Differential Geometric Aspects of Parametric Estimation Theory for States on Finite-Dimensional C^∗-Algebras

The purpose of this work is to present the formulation of estimation theory in the framework of $C^{\ast }$-algebras, with particular attention to differential geometric aspects. Although estimation theory is a well-developed subject both in the classical case of probability distributions [1,2,3,4] and the quantum case of density operators [5,6,7,8,9,10,11,12], and even if quantum estimation theory builds upon classical estimation theory, there is no unifying picture for these subjects. By unifying picture, we mean a mathematical framework in which estimation theory is placed in such a way that the classical and quantum cases appear as particular instances of the general theory. We believe that such a formulation may be helpful in obtaining a better understanding of the similarities and differences of classical and quantum estimation theory. This idea may be considered as the driving force of this work.

Roughly speaking, the main problem estimation theory tries to address is to infer the value of some parameters characterizing the state of the “physical system” under investigation by the theoretical manipulation of the outcomes of experiments performed on such system.

In the classical case, the state of the system is described by a probability distribution on the space of outcomes of the experiment, and the goal of estimation theory is to infer the value of some parameters characterizing the true probability distribution describing the system (e.g., the mean and/or variance for the case of Gaussian distributions) from the outcomes of the experiment. As such, estimation theory is well developed both in its asymptotic and non-asymptotic regimes. Arguably, one little black spot of the theory is that the parameter spaces characterizing the probability distributions under study are usually taken to be homeomorphic to open subsets of some finite-dimensional Euclidean space. Even if this assumption is justified in most of the models, it necessarily introduces some simplifications related with the “nice” structure of the parameter spaces as smooth manifolds. As an example, the existence of global coordinates often lead to the definition of objects that are coordinate-dependent (see, for instance, in ([3] ch. 4), where it is clearly stated that the notion of unbiased estimator developed there is coordinate-dependent). We believe it is healthy to formulate the theory in order to avoid these issues and better comprehend the coordinate-independent aspects of the theory. This geometric attitude already proved itself useful in classical Newtonian, Lagrangian, and Hamiltonian mechanics [13,14,15,16,17,18], in thermodynamics and statistical physics [19,20,21,22], and in quantum mechanics [23,24,25,26]. Clearly, there already have been efforts to formulate classical estimation theory in this direction [27,28,29,30], and here we try to encapsulate the spirit of these works in our formulation of estimation theory on $C^{\ast }$-algebras.

On the other hand, in the quantum case, the state of the system is no longer a probability distribution, it is a density operator on the Hilbert space associated with the system. This adds, at the same time, complexity and richness to the problem of estimation. A first layer of added complexity refers to the need of a statistical interpretation of a given quantum state. Since the dawn of quantum mechanics, the issue of the physical interpretation of Schrödinger’s wavefunction was recognized to be a fundamental question. The idea of interpreting the square modulus of the wavefunction as a probability distribution paved the way to the statistical interpretation of quantum states through what is now called the Born rule [31]. Essentially, the Born rule describes a “procedure” to associate a probability distribution on a suitable outcome space with a given quantum state. Clearly, this depends on both the quantum state and the choice of the outcome space, and this means that there is more than one way to associated probability distributions with quantum states. From the mathematical point of view, the choice of the statistical interpretation is described by a positive operator-valued measure (POVM) on the Hilbert space of the system [10,32]. Accordingly, in order to set up the estimation problem for a given parametric model of quantum states, we need to operate a preliminary choice concerning the POVM “inducing” the statistical interpretation. Of course, this choice influences the estimation problem we set up, and different choices in general lead to different solutions of the associated estimation problem. All this obviously adds a layer of complexity to the estimation problem, but, simultaneously, it opens new possibilities to outperform classical limits of estimation because of the peculiar features of quantum states (e.g., entanglement). Indeed, in the quantum case it is possible to give a precise mathematical meaning to the assertion “measuring one copy N times is less informative than measuring N copies one single time” [12,33,34,35,36]. This assertion relies on the phenomenon of entanglement which is absent in the classical realm, and thus highlights an important difference between the classical and quantum estimation theory.

