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Wigner-Yanase information on quantum state space 1
Wigner-Yanase information on quantum state space:
the geometric approach
Paolo Gibilisco1
Dipartimento di Scienze, Facolta` di Economia, Universita` di Chieti-Pescara “G. D’Annunzio”, Viale
Pindaro 42, I–65127 Pescara, Italy.
Tommaso Isola 2
Dipartimento di Matematica, Universita` di Roma “Tor Vergata”, Via della Ricerca Scientifica, I–00133
Roma, Italy.
Abstract
In the search of appropriate riemannian metrics on quantum state space the concept of statistical
monotonicity, or contraction under coarse graining, has been proposed by Chentsov. The metrics with
this property have been classified by Petz. All the elements of this family of geometries can be seen as
quantum analogues of Fisher information. Although there exists a number of general theorems sheding
light on this subject, many natural questions, also stemming from applications, are still open. In this
paper we discuss a particular member of the family, the Wigner-Yanase information. Using a well-known
approach that mimics the classical pull-back approach to Fisher information, we are able to give explicit
formulae for the geodesic distance, the geodesic path, the sectional and scalar curvatures associated to
Wigner-Yanase information. Moreover we show that this is the only monotone metric for which such an
approach is possible.
1 Introduction
The notion of information proposed by Fisher is fundamental in probability and statistics for a number
of reasons; here we mention only the Cramer-Rao inequality and the asymptotic behaviour of maximum
lilkelihood estimators for exponential models (one can see [5] for unexpected features and applications of
Fisher information). In classical statistics Rao was the first to point out that Fisher information can be seen
as a riemannian metric on the space of probability densities. This point of view was nicely complemented by
the results of Chentsov saying that (on the simplex of probability vectors) Fisher information is the unique
riemannian metric contracting under Markov morphisms. This can be rephrased in a more suggestive way.
Markov morphisms, or positive mappings, are the mathematical counterpart of the notion of noise. Now
suppose that we want to use a distance to distinguish different states (probability densities) in a statistically
relevant way. Then the effect of noise must be that of contracting the metric. Chentsov theorem says therefore
that in the classical case there is only one choice, the Fisher information (another argument producing Fisher
information can be found in [43]).
In the quantum case one deals with density operators instead of density vectors and completely positive
mappings play the role of Markov morphisms. As often happens in the quantum counterpart of a classical
theory, instead of a uniqueness result, one has a classification theorem, due to Petz. This result states that
there is bijection between statistically monotone metrics on quantum state space and the operator monotone
functions: we have therefore a rich “garden” of candidates for the role of Fisher information in quantum
physics. Among the elements of this family of metrics one can find, in a certain sense, the most relevant
riemannian metrics appeared in the literature [37].
1Electronic mail: gibilisc@sci.unich.it
2Electronic mail: isola@mat.uniroma2.it

Wigner-Yanase information on quantum state space 2
Despite the existence of general results for the theory [31, 13, 17, 28, 29, 27, 19] a number of open
problems resists to investigation. For example it does not exist yet a general formula for the geodesic path
and the geodesic distance associated to an arbitrary monotone metric. For the use of this kind of distances
see for example [34]. Because of the absence of a general formula, inequalities (giving bounds for the geodesic
distance) have been proved [41].
In this paper we discuss the Wigner-Yanase skew information. To find the formulae for geodesic path and
geodesic distance we mimic the classical approach to Fisher information via sphere geometry (one should
note the importance of determining geodesic path in the study of the 2-Wasserstein metric [6]). Indeed
Wigner-Yanase information appears as the pull-back of the square root map [18]. Next we prove the formula
for the scalar curvature. One proof, due to J.Dittmann, uses the general formula [13] and requires a long
calculation. The second one just uses the pull-back approach. One should emphasize that, since the scalar
curvature determines the asymptotic behaviour of the volume (for a riemannian metric) then it has also a
statistical meaning in relation to the quantum analogue of Jeffrey’s rule for determining prior probability
distributions (see [37]). Finally we prove, as a corollary of the results in [26, 27, 19] that the Wigner-Yanase
information is the only monotone metric that can be seen as a pull-back metric.
