# Exponential arcs in manifolds of quantum states

Jan Naudts
Universiteit Antwerpen
Physics Department, Universiteitsplein 1, 2610 Antwerpen, Belgium

###### Abstract

The manifold under consideration consists of the faithful normal states on a
sigma-finite von Neumann algebra in standard form. Tangent planes and
approximate tangent planes are discussed. A relative entropy/divergence
function is assumed to be given. It is used to generalize the notion of an
exponential arc connecting one state to another. The generator of the
exponential arc is shown to be unique up to an additive constant. In the case
of Araki’s relative entropy every selfadjoint element of the von Neumann
algebra generates an exponential arc. The generators of composed exponential
arcs are shown to add up. The metric derived from Araki’s relative entropy is
shown to reproduce the Kubo-Mori metric. The latter is the metric used in
Linear Response Theory. The e- and m-connections describe a dual pair of
geometries. Any finite number of linearly independent generators determines a
submanifold of states connected to a given reference state. Such a submanifold
is a quantum generalization of a dually flat statistical manifold.

Keywords: exponential arcs, quantum statistical manifold, quantum divergence
function, Araki’s relative entropy, dually flat geometry, Tomita-Takesaki
theory, linear response theory, Kubo-Mori theory.

##  1 Introduction

Goal of the present paper is to show that the theory of quantum statistical
manifolds can be formulated without reference to density matrices. It is
tradition to describe the statistical state of a quantum model by a density
matrix. In many cases this suffices, in particular when the Hilbert space of
wavefunctions is finite-dimensional. However, even simple models such as the
quantum harmonic oscillator or the hydrogen atom require an infinite-
dimensional Hilbert space. This involves handling of unbounded operators which
cause considerable technical complications. These complications are avoided in
the present work.

A one-to-one correspondence between density matrices and quantum states is
usually accepted. The quantum states form the sample space of the statistical
description. An alternative description emerged in the past century. It
introduces the notion of a mathematical state on an algebra of observables
which can be realised as an algebra of bounded operators on a Hilbert space.
See for instance [13, 14, 30, 17, 8, ].

Equilibrium states of Quantum Statistical Mechanics are described by the
quantum analogue of the probability distribution of Gibbs, which is a density
matrix $\rho$ of the form

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{1}{Z}e^{-\beta H}$

with $H$ a Hermitian matrix, $\beta$ a parameter called the inverse
temperature and $Z$ a function of $\beta$ used to normalize the density matrix
$\rho$ so that its trace equals 1. Models described in this way can be said to
belong to a quantum exponential family. They posses an intriguing property
called the Kubo-Martin-Schwinger (KMS) condition [18]. The KMS condition
describes a symmetry property of the time evolution of quantum states. This
symmetry coincides with the symmetry between left and right multiplication of
operators which is studied in Tomita-Takesaki theory[33]. The book of
Bratteli-Robinson[8] can be used as a reference text for this theory.

The notion of a statistical manifold is studied in Information Geometry[9, 15,
1, 2, 7]. It is a manifold of probability distributions. The quantum analogue
is described in Chapter 7 of [2, ] as a manifold of $k$ by $k$ density
matrices. The book of Petz[26] reviews several aspects of quantum statistics,
including the basics of Quantum Information and Quantum Information Geometry.

The relative entropy of Umegaki[34] is the starting point to implement Amari’s
dually flat geometry on the quantum manifold. Note that a relative entropy is
called a divergence function in the mathematical literature. Araki[3, 4, 5]
generalized Umegaki’s relative entropy to the context of mathematical states
on an algebra of bounded operators on a Hilbert space. The use of Araki’s
relative entropy replacing that of Umegaki is the core of the present work.

Exponential arcs were introduced in [28, 27, ] and used in [31, ]. These arcs
can be considered as one-parameter exponential families embedded in the
manifold. The maximal exponential model centered at a given probability
distribution $p$ equals the set of all probability distributions connected to
$p$ by an open exponential arc. Exponential arcs were studied in the quantum
setting by the author[23]. Here the definition is generalized. The exponential
arcs are used to define quantum statistical manifolds as submanifolds of the
manifold of all quantum states.

The Radon-Nikodym theorem plays an important role in probability theory. For
each measure absolutely continuous w.r.t. the reference measure there exists
an essentially unique probability distribution function. The problem that
arises in the non-commutative context is the non-uniqueness of the definition
of the Radon-Nikodym derivative. This leads to different definitions of the
relative entropy and of the exponential arcs. First attempts to reformulate
the theory of the quantum statistical manifold in terms of states on a
$C^{*}$-algebra are found in [21, 22, ] and in [23, ]. These two approaches
differ in the choice of Radon-Nikodym derivative. In the present work the
definition of an exponential arc is generalized so that it depends explicitly
on the choice of relative entropy and in that way on the choice of Radon-
Nikodym derivative.

The alternative approach of [10, ] relies on Lie theory for the group of
bounded operators with bounded inverse. The state space is partitioned into
the disjoint union of the orbits of an action of the Lie group. Under mild
conditions it is shown that the orbits are Banach manifolds. The restriction
to bounded operators implies that the orbits do not connect quasi-equivalent
states when the Radon-Nikodym derivatives are unbounded operators.

##  2 KMS states

Equilibrium states of Quantum Statistical Mechanics satisfy the KMS condition.
In the GNS representation an equilibrium state becomes a faithful state on a
$\sigma$-finite von Neumann algebra of operators on a complex Hilbert space.
The state is defined by a normalized cyclic and separating vector in the
Hilbert space.

The state of a model of Statistical Physics can be described by a mathematical
state on a $C^{*}$-algebra $\mathfrak{A}$. It can be represented by a
normalized vector $\Omega$ (a wave function) in a Hilbert space $\mathscr{H}$.
This is known as the GNS (Gelfand Naimark Segal) representation theorem.
Observable quantities are represented by self-adjoint operators on
$\mathscr{H}$. The quantum expectation $\langle x\rangle$ of the operator $x$
is then given by

$\displaystyle\langle x\rangle$ $\displaystyle=$
$\displaystyle(x\Omega,\Omega),$ (1)

with in the r.h.s. the scalar product of the two vectors $x\Omega$ and
$\Omega$. Note that the mathematics convention is followed that the scalar
product (inner product) is linear in its first argument and conjugate-linear
the second argument. In Dirac’s bra-ket notation it reads

$\displaystyle\langle x\rangle$ $\displaystyle=$
$\displaystyle\langle\Omega|x\Omega\rangle.$

For convenience one works with a von Neumann algebra $\mathfrak{M}$ of bounded
operators on the Hilbert space $\mathscr{H}$. Observables of interest, when
unbounded, are represented by operators affiliated with $\mathfrak{M}$. The
state $\omega$ on the $C^{*}$-algebra extends to a vector state on
$\mathfrak{M}$ again denoted $\omega$. It is given by

$\displaystyle\omega(x)$ $\displaystyle=$
$\displaystyle(x\Omega,\Omega),\qquad x\in\mathfrak{M}.$

The vector $\Omega$ is cyclic for $\mathfrak{M}$, which means that the
subspace $\mathfrak{M}\Omega$ is dense in the Hilbert space $\mathscr{H}$. It
is also assumed in what follows that the state $\omega$ is faithful, i.e.
$\omega(x^{*}x)=0$ implies $x=0$. This implies that $\Omega$ is a separating
vector for $\mathfrak{M}$, i.e. $x\Omega=0$ implies $x=0$ for any $x$ in
$\mathfrak{M}$, and hence it is a cyclic vector for the commutant
$\mathfrak{M}^{\prime}$ of $\mathfrak{M}$, the algebra of all operators
commuting with all of $\mathfrak{M}$.

Equilibrium states of Statistical Mechanics are characterised by the KMS (Kubo
Martin Schwinger) condition [18]. Roughly spoken, this condition states that
the quantum time evolution of the model has an analytic extension into the
complex plane. This is made more precise in what follows.

The time evolution is described by a strongly continuous one-parameter group
$t\in{\mathbb{R}}\mapsto u_{t}$ of unitary operators which leave the algebra
$\mathfrak{M}$ unchanged, i.e. $x\in\mathfrak{M}$ implies that
$x_{t}=u_{t}^{*}xu_{t}$ belongs to $\mathfrak{M}$ for all $t$. The operators
$u_{t}$ are determined by a self-adjoint operator $H$

$\displaystyle u_{t}$ $\displaystyle=$ $\displaystyle e^{-itH}$

which is the generator of the time evolution in the GNS representation. The
time derivative of $x_{t}$ satisfies

$\displaystyle i\frac{{\rm d}\,}{{\rm d}t}x_{t}$ $\displaystyle=$
$\displaystyle[x_{t},H]_{-}.$

This equation has the same form as Heisenberg’s equation of motion.

The KMS condition requires that for any pair $x,y$ of operators in
$\mathfrak{M}$ there exists a complex function $F(w)$, defined and continuous
on the strip $-\beta\leq\mbox{ Im }w\leq 0$ and analytic inside with boundary
values

$\displaystyle F(t)=(x_{t}y\Omega,\Omega)\quad\mbox{ and }\quad
F(t-i\beta)=(yx_{t}\Omega,\Omega),\qquad t\in{\mathbb{R}}.$

In the Mathematics Literature the parameter $\beta$, which is the inverse
temperature of the model, is usually taken equal to 1 or -1.

An immediate consequence of the KMS condition being satisfied is that the
state $\omega$ is invariant. Indeed, take $y$ equal to the identity operator.
Then one has $F(t-i\beta)=F(t)$ for all $t$ in ${\mathbb{R}}$. If in addition
$x$ is self-adjoint then $F(t)$ is a real function. From Schwarz reflection
principle one then concludes that $F(w)$ is a constant function. This implies
$\omega(x_{t})=\omega(x)$ for all self-adjoint $x$ and hence for all $x$. The
GNS theorem then guarantees that the vector $\Omega$ can be taken to be
invariant, i.e. $u_{t}\Omega=\Omega$ for all $t$.

##  3 The modular operator

The quantum-mechanical time evolution coincides with the modular automorphism
group of Tomita-Takesaki theory.

The KMS condition when satisfied expresses a symmetry which is present in the
context of non-commuting operators. The symmetry is the inversion of the order
of multiplication of operators. In non-commutative groups the modular function
links left and right Haar measures. The analogue in Functional Analysis is
studied in the theory of the modular operator, also called Tomita-Takesaki
theory [33].

The operator $e^{-\beta H}$, with $H$ the generator of the quantum time
evolution is traditionally denoted $\Delta_{{}_{\Omega}}$. It is the modular
operator of Tomita-Takesaki theory. It is in general an unbounded operator
such that $\mathfrak{M}\Omega$ is in the domain of definition of the square
root $\Delta_{{}_{\Omega}}^{1/2}$ of $\Delta_{{}_{\Omega}}$. Hence, the
expression

$\displaystyle F(w)$ $\displaystyle=$
$\displaystyle(x\Delta_{{}_{\Omega}}^{iw/\beta}y\Omega,\Omega),\qquad
x,y\in\mathfrak{M},$ (2)

is well-defined for $0\geq\mbox{ Im }w\geq-\beta/2$. The other half of the
strip $0\geq\mbox{ Im }w\geq-\beta$ is covered by the Schwarz reflection
principle. Indeed, if $x$ and $y$ are selfadjoint then one can show with
Tomita-Takesaki theory that the map $t\mapsto F(t-i\beta/2)$ is a real
function. Hence, the principle can be applied to obtain
$F(w)=\overline{F(w-i\beta)}$.

The unitary time evolution operators $u_{t}$ can be written as

$\displaystyle u_{t}$ $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{it/\beta}.$

The time evolution of an operator $x$ in the Heisenberg picture is then given
by

$\displaystyle x_{t}=\tau_{t}^{\scriptscriptstyle\Omega}x$ $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{-it/\beta}\,x\,\Delta_{{}_{\Omega}}^{it/\beta}.$

The action $t\mapsto\tau_{t}^{\scriptscriptstyle\Omega}$ of the group
${\mathbb{R}},+$ is called the modular automorphism group.