As mentioned before, the goal of this article is to introduce a theoretical framework that allows us to treat the classical and quantum case simultaneously. Specifically, our choice is to consider the theory of $C^{\ast }$-algebras as the backbone of our construction because both probability distributions and quantum states may be realized as linear functionals on suitable $C^{\ast }$-algebras. In the case of probability distributions, this is basically the duality between probability measures and functions given by the Riesz theorem. For quantum states, this comes directly from the axiomatic structure of the theory. The main difference between the two cases is that the algebras involved are commutative in the former case and non-commutative in the latter. In this general framework, probability distributions and quantum states represent different realizations of the notion of state on a $C^{\ast }$-algebra $\mathcal{A}$. The space of states $\mathcal{S}$ is a convex subset of the dual space of $\mathcal{A}$, and the study of its differential geometry is a fascinating subject. The rich algebraic structure of $C^{\ast }$-algebras translates into a rich geometrical structure for their spaces of states that is perfectly suited for the formulation of parametric estimation theory.

The use of $C^{\ast }$-algebras as a theoretical framework to study the geometry of quantum states is not new [37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56]. However, the focus was essentially always on the algebra of bounded linear operators on the Hilbert space of the quantum system, and not on a generic $C^{\ast }$-algebra. While this restriction may seem not particularly relevant for most practical purposes, it is certainly so from the theoretical point of view. Indeed, some recent developments [57,58,59,60,61,62] point out the possibility of describing quantum systems whose associated $C^{\ast }$-algebras are groupoid algebras, and thus are in principle more general than the algebra of bounded linear operators. Consequently, a reformulation of the well-known results for an arbitrary $C^{\ast }$-algebra appears to be useful.

On the other hand, in the classical case, the explicit use of $C^{\ast }$-algebras to investigate the geometry of probability distributions is essentially absent. To the best of our knowledge, the only (very nice) exceptions are the works in [63,64,65]. However, the point of view of these works is different from ours because they consider probability distributions as particular elements of a $C^{\ast }$-algebras, while we consider them as particular linear functionals.

Another reason why we believe it would be useful to consider the framework of $C^{\ast }$-algebras is that the space of states of a $C^{\ast }$-algebra is an example of space of states of general probabilistic theories [66,67]. Therefore, the study of the differential geometry of the space of states of $C^{\ast }$-algebras, and in particular the study of parametric estimation theory in this context, represents a first step toward the study of these subjects in the more comprehensive framework of general probabilistic theories. This intermediate step may be useful because states on $C^{\ast }$-algebras benefit from the rich algebraic structure of the algebras they act upon, while states in general probabilistic theories do not necessarily have such a rich algebraic background to rely on. Consequently, a first study of the richer case may lead to results that can be later generalized to the less rich case once an appropriate and judicious process of extrapolation is pursued.

We confine ourselves to the case of finite-dimensional $C^{\ast }$-algebras because, at this preliminary stage, we want to avoid the technical difficulties with which the infinite-dimensional case is filled. Indeed, we are now interested in exposing the basic aspects of the theory in order to have a solid background on which future works can rely on. In the infinite-dimensional case, the technical difficulties would often obscure the conceptual aspects, and this unavoidably leads to be less communicative. Moreover, it is even not yet clear what are the geometrical players on the fields when infinite dimensions are considered because there is no general consensus on which are the most appropriate manifolds of states to consider in this case (see the works in [4,23,68,69,70,71,72,73,74,75,76,77,78] for some examples).

Incidentally, the restriction to the finite-dimensional case seems to affect more the classical case, rather than the quantum case. Indeed, classical estimation theory essentially deals with parametric models of probability distributions on spaces which are neither discrete nor finite (think for instance to normal distributions), and these cases are naturally associated with infinite-dimensional $C^{\ast }$-algebras. The case of parametric models of probability distributions on discrete and finite spaces is usually less studied because it seldom presents itself in applications. In the context of quantum information theory, the situation is quite the opposite, and the vast majority of the models considered refer to quantum system with a finite-dimensional Hilbert space, and thus with an associated finite-dimensional $C^{\ast }$-algebra. The infinite-dimensional case usually deals only with pure-state models for which the underlying manifold of states is rather friendly, being the Hilbert manifold of a complex projective space associated with a separable, complex Hilbert space.