The paper is organised as follows. In section II we review the geometric approach to Fisher information.
In section III one finds an introduction to the general theory of statistical monotone metrics. Section IV
shows how the Wigner-Yanase information can be seen as a monotone riemannian metric. In section V
we show that the Wigner-Yanase geometry can be seen as the sphere geometry transposed on the space
of density matrices; moreover we characterise it as the unique pull-back metric. Section VI contains some
comments on the main results and on some open problems.
2 Fisher information and its geometry
The classical definition of Fisher information for an indexed family of densities pθ is given by the variance
of the score. In the case of a family indexed by only one parameter θ it is the number
I(θ) = Eθ
[(
∂
∂θ
log p θ
)2]
(2.1)
assigned to the parameter θ. For n parameters, say θ =
(
θ1, . . . , θn
)
, it is a matrix defined on the parameter
manifold given by
I(θ)i j = Eθ
[(
∂
∂θi
log p θ
)(
∂
∂θj
log p θ
)]
. (2.2)
Geometrically this means, that I(θ) is a symmetric bilinear form on the tangent spaces of the parameter
manifold. In a coordinate free language it reads as
I(θ)(U, V ) = Eθ [U (log p θ) V (log p θ)] , (2.3)
where U and V are vectors tangent to the parameter manifold and U (log p θ) is the derivative of log p θ along
the direction U , that means U (log p θ) = ddt log p θ+tU |t=0.
I(θ) is a measure for the statistical distinguishabilty of distribution parameters. Under certain regularity
conditions for θ 7→ p θ the image of this mapping is a manifold of distributions. This manifold is the actual
object of interest in information geometry rather than the space of distribution parameters and formula

Wigner-Yanase information on quantum state space 3
(2.3) defines a Riemannian metric g on it (for a general reference see [1]). Indeed, a vector u tangent to this
manifold is of the form
u =
d
dt
p θ+tU
|t=0
and the right hand side of (2.3) now reads as
g(u, v) := Ep
[
u
p
v
p
]
(2.4)
defining the Fisher metric on the manifold of densities. If the differential of θ 7→ p θ is not injective, than
there is some parameter redundance or ambiguity in the choice of U and V , and therefore the right hand
side of (2.3) does not depend on this choice.
We restrict now to Pn ⊂ Rn, the simplex of strictly positive probability vectors, that is Pn := {ρ ∈
Rn :
∑n
i=1 ρi = 1, ρi > 0, i = 1, . . . , n}. An element ρ ∈ Pn is a density on the n-point set {1, . . . , n} with
ρ(i) = ρi. We regard an element u of the tangent space TρPn ≡ {u ∈ Rn :
∑n
i=1 ui = 0} as a function u on
{1, . . . , n} with u(i) = ui.
Definition 2.1. The Fisher-Rao Riemannian metric on TρPn is given by
〈u, v〉Fρ :=
n∑
i=1
uivi
ρi
(2.5)
for u, v ∈ TρPn.
To see the relation between this metric and the Fisher metric, let u, v ∈ TρPn. We obtain from (2.4)
g(u, v) =
n∑
i=1
u(i)
ρi
v(i)
ρi
ρi =
n∑
i=1
uivi
ρi
in accordance with (2.5).
The following result is well known and is a very special case of a far more general situation (see [15] for
example).
Theorem 2.2. The manifold Pn equipped with the Fisher-Rao Riemannian metric 〈·, ·〉F is isometric with
an open subset of the sphere of radius 2 in Rn.
Proof. We consider the mapping ϕ : Pn → Sn−12 ⊂ Rn,
ϕ(ρ) := 2 (
√
ρ1, . . . ,
√
ρn) .
Then Dρϕ(u) =
(
u1√
ρ1
, . . . , un√ρn
)
and we get
Dρϕ
(
〈·, ·〉F
)
(u, v) := 〈Dρϕ(u), Dρϕ(v)〉R
n
=
n∑
i=1
ui vi
ρi
= 〈u, v〉Fρ .
Hence the standard metric on the sphere of radius 2 is pulled back to the Fisher-Rao Riemannian metric.

