063/1.533053.
  * [42] J. Liu, H. Yuan, X.-M. Lu, and X. Wang.  Quantum Fisher information matrix and multiparameter estimation.  Journal of Physics A: Mathematical and Theoretical, 53(2):023001–69, 2020.  DOI: 10.1088/1751-8121/ab5d4d, arXiv:1907.08037 [quant-ph].
  * [43] T. Matumoto.  Any statistical manifold has a contrast function: on the $C^{3}$-functions taking the minimum at the diagonal of the product manifold.  Hiroshima mathematical journal, 23(2):327 – 332, 1993.
  * [44] H. Nagaoka.  Differential Geometrical Aspects of Quantum State Estimation and Relative Entropy.  In V. P. Belavkin, O. Hirota, and R. L. Hudson, editors, Quantum Communications and Measurement, Springer US, pages 449–452, 1994.  DOI: 10.1007/978-1-4899-1391-3_44.
  * [45] J. Naudts, A. Verbeure, and R. Weder.  Linear Response Theory and the KMS Condition.  Communications in Mathematical Physics, 44:87–99, 1975.  DOI: 10.1007/BF01609060.
  * [46] M. G. A. Paris.  Quantum Estimation for Quantum Technology.  International Journal of Quantum Information, 7(1):125–137, 2009\.  DOI: 10.1142/S0219749909004839, arXiv:0804.2981 [quant-ph].
  * [47] D. Petz.  Geometry of canonical correlation on the state space of a quantum system.  Journal of Mathematical Physics, 35(2):780–795, 1994.  DOI: 10.1063/1.530611.
  * [48] D. Petz.  Monotone metrics on matrix spaces.  Linear Algebra and its Applications, 244:81–96, 1996.  DOI: 10.1016/0024-3795(94)00211-8.
  * [49] D. Petz and G. Toth.  The Bogoliubov Inner Product in Quantum Statistics.  Letters in Mathematical Physics, 27:205–216, 1993.  DOI: 10.1007/BF00739578.
  * [50] H. Shima.  The geometry of Hessian structures.  World Scientific Publishing Company Pte. Ltd., 2007.  DOI: 10.1142/9789812707536.
  * [51] J. Suzuki.  Information Geometrical Characterization of Quantum Statistical Models in Quantum Estimation Theory.  Entropy, 21(7):703, 2019.  DOI: 10.3390/e21070703.
  * [52] J. Suzuki, Y. Yang, and M. Hayashi.  Quantum state estimation with nuisance parameters.  Journal of Physics A: Mathematical and Theoretical, 53(45):453001, 2020.  DOI: 10.1088/1751-8121/ab8b78, arXiv:1911.02790 [quant-ph].
  * [53] M. Tsang, F. Albarelli, and A. Datta.  Quantum Semiparametric Estimation.  Physical Review X, 10(3):031023, 2020.  DOI: 10.1103/PhysRevX.10.031023.
  * [54] A. Uhlmann.  The Metric of Bures and the Geometric Phase.  In R. Gielerak, J. Lukierski, and Z. Popowicz, editors, Groups and Related Topics, pages 267–274. Springer, Dordrecht, 1992.  DOI: 10.1007/978-94-011-2801-823.
  * [55] H. Umegaki.  Conditional expectation in an operator algebra IV: Entropy and information.  Kodai Mathematical Seminar Reports, 14(2):59–85, 1962.  DOI: 10.2996/kmj/1138844604.
  * [56] N. N. Čencov.  Statistical Decision Rules and Optimal Inference.  American Mathematical Society, Providence, RI, 1982.  DOI: 10.1090/mmono/053.
  * [57] J. von Neumann.  Mathematical Foundations of Quantum Mechanics.  Princeton University Press, Princeton, NJ, 1955.
  * [58] D. Šafránek.  Simple expression for the quantum Fisher information matrix.  Physical Review A, 97:042322–06, 2018.  DOI: 10.1103/PhysRevA.97.042322, arXiv:1801.00945 [quant-ph].
  * [59] J. Zhang and G. Khan.  From Hessian to Weitzenböck: manifolds with torsion-carrying connections.  Information Geometry, 8, 2019.  DOI: 10.1007/s41884-019-00018-x.
  * [60] J. Zhang and G. Khan.  New Geometry of Parametric Statistical Models.  In F. Nielsen and F. Barbaresco, editors, Geometric Science of Information, pages 288–296. Springer International Publishing, 2019.  DOI: 10.1007/978-3-030-26980-7_30.
  * [61] J. Zhang and G. Khan.  Statistical Mirror Symmetry.  Differential Geometry and its Applications, 73:101678, 2020.  DOI: 10.1016/j.difgeo.2020.101678.