The modular conjugation operator $J$ of Tomita-Takesaki theory represents the
symmetry which is at the basis of the theory. It is a conjugate-linear
operator satisfying $J=J^{*}$ and $J^{2}={\mathbb{I}}$. An operator $x$
belongs to the von Neumann algebra $\mathfrak{M}$ if and only if $JxJ$ belongs
to the commutant algebra $\mathfrak{M}^{\prime}$. The latter is the space of
operators commuting with all operators in $\mathfrak{M}$. The product
$J\Delta_{{}_{\Omega}}^{1/2}$ is denoted $S_{{}_{\Omega}}$ and has the
property that

$\displaystyle S_{{}_{\Omega}}x\Omega=x^{*}\Omega,\qquad x\in\mathfrak{M}.$

##  4 Dual cones

The natural positive cone $\mathscr{P}_{{}_{\Omega}}$ is needed in subsequent
sections. One reason for making use of it is that there exists a one-to-one
correspondence between normal states on $\mathfrak{M}$ and normalized vectors
in $\mathscr{P}_{{}_{\Omega}}$.

Section 4 of [4, ] introduces the cones $V^{\alpha}_{{}_{\Omega}}$,
$0\leq\alpha\leq 1/2$, of vectors in $\mathscr{H}$. The self-dual cone
$V^{1/4}_{{}_{\Omega}}$ is called the natural positive cone and is denoted
$\mathscr{P}_{{}_{\Omega}}$.

By definition is $V^{\alpha}_{{}_{\Omega}}$ the closure of the cone

$\displaystyle\\{\Delta_{{}_{\Omega}}^{\alpha}x\Omega:\,x\in\mathfrak{M},x\geq
0\\}.$

The cone $V^{1/2}_{{}_{\Omega}}$ is used in [23, ] to introduce exponential
arcs. It is equal to the closure of the set

$\displaystyle\\{y\Omega:\,y\in\mathfrak{M}^{\prime},y\geq 0\\}.$

To see this note that

$\displaystyle\Delta_{{}_{\Omega}}^{1/2}x\Omega$ $\displaystyle=$
$\displaystyle JS_{{}_{\Omega}}x\Omega$ (3) $\displaystyle=$ $\displaystyle
Jx^{*}\Omega$ $\displaystyle=$ $\displaystyle y\Omega$

with $y=Jx^{*}J$. The latter is an arbitrary element of the commutant
$\mathfrak{M}^{\prime}$.

The following characterisation of the natural positive cone
$\mathscr{P}_{{}_{\Omega}}$ is found in Section 2.5 of [8, ].

###### Proposition 1

The cone $\mathscr{P}_{{}_{\Omega}}=V^{1/4}_{{}_{\Omega}}$ equals the closure
of the set of vectors

$\displaystyle\\{xJx\Omega:\,x\in{\mathfrak{M}}\\}.$ (5)

This result can be understood as follows. Take $\Phi$ in $\mathscr{P}$ of the
form (5), i.e. $\Phi=xJx\Omega$ with $x$ in $\mathfrak{M}$. Let

$\displaystyle y$ $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{-1/4}x\Delta_{{}_{\Omega}}^{1/4}.$ (6)

This expression can be inverted to

$\displaystyle x$ $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{1/4}y\Delta_{{}_{\Omega}}^{-1/4}$

so that

$\displaystyle\Phi=xJx\Omega$ $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{1/4}y\Delta_{{}_{\Omega}}^{-1/4}\,J\,\Delta_{{}_{\Omega}}^{1/4}y\Delta_{{}_{\Omega}}^{-1/4}\Omega$
(7) $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{1/4}yJ\,\Delta_{{}_{\Omega}}^{1/2}y\Omega$
(8) $\displaystyle=$
$\displaystyle\Delta_{{}_{\Omega}}^{1/4}yS_{{}_{\Omega}}y\Omega$
$\displaystyle=$ $\displaystyle\Delta_{{}_{\Omega}}^{1/4}yy^{*}\Omega.$

Assume now that one could prove that the operator $y$ defined by (6) belongs
to $\mathfrak{M}$ then the above calculation would show that $\Phi$ is of the
form $\Phi=\Delta_{{}_{\Omega}}^{1/4}a\Omega$ with $a=yy^{*}$ a positive
element of $\mathfrak{M}$. The actual proof of the proposition uses that
$\tau^{\Omega}_{t}x=\Delta_{{}_{\Omega}}^{-it}x\Delta_{{}_{\Omega}}^{it}$
belongs to $\mathfrak{M}$.

The cone $\mathscr{P}_{{}_{\Omega}}$ is independent [4] of the choice of the
cyclic and separating vector $\Omega$ in ${\mathscr{P}_{{}_{\Omega}}}$ and the
isometry $J$ is the same for all these choices. For this reason it is said to
be universal.

From (5) it is easy to see that each vector in $\mathscr{P}_{{}_{\Omega}}$ is
an eigenvector with eigenvalue 1 of the modular conjugation operator $J$.
Indeed, one has

$\displaystyle xJx\Omega=x(JxJ)\Omega=(JxJ)x\Omega=J(xJx\Omega).$

Here, use is made of $J\Omega=\Omega$ and the fact that the operators $x$ and
$JxJ$ commute with each other.

##  5 A manifold of quantum states

A manifold ${\mathbb{M}}$ of vector states on the von Neumann algebra
$\mathcal{M}$ is defined. Tangent vector fields are Fréchet-derivatives of
paths in ${\mathbb{M}}$.

Introduce the notation $\omega_{{}_{\Phi}}$ for the vector state defined by
the normalized vector $\Phi$ in $\mathscr{H}$. It is given by

$\displaystyle\omega_{{}_{\Phi}}(x)$ $\displaystyle=$
$\displaystyle(x\Phi,\Phi),\qquad x\in\mathfrak{M}.$

A manifold ${\mathbb{M}}$ of states on the von Neumann algebra $\mathfrak{M}$
is defined by

$\displaystyle{\mathbb{M}}$ $\displaystyle=$
$\displaystyle\\{\omega_{\Phi}:\,\Phi\in{\mathscr{P}}_{{}_{\Omega}},\mbox{
normalized, cyclic and separating for }\mathfrak{M}\\}.$

The equilibrium state $\omega=\omega_{{}_{\Omega}}$ is taken as a reference
point in ${\mathbb{M}}$. The subset ${\mathscr{P}}_{{}_{\Omega}}$ of
$\mathscr{H}$ is the natural positive cone introduced in the previous section.

The topology on the manifold ${\mathbb{M}}$ is that of the operator norm. One
has

$\displaystyle||\omega_{{}_{\Phi}}-\omega_{{}_{\Psi}}||$ $\displaystyle=$
$\displaystyle\sup\\{|\omega_{{}_{\Phi}}(x)-\omega_{{}_{\Psi}}(x)|:\,x\in\mathfrak{M},||x||\leq
1\\}.$

Several topologies can be defined on the algebra $\mathfrak{M}$. Particularly
relevant is the $\sigma$-weak topology. For what follows it is important to
know that in the present context a state $\omega$ on $\mathfrak{M}$ is said to
be normal if and only if it is $\sigma$-weakly continuous and if and only if
it is a vector state. See for instance Theorems 2.4.21 and 2.5.31 of [8, ].

Any tangent vector is a $\sigma$-weakly continuous linear functional on the
von Neumann algebra $\mathfrak{M}$. Let $t\mapsto\gamma_{t}$ be a Fréchet-
differentiable map defined on an open interval of ${\mathbb{R}}$ with values
in the manifold ${\mathbb{M}}$. The derivative

$\displaystyle\dot{\gamma}_{t}=\frac{{\rm d}\,}{{\rm d}t}\gamma_{t}$

is required to exist as a Fréchet derivative, i.e. it satisfies

$\displaystyle||\gamma_{t}-\gamma_{s}-(t-s)\dot{\gamma}_{t}||$
$\displaystyle=$ $\displaystyle\mbox{o}(t-s).$

From the normalization $\gamma_{t}(1)=1$ for all $t$ in the domain of the map
one obtains $\dot{\gamma}_{t}(1)=0$. From
$\gamma_{t}(x^{*})=\overline{\gamma_{t}(x)}$ one obtains
$\dot{\gamma}_{t}(x^{*})=\overline{\dot{\gamma}_{t}(x)}$. Hence, the linear
functional $\dot{\gamma}_{t}$ is Hermitian.

There are several ways to define the tangent space $T_{\omega}{\mathbb{M}}$ at
the point $\omega$ in ${\mathbb{M}}$. The easiest way is to consider the
manifold ${\mathbb{M}}$ as embedded in the Banach space $\mathfrak{M}_{*}$ of
$\sigma$-continuous Hermitian linear functionals on $\mathfrak{M}$ (see
Proposition 2.4.18 of [8, ]). Then any Hermitian $\sigma$-weakly continuous
linear function $\chi$ satisfying $\chi(1)=0$ defines a tangent curve
$t\mapsto\omega+t\chi$ with Fréchet derivative $\chi$. This justifies the
definition that $T_{\omega}{\mathbb{M}}$ consists of all $\sigma$-weakly
continuous Hermitian linear functions $\chi$ on $\mathfrak{M}$ satisfying
$\chi(1)=0$.

##  6 Approximate tangents

Approximate tangent vectors can be defined in an intrinsic manner.

An alternative definition of the tangent space starts from the following
observation.

###### Proposition 2

The set ${\mathscr{T}}_{\omega}$ defined by

$\displaystyle{\mathscr{T}}_{\omega}$ $\displaystyle=$
$\displaystyle\\{\lambda(\phi-\psi):\,\phi,\psi\in{\mathbb{M}},\lambda\in{\mathbb{R}}\mbox{
and }\omega=\frac{1}{2}(\phi+\psi)\\}.$

is a linear subspace of the tangent space $T_{\omega}{\mathbb{M}}$.

Proof

Let $\phi$ and $\psi$ be two states in ${\mathbb{M}}$ such that
$\omega=\frac{1}{2}(\phi+\psi)$. Construct a Fréchet-differentiable path
$\gamma$ by

$\displaystyle\gamma_{t}$ $\displaystyle=$
$\displaystyle(1-t)\psi+t\phi,\qquad t\in(0,1).$

The state $\gamma_{t}$ belongs to the manifold ${\mathbb{M}}$ because the
latter is a convex set. In particular one has $\omega=\gamma_{1/2}$ and
$\phi-\psi=\dot{\gamma}_{1/2}$ is a tangent vector. This shows that
$\phi-\psi$ and hence also $\lambda(\phi-\psi)$ belongs to
$T_{\omega}{\mathbb{M}}$. One concludes that ${\mathscr{T}}_{\omega}\subset
T_{\omega}{\mathbb{M}}$.

Assume now that $\lambda(\phi-\psi)$ and
$\lambda^{\prime}(\phi^{\prime}-\psi^{\prime})$ both belong to
${\mathscr{T}}_{\omega}$. We have to show that

$\displaystyle\lambda(\phi-\psi)+\lambda^{\prime}(\phi^{\prime}-\psi^{\prime})$

belongs to ${\mathscr{T}}_{\omega}$. If $\lambda=0$ or $\lambda^{\prime}=0$
then the claim is clearly satisfied. Without restriction assume $\lambda=1$.

If $\lambda^{\prime}>0$ then choose

$\displaystyle\phi^{\prime\prime}=\frac{1}{1+\lambda^{\prime}}(\phi+\lambda^{\prime}\phi^{\prime})\quad\mbox{
and
}\quad\psi^{\prime\prime}=\frac{1}{1+\lambda^{\prime}}(\psi+\lambda^{\prime}\psi^{\prime}).$

Both $\phi^{\prime\prime}$ and $\psi^{\prime\prime}$ belong to ${\mathbb{M}}$
because the latter is a convex set. One verifies that
$\phi^{\prime\prime}+\psi^{\prime\prime}=2\omega$ and

$\displaystyle(1+\lambda^{\prime})(\phi^{\prime\prime}-\psi^{\prime\prime})$
$\displaystyle=$
$\displaystyle\phi-\psi+\lambda^{\prime}(\phi^{\prime}-\psi^{\prime}).$

This shows that the latter sum belongs to ${\mathscr{T}}_{\omega}$.