Wigner-Yanase information on quantum state space 4
This identification of Pn with an open subset of a radius 2 sphere allows for obtaining differential geo-
metrical quantities of the Riemannian manifold
(
Pn, 〈·, ·〉F
)
. From the very definition of geodesic distance,
geodesic path and scalar curvature, one has for Sn−1r , with P1, P2 ∈ Sn−1r ,
1) geodesic distance
d(P1, P2) = r · arcos
( 〈P1, P2〉
r2
)
2) geodesic path connecting P1 and P2 :
γP1,P2(t) = r
(1− t)P1 + tP2
||(1− t)P1 + tP2||
(of course, t is not the arc length parameter);
3) scalar curvature
Scal(v) =
1
r2
(n− 1)(n− 2)
because Sn−1r has constant sectional curvature equal to
1
r2 .
Let us denote by dF , γF , ScalF respectively the corresponding quantities for the Fisher information. The
above considerations give, for ρ, σ ∈ Pn,
1) Bhattacharya distance
dF(ρ, σ) = 2arccos
(
∑
i
ρ1/2i σ
1/2
i
)
2) geodesic path connecting ρ and σ:
γρ,σF (t) = 2
((1 − t)√ρ + t√σ)2∑
i((1− t)
√
ρi + t
√
σi)2
3) scalar curvature
ScalF(ρ) =
1
4
(n− 1)(n− 2) ∀ρ ∈ Pn.
The Levi-Civita connection associated to Fisher metric can be decomposed using the geometry of mixture
and exponential models. The rest of the section explains how.
Definition 2.3. A dualistic structure on a manifold M is a triple (〈·, ·〉,∇, ∇˜) where 〈·, ·〉 is a riemannian
metric on M and ∇, ∇˜ are affine connections on M such that
X〈Y, Z〉 = 〈∇XY, Z〉+ 〈Y, ∇˜XZ〉
where X,Y, Z are vector fields. If U∇, U ∇˜ are the parallel transport associated to ∇, ∇˜ then the above
equation is equivalent to
〈U∇(u), U ∇˜(v)〉 = 〈u, v〉.
A divergence on a manifold is a smooth nonnegative function D : M × M → R such that D(ρ, σ) = 0 iff
ρ = σ. To each divergence D one may associate a dualistic structure (〈·, ·〉,∇, ∇˜) (see [1, 14]).

Wigner-Yanase information on quantum state space 5
Let ∇2 be the Levi-Civita connection of Fisher information. The Kullback-Leibler relative entropy
K(ρ, σ) =
∑
i ρi(log ρi − log σi) gives a dualistic structure (〈·, ·〉F ,∇m,∇e) such that
∇2 = 1
2
(∇m +∇e)
where ∇m,∇e are the mixture and exponential connections. These connections are torsion free and flat:
once the representation by scores is used for the tangent spaces, the associated parallel transports are given
by
Umρσ : TρP → TσP Umρσ(u) =
ρ
σ
u
Ueρσ : TρP → TσP Ueρσ(u) = u− Eσ(u).
The geodesics of ∇m,∇e are, respectively, the mixture and exponential models.
3 Metric contraction under coarse graining
In the commutative case a Markov morphism (or stochastic map) is a positive operator T : Rn → Rk. In the
noncommutative case a stochastic map is a completely positive and trace preserving operator T : Mn → Mk
where Mn denotes the space of n by n complex matrices. We shall denote by Dn the manifold of strictly
positive elements of Mn and by D1n ⊂ Dn the submanifold of density matrices.
In the commutative case a monotone metric is a family of riemannian metrics g = {gn} on {Pn}, n ∈ N
such that
gmT (ρ)(TX, TX) ≤ gnρ (X,X)
holds for every stochastic mapping T : Rn → Rm and all ρ ∈ Pn and X ∈ TρPn.
In perfect analogy, a monotone metric in the noncommutative case is a family of Riemannian metrics
g = {gn} on {D1n}, n ∈ N such that
gmT (ρ)(TX, TX) ≤ gnρ (X,X)
holds for every stochastic mapping T : Mn → Mm and all ρ ∈ D1n and X ∈ TρD1n.
