Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings

We discuss the fundamental theoretical framework together with numerous results obtained by the authors and colleagues over an extended period of investigation on the Information Geometric Approach to Chaos (IGAC).

indicators mentioned above, the notion of complexity within the IGAC framework can also be characterized by the Information Geometric Entropy (IGE), originally proposed in [3]. We make reference to the Ali-Cafaro effort in Ref. [8] for a more extensive summary of the IGAC framework that incorporates a set of remarks on entropic evolution and the MrE algorithm. For a presentation of alternative information geometric descriptions of complexity, we suggest the investigation by Felice, Cafaro and Mancini in Ref. [5]. While we certainly appreciate the power of the synthetic, non-component approach to tensors analysis commonly used in theoretical physics (for instance, see Refs. [9]), we have nevertheless chosen to employ the component approach in the present paper. In our opinion, the applied nature of our works can be formulated and analyzed more efficiently (and transparently) within the component approach to tensor calculus.
In the next section, we introduce suitable indicators of complexity within the IGAC.

II. INDICATORS OF COMPLEXITY
In this section, we introduce three indicators of complexity within the context of IGAC framework. We present the IGE, the curvature of the statistical manifold and finally, the notion of Jacobi fields arising from the equation of geodesic deviation.

A. Information geometric entropy
We begin this subsection by discussing the IGE. Assuming the elements {p (x; θ)} of an n-dimensional statistical manifold M s are parametrized by n, real valued variables θ 1 ,..., θ n , the statistical manifold is defined by the set, (1) We point out that the quantity x in Eq. (1) are microvariables belonging to the microspace X , whereas the macrovariables θ appearing in Eq. (1) are elements of the parameter space D The quantity I θ j with 1 ≤ j ≤ n in Eq. (2) is a subset of R n that denotes the full range of admissible values of the statistical macrovariables θ j . The IGE serves as an indicator of temporal complexity associated with geodesic paths in the IGAC framework. The IGE is given by, where the average dynamical statistical volume vol [D θ (τ )] is defined as, We remark that the temporal averaging operation is denoted by the tilde symbol in Eq. (4). Furthermore, the volume vol [D θ (τ )] appearing on the RHS of Eq. (4) is given by, ρ θ 1 ,..., θ n d n θ, where ρ θ 1 ,..., θ n is known as the Fisher density and is equal to the square root of the determinant g (θ) of the Fisher-Rao information metric tensor g µν (θ), ρ θ 1 ,..., θ n def = g (θ).
The Fisher-Rao information metric tensor g µν (θ) is defined as, where µ, ν = 1,..., n for an n-dimensional manifold and ∂ µ def = ∂ ∂θ µ . The volume vol [D θ (τ )] in Eq. (5) can be recast in a more crystalline manner for cases involving statistical manifolds whose information metric tensor has a determinant that can be expressed in a factorized form as follows, g (θ) = g θ 1 ,..., θ n = n j=1 g j θ j . (8) By using the factorized form of the determinant, the IGE appearing in Eq. (3) can be expressed as Within the IGAC framework, the leading asymptotic behavior of S Ms (τ ) in Eq. (9) is employed to specify the complexity of the statistical model under investigation. Therefore, it is quite instructive to take into consideration the quantity that is, the leading asymptotic term in the expression of the IGE. The integration space D θ (τ ) in Eq. (5) is defined by where θ j = θ j (ξ) with τ 0 ≤ ξ ≤ τ 0 + τ and τ 0 denotes the initial value of the affine parameter ξ such that, The domain of integration D θ (τ ) in Eq. (11) is an n-dimensional subspace of D are ndimensional macrovariables {θ} whose components θ j are bounded by the limits of integration θ j (τ 0 ) and θ j (τ 0 + τ ). The temporal functional form of such limits is determined by the integration of the geodesic equations in Eq. (12).
The IGE evaluated at a particular instant is specified by the logarithm of the volume of the effective parameter space probed by the system at that instant. In order to coarse-grain the possibly very complex details of the entropic dynamical characterization of the system however, the process of temporal averaging has been employed. Moreover, in order to remove the effects of potential transient features that may enter the calculation of the expected value of the volume of the effective parameter space, only its asymptotic temporal behavior is taken into consideration. For these reasons, it is evident that the IGE serves as an asymptotic, coarse-grained complexity indicator of dynamical systems in the presence of partial information. For additional specifics concerning the IGE, we refer the interested reader to Refs. [8,10].
We emphasize that it would be interesting to characterize the tendency to increase of the entropy of a physical system that approaches equilibrium as specified by the Boltzmann H theorem and the second law of thermodynamics [11] from an information geometric perspective. For a recent information geometric interpretation of the entropy production, we refer to Ref. [12]. In particular, to understand the possible link between the IGE and the Boltzmann-Shannon entropy, it would be important to study the Kaniadakis S κ entropy (with κ being the so-called deformation parameter) and comprehend how the statistical mechanics based on S κ can be regarded as a natural generalization of the equilibrium Boltzmann-Gibbs statistical mechanics [13]. We leave these intriguing lines of investigations to future efforts.