In the case that $\lambda^{\prime}<0$ one chooses

$\displaystyle\phi^{\prime\prime}=\frac{1}{1-\lambda^{\prime}}(\phi-\lambda^{\prime}\psi^{\prime})\quad\mbox{
and
}\quad\psi^{\prime\prime}=\frac{1}{1-\lambda^{\prime}}(\psi-\lambda^{\prime}\phi^{\prime})$

to reach the same conclusion. This finishes the proof that
${\mathscr{T}}_{\omega}$ is a linear subspace of $T_{\omega}{\mathbb{M}}$.

$\square$

Introduce the notations

$\displaystyle{\mathscr{R}}_{\omega,\epsilon}$ $\displaystyle=$
$\displaystyle\bigcup{\mathscr{T}}_{\phi}\mbox{ with
}\phi\in{\mathbb{M}}\mbox{ such that }||\phi-\omega||<\epsilon\mbox{.}$

and

$\displaystyle{\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$ $\displaystyle=$
$\displaystyle\bigcap_{\epsilon>0}\overline{{\mathscr{R}}_{\omega,\epsilon}}.$

The construction of ${\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$ is
analoguous to the construction of the approximate tangent space in Chapter 3
of [32, ]. Clearly is
${\mathscr{T}}_{\omega}\subset{\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$.
Further properties are derived below.

###### Proposition 3

If $\gamma$ is a Fréchet-differentiable path in ${\mathbb{M}}$ then
$\dot{\gamma}_{t}$ belongs to ${\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$
with $\omega=\gamma_{t}$.

Proof

Let $\gamma$ be a Fréchet-differentiable path in ${\mathbb{M}}$. Without
restriction of generality assume that $\gamma_{0}=\omega$. For any
$\epsilon>0$ and $\delta>0$ there exists $t\not=0$ such that

$\displaystyle||(\gamma_{t}+\gamma_{-t})/{2}-\gamma_{0}||$ $\displaystyle<$
$\displaystyle\epsilon.$

and

$\displaystyle||(\gamma_{t}-\gamma_{-t})/{2t}-\dot{\gamma}_{0}||$
$\displaystyle<$ $\displaystyle\delta.$ (10)

Then $\phi$ defined by

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\gamma_{t}+\gamma_{-t})$

satisfies $||\phi-\omega||<\epsilon$ and $(\gamma_{t}-\gamma_{-t})/{2t}$
belongs to ${\mathscr{R}}_{\omega,\epsilon}$. Hence, (10) shows that the
tangent vector $\dot{\gamma}_{0}$ belongs to the closure of
${\mathscr{R}}_{\omega,\epsilon}$. Because $\epsilon>0$ is arbitrary it
belongs also to the intersection, which is
${\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$.

$\square$

###### Lemma 1

${\mathscr{R}}_{\omega,\epsilon}$ is a linear subspace of
$T_{\omega}{\mathbb{M}}$.

Proof

Take $\chi$ and $\xi$ in ${\mathscr{R}}_{\omega,\epsilon}$. There exist $\phi$
and $\psi$ in ${\mathbb{M}}$ such that $\chi\in{\mathscr{T}}_{\phi}$ and
$\xi\in{\mathscr{T}}_{\psi}$ with $||\phi-\omega||<\epsilon$ and
$||\psi-\omega||<\epsilon$. Therefore there exist real $\lambda,\mu$ and
states $\phi_{1},\Phi_{2},\Psi_{1},\Psi_{2}$ in ${\mathbb{M}}$ such that

$\displaystyle\chi=\lambda(\phi_{1}-\phi_{2})\quad\mbox{ and
}\quad\phi=\frac{1}{2}(\phi_{1}+\phi_{2})$

and

$\displaystyle\xi=\mu(\psi_{1}-\psi_{2})\quad\mbox{ and
}\quad\psi=\frac{1}{2}(\psi_{1}+\psi_{2}).$

If $\lambda=0$ or $\mu=0$ then $\chi+\xi$ belongs to
${\mathscr{R}}_{\omega,\epsilon}$ without further argument. Assume therefore
that $\lambda\not=0$ and $\mu\not=0$. If $\lambda\mu>0$ then $\chi+\xi$
belongs to ${\mathscr{T}}_{\pi}$ with $\pi=(1-\alpha)\phi+\alpha\psi$ and
$\alpha$ given by

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{\mu}{\lambda+\mu}.$

Indeed, let

$\displaystyle\pi_{1}$ $\displaystyle=$
$\displaystyle(1-\alpha)\phi_{1}+\alpha\psi_{1},$ (11) $\displaystyle\pi_{2}$
$\displaystyle=$ $\displaystyle(1-\alpha)\phi_{2}+\alpha\psi_{2}.$

Then both $\pi_{1}$ and $\pi_{2}$ belong to ${\mathbb{M}}$ and satisfy

$\displaystyle\pi_{1}+\pi_{2}$ $\displaystyle=$ $\displaystyle
2(1-\alpha)\phi+2\alpha\psi$ $\displaystyle=$ $\displaystyle 2\pi$

and

$\displaystyle(\lambda+\mu)(\pi_{1}-\pi_{2})$ $\displaystyle=$
$\displaystyle(\lambda+\mu)\,[\frac{1-\alpha}{\lambda}\chi+\frac{\alpha}{\mu}\xi]$
$\displaystyle=$ $\displaystyle\chi+\xi.$

In addition is

$\displaystyle||\pi-\omega||$ $\displaystyle=$
$\displaystyle||(1-\alpha)(\phi-\omega)+\alpha(\psi-\omega)||$ (14)
$\displaystyle\leq$
$\displaystyle||(1-\alpha)(\phi-\omega)||+||\alpha(\psi-\omega)||$
$\displaystyle<$ $\displaystyle\epsilon.$

One concludes that in this case $\chi+\xi$ belongs to
${\mathscr{R}}_{\omega,\epsilon}$.

The case that $\lambda\mu<0$ is similar. That
$\chi\in{\mathscr{R}}_{\omega,\epsilon}$ implies
$\lambda\chi\in{\mathscr{R}}_{\omega,\epsilon}$ is straightforward. One
concludes that ${\mathscr{R}}_{\omega,\epsilon}$ is a linear space. It clearly
is a subspace of $T_{\omega}{\mathbb{M}}$.

$\square$

###### Proposition 4

${\mathscr{T}}_{\omega}^{\mbox{\tiny approx}}$ is a closed linear subspace of
$T_{\omega}{\mathbb{M}}$.

Proof

The lemma shows that ${\mathscr{R}}_{\omega,\epsilon}$ is a linear subspace of
$T_{\omega}{\mathbb{M}}$, which is a space closed in norm. Hence, also the
norm closure of ${\mathscr{R}}_{\omega,\epsilon}$ is a subset of this space
and therefore also of $T_{\omega}{\mathbb{M}}$.

$\square$

##  7 Majorized states

The submanifold of states majorized by a multiple of the reference state
$\omega$ is considered.

###### Definition 1

A state $\phi$ on $\mathfrak{M}$ is said to be majorized by a multiple of the
state $\omega$ if there exists a positive constant $\lambda$ such that

$\displaystyle\phi(x^{*}x)$ $\displaystyle\leq$
$\displaystyle\lambda\omega(x^{*}x)\quad\mbox{ for all }x\in\mathfrak{M}.$

Take $a^{\prime}\not=0$ in the commutant algebra $\mathfrak{M}^{\prime}$ and
let

$\displaystyle\Phi=\frac{1}{||a^{\prime}\Omega||}a^{\prime}\Omega.$

Then the state $\omega_{{}_{\Phi}}$ is majorized by a multiple of the state
$\omega$. Indeed, one has for any positive $x$ in $\mathfrak{M}$

$\displaystyle\omega_{{}_{\Phi}}(x)$ $\displaystyle=$
$\displaystyle\frac{(xa^{\prime}\Omega,a^{\prime}\Omega)}{(a^{\prime}\Omega,a^{\prime}\Omega)}$
(16) $\displaystyle=$
$\displaystyle\frac{(a^{\prime*}a^{\prime}x^{1/2}\Omega,x^{1/2}\Omega)}{(a^{\prime}\Omega,a^{\prime}\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{||a^{\prime*}a^{\prime}||}{(a^{\prime}\Omega,a^{\prime}\Omega)}\,\omega(x).$

It is well-known that all states majorized by a multiple of the state $\omega$
are obtained in this way. This is the content of the following proposition.

###### Proposition 5

If the vector state $\omega_{{}_{\Phi}}$ is majorized by a multiple of the
state $\omega$ then there exists a unique element $a^{\prime}$ of the
commutant $\mathfrak{M}^{\prime}$ such that $\Phi=a^{\prime}\Omega$.

Proof

An operator $a^{\prime}$ is densely defined by

$\displaystyle a^{\prime}x\Omega$ $\displaystyle=$ $\displaystyle x\Phi,\qquad
x\in\mathfrak{M}.$

It satisfies $a^{\prime}\Omega=\Phi$. It is well-defined because $x\Omega=0$
implies

$\displaystyle||x\Phi||^{2}=\omega_{{}_{\Phi}}(x^{*}x)\leq\mbox{constant
}\omega(x^{*}x)=\mbox{constant }||x\Omega||^{2}=0$

so that $x\Phi=0$.

The operator $a^{\prime}$ is bounded because

$\displaystyle||a^{\prime}x\Omega||^{2}=\phi(x^{*}x)\leq\mbox{constant
}\omega(x^{*}x)=\mbox{constant }||x\Omega||^{2}.$

The operator $a^{\prime}$ commutes with any $x$ in $\mathfrak{M}$ because

$\displaystyle
a^{\prime}x(y\Omega)=xy\Phi=x(a^{\prime}y\Omega)=xa^{\prime}(y\Omega)$

and $\Omega$ is cyclic for $\mathfrak{M}$.

The operator $a^{\prime}$ is unique. Indeed, assume $b^{\prime}$ in
$\mathfrak{M}^{\prime}$ satisfies $\Phi=b^{\prime}\Omega$. Then one has for
all $x$ in $\mathfrak{M}$

$\displaystyle
0=x(a^{\prime}-b^{\prime})\Omega=(a^{\prime}-b^{\prime})x\Omega.$

Hence, $a^{\prime}-b^{\prime}$ vanishes on $\mathfrak{M}\Omega$ which is dense
in the Hilbert space because $\Omega$ is cyclic for $\mathfrak{M}$. Because
$a^{\prime}-b^{\prime}$ is a bounded and hence continuous operator it vanishes
everywhere so that $a^{\prime}=b^{\prime}$.

$\square$

Item (8) of Theorem 3 of [4, ] implies the following.

###### Proposition 6

If a vector state $\omega_{{}_{\Phi}}$, defined by a vector $\Phi$ in the
natural positive cone $\mathscr{P}_{{}_{\Omega}}$, is dominated by a multiple
of the state $\omega$ then there exists a unique element $a$ in the algebra
$\mathfrak{M}$ such that $\Phi=a\Omega$ and

$\displaystyle\omega_{{}_{\Phi}}(x)=\omega(a^{*}xa),\qquad x\in\mathfrak{M}.$

Proof

Proposition 5 shows that $a^{\prime}$ in the commutant $\mathfrak{M}^{\prime}$
exists such that $\Phi=a^{\prime}\Omega$. Because $\Phi$ and $\Omega$ both
belong to $\mathscr{P}_{{}_{\Omega}}$ one has $\Phi=J\Phi=Ja^{\prime}J\Omega$.

Let $a=Ja^{\prime}J$. From $J\mathfrak{M}^{\prime}J=\mathfrak{M}$ it follows
that $a$ belongs to $\mathfrak{M}$. This shows the existence.

The element $a$ is unique because the correspondence between vector states on
$\mathfrak{M}$ and vectors in $\mathscr{P}_{{}_{\Omega}}$ is one-to-one and
$\Omega$ is a separating vector for $\mathfrak{M}$.

$\square$

If $\mathfrak{M}$ is a commutative algebra then $a^{*}a$ is the Radon-Nikodym
derivative of the state $\omega_{{}_{\Phi}}$ w.r.t. the reference state
$\omega$.

The submanifold of states of ${\mathbb{M}}$ majorized by a multiple of the
state $\omega$ is dense in ${\mathbb{M}}$ in the sense that for any state
$\phi$ in ${\mathbb{M}}$ there exists a sequence $(a_{n})_{n}$ of elements of
$\mathfrak{M}$ with the property that $a_{n}\Omega$ is a Cauchy sequence and

$\displaystyle\phi(x)$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}\omega(a_{n}^{*}xa_{n}),\qquad
x\in\mathfrak{M}.$

See Propositions 1.5 and 2.5 of [25, ].

