Characterizations of Positive Operator-Monotone Functions and Monotone Riemannian Metrics via Borel Measures

: We show that there is a one-to-one correspondence between positive operator-monotone functions on the positive reals, monotone Riemannian metrics, and ﬁnite positive Borel measures on the unit interval. This correspondence appears as an integral representation of weighted harmonic means with respect to that measure on the unit interval. We also investigate the normalized/symmetric conditions for operator-monotone functions. These conditions turn out to characterize monotone metrics and Morozowa–Chentsov functions as well. Concrete integral representations of such functions related to well-known monotone metrics are also provided. Moreover, we use this integral representation to decompose positive operator-monotone functions. Such decomposition gives rise to a decomposition of the associated monotone metric.


Introduction
This paper is motivated from functional analysis aspects in quantum statistical mechanics. In classical statistics, the (Fisher) information is a measurement of the amount of information that an observable random variable conveys about an unknown parameter of its distribution. The quantum Fisher information in quantum statistics is an analogous concept to the classical one; see e.g., [1]. Recall that a physical observable of a quantum mechanical system is represented by a self-adjoint operator A acting on a Hilbert space (H, ·, · ). The state of the physical system is often modeled by a unit vector x in H. In this case, the expectation of A in that state is given by the inner product Ax, x . If dim H < ∞, the states (i.e., the expectation of the states) can be realized as Tr(DA) where D is the density matrix associated with the state. In order for a difficult-to-measure observable to measure a conserved quantity, Wigner and Yanase [2] proposed the so-called skew information defined by  Recall that the t-weighted harmonic mean ! t is defined by This paper focuses on a one-to-one correspondence between four kind objects: We show that there is a bijection between the finite Borel measures µ on [0, 1] and the positive operator-monotone functions f via a canonical representation Moreover, the map µ → f is bijective, affine, and order-preserving. This means that the functions x → 1 ! t x for t ∈ [0, 1] form building blocks for the set OM(R + ). This integral representation reflects some interesting information of operator-monotone functions. In fact, a function f ∈ OM(R + ) is normalized if and only if its associated measure µ is a probability measure. We also show that an f ∈ OM(R + ) is symmetric (in the sense that f (x) = x f (1/x) for all x > 0) if and only if the corresponding measure µ is invariant under the function t → 1 − t on [0, 1]. The normalized/symmetric conditions on f ∈ OM(R + ) turn out to characterize such conditions for the associated monotone metrics and the associated Morozowa-Chentsov functions as well.
The canonical representation (1) also reflects the geometry of the set of (symmetric) normalized operator-monotone functions. More precisely, the extreme points of the convex set of such functions are obtained via the affinity of the map µ → f . Furthermore, the representation (1) has benefits in decomposing positive operator-monotone functions as the sum of three explicit parts, namely, its singularly-discrete part, its absolutely-continuous part, and its singularly-continuous part. Such decomposition leads to a decomposition of the associated monotone metrics as well.
The rest of this paper is organized as follows. In Section 2, we recall fundamental results about monotone Riemannian metrics on the smooth manifold of invertible density matrices. Then, in Section 3, we establish an integral representation for positive operator-monotone functions with respect to a Borel measure on the unit interval. Moreover, we investigate some attractive properties from such representations. In Sections 4 and 5, we illustrate monotone metrics of type singularly-discrete and of type absolutely-continuous, respectively. Section 6 deals with decompositions of operator-monotone functions. We summarize the paper in Section 7.

Monotone Riemannian Metrics on the Smooth Manifold of Invertible Density Matrices
We denote the set of (n × n) complex matrices by M n . Recall that a density matrix is a positive semidefinite matrix with trace 1. The set D n of all (n × n) invertible density matrices is an open subset of the set of (n × n) Hermitian matrices. This is because the function A → x * Ax is continuous for each x ∈ C n . Hence, the set D n forms a smooth manifold.
A metric K on M n is a parametrized family {K D } D∈D n of sesquilinear forms K D : M n × M n → C such that The metric K is said to be monotone if for every D ∈ D n , A ∈ M n and stochastic map T : M n → M n , we have Here, recall that a linear map T : M n → M n is said to be stochastic if T is completely positive and T preserves invertible density matrices. It turns out that a differentiable monotone metric on D n determines a Riemannian metric; see more information in [10].
Let ·, · be the Hilbert-Schmidt inner product on M n , i.e., A, B = Tr(A * B) for any A, B ∈ M n .
For each D ∈ D n , let L D and R D be the left (right) multiplication operators from M n to itself, i.e., L D : X → DX and R D : X → XD. Then, (L D , R D ) is a pair of commuting invertible positive operators such that L t D = L D t and R t D = R D t for any t ∈ R. Morozowa and Chentsov [5] gave an explicit form of a monotone metric K. Indeed, for each D ∈ D n and A ∈ M n , the value K D (A, A) appears in terms of the so-called associated Morozowa-Chentsov function, and we obtain K D (A, B) by means of polarization. Petz [6,7] improved this representation to the Hilbert-Schmidt inner product and operator-monotone function on R + as follows: [6,7]) There is a one-to-one correspondence between operator-monotone function f : R + → R + and monotone metric K such that for any D ∈ D n and A, B ∈ M n , where K D = R for any x, y > 0.
Here, g(L D , R D ) is computed by applying functional calculus on the pair of commuting operators L D and R D .

