Abstract
The seminal work of Kubo and Ando (Math Ann 246:205–224, 1979/80) provided us with an axiomatic approach to means of positive operators. As most of their axioms are algebraic in nature, this approach has a clear algebraic flavour. On the other hand, it is highly natural to take the geomeric viewpoint and consider a distance (understood in a broad sense) on the cone of positive operators, and define the mean of positive operators by an appropriate notion of the center of mass. This strategy often leads to a fixed point equation that characterizes the mean. The aim of this survey is to highlight those cases where the algebraic and the geometric approaches meet each other.
Keywords: Operator mean, Barycenter, Fixed point equation
An algebraic approach to operator menas
Although the paralell sum of positive definite matrice was considered already in 1969 by Anderson and Duffin [2], who used this operation to study electrical networks, and the geometric mean was introduced by Pusz and Woronowicz in 1975 [23], the first systematic study of means of positive operators is due to Kubo and Ando from 1980 [17], who provided an axiomatic approach to operator means which proved extremely influential in the last decades. The binary operations characterized by their axioms are now called Kubo-Ando connections and means, and the study of these means is flourishing research field within operator theory with intimate connections to quantum information theory.
This first section is very brief introduction to the Kubo-Ando theory which, however, aims to cover the most important concepts and phenomena. First, we need to fix some notation. Let be a complex Hilbert space, either finite or infinite dimensional, and let denote the set of all bounded linear operators on Let and stand for the set of all bounded self-adjoint, positive semidefinite, and positive definite (i.e., positive semidefinite and invertible) operators, respectively. The symbol I stands for the identity of and we consider the Löwner order induced by positivity on that is, by we mean that is positive semidefinite, and means that is positive definite. The spectrum of is denoted by The symbols and denote the first and second Fréchet derivatives, respectively.
Definition 1
(Operator connections and means) A binary operation
is called an operator connection if it satisfies the following three properties:
- (P1) Monotonocity in both variables:
- (P2) Transformer inequality:
- (P3) Continuity for decreasing sequences: if and in the strong operator topology, and similarly, and strongly, then
Furthermore, operator means are those operator connections that satisfy the
(P4) Normalization condition:
One of the most striking results of [17] is that there is one-by-one correspondence between operator connections and operator monotone functions mapping the positive half-line into itself. Namely, for every operator connection there is a unique operator monotone function such that
1 |
The correspondence described in (1) is in fact an affine order-isomorphism. It is instructive to take a look at those particular connections that have already been mentioned, and note that the parallel sum corresponds to the operator monotone function while the geometric mean corresponds to
Operator monotone functions mapping the positive half-line into itself admit a transparent integral-representation by Löwner’s theory. In [17], the following integral representation was considered:
2 |
where m is a positive Radon measure on the extended half-line There is another integral-representation formula for operator monotone functions which may be even more convenient than (2). By a simple push-forward of m by the transformation we get the following integral-representation of positive operator monotone functions on
3 |
where that is, for every Borel set This representation is also well-known and appears — among others — in [10]. Note that if m is absolutely continuous with respect to the Lebesgue measure and then the density of is given by
In particular, by (1) and (3), there is an affine isomorphism between operator connections and positive Radon measures on [0, 1]. We note that the normalization condition is satisfied if and only if
We denote by the set of all Borel probability measures on [0, 1], and by the center of mass of There is a natural way to assign a weight parameter to a mean namely, where f generates in the sense of (1) and generates f in the sense of (3). This weight parameter will play an essential role later when we turn to the discussion of generalized Hellinger distances. Here we only mention that for the weighted arithmetic, geometric, and harmonic means generated by the functions
respectively, we have That is, this weight parameter coincides with the usual one in the most important special cases.
The convex order is a well-known relation between probability measures; for we say that if for all convex functions we have It is clear that for all with we have where denotes the Dirac mass concentrated on x. For any fixed the map —which is the core of the integral representation (3) — is convex. Therefore, if then for all and hence for all where denotes the generating function corresponding to via (3) and denotes the operator mean corresponding to via (1). Consequently, if then is always positive, in particular, This quantity is exactly the generalized quantum Hellinger divergence we will discuss in Sect. 2.3.
Let us turn to the description of two interesting involutive operations on operator connections. The adjoint of a conncetion is denoted by and is defined by
4 |
for If f is the generating function of in the sense of (1), then is generated by Note that taking the inverse on positive definite operators is an order-reversing operation, and hence is operator monotone whenever f is so. Important examples of self-adjoint connections are the weighted geometric means generated by the power functions for The transpose of is denoted by and is defined by
5 |
An important fact is that if is represented by f, then its transpose is represented by xf(1/x). Therefore, a connection is symmetric if and only if its generator f satisfies
All in all, the Kubo-Ando theory is a beautiful and satisfactory theory of two-variable operator means. However, it leaves the problem multivariate operator means untouched. A clear advantage of the geometric approach to be presented in the next section is that it produces natural candidates for means of several positive operators. Apparently, a substantial part of the studies of Riemannian geometries on positive operators was motivated by the problem of finding appropriate multivariate counterparts of well-established bi-variate operator means.