###### Proposition 7

A tangent vector $\chi$ belongs to the subspace ${\mathscr{T}}_{\omega}$ of
the tangent space $T_{\omega}{\mathbb{M}}$ if and only if it is proportional
to the difference of two states $\phi$ and $\psi$ in ${\mathbb{M}}$ both
majorized by a multiple of the state $\omega$.

Proof

If $\chi$ belongs to ${\mathscr{T}}_{\omega}$ then by definition there exists
states $\phi$ and $\psi$ in ${\mathbb{M}}$ such that $\chi=\lambda(\phi-\psi)$
and $\phi+\psi=2\omega$. The latter implies that both $\phi$ and $\psi$ are
majorized by $2\omega$.

Conversely, assume that $\phi$ and $\psi$ in ${\mathbb{M}}$ both majorized by
a multiple of the state $\omega$ and let $\chi=\lambda(\phi-\psi)$. This
implies the existence of $\mu\geq 1$ and $\nu\geq 1$ such that
$\phi\leq\mu\omega$ and $\psi\leq\nu\omega$.

Without restriction assume that $\lambda>0$.

Introduce

$\displaystyle\phi^{\prime}=\omega+\rho\chi\quad\mbox{ and
}\quad\psi^{\prime}=\omega-\rho\chi$

with $\rho$ still to be chosen. By construction is
$\phi^{\prime}+\psi^{\prime}=2\omega$ and
$\phi^{\prime}-\psi^{\prime}=2\rho\chi$. Hence, if $\phi^{\prime}$ and
$\psi^{\prime}$ are states in ${\mathbb{M}}$ and $\rho\not=0$ then one can
conclude that $\chi$ belongs to ${\mathscr{T}}_{\omega}$.

From

$\displaystyle\chi(x^{*}x)\leq\lambda\phi(x^{*}x)\leq\lambda\mu\omega(x^{*}x)$

and

$\displaystyle\chi(x^{*}x)\geq-\lambda\psi(x^{*}x)\geq-\lambda\nu\omega(x^{*}x)$

one obtains

$\displaystyle\phi^{\prime}(x^{*}x)$ $\displaystyle\geq$
$\displaystyle[1-\rho\lambda\nu]\,\omega(x^{*}x),$ (18)
$\displaystyle\psi^{\prime}(x^{*}x)$ $\displaystyle\geq$
$\displaystyle[1-\rho\lambda\mu]\,\omega(x^{*}x).$

Choose now $\rho$ equal to the inverse of the maximum of $\lambda\mu$ and
$\lambda\nu$ to prove the positivity of the functionals $\phi^{\prime}$ and
$\psi^{\prime}$. Normalization $\phi^{\prime}(1)=\psi^{\prime}(1)=1$ follows
from $\chi(1)=0$. The functions are $\sigma$-weakly continuous as well. Hence
they are states in ${\mathbb{M}}$. This ends the proof that $\chi$ belongs to
${\mathscr{T}}_{\omega}$.

$\square$

##  8 Exponential arcs

The author introduces[23] the notion of an exponential arc in Hilbert space,
inspired by the notion of exponential arcs in probability space as introduced
by Pistone et al.[28, 27]. Here, a definition is given which depends on the
choice of a relative entropy.

In the present context a divergence function $D(\phi||\psi)$ is a real
function of two states $\phi$ and $\psi$ in the manifold ${\mathbb{M}}$. It
cannot be negative and it vanishes if and only if the two arguments are equal.
A value of $+\infty$ is allowed. An energy function is an affine function
$\mathfrak{h}$ defined on a convex subset of the set of normal states on the
algebra $\mathfrak{M}$.

The following definition of an exponential arc in the manifold ${\mathbb{M}}$
assumes that a divergence function $D(\phi||\psi)$ is given.

###### Definition 2

An exponential arc $\gamma$ is a path in the manifold

$\displaystyle t\in[0,1]\mapsto\gamma_{t}\in{\mathbb{M}}$

for which there exists an energy function $\mathfrak{h}$ such that

  * •

$\gamma_{t}$ is in the domain of $\mathfrak{h}$;

  * •

The divergence $D(\gamma_{s}||\gamma_{t})$ between any two points of the arc
is finite;

  * •

For any state $\psi$ in the domain of $\mathfrak{h}$ one has

$\displaystyle D(\psi||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\psi||\gamma_{0})+D(\gamma_{0}||\gamma_{t})+t\,(\mathfrak{h}(\gamma_{0})-\mathfrak{h}(\psi)),\qquad
0\leq t\leq 1.$ (19)

The energy function $\mathfrak{h}$ is the generator of the exponential arc.
The arc is said to connect the state $\gamma_{1}$ to the state $\gamma_{0}$.

A subclass of energy functions is formed by functions $\mathfrak{h}$ for which
there exists a self-adjoint operator $h$ in the von Neumann algebra
$\mathfrak{M}$ so that

$\displaystyle\mathfrak{h}(\psi)$ $\displaystyle=$
$\displaystyle\psi(h),\qquad\psi\in{\mathbb{M}}.$ (21)

In such a case $h$ is called the generator as well. The exponential arcs
defined in [23, ] agree with the above definition with a generator defined by
an unbounded operator affiliated with the commutant algebra
$\mathfrak{M}^{\prime}$.

###### Proposition 8

Expression (LABEL:arcs:arcdef) implies

$\displaystyle D(\gamma_{s}||\gamma_{0})+D(\gamma_{0}||\gamma_{s})$
$\displaystyle=$ $\displaystyle
s\,(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\gamma_{0})).$ (22)

and

$\displaystyle D(\psi||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\psi||\gamma_{s})+D(\gamma_{s}||\gamma_{t})+(t-s)\,(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\psi)).$
(23)

Note that with $s=0$ expression (23) reduces to (LABEL:arcs:arcdef).

Proof

Take $\psi=\gamma_{s}$ in (LABEL:arcs:arcdef) to find

$\displaystyle D(\gamma_{s}||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\gamma_{s}||\gamma_{0})+D(\gamma_{0}||\gamma_{t})+t\,(\mathfrak{h}(\gamma_{0})-\mathfrak{h}(\gamma_{s})),\qquad
0\leq s,t\leq 1.$ (24)

In particular, with $s=t$ this implies (22).

To prove (23) use (LABEL:arcs:temp) to write the r.h.s. as

r.h.s. $\displaystyle=$ $\displaystyle
D(\psi||\gamma_{s})+D(\gamma_{s}||\gamma_{0})+D(\gamma_{0}||\gamma_{t})$
$\displaystyle\quad+t\,(\mathfrak{h}(\gamma_{0})-\mathfrak{h}(\psi))-s\,(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\psi)).$

Next, eliminate $D(\gamma_{0}||\gamma_{t})$ and $D(\psi||\gamma_{s})$ with the
help of (LABEL:arcs:arcdef). This gives

r.h.s. $\displaystyle=$ $\displaystyle
D(\psi||\gamma_{s})+D(\gamma_{s}||\gamma_{0})+D(\psi||\gamma_{t})-D(\psi||\gamma_{0})-s\,(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\psi))$
(27) $\displaystyle=$ $\displaystyle
D(\gamma_{s}||\gamma_{0})+D(\psi||\gamma_{t})+D(\gamma_{0}||\gamma_{s})+s\,(\mathfrak{h}(\gamma_{0})-\mathfrak{h}(\gamma_{s}))$
$\displaystyle=$ $\displaystyle D(\psi||\gamma_{t}).$

To obtain the last line use (22).

$\square$

###### Corollary 1

If $t\mapsto\gamma_{t}$ is an exponential arc with generator $\mathfrak{h}$
that connects $\gamma_{1}$ to $\gamma_{0}$ then for any $s,t$ in $[0,1]$ the
map $\epsilon\mapsto\gamma_{(1-\epsilon)s+\epsilon t}$ is an exponential arc
with generator $(t-s)\mathfrak{h}$ that connects $\gamma_{t}$ to $\gamma_{s}$.

###### Corollary 2

If $t\mapsto\gamma_{t}$ is an exponential arc with generator $\mathfrak{h}$
that connects $\gamma_{1}$ to $\gamma_{0}$ then $t\mapsto\gamma_{1-t}$ is an
exponential arc with generator $-\mathfrak{h}$, connecting the state
$\gamma_{0}$ to the state $\gamma_{1}$.

The following two propositions deal with uniqueness of an exponential arc and
of its generator.

###### Proposition 9

Let $\omega$ and $\phi$ be two states in ${\mathbb{M}}$. Fix an energy
function $\mathfrak{h}$. There is atmost one exponential arc
$t\mapsto\gamma_{t}$ with generator $\mathfrak{h}$ that connects $\phi$ to
$\omega$.

Proof

Assume both $t\mapsto\gamma_{t}$ and $t\mapsto\delta_{t}$ are exponential arcs
connecting the state $\phi$ to the state $\omega$. Subtract
(LABEL:arcs:arcdef) from the same expression with $\gamma_{t}$ replaced by
$\delta_{t}$ and take $s=0$. This gives

$\displaystyle D(\psi||\delta_{t})-D(\psi||\gamma_{t})$ $\displaystyle=$
$\displaystyle D(\omega||\delta_{t})-D(\omega||\gamma_{t}).$ (29)

Take $\psi$ equal to $\delta_{t}$. Then one obtains

$\displaystyle
0\geq-D(\delta_{t}||\gamma_{t})=D(\omega||\delta_{t})-D(\omega||\gamma_{t}).$

On the other hand with $\psi=\gamma_{t}$ one obtains

$\displaystyle 0\leq
D(\gamma_{t}||\delta_{t})=D(\omega||\delta_{t})-D(\omega||\gamma_{t}).$

The two expressions together yield

$\displaystyle D(\omega||\delta_{t})-D(\omega||\gamma_{t})$ $\displaystyle=$
$\displaystyle 0.$

This implies $D(\gamma_{t}||\delta_{t})=0$. By the basic property of a
divergence one concludes that $\gamma_{t}=\delta_{t}$.

$\square$

###### Proposition 10

If the exponential arc $t\mapsto\gamma_{t}$ has two generators $\mathfrak{h}$
and $\mathfrak{k}$ then these generators differ by a constant on their common
domain of definition.

Proof

It follows from (LABEL:arcs:arcdef) that

$\displaystyle\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\psi)$ $\displaystyle=$
$\displaystyle\mathfrak{k}(\gamma_{s})-\mathfrak{k}(\psi),\qquad s\in[0,1]$
(30)

for all states $\psi$ in the intersection of the domains of $\mathfrak{h}$ and
$\mathfrak{k}$. This implies that a constant $c$ exists so that

$\displaystyle\mathfrak{k}(\psi)$ $\displaystyle=$
$\displaystyle\mathfrak{h}(\psi)+c$

for all $\psi$ in the common domain.

$\square$

The requirement (LABEL:arcs:arcdef) is a stability condition. The generator
$\mathfrak{h}$ is a perturbation which shifts the state $\gamma_{0}$ to the
state $\psi$. This interpretation will become clear further on. The effect on
the relative entropy of the shift along the arc $t\mapsto\gamma_{t}$ is
linear. In the standard case the relative entropy is based on the logarithmic
function. This justifies to call the path $t\mapsto\gamma_{t}$ an exponential
arc.

Note that the Pythagorean relation [11, 12]

$\displaystyle D(\psi||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\psi||\gamma_{s})+D(\gamma_{s}||\gamma_{t})$

is satisfied for all $\psi$ with the same energy as the state $\gamma_{s}$,
i.e. with

$\displaystyle\mathfrak{h}(\psi)=\mathfrak{h}(\gamma_{s}).$

If the divergence function is interpreted as the square of a pseudo-distance
then the above relation states that for an arbitrary state $\psi$ the point
$\gamma_{s}$ of the arc which has the same energy is the point with minimal
distance.

##  9 The scalar potential

The exponential arc has a dual structure similar to that found in Information
Geometry [1, 2].