The content of this work builds on well-known and established results in the context of both classical and quantum estimation theory. However, the presentation of these results in the unifying framework of $C^{\ast }$-algebras is essentially new, as are the proofs of some results. We believe that this attitude may be particularly useful in future research dealing with the infinite-dimensional case and in dealing with the comparison of classical and quantum methods. Accordingly, this work should be considered more as a first, preliminary step in a research program aimed at the understanding of the unification of classical and quantum estimation theory rather than an exposition of a finite theory, and the focus of the work is more on the discussion of general structures rather than on the presentation of specific examples.

The article is structured as follows. In Section 2, some differential geometric aspects of finite-dimensional $C^{\ast }$-algebras and of their spaces of states are recalled. In Section 3, the notion of parametric model of states on a $C^{\ast }$-algebra $\mathcal{A}$ is introduced and the notion of Symmetric Logarithmic Derivative used in quantum information theory is generalized to the $C^{\ast }$-algebraic setting. In Section 4, the notion of parametric statistical model associated with a given parametric model of states is introduced. This notion represents the bridge between the models of states on a possibly noncommutative $C^{\ast }$-algebra and the models of probability distributions used in classical estimation theory. Moreover, the notion of multiple round model and its geometrical properties are briefly discussed. In Section 5, the problem of estimation theory is formulated in the $C^{\ast }$-algebraic framework and the notion of manifold-valued estimator is recalled. In Section 6, a proof of the Cramer–Rao bound for manifold-valued estimators on finite outcome spaces is given following the work of Hendriks [29]. Finally, in Section 7, the generalization of the Helstrom bound used in quantum information theory to the $C^{\ast }$-algebraic framework is given.

We start with a brief summary of $C^{\ast }$-algebras [79,80,81,82]. Let A be a complex algebra with identity $\mathbb{I}$. If there is an anti-linear map $†:A\to A$ such that ${\left({\mathbf{a}}^{†}\right)}^{†}=\mathbf{a}$ for all $\mathbf{a}\in A$, and such that ${\left(\mathbf{ab}\right)}^{†}={\mathbf{b}}^{†}{\mathbf{a}}^{†}$ for all $\mathbf{a},\mathbf{b}\in A$, then † is called an involution and $(A,†)$ an involutive algebra. If there is a norm $\parallel ·\parallel$ on A turning it into a Banach space satisfying the additional relations $\parallel \mathbf{ab}\parallel \le \parallel \mathbf{a}\parallel \parallel \mathbf{b}\parallel$ and $\parallel \mathbf{a}{\mathbf{a}}^{†}{\parallel \le \parallel \mathbf{a}\parallel }^2$ for all $\mathbf{a},\mathbf{b}\in A$, then $(A,†,\parallel ·\parallel )$ is called a $C^{\ast }$-algebra, and, for the sake of notational simplicity, it will be denoted simply by $\mathcal{A}$.

An element $\mathbf{a}\in \mathcal{A}$ is called self-adjoint if $\mathbf{a}={\mathbf{a}}^{†}$. The space of self-adjoint elements in $\mathcal{A}$ is denoted as ${\mathcal{A}}_{sa}$. It is a real Banach space whose dual space is denoted as $\mathcal{V}$, and there is a direct sum decomposition where ı is the imaginary unit.

An element $\mathbf{b}\in \mathcal{A}$ is called positive if there exists $\mathbf{a}\in \mathcal{A}$ such that $\mathbf{b}={\mathbf{a}}^{†}\mathbf{a}$. Clearly, a positive element $\mathbf{b}$ is self-adjoint, and it can be proved that there is a unique self-adjoint element $\mathbf{s}$ such that $\mathbf{b}={\mathbf{s}}^2$.

An element $\mathrm{g}\in \mathcal{A}$ is called invertible if there is another element written as ${\mathrm{g}}^{-1}$ such that $\mathrm{g}{\mathrm{g}}^{-1}={\mathrm{g}}^{-1}\mathrm{g}=\mathbb{I}$. The set of invertible elements in $\mathcal{A}$ is denoted as $\mathcal{G}$, and it is a real Banach–Lie group; the Banach–Lie algebra of which is $\mathcal{A}$ endowed with the commutator [83,84]. An element $\mathbf{u}\in \mathcal{G}$ is called unitary if ${\mathbf{u}}^{-1}={\mathbf{u}}^{†}$. The set of unitary elements in $\mathcal{A}$ is denoted as $\mathcal{U}$ and it is a real Banach–Lie subgroup of $\mathcal{G}$, called the unitary group of $\mathcal{A}$, whose Banach–Lie algebra is the subspace $ı{\mathcal{A}}_{sa}$ in the decomposition (1) endowed with the commutator inherited from $\mathcal{A}$ [83,84].