Characterizations of Positive Operator-Monotone Functions and Monotone Metrics
In this section, we characterize operator-monotone functions from R + to R + in terms of finite positive Borel measures on the unit interval. These results give rise to characterizations of monotone metrics as well. The normalized/symmetric conditions for monotone metrics and operator-monotone functions are also considered.
For real sequences, we use the notation x n ↑ x for the case that (x n ) is an increasing sequence converging to x. The expression x n ↓ x is used for the decreasing case.
is well-defined and continuous.
For each x > 0, the positivity of the function t → 1 ! t x implies that the resulting integral f (x) is positive.
We shall show that f is left and right continuous. First, note that the increasingness of function be a sequence in R + such that x n ↑ x. For convenience, put φ(t) = 1 ! t x and φ n (t) = 1 ! t x n for each n ∈ N and t ∈ [0, 1]. Then (φ n (t)) ∞ n=1 is a increasing sequence of positive real numbers such that φ n (t) → φ(t) as n → ∞ for each fixed t. It follows that the sequence ( f (x n )) = ( φ n ) is increasing. Moreover, the monotone convergence theorem implies that This means that f (x n ) ↑ f (x). Thus, f is left continuous. For the right continuity of f , let x > 0 and consider a sequence (x n ) ∞ n=1 in R + such that x n ↓ x. For convenience, put φ(t) = 1 ! t x and φ n (t) = 1 ! t x n for each n ∈ N and t ∈ [0, 1]. Then, for each fixed . Therefore, f is right continuous.

Lemma 2.
A necessary and sufficient condition for a continuous function f : R + → R + to be operator-monotone is that there is a unique finite Borel measure ν on [0, ∞] such that Proof. See, e.g., ([20], Theorem 2.7.11).