Given an exponential arc $t\mapsto\gamma_{t}$ with generator $\mathfrak{h}$
introduce the potential $\Phi_{\gamma}$ defined by

$\displaystyle\Phi_{\gamma}(t)$ $\displaystyle=$ $\displaystyle
D(\gamma_{0}||\gamma_{t})+t\mathfrak{h}(\gamma_{0}).$

Its Legendre transform is given by

$\displaystyle\Phi^{*}_{\gamma}(\alpha)$ $\displaystyle=$
$\displaystyle\sup\\{\alpha t-\Phi_{\gamma}(t):\,0\leq t\leq 1\\}.$

###### Proposition 11

For any exponential arc $t\mapsto\gamma_{t}$ with generator $\mathfrak{h}$ one
has

  * (a)

The function $t\mapsto\mathfrak{h}(\gamma_{t})$ is strictly increasing;

  * (b)

$\Phi_{\gamma}(t)=\Phi_{\gamma}(s)+D(\gamma_{s}||\gamma_{t})+(t-s)\mathfrak{h}(\gamma_{s})$;

  * (c)

The line $t\mapsto\Phi_{\gamma}(s)+(t-s)\mathfrak{h}(\gamma_{s})$ is tangent
to the potential $\Phi_{\gamma}$ at the point $t=s$; this implies that the
potential $\Phi_{\gamma}(s)$ is a strictly convex function, continuous on the
open interval $(0,1)$;

  * (d)

The following identity holds:

$\displaystyle\Phi_{\gamma}(s)+\Phi^{*}_{\gamma}(\mathfrak{h}(\gamma_{s}))=s\mathfrak{h}(\gamma_{s}),\qquad
s\in[0,1].$

Proof

(a)

Take $\psi=\gamma_{t}$ in (LABEL:arcs:arcdef). This gives

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
D(\gamma_{t}||\gamma_{s})+D(\gamma_{s}||\gamma_{t})+(t-s)\left(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\gamma_{t})\right).$

Because divergences cannot be negative this implies that
$t\mapsto\mathfrak{h}(\gamma_{t})$ is non-decreasing. Assume now that
$\mathfrak{h}(\gamma_{s})=\mathfrak{h}(\gamma_{t})$. Then it follows that

$\displaystyle 0=D(\gamma_{t}||\gamma_{s})=D(\gamma_{s}||\gamma_{t}).$

The latter implies that $s=t$. One concludes that $s<t$ implies a strict
inequality $\mathfrak{h}(\gamma_{s})<\mathfrak{h}(\gamma_{t})$.

(b)

From the definition of the exponential arc one obtains

$\displaystyle D(\gamma_{s}||\gamma_{t})+(t-s)\mathfrak{h}(\gamma_{s})$
$\displaystyle=$ $\displaystyle
D(\psi||\gamma_{t})-D(\psi||\gamma_{s})+(t-s)\mathfrak{h}(\psi).$

Take $\psi=\gamma_{0}$ in this expression to find

$\displaystyle D(\gamma_{s}||\gamma_{t})+(t-s)\mathfrak{h}(\gamma_{s})$
$\displaystyle=$ $\displaystyle
D(\gamma_{0}||\gamma_{t})-D(\gamma_{0}||\gamma_{s})+(t-s)\mathfrak{h}(\gamma_{0})$
$\displaystyle=$ $\displaystyle\Phi_{\gamma}(t)-\Phi_{\gamma}(s).$

(c)

From (b) one obtains

$\displaystyle\Phi_{\gamma}(t)-t\mathfrak{h}(\gamma_{s})$ $\displaystyle\geq$
$\displaystyle\Phi_{\gamma}(s)-s\mathfrak{h}(\gamma_{s}),\qquad 0\leq t\leq 1$
(32)

because $D(\gamma_{s}||\gamma_{t})\geq 0$ with equality if and only if $s=t$.
This implies that $t\mapsto\Phi_{\gamma}(s)+(t-s)\mathfrak{h}(\gamma_{s})$ is
a line tangent to the potential $\Phi_{\gamma}(s)$. By (a) the slope of this
line is a strictly increasing function of $s$. Hence, the potential
$\Phi_{\gamma}(s)$ is a strictly convex function, continuous on the open
interval $(0,1)$.

(d)

(32) implies that

$\displaystyle\Phi^{*}_{\gamma}(\mathfrak{h}(\gamma_{s}))$ $\displaystyle=$
$\displaystyle\sup_{t}\,t\mathfrak{h}(\gamma_{s})-\Phi_{\gamma}(t)$
$\displaystyle\leq$ $\displaystyle
s\mathfrak{h}(\gamma_{s})-\Phi_{\gamma}(s).$

On the other hand one has using (b)

$\displaystyle\Phi_{\gamma}(s)+\Phi^{*}_{\gamma}(\mathfrak{h}(\gamma_{s}))$
$\displaystyle\geq$
$\displaystyle\Phi_{\gamma}(s)+t\mathfrak{h}(\gamma_{s})-\Phi_{\gamma}(t)$
(34) $\displaystyle=$ $\displaystyle
t\mathfrak{h}(\gamma_{s})-\left[D(\gamma_{s}||\gamma_{t})+(t-s)\mathfrak{h}(\gamma_{s})\right]$
$\displaystyle=$
$\displaystyle-D(\gamma_{s}||\gamma_{t})+s\mathfrak{h}(\gamma_{s}).$

The optimal choice $t=s$ yields the lower bound $s\mathfrak{h}(\gamma_{s})$.

$\square$

A dual parameter $\eta$ of the exponential arc $\gamma$, dual to the parameter
$t$, is the value $\mathfrak{h}(\gamma_{t})$ of the generator $\mathfrak{h}$.
By item (a) of the proposition it is a strictly increasing function of $t$. It
is almost everywhere equal to the derivative $\dot{\Phi}_{\gamma}(t)$ of the
value of the potential along the path.

##  10 The matrix case

If $\rho$ and $\sigma$ are two density matrices then the obvious definition of
an exponential arc connecting $\sigma$ to $\rho$ is

$\displaystyle t$ $\displaystyle\mapsto$
$\displaystyle\sigma_{t}=\exp\left(\log\rho+t(\log\sigma-\log\rho)-\zeta(t)\right)$

with normalization $\zeta(t)$ given by

$\displaystyle\zeta(t)$ $\displaystyle=$ $\displaystyle\log\,\mbox{ Tr
}\exp\left(\log\rho+t(\log\sigma-\log\rho)\right).$

It is shown below that the corresponding states given by

$\displaystyle\phi_{t}(x)$ $\displaystyle=$ $\displaystyle\mbox{ Tr
}\sigma_{t}x,\qquad x\in\mathfrak{A}$

form an exponential arc for the relative entropy of Umegaki [34] in the GNS-
representation of the state $\omega_{0}$.

Fix a non-degenerate density matrix $\rho$ of size $n$-by-$n$. It is a
positive-definite matrix with trace $\mbox{ Tr }\rho$ equal to 1.

Umegaki’s relative entropy for the pair of density matrices $\sigma$, $\tau$
is given by

$\displaystyle D(\sigma||\tau)$ $\displaystyle=$ $\displaystyle\mbox{ Tr
}\sigma(\log\sigma-\log\tau).$

Assume now a map

$\displaystyle t\mapsto\sigma_{t}$ $\displaystyle=$
$\displaystyle\exp\left(\log\rho+th-\zeta(t)\right)$ (36)

with normalization $\zeta(t)$ and $h$ given by

$\displaystyle h$ $\displaystyle=$ $\displaystyle\log\sigma-\log\rho.$

This is the obvious definition of an exponential arc in terms of density
matrices. The corresponding potential is

$\displaystyle\Phi_{\sigma}(t)$ $\displaystyle=$ $\displaystyle
D(\sigma_{0}||\sigma_{t})+t\mathfrak{h}(\sigma_{0})$ $\displaystyle=$
$\displaystyle\zeta(t).$

The map (36) is also an exponential arc in the sense of Definition 2. To see
this consider any density matrix $\tau$ and calculate

$\displaystyle
D(\tau||\sigma_{t})-D(\tau||\sigma_{s})-D(\sigma_{s}||\sigma_{t})$
$\displaystyle=$ $\displaystyle\mbox{ Tr }\tau(\log\tau-\log\sigma_{t})$ (40)
$\displaystyle-\mbox{ Tr }\tau(\log\tau-\log\sigma_{s})$ $\displaystyle-\mbox{
Tr }\sigma_{s}(\log\sigma_{s}-\log\sigma_{t})$ $\displaystyle=$
$\displaystyle-(t-s)\mbox{ Tr }(\tau-\sigma_{s})h$ $\displaystyle=$
$\displaystyle(t-s)\,(\mathfrak{h}(\gamma_{s})-\mathfrak{h}(\tau))$

with

$\displaystyle\mathfrak{h}(\tau)=\mbox{ Tr }\tau h=\mbox{ Tr
}\tau(\log\sigma-\log\rho).$

This is of the form (LABEL:arcs:arcdef) except that the relative entropy is
expressed in terms of density matrices in $\mathfrak{M}$ instead of vector
states in the GNS representation of the state defined by the density matrix
$\rho$.

An explicit construction of the GNS representation is possible. See for
instance the appendix of [21, ]. Let $\omega$ denote the state determined by
the density matrix $\rho$

$\displaystyle\omega(A)$ $\displaystyle=$ $\displaystyle\mbox{ Tr }\rho A$

for any $n$-by-$n$ matrix $A$ with entries in ${\mathbb{C}}$. Such a matrix
$A$ is represented on the Hilbert space
$\mathscr{H}={\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$ by the operator
$A\otimes{\mathbb{I}}$, where ${\mathbb{I}}$ is the $n$-by-$n$ identity
matrix. The von Neumann algebra $\mathfrak{M}$ is the space of operators
$A\otimes{\mathbb{I}}$.

The matrix $\rho$ can be diagonalized. This gives the spectral representation

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\sum_{i}\,p_{i}e_{i},$

where $(e_{i})i_{i}$ is an orthonormal basis in ${\mathbb{C}}^{n}$. Let

$\displaystyle\Omega$ $\displaystyle=$
$\displaystyle\sum_{i}\sqrt{p_{i}}e_{i}\otimes e_{i}.$

It is a normalized vector in $\mathscr{H}$. One readily verifies that

$\displaystyle\omega(A)$ $\displaystyle=$
$\displaystyle(A\otimes{\mathbb{I}}\,\Omega,\Omega)$

for any $n$-by-$n$ matrix $A$. In this way any density matrix $\rho$ defines a
vector $\Omega$ in $\mathscr{H}$. The vector $\Omega$ is cyclic and separating
for $\mathfrak{M}$ if $\rho$ is non-degenerate. Hence, there is a one-to-one
correspondence between non-degenerate density matrices and states in the
manifold ${\mathbb{M}}$. It is then straightforward to replace density
matrices by states in the expressions obtained in the first part of this
section.

##  11 The relative modular operator

Araki[6] introduces the relative modular operator $\Delta_{\Phi,\Psi}$ for any
pair of vectors $\Phi$, $\Psi$ in the natural positive cone $\mathscr{P}$.

Assume that $\Phi$ and $\Psi$ are vectors in $\mathscr{P}$ which are
separating for the algebra $\mathfrak{M}$. Then a conjugate-linear operator is
defined by

$\displaystyle x\Psi\mapsto x^{*}\Phi,\qquad x\in\mathfrak{M}.$

It is well-defined because by assumption $x\Psi=0$ implies that $x=0$ so that
also $x^{*}\Phi=0$. It is a closable operator. Indeed, assume the sequence
$x_{n}\Psi$ converges to 0. Then one has for any $y$ in the commutant
$\mathfrak{M}^{\prime}$ that

$\displaystyle(x_{n}\Psi,y\Phi)=(y\Psi,x_{n}^{*}\Phi)$

converges to zero. By assumption is $\Psi$ separating for $\mathfrak{M}$ so
that it is cyclic for the commutant $\mathfrak{M}^{\prime}$. Hence, if the
sequence $x_{n}^{*}\Phi$ converges then it converges to 0. This shows
closability of the operator.

Let $S_{\Phi,\Psi}$ denote the closure of this operator. It satisfies

$\displaystyle S_{\Phi,\Psi}x\Psi$ $\displaystyle=$ $\displaystyle
x^{*}\Phi,\qquad x\in\mathfrak{M}.$

Its inverse equals $S_{\Psi,\Phi}$.