B. Curvature
We present here the notion of curvature of statistical manifolds. We begin by recalling that an n-dimensional, C ∞ differentiable manifold is defined by a set of points M endowed with coordinate systems C M fulfilling the following two requirements: 1) each element c ∈ C M is a one-to-one mapping from M to an open subset of R n ; 2) given any one-to-one mapping η : M → R n , we have that ∀c ∈ C M , η ∈ C M ⇔ η • c −1 is a C ∞ diffeomorphism.
In this paper, we focus on Riemannian manifolds (M, g) where the points of M are probability distribution functions. It is worth noting that the manifold structure of M is insufficient to specify in a unique manner the Riemannian metric g. On a formal level, an infinite number of Riemannian metrics can be defined on the manifold M . In the context of information geometry however, the selection of the Fisher-Rao information metric (see Eq. (7)) as the metric underlying the Riemannian geometry of probability distributions [2,14,15] serves as a primary working assumption. The characterization theorem attributed to Cencov [16] gives significant support for this particular choice of metric. In this characterization theorem, Cencov demonstrates that, up to any arbitrary constant scale factor, the information metric is the only Riemannian metric that is invariant under congruent embeddings (that is, under a family of probabilistically meaningful mappings) of Markov morphism [16,17].
Upon introducing the Fisher-Rao information metric g µν (θ) in Eq. (7), standard differential geometric techniques can be used on the space of probability distributions to describe the geometry of the statistical manifold M s . The Ricci scalar curvature R Ms is one example of such a geometric property, where R Ms is defined as [18], where g µν g νρ = δ µ ρ and g µν = (g µν ) −1 . The Ricci tensor R µν appearing in Eq. (13) is given as [18], The Christoffel connection coefficients Γ ρ µν of the second kind that specify the Ricci tensor in Eq. (14) are [18], Next we consider geodesic curves on statistical manifolds. A geodesic on an n-dimensional statistical manifold M s can be interpreted as the maximum probability trajectory explored by a complex system during its change from an initial θ initial to final macrostates θ final , respectively. Each point along a geodesic path corresponds to a macrostate specified by the macroscopic variables θ = θ 1 ,..., θ n . In the context of ED, each component θ j with j = 1,..., n is a solution of the geodesic equation [6], At this juncture, we reiterate the fact that each macrostate θ is in one-to-one correspondence with the probability distribution p (x|θ), with the latter characterizing a distribution of the microstates x. It is useful to recognize that the scalar curvature R Ms can be readily recast as the sum of sectional curvatures K (e ρ , e σ ) of all tangent space planes T p M s with p ∈ M s spanned by pairs of orthonormal basis vectors e ρ = ∂ θρ(p) , where K (a, b) is given by [18], with, We observe that the Riemann curvature tensor R αβρσ [18] is fully determined by the sectional curvatures K (e ρ , e σ ) where, The negativity of the Ricci scalar curvature R Ms is a strong (i.e., a sufficient but not necessary) criterion of local dynamical instability. Moreover, the compactness of the manifold M s is required to specify genuine chaotic (that is, temporally complex) dynamical systems. In particular, it is evident from Eq. (17) that the negativity of R Ms imply that negative principal curvatures (i.e., extrema of sectional curvatures) are more dominant than positive ones. For this reason, the negativity of R Ms is a sufficient but not necessary requirement for local instability of geodesic flows on statistical manifolds. It is worth mentioning the possible circumstance of scenarios in which negative sectional curvatures are present, but the positive curvatures dominate in the sum of Eq. (17) such that R Ms is a non-negative quantity despite flow instability in those directions. For additional mathematical considerations related to the concept of curvature in differential geometry, we suggest Ref. [19].