Let ${\mathcal{A}}^{\ast }$ be the complex Banach dual of $\mathcal{A}$. An element $\xi \in {\mathcal{A}}^{\ast }$ is called a self-adjoint linear functional if $\xi \left({\mathbf{a}}^{†}\right)=\overline{\xi \left(\mathbf{a}\right)}$. The set of self-adjoint linear functionals is precisely the real Banach dual $\mathcal{V}$ of ${\mathcal{A}}_{sa}$. A self-adjoint linear functional $\omega$ is called positive if $\omega \left(\mathbf{a}\right)\ge 0$ for every positive element $\mathbf{a}\in \mathcal{A}$. A positive element $\omega$ is called faithful if $\omega \left(\mathbf{a}\right)=0$ implies $\mathbf{a}=\mathbf{0}$ for all positive elements in $\mathcal{A}$. The set of positive elements is denoted as $\mathcal{P}$. A positive linear functional $\rho$ is called a state if $\rho \left(\mathbb{I}\right)=1$. The set of states is denoted as $\mathcal{S}$.

In the following, we will focus only on finite-dimensional $C^{\ast }$-algebras. Given a self-adjoint element $\mathbf{a}\in {\mathcal{A}}_{sa}$, we write $f_{\mathbf{a}}$ for the linear function on $\mathcal{V}$ given by as well as for its restrictions to the various submanifolds of $\mathcal{V}$ we will introduce below (with an evident abuse of notation).

There is a group action of $\mathcal{G}$ on $\mathcal{S}$ given by [41,68] and the space of states $\mathcal{S}$ decomposes into the disjoint union of orbits of the $\mathcal{G}$-action, and evidently, each such orbit is a homogeneous space.

Recalling that $\mathcal{A}$ endowed with the commutator is the Lie algebra of $\mathcal{G}$, the fundamental vector fields of $\mathrm{\Phi }$ are labelled by elements of $\mathcal{A}$. Recalling (1), we write an element in $\mathcal{A}$ as $\mathbf{a}+ı\mathbf{b}$ where $\mathbf{a},\mathbf{b}\in {\mathcal{A}}_{sa}$, and ı is the imaginary unit. Accordingly, we write ${\Gamma }_{\mathbf{ab}}$ for the fundamental vector field associated with $\frac{1}{2}\left(\mathbf{a}+ı\mathbf{b}\right)$. A direct computation shows that the tangent vector ${\Gamma }_{\mathbf{ab}}\left(\rho \right)$, identified with a self-adjoint linear functional in $\mathcal{V}$ because the orbit $\mathcal{O}$ is an immersed submanifold of $\mathcal{V}$, is given by where $\{,\}$ and $\left[\right[,\left]\right]$ denote, respectively, the Jordan product and the Lie product in $\mathcal{A}$ given by Note that $\{,\}$ and $\left[\right[,\left]\right]$ preserve ${\mathcal{A}}_{sa}$, and actually turn it into a Jordan–Lie algebra [85,86].

We set and we call ${\mathbb{Y}}_{\mathbf{a}}$ a gradient vector field (the origin of the name will be explained below) and ${\mathbb{X}}_{\mathbf{b}}$ a Hamiltonian vector field. It is not hard to show that the Hamiltonian vector fields give an anti-representation of the Lie algebra of the group $\mathcal{U}\subset \mathcal{G}$ of unitary element of $\mathcal{G}$ [41,68]. This Lie algebra anti-representation integrates to a left action of $\mathcal{U}$ on $\mathcal{O}$ given by the restriction of $\mathrm{\Phi }$ to $\mathcal{U}$.

If we fix a basis ${\left\{{\mathbf{e}}^j\right\}}_{j=1,...,N}$ of self-adjoint elements in $\mathcal{A}$ (where $\dim \left(\mathcal{A}\right)=N$), we may introduce the structure constants $d_l^{jk}$ and $c_l^{jk}$ of the Jordan and Lie products in Equation (5) by setting Then, the gradient and Hamiltonian vector fields are easily seen to be given by where ${\left\{x^j\right\}}_{j=1,...,N}$ is the Cartesian coordinate system on $\mathcal{V}$ associated with the dual basis ${\left\{{\mathbf{e}}_j\right\}}_{j=1,...,N}$ of ${\left\{{\mathbf{e}}^j\right\}}_{j=1,...,N}$.