The relative modular operator $\Delta_{\Phi,\Psi}$ is defined by

$\displaystyle\Delta_{\Phi,\Psi}$ $\displaystyle=$ $\displaystyle
S^{*}_{\Phi,\Psi}S_{\Phi,\Psi}.$

Important properties of the relative modular operator are

$\displaystyle\Delta_{\Phi,\Phi}=\Delta_{\Phi}\quad\mbox{ and }\quad
S_{\Phi,\Psi}=J\Delta_{\Phi,\Psi}^{1/2}.$

where $J$ is the modular conjugation operator for the vector $\Phi$.

##  12 Araki’s relative entropy

Araki[5, 6] uses the relative modular operator $\Delta_{\Phi,\Psi}$ to define
the relative entropy/divergence $D(\phi||\psi)$ of the corresponding states
$\phi=\omega_{\Phi}$ and $\psi=\omega_{\Psi}$ by

$\displaystyle D(\phi||\psi)$ $\displaystyle=$
$\displaystyle((\log\Delta_{\Phi,\Psi})\Phi,\Phi).$

###### Proposition 12

The divergence $D(\phi||\psi)$ satisfies $D(\phi||\psi)\geq 0$ with equality
if and only if $\phi=\psi$.

Proof

Let

$\displaystyle\Delta_{\Phi,\Psi}$ $\displaystyle=$
$\displaystyle\int\lambda\,{\rm d}E_{\lambda}$

denote the spectral decomposition of the operator $\Delta_{\Phi,\Psi}$. From
the concavity of the logarithmic function it follows that

$\displaystyle D(\phi||\psi)$ $\displaystyle=$
$\displaystyle-((\log\Delta^{-1}_{\Phi,\Psi})\Phi,\Phi)$ (42) $\displaystyle=$
$\displaystyle-\int\log\lambda^{-1}\,{\rm d}(E_{\lambda}\Phi,\Phi)$ (43)
$\displaystyle\geq$ $\displaystyle-\log\int\lambda^{-1}\,{\rm
d}(E_{\lambda}\Phi,\Phi)$ (44) $\displaystyle=$
$\displaystyle-\log(\Delta^{-1/2}_{\Phi,\Psi}\Phi,\Delta^{-1/2}_{\Phi,\Psi}\Phi)$
(45) $\displaystyle=$ $\displaystyle-\log(\Psi,\Psi)$ $\displaystyle=$
$\displaystyle 0.$

This shows that the divergence cannot be negative.

If $\phi=\psi$ then one has

$\displaystyle D(\phi||\phi)=((\log\Delta_{\Phi})\Phi,\Phi)=0$

because $\Delta_{\phi}\Phi=\Phi$.

Finally, $D(\phi||\psi)=0$ implies that $\Phi$ is in the domain of
$\log\Delta_{\Phi,\Psi}$ and that

$\log\Delta_{\Phi,\Psi}\Phi=0.$

The latter implies that

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\Delta^{-1}_{\Phi,\Psi}\Phi\,=\,\Phi.$

This shows that $D(\phi||\psi)=0$ vanishes only when $\Phi=\Psi$.

$\square$

Theorem 2.4 of [6, ] shows that

$\displaystyle\log\Delta_{\Phi,\Psi}+J\log\Delta_{\Psi,\Phi}J$
$\displaystyle=$ $\displaystyle 0.$

Because $\Phi$ belongs by assumption to the natural positive cone
$\mathscr{P}$ it satisfies $\Phi=J\Phi$. Hence one has also

$\displaystyle D(\phi||\psi)$ $\displaystyle=$
$\displaystyle-((\log\Delta_{\Psi,\Phi})\Phi,\Phi).$

##  13 A theorem

Each self-adjoint element $h$ of the von Neumann algebra $\mathfrak{M}$
defines an exponential arc with as generator the energy function defined by
$h$.

Araki[3, ] constructs for each selfadjoint operator $h$ in $\mathfrak{M}$ a
vector $\Phi_{h}$ in the natural positive cone $\mathscr{P}$ and calls $h$ the
relative hamiltonian. Inspection of the explicit expression used in [3, ]
shows that

$\displaystyle\Phi_{h}$ $\displaystyle=$
$\displaystyle\Omega+Xh\Omega+\mbox{O}(h^{2})$ (47)

with the operator $X$ given by

$\displaystyle X$ $\displaystyle=$ $\displaystyle\int_{0}^{1/2}{\rm
d}u\,\Delta_{{}_{\Omega}}^{u}.$

The vector $\Phi_{h}$ defines a state $\phi_{h}$ by

$\displaystyle\phi_{h}(x)$ $\displaystyle=$ $\displaystyle
e^{-\xi(h)}(x\Phi_{h},\Phi_{h}),\qquad x\in\mathfrak{M}.$

Here, $\xi(h)$ is the normalization

$\displaystyle\xi(h)$ $\displaystyle=$ $\displaystyle\log(\Phi_{h},\Phi_{h}).$

Theorem 3.10 of [6, ] implies that the state $\phi_{h}$ obtained in this way
satisfies for all $\psi$ in ${\mathbb{M}}$

$\displaystyle D(\psi||\phi_{h})$ $\displaystyle=$ $\displaystyle
D(\psi||\omega)-\psi(h)+\xi(h).$ (48)

Take $\psi=\phi_{h}$, respectively $\psi=\omega$, to find that the
normalization $\xi(h)$ is given by

$\displaystyle\xi(h)=\phi_{h}(h)-D(\phi_{h}||\omega)=\omega(h)+D(\omega||\phi_{h}).$

Consider now the path $\gamma$ defined by $\gamma_{t}=\phi_{th}$. Then (48)
becomes

$\displaystyle D(\psi||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\psi||\omega)-t\psi(h)+\zeta(t).$ (49)

with

$\displaystyle\zeta(t)\,=\,t\gamma_{t}(h)-D(\gamma_{t}||\omega)\,=\,t\omega(h)+D(\omega||\gamma_{t})=\xi(th).$

From this last expression one obtains

$\displaystyle 0\,\leq\,D(\gamma_{t}||\omega)+D(\omega||\gamma_{t})$
$\displaystyle=$ $\displaystyle t[\gamma_{t}(h)-\omega(h)].$

From (47) it follows that $\gamma_{t}$ converges to $\omega$ as $t\downarrow
0$. Hence, $D(\gamma_{t}||\omega)$ and $D(\omega||\gamma_{t})$ converge to 0
faster than $t$. This implies that the derivative $\dot{\zeta}(0)$ exists and
equals $\omega(h)$. This implies also that

$\displaystyle\frac{{\rm d}\,}{{\rm d}t}\bigg{|}_{t=0}D(\psi||\gamma_{t})$
$\displaystyle=$ $\displaystyle\omega(h)-\psi(h).$ (50)

Elimination of $\zeta(t)$ from (49) yields

$\displaystyle D(\psi||\gamma_{t})$ $\displaystyle=$ $\displaystyle
D(\psi||\omega)+D(\omega||\gamma_{t})+t(\omega(h)-\psi(h)).$

This shows that $\gamma$ is an exponential arc connecting $\gamma_{1}$ to
$\gamma_{0}=\omega$.

###### Proposition 13

One has

$\displaystyle\dot{\gamma}_{0}(x)$ $\displaystyle=$
$\displaystyle(T_{{}_{\Omega}}h\Omega,T_{{}_{\Omega}}[x-\omega(x)]^{*}\Omega)$
(51)

with the operator $T_{{}_{\Omega}}$ given by

$\displaystyle T_{{}_{\Omega}}$ $\displaystyle=$
$\displaystyle\left(\frac{\Delta_{{}_{\Omega}}-1}{\log\Delta_{{}_{\Omega}}}\right)^{1/2}.$
(52)

Note that this operator $T_{{}_{\Omega}}$ was introduced in [24, ].

Proof

From (47) one obtains

$\displaystyle\dot{\gamma}_{0}(x)=\frac{{\rm d}\,}{{\rm
d}t}\bigg{|}_{t=0}\gamma_{t}(x)$ $\displaystyle=$
$\displaystyle(xXh\Omega,\Omega)+(x\Omega,Xh\Omega)-\dot{\zeta}(0)\omega(x).$
(53)

Write

$\displaystyle(xXh\Omega,\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1/2}{\rm
d}u\,(x\Delta_{{}_{\Omega}}^{u}h\Omega,\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1/2}{\rm
d}u\,(\Delta_{{}_{\Omega}}^{u/2}h\Omega,\Delta_{{}_{\Omega}}^{u/2}x^{*}\Omega)$

and

$\displaystyle(x\Omega,Xh\Omega)$ $\displaystyle=$
$\displaystyle\int_{0}^{1/2}{\rm
d}u\,(x\Omega,\Delta_{{}_{\Omega}}^{u}h\Omega)$ (55) $\displaystyle=$
$\displaystyle\int_{0}^{1/2}{\rm
d}u\,(\Delta_{{}_{\Omega}}^{u/2}J\Delta_{{}_{\Omega}}^{1/2}x^{*}\Omega,\Delta_{{}_{\Omega}}^{u/2}J\Delta_{{}_{\Omega}}^{1/2}h\Omega)$
(56) $\displaystyle=$ $\displaystyle\int_{0}^{1/2}{\rm
d}u\,(J\Delta_{{}_{\Omega}}^{(1-u)/2}x^{*}\Omega,J\Delta_{{}_{\Omega}}^{(1-u)/2}h\Omega)$
(57) $\displaystyle=$ $\displaystyle\int_{1/2}^{1}{\rm
d}u\,(J\Delta_{{}_{\Omega}}^{u/2}x^{*}\Omega,J\Delta_{{}_{\Omega}}^{u/2}h\Omega)$
$\displaystyle=$ $\displaystyle\int_{1/2}^{1}{\rm
d}u\,(\Delta_{{}_{\Omega}}^{u/2}h\Omega,\Delta_{{}_{\Omega}}^{u/2}x^{*}\Omega).$

The two contributions to (53) can now be taken together. One obtains

$\displaystyle\dot{\gamma}_{0}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}{\rm
d}u\,(\Delta_{{}_{\Omega}}^{u/2}h\Omega,\Delta_{{}_{\Omega}}^{u/2}x^{*}\Omega)-\dot{\zeta}(0)\omega(x)$
$\displaystyle=$
$\displaystyle(T_{{}_{\Omega}}h\Omega,T_{{}_{\Omega}}x^{*}\Omega)-\dot{\zeta}(0)\omega(x).$

Take $x=1$ to see that

$\displaystyle\dot{\zeta}(0)=(T_{{}_{\Omega}}h\Omega,T_{{}_{\Omega}}\Omega)=\omega(h)$

so that (51) follows.

$\square$

In summary, one has

###### Theorem 1

Let $\omega$ in ${\mathbb{M}}$ be a vector state with cyclic and separating
vector $\Omega$. Choose the divergence function equal to the relative entropy
of Araki as defined by (12). For each self-adjoint element $h$ in
$\mathfrak{M}$ an energy function $\mathfrak{h}$ is defined by
$\mathfrak{h}(\phi)=\phi(h)$ and there exists an exponential arc $\gamma$ with
generator $\mathfrak{h}$ connecting some state $\gamma_{1}$ of ${\mathbb{M}}$
to the state $\gamma_{0}=\omega$. For any state $\psi$ in ${\mathbb{M}}$ the
derivative of $D(\psi||\gamma_{t})$ at $t=0$ exists and is given by
$\omega(h)-\psi(h)$. The derivative of the exponential arc at $t=0$ satisfies
(51).

Further properties hold for the exponential arc of the above theorem.

###### Proposition 14

For any exponential arc $\gamma$ constructed in Theorem 1 the derivative
$\dot{\gamma}_{0}$ is a Fréchet derivative.

Proof

Let $\Xi(h)$ denote the remainder of order $h^{2}$ in (47), i.e.