Let us recall that a function f : (0,∞) → R is called operator monotone if for any n ∈ N, any A, B ∈ Mn
such that 0 ≤ A ≤ B, the inequalities 0 ≤ f(A) ≤ f(B) hold. An operator monotone function is said
symmetric if f(x) := xf(x−1) and normalized if f(1) = 1. In what follows by operator monotone we mean
normalised symmetric operator monotone. With each operator monotone function f one associates also the
so-called Chentsov–Morotzova function
cf (x, y) :=
1
yf(xy )
for x, y > 0.
Define Lρ(A) := ρA, and Rρ(A) := Aρ. Since Lρ, Rρ commute we may define c(Lρ, Rρ). Now we can state
the fundamental theorems about monotone metrics (uniqueness and classification are up to scalars).
Theorem 3.1. [7] There exists a unique monotone metric on Pn given by the Fisher information.
Theorem 3.2. [36] There exists a bijective correspondence between monotone metrics on D1n and operator
monotone functions given by the formula
〈A,B〉ρ,f := Tr(A · cf (Lρ, Rρ)(B)).

Wigner-Yanase information on quantum state space 6
The tangent space to D1n at ρ is given by TρD1n ≡ {A ∈ Mn : A = A∗, T r(A) = 0}, and can be
decomposed as TρD1n = (TρD1n)c ⊕ (TρD1n)o, where (TρD1n)c := {A ∈ TρD1n : [A, ρ] = 0}, and (TρD1n)o is the
orthogonal complement of (TρD1n)c, with respect to the Hilbert-Schmidt scalar product 〈A,B〉 := Tr(A∗B).
Each statistically monotone metric has a unique expression (up to a constant) given by Tr(ρ−1A2), for
A ∈ (TρD1n)c. The following result will be used in Section 5.
Proposition 3.3. (See [3]). Let A ∈ TρD1n be decomposed as A = Ac + i[ρ, U ] where Ac ∈ (TρD1n)c and
i[ρ, U ] ∈ (TρD1n)o. Suppose ϕ ∈ C1(0,+∞). Then
(Dρϕ)(A) = ϕ′(ρ)Ac + i[ϕ(ρ), U ].
As proved by Lesniewski and Ruskai each monotone metric is the hessian of a suitable relative entropy;
to state this result more precisely, we introduce some notation. In what follows g is an operator convex
function defined on (0,+∞) and such that g(1) = 0. The formula
f(x) ≡ fg(x) :=
(x− 1)2
g(x) + xg(x−1)
associates a normalised, symmetric operator monotone function f = fg to each g. We denote by ∆σ,ρ =
LσR−1ρ the relative modular operator. The relative g–entropy of ρ and σ is defined as
Hg(ρ, σ) := Tr(ρ
1
2 g(∆σ,ρ)(ρ
1
2 )).
Hg is a divergence on Dn in the sense of [14, 1]. If ρ, σ are diagonal Hg, reduces to the commutative relative
g–entropy (see [9]).
Theorem 3.4. [31] Let g be operator convex, g(1) = 0, f = fg and ρ ∈ Dn. Then
− ∂
∂t
∂
∂s
Hg(ρ+ tA, ρ + sB)
∣∣∣
t=s=0
= Tr(A · cf (Lρ, Rρ)(B)).
To state the general formula for the scalar curvature of a monotone metric we need some auxiliary
functions. In what follows c′, (log c)′ denote derivatives with respect to the first variable, and c = cf .
h1(x, y, z) :=
c(x, y)− z c(x, z) c(y, z)
(x− z)(y − z)c(x, z)c(y, z) ,
h2(x, y, z) :=
(c(x, z)− c(y, z))2
(x− y)2c(x, y)c(x, z)c(y, z) ,
h3(x, y, z) := z
(ln c)′(z, x)− (ln c)′(z, y)
x− y ,
h4(x, y, z) := z (ln c)′(z, x) (ln c)′(z, y) ,
h := h1 −
1
2
h2 + 2h3 − h4 . (3.1)
The functions hi have no essential singularities if arguments coincide.