According to the work in [41], the gradient vector fields provide an overcomplete basis of the tangent space at each point in every orbit $\mathcal{O}$. Furthermore, on every $\mathcal{O}$ we may define a Riemannian metric tensor $\mathrm{G}$ given by and ${\mathbb{Y}}_{\mathbf{a}}$ is the gradient vector field associated with the smooth function $f_{\mathbf{a}}$ (see Equation (2)) by means of $\mathrm{G}$. This metric tensor is invariant with respect to the action of the unitary group in the sense that where ${\mathrm{\Phi }}_{\mathbf{U}}$ is the diffeomorphism given by However, $\mathrm{G}$ is not invariant under the action of all of $\mathcal{G}$.

The metric tensor $\mathrm{G}$ turns out to be the $C^{\ast }$-algebraic version of some well-known and relevant metric tensors when explicit cases are considered [41]. For instance, if $\mathcal{A}={\mathcal{C}}_n$ and $\mathcal{O}={\Delta }_n^{+}$, then $\mathrm{G}$ coincides with the Fisher–Rao metric tensor. If $\mathcal{A}=\mathcal{B}\left(\mathcal{H}\right)$ and $\mathcal{O}\cong \mathbb{CP}\left(\mathcal{H}\right)$ is the orbit of pure states), then $\mathrm{G}$ coincides with the Fubini–Study metric tensor. If $\mathcal{A}=\mathcal{B}\left(\mathcal{H}\right)$ and $\mathcal{O}$ is the orbit of faithful states, then $\mathrm{G}$ coincides with the Bures–Helstrom metric tensor.

According to the work in [41], the geodesic of $\mathrm{G}$ starting at $\rho \in \mathcal{O}$ with initial tangent vector $\mathbf{v}\in T_{\rho }\mathcal{O}$ reads where The geodesic ${\nu }_{\rho }^{\mathbf{v}}\left(t\right)$ remains inside the space of states $\mathcal{S}$ for all $t\in \mathbb{R}$, but it also exits and enters the orbit $\mathcal{O}$ containing the initial state $\rho$ at multiple times [41].

Motivated by the classical theory of parametric estimation, we will now introduce the notion of a parametric model of states on a finite-dimensional $C^{\ast }$-algebra, and then reformulate the theory of parametric estimation in this theoretical framework. This will allow for the simultaneous handling of the classical and the quantum case.

Some comments are in order. First of all, we fix the codomain of $\mathit{j}$ to be an orbit of states $\mathcal{O}$ because, as will be clear below, we want to exploit the differential geometric aspects of $\mathcal{O}$ itself. In practice, a vast part of the models considered in the literature falls in this category. For instance, in quantum information geometry, it is customary to deal with parametric models consisting only of pure states, or only of invertible density operators. In principle, it would also be possible to consider a more general case in which $\mathit{j}$ is a smooth map of M into the Banach space $\mathcal{V}$ of self-adjoint linear functionals in such a way that $\mathit{j}\left(M\right)\subset \mathcal{S}$, and M intersects different orbits of states. This line of thought would require a different way to handle geometrical properties of the space of states in relation with the parameter manifold, based, for example, on the methodology introduced in [4,71] for the classical case. This line of reasoning may be useful in the transition to the infinite-dimensional case where the smooth structure of the orbits $\mathcal{O}$ is in general not guaranteed, and we plan to address this and related questions in the future.

Concerning the identifiability of a model, it may seem at first glance reasonable to consider only identifiable models, but we will show that there are well-known and “simple” parametric models of quantum states (e.g., qubit models) for which either this assumption is not satisfied, or it leads to difficulties with the statistical interpretation of the model.

Now, we turn our attention to the geometrical objects that M inherits by means of the smooth map $\mathit{j}$. Indeed, once we have the smooth map $\mathit{j}$, a symmetric, covariant $(0,2)$ tensor is naturally obtained on M by considering the Riemannian metric $\mathrm{G}$ on $\mathcal{O}$ introduced before and taking its pullback to M with respect to $\mathit{j}$. This gives a tensor on M which “feels” the possible non-commutativity of $\mathcal{A}$ and gives the “correct” tensor in the classical case.