$\displaystyle\Phi_{h}$ $\displaystyle=$
$\displaystyle\Omega+Xh\Omega+\Xi(h).$

Then one has using (51)

$\displaystyle\gamma_{t}(x)-\gamma_{0}(x)-t\dot{\gamma}_{0}(x)$ (60)
$\displaystyle=$ $\displaystyle
e^{-\zeta(t)}(x\Phi_{th},\Phi_{th})-\omega(x)-t(xXh\Omega,\Omega)-t(x\Omega,Xh\Omega)+t\omega(h)\omega(x)$
(61) $\displaystyle=$
$\displaystyle\left[e^{-\zeta(t)}-1+t\omega(h)\right]\omega(x)$ (62)
$\displaystyle+$ $\displaystyle
t\left[e^{-\zeta(t)}-1\right]\,\left[(xXh\Omega,\Omega)+(x\Omega,Xh\Omega)\right]$
(63) $\displaystyle+$ $\displaystyle
e^{-\zeta(t)}\bigg{[}(x\Xi(th),\Omega)+(x\Omega,\Xi(th))+t^{2}(xXh\Omega,Xh\Omega)$
$\displaystyle\qquad+t(x\Xi(th),Xh\Omega)+t(xXh\Omega,\Xi(th))+(x\Xi(th),\Xi(th))\bigg{]}.$

This yields

$\displaystyle||\gamma_{t}-\gamma_{0}-t\dot{\gamma}_{0}||$ $\displaystyle\leq$
$\displaystyle|e^{-\zeta(t)}-1+t\omega(h)|$
$\displaystyle+2t|e^{-\zeta(t)}-1|\,||Xh\Omega||$
$\displaystyle+2e^{-\zeta(t)}\,||\Xi(th)||$
$\displaystyle+e^{-\zeta(t)}\,\left[||\Xi(th)||+t||Xh\Omega||\right]^{2}.$

Each of the terms in the r.h.s. of this expression is of order less than $t$
as $t$ tends to 0. Hence, $\dot{\gamma}_{0}$ is a Fréchet derivative.

$\square$

######  Proposition 15 (Additivity of generators)

If the state $\phi$ is connected to the state $\omega$ by the exponential arc
with generator $h$ and $\psi$ is connected to $\phi$ by the exponential arc
with generator $k$ then $\psi$ is connected to $\omega$ by the exponential arc
with generator $h+k$ and $\omega$ is connected to $\psi$ by the exponential
arc with generator $-h$.

For the proof see Proposition 4.5 of [3, ].

##  14 The metric

Eguchi[16] introduces the technique of deriving the metric of the tangent
space by taking two derivatives of the divergence. Application here yields the
metric which is used in the Kubo-Mori theory of linear response [19, 20].

Consider two exponential arcs $t\mapsto\gamma_{t}$ and $s\mapsto\eta_{s}$ with
respective generators $h$ and $k$. They connect the states $\gamma_{1}$ and
$\eta_{1}$ to the reference state $\omega$. The tangent vectors at $s=t=0$ are
$\dot{\gamma}_{0}$ and $\dot{\eta}_{0}$. They belong to the tangent space
$T_{\omega}{\mathbb{M}}$. The scalar product between them is by definition
given by

$\displaystyle(\dot{\eta}_{0},\dot{\gamma}_{0})_{\omega}$ $\displaystyle=$
$\displaystyle-\frac{\partial\,}{\partial s}\frac{\partial\,}{\partial
t}\bigg{|}_{s=t=0}D(\eta_{s}||\gamma_{t}).$

Assume now that these exponential arcs are those constructed in Theorem 1.
Then one has

$\displaystyle(\dot{\eta}_{0},\dot{\gamma}_{0})_{\omega}$ $\displaystyle=$
$\displaystyle-\frac{\partial\,}{\partial
s}\bigg{|}_{s=t=0}\left(\omega(h)-\eta_{s}(h)\right)$ (68) $\displaystyle=$
$\displaystyle\dot{\eta}_{0}(h)$ (69) $\displaystyle=$
$\displaystyle(T_{{}_{\Omega}}k\Omega,T_{{}_{\Omega}}(h-\omega(h))\Omega)$
(70) $\displaystyle=$
$\displaystyle(T_{{}_{\Omega}}(k-\omega(k))\Omega,T_{{}_{\Omega}}(h-\omega(h))\Omega),$
(71)

with the operator $T_{{}_{\Omega}}$ defined by (52). Note that in most
applications one assumes that the expectations $\omega(h)$ of the generator
$h$ and $\omega(k)$ of the generator $k$ vanish. Then the result obtained here
coincides with that used in [24, ]. In what follows a non-vanishing
expectation of the generators is taken into account.

Let us now discuss some technical issues. The scalar product is well-defined
by (71). This follows from

###### Lemma 2

If two exponential arcs with initial point $\omega$ with generators $h$,
respectively $k$, both in $\mathfrak{M}$, have the same initial tangent vector
then one has

$\displaystyle T_{{}_{\Omega}}(h-\omega(h))\Omega$ $\displaystyle=$
$\displaystyle T_{{}_{\Omega}}(k-\omega(k))\Omega.$

Proof

Let $\gamma$ and $\eta$ be two exponential arcs with generators $h$,
respectively $k$ in $\mathfrak{M}$ and such that $\gamma_{0}=\eta_{0}=\omega$.
Without restriction assume that $\omega(h)=\omega(k)=0$ and
$\dot{\gamma}_{0}=\dot{\eta}_{0}$. Then (51) implies that

$\displaystyle(T_{{}_{\Omega}}(h-k)\Omega,T_{{}_{\Omega}}x^{*}\Omega)$
$\displaystyle=$ $\displaystyle 0,\qquad x\in\mathfrak{M}.$

Take $x=h-k$. Then it follows that $T_{{}_{\Omega}}(h-k)\Omega=0$.

$\square$

The above lemma shows that the map

$\displaystyle\dot{\gamma}_{0}\mapsto T_{{}_{\Omega}}(h-\omega(h))\Omega$ (72)

is one-to-one and identifies the tangent vector $\dot{\gamma}_{0}$ with the
vector $T_{{}_{\Omega}}h\Omega$ in the Hilbert space $\mathscr{H}$.

Expression (71) defines a bilinear form. This follows from

###### Lemma 3

The map (72) is linear.

Proof

Let $\gamma$ be an exponential arc with generator $h$ in $\mathfrak{M}$. Then
$t\mapsto\gamma_{\epsilon t}$ is an exponential arc with generator $\epsilon
h$ for any $\epsilon$ in $[-1,1]$ and the tangent vector is
$\epsilon\dot{\gamma}_{0}$. Hence, (72) maps $\epsilon\dot{\gamma}_{0}$ onto
$\epsilon T_{{}_{\Omega}}h\Omega$.

Next consider a pair of exponential arcs $\gamma$ and $\eta$ with generators
$k$, respectively $h$, in $\mathfrak{M}$ and with
$\gamma_{0}=\eta_{0}=\omega$. Let $\theta$ denote the exponential arc with
generator $h+k$. It exists by Theorem 1. The state $\theta_{t}$ can then be
written as

$\displaystyle\theta_{t}(x)$ $\displaystyle=$
$\displaystyle(x\Phi_{th+tk},\Phi_{th+tk})$

with $\Phi_{th+tk}$ the unique element in the natural positive cone
representing the state $\theta_{t}$. Use now (47) to write

$\displaystyle\theta_{t}(x)$ $\displaystyle=$
$\displaystyle\omega(x)+\frac{t}{2}\int_{0}^{1}{\rm
d}u\,(x\Delta_{{}_{\Omega}}^{u/2}(h+k)\Omega,\Omega)$
$\displaystyle\qquad+\frac{t}{2}\int_{0}^{1}{\rm
d}u\,(x\Omega,\Delta_{{}_{\Omega}}^{u/2}(h+k)\Omega)+\mbox{ o}(t).$

This implies

$\displaystyle\dot{\theta}_{0}$ $\displaystyle=$
$\displaystyle\dot{\gamma}_{0}+\dot{\eta}_{0}.$

Both observations together prove linearity of the map (72).

$\square$

###### Proposition 16

Expression (71) defines a non-degenerate scalar product on the space of
tangent vectors of the form $\dot{\gamma}_{0}$ with $\gamma$ an exponential
arc as constructed in Theorem 1.

Proof

The two lemmas show that (71) is a well-defined bilinear form. Positivity of
the form is clear. The symmetry follows from (71). It remains to be shown that
it is non-degenerate.

Assume that $(\dot{\gamma}_{0},\dot{\gamma}_{0})=0$. This implies

$\displaystyle T_{{}_{\Omega}}(h-\omega(h))\Omega$ $\displaystyle=$
$\displaystyle 0,$

with $h$ the generator of $\gamma$. The operator $T_{{}_{\Omega}}$ is
invertible — see the proof of Lemma II.2 of [24, ]. Hence, it follows that

$\displaystyle(h-\omega(h))\Omega$ $\displaystyle=$ $\displaystyle 0.$

Because $\Omega$ is separating for $\mathfrak{M}$ it follows that $h$ is a
multiple of the identity. The latter implies that $\dot{\gamma}=0$.

$\square$

##  15 Dual geometries

The geodesics of the e-connection are the exponential arcs. In the
m-connection the geodesics are made up by convex combinations of a pair of
states. The m- and e-connections are each others dual w.r.t. the metric of
Section 14.

Consider two states $\omega$ and $\phi$ in the manifold ${\mathbb{M}}$. The
tangent vector

$\displaystyle\dot{\gamma}_{t}=\frac{{\rm d}\,}{{\rm
d}t}\gamma_{t}=\phi-\omega,\qquad 0<t<1,$

is independent of $t$. Hence, it is a geodesic for the connection in which all
parallel transport operators are taken equal to the identity operator. Note
that the tangent space $T_{\omega}{\mathbb{M}}$ coincides with the space of
$\sigma$-weakly continuous linear functionals $\chi$ satisfying $\chi(1)=0$
and hence is everywhere the same. This connection is by definition the
m-connection.

For $t$ in $(0,1)$ the tangent vector $\dot{\gamma}_{t}$ belongs to the
subspace ${\mathscr{T}}_{\gamma_{t}}$ of the tangent space
$T_{\omega}{\mathbb{M}}$ which is introduced in Section 6. Conversely, every
vector $\chi$ in ${\mathscr{T}}_{\gamma_{t}}$ is the tangent vector of an
m-geodesic passing through the point $\gamma_{t}$. However, this does not
imply that through parallel transport $\Pi(\gamma_{t}\mapsto\gamma_{s})$ the
space ${\mathscr{T}}_{\gamma_{t}}$ maps onto the space
${\mathscr{T}}_{\gamma_{s}}$.

The transport operators $\Pi^{*}$ of the dual geometry are defined by

$\displaystyle\left(\Pi(\phi\mapsto\omega)V,\Pi^{*}(\phi\mapsto\omega)W\right)_{\omega}$
$\displaystyle=$ $\displaystyle\left(V,W\right)_{\phi}.$

In this expression $V$ and $W$ are vector fields and $(\cdot,\cdot)_{\omega}$
is the scalar product defined in the previous section and evaluated at the
point $\omega$ of the manifold ${\mathbb{M}}$.

It is shown below that any exponential arc $\gamma$ is a geodesic for this
dual geometry. To do so we have to show that

$\displaystyle\Pi^{*}(\gamma_{s}\mapsto\gamma_{t})\dot{\gamma}_{s}$
$\displaystyle=$ $\displaystyle\dot{\gamma}_{t}.$

The tangent vector $\dot{\gamma}_{t}$ at $t=0$ is given by (51). Its value for
arbitrary $t$ is given by the following proposition.

###### Proposition 17

Let $\gamma$ denote an exponential arc $\gamma$ with generator $h$ belonging
to $\mathfrak{M}$. Let $\Phi_{t}$ be the normalized vector in the natural
positive cone $\mathscr{P}$ representing the state $\gamma_{t}$. The
derivative $\dot{\gamma}_{t}$ is given by

$\displaystyle\dot{\gamma}_{t}(x)$ $\displaystyle=$
$\displaystyle\left(T_{\Phi_{t}}h{\Phi_{t}},T_{\Phi_{t}}[x-\gamma_{t}(x)]^{*}\right){\Phi_{t}}),\qquad
x\in\mathfrak{M}.$ (74)

Proof

The state ${\gamma_{1}}$ is connected to $\omega$ by the exponential arc with
generator $h$ and $\gamma_{t}$ is connected to $\omega$ by the exponential arc
with generator $th$. Let

$\displaystyle\psi_{s}$ $\displaystyle=$ $\displaystyle\gamma_{(1-s)t+s}.$

It follows from Proposition 8 that $s\mapsto\psi_{s}$ is an exponential arc
with generator $(1-t)h$ connecting $\gamma_{t}$ to $\gamma_{1}$. Application
of (51) to the latter arc gives

$\displaystyle\dot{\psi}_{0}(x)=\frac{{\rm d}\,}{{\rm
d}s}\bigg{|}_{s=0}\psi_{s}(x)$ $\displaystyle=$
$\displaystyle(1-t)\,(T_{\Psi}h\Psi,T_{\psi}(x-\omega(x))^{*}\Psi)$ (75)

with $\Psi=\Phi_{t}$. This implies (74) because
$\dot{\psi}_{0}=(1-t)\dot{\gamma}_{t}$.