Note that 〈A,B〉fρ := Tr(A·cf (Lρ, Rρ)(B)) defines a riemannian metric also over Dn (D1n is a submanifold
of codimension 1). Let Scalf (ρ) be the scalar curvature of (Dn, 〈·, ·〉fρ) at ρ and Scal
1
f (ρ) be the scalar
curvature of (D1n, 〈·, ·〉
f
ρ).

Wigner-Yanase information on quantum state space 7
Theorem 3.5. [13] Let σ(ρ) be the spectrum of ρ. Then
Scalf (ρ) =
∑
x,y,z∈σ(ρ)
h(x, y, z)−
∑
x∈σ(ρ)
h(x, x, x) (3.2)
Scal1f (ρ) = Scalf (ρ) +
1
4
(n2 − 1)(n2 − 2).
4 Wigner-Yanase information as a riemannian metric
Let ρ ∈ D1n be a density matrix and let A be a self adjoint matrix. The Wigner-Yanase information (or skew
information, information content relative to A) was defined as
I(ρ,A) := −Tr([ρ1/2, A]2)
where [·, ·] denotes the commutator(see [42]). Consider now g(x) := gwy(x) := 4(1−
√
x). In this case
Hg(ρ, σ) = 4(1− Tr(ρ
1
2σ
1
2 )).
The associated operator monotone and Chentsov-Morotzova functions are
fwy(x) :=
1
4
(
√
x+ 1)2 cwy(x, y) :=
1
yfwy(xy )
=
4
(
√
x+
√
y)2
Let us consider the monotone metric
〈A,B〉wyρ := Tr(Acwy(Lρ, Rρ)(B)).
A typical element of (TρDn)o has the form i [ρ,A], where A is self-adjoint. We have
〈i [ρ,A], i [ρ,A]〉wyρ = Tr
(
i [ρ,A]4(L1/2ρ +R
1/2
ρ )
−2(i [ρ,A])
)
= −4Tr
(
(L1/2ρ + R
1/2
ρ )
−1([ρ,A]) (L1/2ρ +R
1/2
ρ )
−1([ρ,A])
)
= −4Tr
(
(L1/2ρ + R
1/2
ρ )
−1 ◦ (Lρ −Rρ)(A) (L1/2ρ +R1/2ρ )−1 ◦ (Lρ −Rρ)(A)
)
= −4Tr
(
(L1/2ρ − R1/2ρ )(A) (L1/2ρ −R1/2ρ )(A)
)
= −4Tr
(
[ρ1/2, A]2
)
= 4I(ρ,A)
and this explains why the monotone metric associated with the function 14 (
√
x + 1)2 is called the Wigner-
Yanase monotone metric.
5 The main result
First of all we calculate the scalar curvature of Wigner-Yanase information using Theorem 3.5. If fwy(x) :=
1
4 (
√
x+ 1)2 we write Scal1wy for Scal
1
f .

Wigner-Yanase information on quantum state space 8
Theorem 5.1.
Scal1wy(ρ) =
1
4
(n2 − 1)(n2 − 2).
Proof. Let us calculate the auxiliary functions for cwy(x, y) := 4(
√
x+
√
y)−2. We get
h1(x, y, z) =
√
x
√
y + 3
√
x
√
z + 3
√
y
√
z + z
4
(√
x+
√
y
)2
(
√
x+
√
z)
(√
y +
√
z
)
h2(x, y, z) =
(√
x+
√
y + 2
√
z
)2
4 (
√
x+
√
z)2
(√
y +
√
z
)2
h3(x, y, z) =
√
z(√
x+
√
y
)
(
√
x+
√
z)
(√
y +
√
z
)
h4(x, y, z) =
1
(
√
x+
√
z)
(√
y +
√
z
)
.
Now one can verify by calculation that the symmetrization of h1− 12 h2 and the symmetrization of 2 h3−h4
vanish. Hence, by (3.1), the symmetrization of h vanishes, too. Since we sum up in formula (3.2) over all
triples of eigenvalues we may replace h with its symmetrization without changing the summation result.