Theorem 2.
There is a bijection between the set of finite Borel measure on [0, 1] and the set OM(R + ) that sending a measure µ to an f ∈ OM(R + ) satisfying the representation Moreover, the map µ → f is bijective, affine, and order-preserving.
Proof. The function f in (5) is well-defined and continuous by Lemma 1.
To show that f is operator-monotone, let us consider invertible operators A and B on a Hilbert space such that A B.
The monotonicity of weighted harmonic means and Bochner integrals implies that This means that the map µ → f is well-defined. For the injectivity of this map, let µ 1 and µ 2 be finite Borel measures Then, for each i = 1, 2 and x > 0, we get To show the surjectivity of this map, we consider f ∈ OM(R + ). By Lemma 2, there is a finite Borel measure ν on [0, ∞] such that (4) holds. Define a finite Borel measure µ on [0, 1] by µ = νΨ −1 . A direct computation shows that Hence, the map µ → f is surjective. It is straightforward to show that this map is affine and order-preserving.
From Theorems 1 and 2, we get: The work [7] studied the normalized condition on a monotone metric K D (I, The next result gives a complete characterization of normalized monotone metrics. Corollary 2. Let K be a monotone metric on M n with the associated function f ∈ OM(R + ), the associated Morozowa-Chentsov function g, and the associated measure µ on [0, 1]. Then the following statements are equivalent: Thus, there is a one-to-one correspondence between normalized monotone metrics, normalized positive operator-monotone functions, and probability Borel measures on [0, 1] via the representations (2), (3), and (5).
Proof. The content of ( [7], Corollary 6) indicates that the assertion (i) is equivalent to (ii). The assertion (ii) is clearly equivalent to (iii). The equivalence between (ii) and (iv) follows from the integral representation (5) in Theorem 2.
This corollary asserts that every normalized positive operator-monotone function can be regarded as an average of the special operator-monotone functions x → 1 ! t x for t ∈ [0, 1].
Recall from [21] that if f ∈ OM(R + ), then the transpose of f , defined by x → x f (1/x) for any x > 0, also belongs to OM(R + ). The function f is said to be symmetric if it coincides with its transpose. We say that a Borel measure µ The associated measure of transpose of f ∈ OM(R + ) can be computed as follows.
Proposition 1. Let f ∈ OM(R + ) be a function with associated measure µ. Then the associated measure of the transpose of f is given by µΘ where Θ : Proof. It follows from the integral representation (5) of f that By Theorem 2, the transpose of f has µΘ as its associated measure.
The next result provides a complete characterization of symmetric monotone metrics. (i) K is symmetric.
Thus, there is a one-to-one correspondence between symmetric monotone metrics, symmetric positive operator-monotone functions, and symmetric Borel measures on [0, 1] via the representation (2) and the integral representation Proof. The content of ([7], Theorem 7) indicates the equivalence between (i) and (ii). The latter condition is equivalent to (iii). From the formula (6) in Proposition 1 and the uniqueness of the associated measure (Theorem 2), we have that f (x) = x/ f (1/x) if and only if µ = µΘ. Thus, the assertions (ii) and (iv) are equivalent. Theorem 1 establishes the correspondence between monotone metrics and operator-monotone functions via (2). From the canonical representation (5) and an observation that we can write the function f in a symmetric form (7), relating f to its associated measure.
It follows from Corollaries 2 and 3 that there is a one-to-one correspondence between normalized symmetric positive operator-monotone functions and probability symmetric Borel measures on [0, 1] via the integral representation (7).
Note that the set of normalized (symmetric) operator-monotone functions on R + is a convex set. We denote the Dirac measure concentrated at a point t by δ t . Now, the integral representation (5) also reflects the geometry of this set as follows.  To prove (1), note that the Dirac measures are extreme points of that set. Suppose there is a probability measure µ on [0, 1] which is an extreme point, but µ is not a Dirac measure. Then there is an s ∈ [0, 1] such that 0 < µ({s}) < 1. Define We can verify that ν is a probability positive measure on [0, 1]. It follows that i.e., µ is a non-trivial convex combination of two probability Borel measures. This contradicts the assumption that µ is an extreme point. To prove (2), consider the measure (δ t + δ 1−t )/2 where t ∈ [0, 1]. Suppose that there are a constant α ∈ (0, 1) and probability measures µ 1 , µ 2 on [0, 1], which are invariant under the function Θ, such that For any s ∈ [0, 1] − {t, 1 − t}, we have We now get µ 1 = (δ t + δ 1−t )/2 and, similarly, µ 2 = (δ t + δ 1−t )/2. Hence, the trivial combination is the only convex combination for (δ t + δ 1−t )/2, i.e., this measure is an extreme point of that set.

Singularly-Discrete Monotone Metrics
This section provides typical examples of "singularly-discrete"monotone metrics.

Example 1.
Consider the operator-monotone function f (x) = 1. The associated Morozowa-Chentsov function is given by c(x, y) = 1/y. For each D ∈ D n , we have K D = R D and thus K −1 D = R D −1 . Hence, its associated monotone metric is given by The associated Borel measure on [0, 1] is given by the Dirac measure δ 0 .

Example 2.
Consider the operator-monotone function f (x) = x. The associated Morozowa-Chentsov function is given by c(x, y) = 1/x. For each D ∈ D n , we have K D = L D and thus K −1 D = L D −1 . Hence, its associated monotone metric is given by Its associated Borel measure is given by the Dirac measure δ 1 .

Example 3.
For each t ∈ [0, 1], the operator-monotone function x → 1 ! t x corresponds to the Dirac measure δ t . By affinity of the map µ → f , the measure ∑ n i=1 a i δ t i , where t i ∈ [0, 1] and a i 0, is associated to the function

Example 5.
The largest monotone metric is the metric represented by the Morozowa-Chentsov function This metric is associated to the operator-monotone function f (x) = 2x/(x + 1). It follows from Example 3 that its associated measure of this metric is δ 1/2 .
More generally, let us consider the operator-monotone function where α ∈ [0, 1]. For each D ∈ D n , a direct computation reveals that Its associated measure is given by the probability measure δ α .

Absolutely-Continuous Monotone Metrics
In this section, we illustrate the one-to-one correspondence between positive operator-monotone functions, absolutely-continuous Borel measures, and certain type of monotone metrics. We call such metrics absolutely-continuous monotone metrics. Such functions will be typical examples of absolutely-continuous type in the next section. Example 6. Consider the operator-monotone function x → log (1 + x). Using improper integration, we have where h(t) = (1/t)χ [1/2,1] (t). Here, χ [1/2,1] denotes the characteristic function on the set [1/2, 1]. Thus, its associated measure is given by the absolutely-continuous measure having h as its density function. Hence, the function log (1 + x) gives rise to a monotone metric.