Indeed, if $\mathcal{A}$ is Abelian, then $\mathcal{O}$ is diffeomorphic to the open interior of a suitable simplex, $\mathrm{G}$ is the Fisher–Rao metric tensor [41], and ${\mathrm{G}}^M$ is the pullback of the Fisher–Rao metric tensor to the manifold M seen as a model of probability distributions [3].

On the other hand, if $\mathcal{A}$ is the algebra $\mathcal{B}\left(\mathcal{H}\right)$ of bounded linear operators on a finite-dimensional, complex Hilbert space $\mathcal{H}$ and $\mathcal{O}$ is the manifold of pure states, then $\mathcal{O}$ is diffeomorphic to the complex projective space $\mathbb{CP}\left(\mathcal{H}\right)$ associated with $\mathcal{H}$, $\mathrm{G}$ is the Fubini–Study metric [41] on $\mathcal{O}=\mathbb{CP}\left(\mathcal{H}\right)$, and ${\mathrm{G}}^M$ is the quantum counterpart of the Fisher–Rao metric tensor on the manifold M seen as a model of pure quantum states [89]. Moreover, if $\mathcal{O}$ is the manifold of faithful states, then $\mathrm{G}$ is the Bures–Helstrom metric tensor [41], and ${\mathrm{G}}^M$ may be read as a quantum counterpart of the Fisher–Rao metric tensor on the manifold M seen as a model of faithful quantum states [90].

We will now introduce the $C^{\ast }$-algebraic version of the Symmetric Logarithmic Derivative (SLD) introduced in quantum estimation theory by Helstrom in [6]. For this purpose, note that every tangent vector at $\rho \in \mathcal{O}$ may be expressed in terms of gradient vector fields, that is, given $\rho \in \mathcal{O}$, for every tangent vector $V_{\rho }\in T_{\rho }\mathcal{O}$ there exists a self-adjoint element $\mathbf{a}\in {\mathcal{A}}_{sa}$ depending on $V_{\rho }$ such that Consequently, if we consider a tangent vector $v_m\in T_{m}M$, it makes sense to ask for the gradient vector field ${\mathbb{Y}}_{\mathbf{a}}$ on $\mathcal{O}$ such that where ${\rho }_m:=\rho \left(j\left(m\right)\right)$. The gradient vector field ${\mathbb{Y}}_{\mathbf{a}}$ in general depends on both the point $m\in M$ and the tangent vector $v_m$. The tangent vector ${\mathbb{Y}}_{\mathbf{a}}\left({\rho }_m\right)$ satisfying Equation (23) is called the SLD of $v_m$ at ${\rho }_m$.

To appreciate the link with the standard definition of the SLD, let us consider a parametric model $(M,\mathit{j},\mathcal{O})$ where $\mathcal{A}=\mathcal{B}\left(\mathcal{H}\right)$, $\mathcal{O}$ is the manifold of faithful states (invertible density operators), M is an open submanifold of $\mathbb{R}$, and $\mathit{j}$ is a suitable smooth map. Setting $v_m={\partial }_t\left(m\right)$ where ${\partial }_t$ is the restriction to M of the vector field generating the group structure of $\mathbb{R}$, a direct computation shows that the solution of Equation (23) coincides with the Symmetric Logarithmic Derivative (SLD) of the work in [6]. Indeed, ${\partial }_t$ is the infinitesimal generator of $m_t=m+t$, and considering an arbitrary function $f_{\mathbf{b}}$ on $\mathcal{O}$, we have so that Equation (23) may be alternatively written as where which is precisely the definition of the SLD (see also Equation (3) in [90] and Equations (3.4) and (3.14) in [91] for the multiparametric case). This justifies the interpretation of Equation (23) as the $C^{\ast }$-algebraic generalization of the SLD embracing also the multiparametric quantum and classical cases.