Therefore
Scalwy(ρ) = 0 , Scal
1
wy(ρ) =
1
4
(n2 − 1)(n2 − 2) ∀ρ ∈ D1n.
In what follows we use the pull-back approach to derive (and explain) the above formula in a direct way.
Furthermore we deduce the geodesic distance and geodesic equation.
Let us denote by S the manifold {A ∈ Mn : TrAA∗ = 2, A = A∗}. Clearly, since S is the intersection of
the radius 2 sphere in Cn×n and the subspace of Hermitian matrices, it is isometric with a radius 2 sphere
Sn
2−1
2 .
Now, let ϕ : D1n → S ⊂ Cn×n, ϕ(ρ) := 2
√
ρ. Then we have the following result (see [26, 18, 28, 21]).
Theorem 5.2. The pull-back by the map ϕ of the natural metric on S ≡ Sn2−12 coincides with the Wigner-
Yanase monotone metric.
Proof. Let A and B be vectors tangent to D1n at ρ. Because ϕ(ρ)ϕ(ρ) = 4 ρ we get from the Leibniz rule
Dρϕ(A)
√
ρ +
√
ρDρϕ(A) = 2A Thus, the differential of ϕ at the point ρ is given by
Dρϕ(A) = 2
(
L1/2ρ +R
1/2
ρ
)−1
(A) .
Therefore the pull-back of the real part of the Hilbert-Schmidt metric yields
Dρϕ(Re 〈·, ·〉)(A,B) = Re 〈Dρϕ(A), Dρϕ(B)〉
= 4Re 〈(L1/2ρ +R1/2ρ )−1(A), (L1/2ρ +R1/2ρ )−1(B)〉
= 4 〈A, (L1/2ρ +R1/2ρ )−2(B)〉
= 4TrA (L1/2ρ +R
1/2
ρ )
−2(B)
= TrAcwy(Lρ, Rρ)(B) = 〈A,B〉wyρ
which was to be proved.

Wigner-Yanase information on quantum state space 9
From this result one can deduce the following
Theorem 5.3. For the geodesic distance, geodesic path and the scalar curvature of Wigner-Yanase infor-
mation the following formulae hold
1) geodesic distance
dwy(ρ, σ) = 2arccos(Tr(ρ1/2σ1/2)) (5.1)
2) geodesic path
γρ,σwy(t) = 2
((1 − t)√ρ + t√σ)2
Tr(((1 − t)√ρ + t√σ)2) (5.2)
3) scalar curvature
Scal1wy(ρ) =
1
4
(n2 − 1)(n2 − 2). (5.3)
Proof. The formulae are immediate consequences of the preceding theorem and of sphere geometry. Indeed
by the pull-back argument the Wigner-Yanase metric looks locally like a sphere of radius 2 of dimension
(n2 − 1). But for a sphere of this kind the sectional curvatures are all equal to 14 and therefore the scalar
curvature is given by 14 (n
2 − 1)(n2 − 2).
One may ask if other monotone metrics are the pull-back of some function ϕ different from the square
root. The rest of the section answers this question.
Definition 5.4. A monotone metric 〈·, ·〉ρ,f is a pull-back metric if there exists a manifold S ⊂ Mn and a
function ϕ ∈ C1(0,+∞) such that the pull-back metric of ϕ : D1n → S ⊂ Mn coincides with 〈·, ·〉ρ,f .
Proposition 5.5. Let 〈·, ·〉ρ,f be a monotone metric, let c = cf be the associated CM -function and let
ϕ ∈ C1(0,+∞). We have that 〈·, ·〉ρ,f is a pull-back metric by ϕ if and only if
(
ϕ(x) − ϕ(y)
x− y
)2
= c(x, y). (5.4)
Proof. Apply the formula (3.3) to tangent vectors in (TρD1n)o.
Definition 5.6. Let ϕ, χ ∈ C1(0,+∞). We say that (ϕ, χ) is a dual pair if there exist an operator monotone
f such that
ϕ(x) − ϕ(y)
x− y ·
χ(x) − χ(y)
x− y = c(x, y). (5.5)
where c = cf is the CM -function associated to f .