Example 7.
For each 0 < α < 1, consider the Morozowa-Chentsov function c(x, y) = x −α y α−1 . This function is associated to the operator-monotone function f (x) = x α . For each D ∈ D n , we have Thus, the associated monotone metric is given by To compute its associated measure, we recall a standard result from contour integrations: Denoting Ψ(t) = t/(1 − t), we have where the associated measure µ is given by This means that the associated measure of f is given by dµ(t) = h(t) dt, where Using integration by parts twice, we have Hence, .

Example 9.
Consider the dual 1/ f (1/x) of the function f in Example 8. We have the integral representation that is, this function corresponds to the Lebesgue measure.

Explicit Descriptions of Positive Operator-Monotone Functions
In this section, we give an explicit description of arbitrary operator-monotone functions on R + by decomposing them into typical concrete ones we have encountered in Sections 4 and 5. It is important to note that the one-to-one correspondences (2) and (5) are both affine. Thus, if we can decompose an operator-monotone function, then it gives rise to a decomposition of the associated monotone metric as well. We also investigate such decomposition when such functions are normalized or symmetric. For this section, we denote Lebesgue measure by m.
(iii) its associated measure of f sc is continuous and mutually singular to m.
Moreover, the associated measure of f sd is given by ∑ t∈D a t δ t .
Proof. Let µ be the associated measure of f . By a standard result in measure theory (e.g., [23]), there is a unique triple (µ ac , µ sc , µ sd ) of finite Borel measures on [0, 1] such that µ = µ ac + µ sc + µ sd where (I) µ sd is a discrete measure (II) µ ac is absolutely continuous with respect to m (III) µ sc is a continuous measure mutually singular to m. Then f ac , f sd , f sc ∈ OM(R + ) and f = f ac + f sd + f sc . The condition (I) means precisely that there are a countable set D ⊆ [0, 1] and a family {a λ } λ∈D in R + such that ∑ λ∈D a λ < ∞ and µ sd = ∑ t∈D a t δ t . Hence, we arrive at the formula (8). Note that this series converges since The condition (II) means precisely the condition (ii) by Radon-Nikodym theorem. The uniqueness of ( f ac , f sc , f sd ) follows from the uniqueness of (µ ac , µ sc , µ sd ) and the correspondence between operator-monotone functions and measures. The measure ∑ t∈D a t δ t is associated to f sd since the associated measure of x → 1 ! t x is δ t for each t ∈ [0, 1] by Example 3.
Theorem 4 asserts that every f ∈ OM(R + ) consists of three parts. The singularly-discrete part f sd is a countable sum of x → 1 ! t x for t ∈ [0, 1], given by (8). Such type of functions include the straight lines with positive slopes, the constant functions, the multiple functions x → kx, and the examples in Section 4. The absolutely-continuous part f ac arises explicitly as an integral with respect to Lebesgue measure given by (9). Typical examples of such functions are already provided in Section 5. The singularly-continuous part f sc admits an integral representation with respect to a continuous measure mutually singular to Lebesgue measure.  Proof. It follows from Corollary 2 and Theorem 3.
We say that a density function h : [0, 1] → R + is symmetric if h • Θ = h. Next, we decompose a normalized operator-monotone function as a convex combination of normalized operator-monotone functions. such that f = k ac f ac + k sc f sc + k sd f sd , k ac + k sc + k sd = 1, and (i) there are a countable set D ⊆ [0, 1] and a family {a t } t∈D ⊆ [0, 1] such that ∑ t∈D a t = 1 and f sd (x) = ∑ λ∈D a t (1 ! t x) for each x ∈ R + ; (ii) there is a (unique m-a.e.) integrable function h : [0, 1] → R + with average 1 such that f ac (x) = 1 0 h(t)(1 ! t x) dm(t) for x ∈ R + ; (iii) its associated measure of f sc is continuous and mutually singular to m.

Conclusions
There are strongly connections between positive operator-monotone functions on the positive reals, monotone (Riemannian) metrics, Morozowa-Chentsov functions, and finite Borel measures on the unit interval. Indeed, there are one-to-one correspondences between the four kind objects. It follows that certain properties (e.g., symmetry, normalization) of monotone metrics can be investigated through the associated properties of the other objects. Moreover, we can decompose the operator-monotone functions (thus, the other objects) into three parts, namely, its singularly-discrete part, its absolutely-continuous part, and its singularly-continuous part. Concrete monotone metrics in quantum Fisher information theory are illustrated with the associated operator-monotone functions, the associated Morozowa-Chentsov functions, and the associated measures.
Author Contributions: Project administration, P.C.; Writing -original draft, P.C.; Writing -review and editing, S.V.S. All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.