When an experiment is performed on a system in a given state $\rho$, we obtain an outcome lying in a given outcome space $\mathcal{X}$ which is associated with the measurement procedure. The state $\rho$ is then “transformed” into a probability distribution on $\mathcal{X}$ in the sense that different repetitions of the same experimental procedure (i.e., preparation of the system in the state $\rho$ followed by the measurement procedure with outcome space $\mathcal{X}$) will produce in general different outcomes characterized by a probability distribution which is associated with the state $\rho$ and with the measurement procedure adopted. In this work, we will always consider outcome spaces which are discrete and finite.

Given a discrete and finite outcome space ${\mathcal{X}}_n$ with n elements, the statistical interpretation of the state $\rho$ is encoded in a map ${\mathfrak{m}}^{\ast }:\mathcal{S}⟶\mathrm{P}\left({\mathcal{X}}_n\right)\equiv {\Delta }_n$, which we will assume to be convex in order to preserve one of the basic features of probabilities and states. From this, it follows that ${\mathfrak{m}}^{\ast }$ can be extended to a linear map ${\mathfrak{m}}^{\ast }:{\mathcal{A}}^{\ast }⟶\mathrm{S}\left({\mathcal{X}}_n\right)$, where $\mathrm{S}\left(\mathcal{X}\right)$ is the vector space of signed measures on ${\mathcal{X}}_n$. From the $C^{\ast }$-algebraic perspective, $\mathrm{S}\left({\mathcal{X}}_n\right)$ is the space of self-adjoint linear functionals on the Abelian $C^{\ast }$-algebra ${\mathcal{C}}_n:=C\left({\mathcal{X}}_n\right)$ of complex-valued, continuous functions on ${\mathcal{X}}_n$, and thus, as ${\mathfrak{m}}^{\ast }$ is continuous because ${\mathcal{A}}^{\ast }$ and $\mathrm{S}\left({\mathcal{X}}_n\right)$ are finite-dimensional, we immediately obtain that there is a continuous linear map $\mathfrak{m}:{\mathcal{C}}_n⟶\mathcal{A}$ of which ${\mathfrak{m}}^{\ast }$ is the dual map. By construction, the map $\mathfrak{m}$ must be such that its dual map ${\mathfrak{m}}^{\ast }$ sends the space of states of $\mathcal{A}$ into the space of states of ${\mathcal{C}}_n$. One way to implement this condition is to require $\mathfrak{m}:{\mathcal{C}}_n⟶\mathcal{A}$ to be a unital, positive map between $C^{\ast }$-algebras, that is, a linear map preserving the identity and sending positive elements into positive elements (clearly, any such map sends self-adjoint elements into self-adjoint elements).

Specifically, given a finite and discrete outcome space ${\mathcal{X}}_n$, we can always consider the basis of ${\mathcal{C}}_n$ given by the elements ${\left\{{\mathbf{e}}^j\right\}}_{j=1,...,n}$, where ${\mathbf{e}}^j$ is the “delta function” at the j-th element of ${\mathcal{X}}_n$. The measurement procedure $\mathfrak{m}$ amounts to define the elements in such a way that they satisfy and Essentially, we are considering a (discrete) POVM in the $C^{\ast }$-algebraic framework. The probability distribution ${\mathfrak{m}}^{\ast }\left(\rho \right)$ associated with the state $\rho$ is characterized by the numbers

Once a parametric model $(M,\mathit{j},\mathcal{O})$ is chosen, we immediately have the map We require this map to lie entirely in a given fixed orbit of states inside ${\Delta }_n$. Clearly, as every orbit in ${\Delta }_n$ is diffeomorphic to ${\Delta }_k^{+}$ for some $k\ne n$ (see Example 1), there is no loss of generality in requiring the codomain of ${\mathit{j}}^c$ to lie entirely inside the manifold ${\Delta }_n^{+}$ of faithful states on ${\mathcal{C}}_n$. Indeed, if this is not the case, it suffices to redefine ${\mathcal{X}}_n$ to be the subset $I_k$, exchange ${\mathcal{C}}_n$ with $C\left(I_k\right)$, and relabel k as n.

Once a regular measurement procedure $\mathfrak{m}$ for $(M,\mathit{j},\mathcal{O})$ is chosen, we are ready to build a parametric statistical model (in the sense of information geometry [2,3,4]) associated with the parametric model $(M,\mathit{j},\mathcal{O})$.