$\square$

###### Theorem 2

Any exponential arc $\gamma$ with generator $h$ in $\mathfrak{M}$ is a
geodesic for the dual of the m-connection w.r.t. the metric introduced in
Section 14.

Proof

Let $t\mapsto\phi_{t}$ be an exponential arc with generator $k$ in
$\mathfrak{M}$ such that $\phi_{0}=\gamma_{t}$. Fix $t$ in $[0,1]$ and let
$\Phi_{t}$ denote the normalized element of the natural positive cone
$\mathscr{P}$ representing the state $\gamma_{t}$. Let $\eta$ be an
exponential arc with generator $k$ starting at $\gamma_{t}$, i.e.
$\eta_{0}=\gamma_{t}$. Because $\Pi(\gamma_{s}\mapsto\gamma_{t})$ is the
identity the definition of the dual transport operator yields

$\displaystyle\left(\dot{\eta}_{0},\Pi^{*}(\gamma_{s}\mapsto\gamma_{t})\dot{\gamma}_{s}\right)_{\gamma_{t}}$
$\displaystyle=$
$\displaystyle\left(\dot{\eta}_{0},\dot{\gamma}_{s}\right)_{\gamma_{s}}$
$\displaystyle=$
$\displaystyle\left(T_{\Phi_{t}}(k-\gamma_{t}(k))\Phi_{t},T_{\Phi_{t}}(l-\gamma_{t}(l))\Phi_{t}\right),$

with $l$ the generator of the arc $s\mapsto\gamma_{(1-s)t+s}$. It equals
$l=(1-t)h$. This last expression equals

$\displaystyle=$
$\displaystyle\left(\dot{\eta}_{0},\dot{\gamma}_{t}\right)_{\gamma_{t}}.$

By proposition 16 the scalar product $(\cdot,\cdot)_{\gamma_{t}}$ is non-
degenerate. Therefore one can conclude that

$\displaystyle\Pi^{*}(\gamma_{s}\mapsto\gamma_{t})\dot{\gamma}_{s}$
$\displaystyle=$ $\displaystyle\dot{\gamma}_{t}.$

This shows that the exponential arc $\gamma$ is a geodesic for the dual of the
m-connection.

$\square$

##  16 Finite-dimensional submanifolds

A finite set of linearly independent generators is shown to define a finite-
dimensional submanifold in which all states are connected to the reference
state by an exponential arc. The submanifold defined in this way is a dually
flat quantum statistical manifold.

Let $\omega$ be the reference state of ${\mathbb{M}}$. It is a vector state
with cyclic and separating vector $\Omega$. Choose an independent set of
selfadjoint operators $h_{1},\cdots,h_{n}$ in $\mathfrak{M}$. By Theorem 1
there exists an exponential arc $\gamma$ with generator $h=\theta^{i}h_{i}$
connecting some state $\gamma_{1}$ in ${\mathbb{M}}$ to the state
$\gamma_{0}=\omega$. A parameterized family of states $\omega_{\theta}$,
$\theta\in{\mathbb{R}}^{n}$ is now defined by putting
$\omega_{\theta}=\gamma_{1}$. These states form a submanifold of
${\mathbb{M}}$.

From the definition of an exponential arc one obtains immediately that for any
$\psi$ in ${\mathbb{M}}$

$\displaystyle D(\psi||\omega_{\theta})$ $\displaystyle=$ $\displaystyle
D(\psi||\omega)+D(\omega||\omega_{\theta})+\theta^{i}\left(\omega(h_{i})-\psi(h_{i})\right).$
(77)

Take $\psi=\omega_{\theta}$ in this expression to find

$\displaystyle\theta^{i}\omega(h_{i})\,\leq\,D(\omega||\omega_{\theta})+\theta^{i}\omega(h_{i})$
$\displaystyle=$
$\displaystyle\theta^{i}\omega_{\theta}(h_{i})-D(\omega_{\theta}||\omega)\,\leq\,\theta^{i}\omega_{\theta}(h_{i}).$
(78)

Hence, the quantity $\theta^{i}\omega(h_{i})$ is maximal if and only if
$\omega_{\theta}$ equals the reference state $\omega$.

###### Proposition 18

Dual coordinates $\eta_{i}$ are defined by

$\displaystyle\eta_{i}$ $\displaystyle=$
$\displaystyle\omega_{\theta}(h_{i}).$

They satisfy

$\displaystyle\frac{\partial\eta_{i}}{\partial\theta^{j}}$ $\displaystyle=$
$\displaystyle(\partial_{i},\partial_{j})_{\theta}$

with $(\cdot,\cdot)_{\theta}$ equal to the scalar product
$(\cdot,\cdot)_{\omega_{\theta}}$ introduced in Section 14 and with basis
vectors $\partial_{i}$ equal to $\partial\omega_{\theta}/\partial\theta^{i}$.

Proof

Introduce the path $\gamma^{(i)}$ defined by

$\displaystyle\gamma^{(i)}:\,t\mapsto\omega_{\theta+tg_{i}}.$

It satisfies

$\displaystyle\frac{\partial\,}{\partial\theta^{i}}\omega_{\theta}=\partial_{i}=\dot{\gamma}^{(i)}(0).$

By definition is $\omega_{\theta+g_{i}}$ the end point of the exponential arc
with generator $(\theta^{j}+g^{j}_{i})h_{i}$. From Proposition 15 it then
follows that $\gamma^{(i)}$ is an exponential arc with generator $h_{i}$
connecting $\omega_{\theta+g_{i}}$ to $\omega_{\theta}$. These arcs
$\gamma^{(i)}$ are used in the calculation that follows.

The definition of the scalar product at the beginning of Section 14 gives

$\displaystyle(\partial_{i},\partial_{j})_{\omega_{\theta}}$ $\displaystyle=$
$\displaystyle(\dot{\gamma}^{(i)}(0),\dot{\gamma}^{(j)}(0))_{\omega_{\theta}}$
(79) $\displaystyle=$ $\displaystyle-\frac{\partial\,}{\partial
s}\frac{\partial\,}{\partial
t}\bigg{|}_{s=t=0}D(\gamma^{(i)}_{s}||\gamma^{(j)}_{t})$ (80) $\displaystyle=$
$\displaystyle-\frac{\partial\,}{\partial
s}\bigg{|}_{s=0}\frac{\partial\,}{\partial\theta^{j}}D(\gamma^{(i)}_{s}||\omega_{\theta})$
(81) $\displaystyle=$ $\displaystyle\frac{\partial\,}{\partial
s}\bigg{|}_{s=0}\gamma^{(i)}_{s}(h_{j})$ (82) $\displaystyle=$
$\displaystyle\partial_{i}(h_{j})$ $\displaystyle=$
$\displaystyle\frac{\partial\eta_{j}}{\partial\theta^{i}}.$

$\square$

###### Corollary 3

There exists a potential $\Phi(\theta)$ such that

$\displaystyle\eta_{i}$ $\displaystyle=$
$\displaystyle\frac{\partial\Phi}{\partial\theta^{i}}.$ (84)

This follows because the scalar product is symmetric so that

$\displaystyle\frac{\partial\eta_{j}}{\partial\theta^{i}}=(\partial_{i},\partial_{j})_{\theta}=(\partial_{j},\partial_{i})_{\theta}=\frac{\partial\eta_{i}}{\partial\theta^{j}}.$

This symmetry is a sufficient condition for the potential $\Phi(\theta)$ to
exist.

Consider the following generalization of the potential introduced in Section
9.

$\displaystyle\Phi(\theta)$ $\displaystyle=$ $\displaystyle
D(\omega||\omega_{\theta})+\theta^{i}\omega(h_{i}).$ (85)

Apply (50) to the exponential arc $\gamma^{(i)}$ which connects
$\omega_{\theta+g_{i}}$ to $\omega_{\theta}$ to find

$\displaystyle\frac{\partial\,}{\partial\theta^{i}}D(\omega||\omega_{\theta})$
$\displaystyle=$ $\displaystyle\omega_{\theta}(h_{i})-\omega(h_{i}).$

This implies that $\Phi(\theta)$ satisfies (84).

One concludes that the selection of an independent set of selfadjoint
operators $h_{1},\cdots,h_{n}$ in $\mathfrak{M}$ defines a parameterized
statistical model $\theta\mapsto\omega_{\theta}$ of states on the von Neumann
algebra $\mathfrak{M}$. An obvious basis in the tangent plane
$T_{\omega_{\theta}}{\mathbb{M}}$ is formed by the derivative operators
$\partial_{i}$. The scalar product
$(\partial_{i},\partial_{j})_{\omega_{\theta}}$ introduced in Section 14
starting from the relative entropy of Araki defines a Hessian metric on the
tangent planes. Exponential arcs are geodesics for the e-connection which is
the dual of the m-connection.

##  17 Discussion

  * •

The manifold ${\mathbb{M}}$ under consideration consists of vector states on a
sigma-finite von Neumann algebra $\mathfrak{M}$ in its standard
representation. Such a manifold has nice properties described by Tomita-
Takesaki theory and hence is an obvious study object when exploring quantum
statistical manifolds in an infinite-dimensional setting. Particular attention
is given in the present work on the definition of the tangent planes. This is
also a point of concern in the commutative context of manifolds of probability
measures. See for instance the approach of [29, ]. A convenient choice for the
tangent space $T_{\omega}{\mathbb{M}}$ at the state $\omega$ in the manifold
${\mathbb{M}}$ is to take it equal to the space of all $\sigma$-weakly
continuous Hermitian linear functionals $\chi$ on $\mathfrak{M}$ vanishing on
the identity operator ${\mathbb{I}}$. However, it is well possible that the
equivalence class of smooth curves through $\omega$ with initial tangent equal
to a given $\chi$ is empty. Approximate tangent planes are considered as an
alternative in Section 6. They form a subspace of $T_{\omega}{\mathbb{M}}$ as
defined above. Never the less, the initial tangent vectors of Fréchet-
differentiable paths starting at $\omega$ belong to the approximate tangent
space. It is not clear whether the initial tangents of exponential arcs are
dense in the approximate tangent space w.r.t. the inner product of Section 14.
Further research is needed at this point.

  * •

A new definition of exponential arcs is given. It depends on the choice of a
divergence function/relative entropy defined on pairs of points in the
manifold and on the choice of a generator which is a linear functional defined
on a domain in the manifold. It is general enough to cover different
approaches that one can follow to solve the non-uniqueness problem of the
Radon-Nikodym derivative in the context of non-commutative probability. Never
the less one can prove in full generality nice properties such as uniqueness
of the generator, existence of a scalar potential, Pythagorean relations.
Additivity of generators when composing exponential arcs is shown in the
specific context of Araki’s relative entropy. See Proposition 15.

  * •

The second half of the paper focuses on the relative entropy of Araki. Only
exponential arcs with bounded generators belonging to the von Neumann algebra
are considered. This suffices to reach the goal of replacing the existing
approach based on density matrices and Umegaki’s relative entropy. However,
the solution of the problem mentioned above regarding the extent of the
tangent spaces most likely requires the handling of unbounded generators.

  * •

The scalar product of Bogoliubov presented in Section 14 is used extensively
in Linear Response Theory, also known as Kubo-Mori theory. Its link with the
KMS condition of Section 2 is not highlighted in the present text. It is
tradition in Kubo-Mori theory and more generally in Statistical Mechanics to
focus on a small number of variables. It is shown in Section 16 that the
selection of a finite number of variables defines a quantum statistical
manifold supporting Amari’s dually flat geometry.

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