In such a case we say that f (or cf ) is a dual function. If (ϕ,ϕ) is a dual pair with respect to f (or cf )
we say that f (or cf ) is self-dual. Obviously one has
Proposition 5.7. To say that 〈·, ·〉ρ,f is a pull-back metric by ϕ it is equivalent to say that f (or cf ) is
self-dual with respect to ϕ.

Wigner-Yanase information on quantum state space 10
Definition 5.8. Two dual pairs (ϕ, χ), (ϕ˜, χ˜) are equivalent if there exist constants A1, A2, B1, B2 such that
A1A2 = 1
ϕ˜ = A1ϕ+B1
χ˜ = A2χ+B2.
Obviously equivalent pairs define the same CM -function. In what follows we consider dual pairs up to
this equivalence relation with the traditional abuse of language. We are ready to state the fundamental
result of the theory that classifies dual pairs.
Theorem 5.9. ([26, 27, 19]) Let ϕ, χ ∈ C1(0,+∞). Then (ϕ, χ) is a dual pair if and only if one of the
following two possibilities hold
(ϕ(x), χ(x)) = (
xp
p
,
x1−p
1− p) p ∈ [−1, 2] \ {0, 1}
(ϕ(x), χ(x)) = (x, log(x)).
Corollary 5.10. The function f(x) = 14 (
√
x+1)2 is the only self-dual operator monotone function, that is:
the Wigner-Yanase metric is the only pull-back metric among statistically monotone metrics.
6 Conclusions
Remark 6.1. Note that the formula (5.1) implies dwy(ρ, σ) ≤ 2pi. An analogous inequality holds for the
Bures metric (see [10], p.311). It seems that in the literature there are no other explicit formulas for the
geodesics distance. For example it is known that the formula
db(ρ, σ) =
√
2− 2Tr(ρ 12 σρ 12 ) 12 (6.1)
defines a metric on the state space whose infinitesimal counterpart (say the hessian) is the SLD-metric (that
is f(x) = 12 (1+x)). But this does not imply that Equation (6.1) is the geodesic distance of the SLD-metric.
Remark 6.2. In general it is difficult to give explicit formulae for geodesic paths of monotone metrics. In
the case of the Bures metric these geodesics can be given because they are projections of large circles on a
sphere in the purifying space (see [10] p.311 and [12][4]). For a discussion of geodesics for α-connections see
[28, 29].
Remark 6.3. A classical theorem classifies the spaces of costant curvature [30]. It is not known at the moment
if there are other monotone metrics of costant sectional and scalar curvature.
Remark 6.4. We have seen in the commutative case that for the Levi-Civita connection of the pull-back of
the square root it is available the decomposition
∇2 = 1
2
(∇m +∇e).
In the non-commutative case an analogous decomposition for the pull-back of the square root no longer
holds. Indeed, on one hand, the use of Umegaki relative entropy H(ρ, σ) = Tr(ρ(log ρ− log σ)) produces a
similar decomposition, but for the Bogoliubov-Kubo-Mori metric [33, 1, 23]. On the other hand, if one uses

Wigner-Yanase information on quantum state space 11
Hwy(ρ, σ) = 4(1 − Tr(ρ1/2σ1/2)) as a divergence on D1n and constructs the associated dualistic structure
(〈·, ·〉Hwy ,∇Hwy ,∇Hwy) (again following the lines of [14, 1]), then the construction is trivial, namely the
dual connections both coincide with the Levi-Civita connection of the Wigner-Yanase information. This is
easily seen on Pn where Hg(ρ, σ) reduces to Csiszar relative g-entropy: it is known that such an entropy
induces the α-geometry where α is given by the formula α = 3+2g′′′(1)/g′′(1) (see [1] p.57). For g = 4(1−√x)
this gives α = 0 that is the Fisher information case (see also [21]).
ACKNOWLEDGEMENTS
This research has been supported by the italian MIUR program ”Quantum Probability and Infinite Di-
mensional Analysis” 2001-2002. It is a pleasure to thank J. Dittmann for a number of valuable conversations
on this subject and for permitting us to reproduce in this paper the proof of Theorem 5.1.
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