The open interior of the simplex ${\Delta }_n^{+}$ coincides with the space of faithful states on the finite-dimensional, commutative $C^{\ast }$-algebra ${\mathcal{C}}_n$ of complex-valued continuous functions on the discrete n-point space $\mathcal{X}n$, and thus the Radon–Nikodym derivative of $\mathbf{p}\in {\Delta }_n^{+}$ with respect to the counting measure on ${\mathcal{X}}_n$ is well defined as a function on ${\mathcal{X}}_n$, and it is called the probability density function of $\mathbf{p}$. Clearly, being ${\mathit{j}}^c\left(M\right)\subseteq {\Delta }_n^{+}$, every element $m\in M$ may be uniquely associated with the probability density function of ${\mathbf{p}}_m={\mathit{j}}^c\left(m\right)$. Moreover, for every $x\in {\mathcal{X}}_n$, the function $p(m,x)={\mathbf{p}}_m\left(\left\{x\right\}\right)$ is a smooth function on M because ${\mathbf{p}}_m$ is a linear functional on ${\mathcal{C}}_n$ and ${\mathit{j}}^c$ is smooth, and its support does not depend on the chosen $x\in {\mathcal{X}}_n$ because ${\mathit{j}}^c\left(M\right)\subseteq {\Delta }_n^{+}$. These regularity properties are particularly meaningful with respect to the Cramer–Rao bound discussed in Section 6.

Exploiting the Riemannian geometry of ${\Delta }_n^{+}$, the parameter manifold M may be endowed with another symmetric, covariant $(0,2)$ tensor which is in general different from the metric ${\mathrm{G}}^M$ introduced before. Indeed, we may consider the Fisher–Rao Riemannian metric ${\mathrm{G}}_{FR}$ on ${\Delta }_n^{+}$, which is the Riemannian metric tensor $\mathrm{G}$ associated with the Jordan product of the self-adjoint part of ${\mathcal{C}}_n$ as described in Section 2, and then take its pullback to M (the “c” stands for classical, or commutative). In this case, we obtain a symmetric covariant tensor on M which, unlike ${\mathrm{G}}^M$ given by Equation (21), cannot feel the possible non-commutativity of $\mathcal{A}$, and which is the pullback of the Fisher–Rao metric tensor on M thought of as a parametric statistical model in ${\Delta }_n^{+}$ along the lines of classical information geometry.

To accommodate multiple runs, say N, of the same experimental procedure on N identical and independent copies of the initial state, we introduce the parametric model $(M,{\mathit{j}}^N,{\mathcal{O}}^N)$, where ${\mathcal{O}}^N$ is the manifold of states on the tensor product algebra containing the product states of the form ${\rho }_1\otimes \cdots \otimes {\rho }_N$ with ${\rho }_j\in \mathcal{O}$ for every $j=1,...,N$, and ${\mathit{j}}^N:M⟶{\mathcal{O}}^N$ is given by Clearly, we may endow M with the Riemannian metric ${\mathrm{G}}^{MN}$ defined by where ${\mathrm{G}}^N$ denotes the canonical Riemannian metric on ${\mathcal{O}}^N$ associated with the Jordan product on ${\mathcal{A}}^{\otimes N}$. As the smooth embedding ${\mathit{j}}^N$ has been defined in terms of a “multiplicative object”, namely, the tensor product, it is reasonable to expect that this multiplicative feature reflects also in the pullback metric. Indeed, below we will prove that

Performing N runs of an experiment provides us with a list of N outcomes, and we consider the outcome space At this point, we must choose a measurement procedure ${\mathfrak{m}}^N:{\mathcal{C}}_n^{\otimes N}=C\left({\mathcal{X}}^N\right)⟶{\mathcal{A}}^{\otimes N}$ so that, setting ${\mathit{j}}^{cN}={\mathfrak{m}}^N\circ {\mathit{j}}^N$, we can build a statistical model $(M,{\mathit{j}}^{cN},{\Delta }_{Nn}^{+})$ in the obvious way. We may endow M with the Riemannian metric ${\mathrm{G}}^{McN}$ defined by where $N{\mathrm{G}}_{FR}$ is the Fisher–Rao metric tensor on ${\Delta }_{nN}^{+}$ (this either follows from standard arguments in classical information geometry, or by Proposition 1 below applied to the case where $\mathcal{A}={\mathcal{C}}_n$).