A geometric approach to operator means
The notion of barycenter—or least squares mean—plays a central role in averaging procedures related to various topics in mathematics and mathematical physics. Given a metric space and an m-tuple in X with positive weights such that the barycenter is defined to be
6 |
The object defined above by (6) plays an important role in various areas of mathematics, and hence can be found under various names: it is sometimes called Fréchet mean or Karcher mean or Cartan mean. It is also called mean squared error estimator for the following reason: imagine that we want to determine an unknown element of the metric space and we can perform imperfect measurements many times. If we measure with relative frequency then an attractive estimation of the unknown object is (6) which minimizes the weighted mean squared error from the measured objects.
In the sequel, we review two distinguished metrics on the cone of positive definite operators on finite dimensional Hilbert spaces, namely the Riemannian trace metric and the Bures-Wasserstein metric. The focus of these review will be on the barycenters determined by these metrics.
The Riemannian trace metric
The convex Boltzmann entropy (or H-functional) of a random variable X with probability density is given by
7 |
This is a particularly important functional; for instance, the heat equation
can be seen as the gradient flow for the Boltzmann entropy as potential (or “energy”) in the differential structure induced by optimal transportation [11–13].
Let us restrict our attention to special random variables. A circularly-symmetric centered complex Gaussian distribution on is completely described by its covariance matrix The probability density of such a zero-mean complex Gaussian is given by
Consequently, the Boltzmann entropy of can be given in the following simple closed form:
8 |
where C(N) is an irrelevant constant depending only on the dimension N. So the Boltzmann entropy is a smooth convex functional on the sub-manifold of centered, circularly-symmetric, non-degenerate complex Gaussians. We identify this sub-manifold with the cone of positive definite matrices by the convenient identification of the random variable with its covariance matrix.
Direct computations shows that the Hessian (that is, the second derivative) of the Boltzmann entropy (8) on these appropriate Gaussians is given by
9 |
This is a collection of positive definite bilinear forms on the tangent spaces
that depends smoothly on the foot point A. Therefore, (9) is a Riemannian metric tensor field. The global metric induced by this Riemannian tensor field
is called the Riemannian trace metric (RTM).
A particularly nice feature of the Riemannian trace metric is that the geodesic curve connecting the points has the following simple closed form:
10 |
That is, the geodesic consists of the weighted geometric means where the weight parameter t runs from 0 to 1. Consequently, the derivative of the geodesic reads as
11 |
and the RTM has the following simple closed form:
12 |
The notation stands here and throughout this paper for the Hilbert-Schmidt norm
Therefore, the barycenter (6) of the positive definite operators with probability weights is
13 |
One can use Karcher’s formula [14, Theorem 2.1] to compute the gradient of the objective function
14 |
and deduce that the barycenter (13)—which is more often called Karcher mean in this context—coincides with the unique positive definite solution of the Karcher equation
15 |
See [18] and [4] for alternative approaches on the derivation of the Karcher Eq. (15).
The Bures-Wasserstein metric
The classical optimal transport (OT) problem is to arrange the transportation of goods from producers to consumers in an optimal way, given the distribution of the sources and the needs (described by probability measures and ), and the cost c(x, y) of transporting a unit of goods from x to y. Accordingly, a transport plan is modeled by a probability distribution on the product of the initial and the target spaces, where is the amount of goods to be transferred from x to y, and hence the marginals of are and So the optimal transport cost is the minimum of a convex optimization problem with linear loss function:
16 |
where denotes the ith marginal of and X is the initial and Y is the target space.
OT costs (16) give rise to OT distances (Wasserstein distances) on measures for certain cost functions c(., .). A prominent example is the quadratic Wasserstein distance between probabilities on having finite second moment defined by
17 |
The above definition (17) of the 2-Wasserstein metric is static in nature. It refers only to the initial and final distributions of the mass to be transported. The dynamical theory of mass transport concerns on the contrary flows of measures connecting the initial and final states. The optimization problem is minimizing the total kinetic energy needed to perform the transport. More precisely, the task is to minimize the kinetic energy over flows of measures connecting the initial and final distributions and time-depending velocity fields governing the flows—we say that the velocity field governs the flow if they satisfy the linear transport equation (or continuity equation)
18 |
Accordingly, the formula for the minimal kinetic energy (MKE) needed to transform to if the total time allocated for the transport is T, reads as follows:
19 |
A seminal result of Benamou and Brenier [3] connects the static and the dynamic theory beautifully: the static 2-Wasserstein distance (17) and the minimal kinetic energy needed to perform the dynamics are essentially the same. More precisely, the Benamou-Brenier formula [3] tells us that the minimizing flow of measures in (19) is given by the displacement interpolation, and
20 |
Now, if we take another look at (19), keeping (20) in mind, we may observe that the 2-Wasserstein distance is given by a formula that looks very much like a Riemannian geodesic formula. And indeed, there is a Riemannian metric tensor field on the space of probabilities that gives rise to the 2-Wasserstein metric. The discussion of this Riemannian metric in general is beyond the scope of this survey—we will be interested only in the special case of centered Gaussian measures. We only mention that this line of research was pioneered by Otto [20] and we refer the interested reader to Subsection 8.1.2. of [25] for a detailed description of the theory. We must note, however, the groundbreaking discovery of Jordan et al. [11–13] who proved that the heat flow is the gradient flow of the Boltzmann entropy with respect to the 2-Wasserstein Riemannian metric on probability densities.
We shall restrict our attention to non-degenerate centered Gaussian measures on that we identify with their non-singular covariance matrices. If is the law of the random variable and is the law of then the quadratic Wasserstein distance between and admits the following closed form that refers only to the covariance matrices [1, 24]:
21 |
The distance between the positive definite operators acting on the Hilbert space that appears on the right-hand side of (21) has a quantum information theoretic interpretation, as well. In that context, the name Bures distance is used more frequently. We refer the reader to [8] for a thorough study of this Bures-Wasserstein distance given for by
22 |
We note furthermore that the isometries of the density spaces of -algebras with respect to (22) have been determined by Molnár in [19]. The geodesic line segment in the Bures-Wasserstein metric connecting A with B has the following simple closed form [7, 8]:
23 |
where t runs from 0 to 1, and the square roots of the non-Hermitian operators are understood as follows: and
The elements of the geodesic segment (23) are Bures-Wasserstein barycenters with appropriate weights [7], namely,
24 |
An attractive feature of the Bures-Wasserstein barycenter is that it is characterized by a rather simple fixed point equation: the unique minimizer of the functional
25 |
on the positive definite cone coincides with the unique positive definite solution of the operator equation
26 |
This striking result was first proved in [1] and a very transparent presentation of the proof can be found, e.g., in [8, Section 6].
Barycenters for generalized quantum Hellinger distances
As one can see in (22), the Bures-Wasserstein distance is the square root of the distance between the trace of the arithmetic mean of A and B and the trace of a certain geometric mean of the same operators. Therefore it is a non-commutative version of the Hellinger distance of probability vectors defined by
27 |
A thorough study of a variety of quantum Hellinger distances is presented in [5]. Somewhat later, generalized quantum Hellinger distances were introduced and studied with a strong emphasis on the characterization of the barycenter [21]. Very recently, a one-parameter family of distances including the Bures-Wasserstein distance and a certain Hellinger distance was proposed and studied in [16].
Given a Borel probability measure the corresponding generalized quantum Hellinger divergence is given by
28 |
where is the center of mass of and is the Kubo-Ando mean generated by the operator monotone function in the sense of (1) and is determined by in the sense of (3). We believe that the characterization of the barycenter of finitely many positive operators by a fixed point equation is instructive, especially as we did not present the proof of the analogous result for the Bures-Wasserstein metric. The following result and its proof appeared originally in [21].
Theorem 1
Let and let be the generalized quantum Hellinger divergence generated by given in (28). The barycenter of the positive definite operators with positive weights with respect to i.e.,
29 |
coincides with the unique positive definite solution of the fixed point equation
30 |
where stands for the absolute value of an operator, that is,
Proof
Assume that the positive definite operators and the weights are given. By the strict concavity of the function
is strictly convex on see, e.g., [9, 2.10. Thm.]. Therefore, there is a unique solution of (29), and it is necessarily a critical point of the function That is, it satisfies
31 |
Easy computations give that
32 |
where for a positive definite operator A, the map is defined by
33 |
By differentiating (3), we have
34 |
for Consequently,
35 |
By the linearity and the cyclic property of the trace, we get from (32) and (35) that (31) is equivalent to
36 |
This latter equation amounts to
37 |
Multiplying by from both left and right gives the desired operator Eq. (30).
Connections between the algebraic and the geometric approaches
This section is devoted to the phenomenon when the algebraic and the geometric approach to operator means meet each other, that is, when Kubo-Ando means admit barycentric interpretation.
It is well known that special Kubo-Ando operator means, namely, the arithmetic and the geometric means admit divergence center interpretations. The arithmetic mean is clearly the barycenter for the Euclidean metric on positive operators:
A much more interesting fact is that the geometric mean is the barycenter for the Riemannian trace metric that is,
38 |
The barycentric representation (38) of the bivariate geometric mean opened the gate for the definition of the multivariate geometric mean as the barycenter with respect to the Riemannian trace metric. This definition was introduced by Moakher [18] and Bhatia-Holbrook [6]. A recent result in [22] tells us that every symmetric Kubo-Ando means admits a divergence center interpretation. Now we turn to the detailed explanation of this latter result, and we follow the presentation of [22].
Let be a symmetric Kubo-Ando operator mean, and let be the operator monotone function representing in the sense that
39 |
Clearly, and the symmetry of implies that for and hence . We define
by
40 |
Obviously, , and as . Since is strictly monotone increasing, so is , and hence is strictly convex on its domain. Now we define the quantity
41 |
for positive definite operators such that the spectrum of is contained in We define if It will be important in the sequel that by [9, 2.10. Thm.] the strict convexity of implies that is strictly convex (whenever finite) for every and A. Now we are in the position to formalize the divergence center interpretation of symmetric Kubo-Ando means. The precise statement reads as follows.
Theorem 2
For any
42 |
That is, is a unique minimizer of the function on .
Proof
By the strict convexity of it is sufficient to show that is a critical point, and therefore a unique minimizer. First we compute the derivative
43 |
for all . Since , we get
44 |
for all . Substituting into the derivative above, the right hand side of (44) becomes
45 |
Since the operator mean is symmetric, that is
a similar computation for the derivative
at gives
46 |
for all . Using (45) and (46) we get for the derivative
for all . So we obtained that is a critical point and hence a unique minimizer of
The above characterization of symmetric Kubo-Ando means as barycenters (Theorem 2) naturally leads to the idea of defining weighted and multivariate versions of Kubo-Ando means as minimizers of appropriate loss functions derived from the divergence Given a symmetric Kubo-Ando mean a finite set of positive definite operators and a discrete probability distribution with we define the corresponding loss function by
47 |
where is defined by (41).
However, in the weighted multivariate setting, when is smaller than the whole positive half-line then some undesirable phenomena occur which are illustrated by the next example.
Consider the arithmetic mean generated by with Let satisfy In this case, for any the loss function is finite only if So the barycenter of and with weights is separated from for every even for values very close to 0.
To exclude such phenomena, from now on, we assume that the range of is maximal, that is, and hence is defined on the whole positive half-line Consequently, is always finite, and hence so is on the whole positive definite cone
Definition 2
Let be a symmetric Kubo-Ando operator mean such that which is the operator monotone function representing in the sense of (39), is surjective. Let be defined as in (40), and be defined as in (41). We call the optimizer
48 |
the weighted barycenter of the operators with weights .
By Theorem 2, this barycenter may be considered as a weighted multivariate version of Kubo-Ando means. To find the barycenter we have to solve the critical point equation
49 |
for the strictly convex loss function where the symbol
stands for the Fréchet derivative of at the point For any we have
that is, the equation to be solved is
50 |
By the definition of see (40), for and hence the critical point of the loss function is described by the equation
51 |
For the generating function is and hence the inverse is In this case, the critical point equation (51) describing the barycenter reads as follows:
52 |
Note that (52) may be considered as a generalized Riccati equation, and in the special case the solution of (52) is the symmetric geometric mean
More generally, if and then (52) has the following form:
or equivalently
53 |
Recall that for positive definite A and B, the Riccati equation
has a unique positive definite solution, that is the geometric mean
We can observe that (53) is the Riccati equation for the weighted harmonic mean
and the weighted arithmetic mean , ie
54 |
Hence the solution of (53) is the geometric mean of the weighted harmonic and the weighted arithmetic mean
55 |
It means that in this case the weighted barycenter with respect to does not coincide with the weighted geometric mean, nevertheless
that is, is the Kubo-Ando mean of and with representing function
These means were widely investigated in [15].
We note that the critical point equation (52) can be rearranged as
56 |
This is the Ricatti equation for the weighted multivariate harmonic mean and arithmetic mean hence the barycenter coincides with the weighted -mean of Kim, Lawson, and Lim [15], that is,
57 |
Acknowledgements
I am grateful to the anonymous reviewer for his/her valuable comments and recommendations.
Funding
Open access funding provided by HUN-REN Alfréd Rényi Institute of Mathematics.
Declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no Conflict of interest.
Footnotes
D. Virosztek is supported by the Momentum Program of the Hungarian Academy of Sciences under grant agreement no. LP2021-15/2021, and partially supported by the ERC Synergy Grant No. 810115.
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