C. Jacobi fields
We introduce here the concept of the Jacobi vector field. It is worth noting that the analysis of stability/instability arising in natural (geodesic) evolutions is readily accomplished by means of the Jacobi-Levi-Civita (JLC) equation for geodesic deviation. This equation is familiar in both theoretical physics (for example, in the case of General Relativity) as well as in Riemannian geometry. The JLC equation describes in a covariant manner, the degree to which neighboring geodesics locally scatter. In particular, the JLC equation effectively connects the curvature properties of an underlying manifold to the stability/instability of the geodesic flow induced thereupon. Indeed, the JLC equation provides a window into a diverse and mostly unexplored field of study concerning the connections among topology, geometry and geodesic instability, and thus to complexity and chaoticity. The use the JLC equation in the setting of information geometry originally appeared in Ref. [20].
In what follows we take into consideration two neighboring geodesic paths θ α (ξ) and θ α (ξ) + δθ α (ξ), where the quantity ξ denotes an affine parameter satisfying the geodesic equations, and respectively. Noting that to first order in δθ α , after some algebraic calculations, to first order in δθ α Eq. (22) becomes The equation of geodesic deviation can be found by subtracting Eq. (21) from Eq. (24), Equation (25) can be conveniently recast via the covariant derivatives (see Ref. [21], for instance) along the curve θ α (ξ), The covariant derivative is defined as = ∂/∂ξ, respectively. By combining Eqs. (25) and (26), and performing some tensor algebra calculations, we get Finally, the geodesic deviation equation expressed in component form becomes where J α def = δθ α is the α-component of the Jacobi vector field [18]. Equation (28) is known formally as the JLC equation. We observe from the JLC equation in Eq. (28) that neighboring geodesics accelerate relative to each other at a rate measured in a direct manner by the Riemannian curvature tensor R αβγδ . The quantity J α is defined as, Surface Curvature Jacobi Field Behavior IGE Behavior  sphere  positive  oscillatory  sublogarithmic  cylinder  zero  linear  logarithmic  hyperboloid negative  exponential  linear   TABLE I: Schematic description of the behavior of the IGE and the Jacobi field for different types of two-dimensional surfaces characterized by distinct constant values of their Gaussian curvature. For such surfaces, the sectional and the scalar curvatures coincide, while the Gaussian curvature is simply one-half of the scalar curvature. In particular, positive curvature causes geodesics to converge while negative curvature causes geodesics to spread out. More specifically, in flat, positively, and negatively curved manifolds, the geodesic deviation equation yields deviations of nearby geodesics that exhibit linear, oscillatory, and exponential behaviors, respectively. Moreover, the volumes of the manifolds regions explored during the entropic motion tend to increase while transitioning from positively to negatively curved manifolds. Correspondingly, the IGE exhibits its maximum growth (that is, linear growth) in the presence of exponential instability on negatively curved manifolds.
where {θ µ (ξ; φ)} denotes the one-parameter φ family of geodesics whose evolution is specified by means of the affine parameter ξ. The Jacobi vector field intensity J Ms on the manifold M s is given by, In general, the JLC equation is intractable even at low dimensions. However, in the case of isotropic manifolds, it reduces to The sectional curvature K in Eq. (31) assumes a constant value throughout the manifold. In particular, when K < 0, unstable solutions of Eq. (31) become with initial conditions J µ (0) = 0 and dJ µ (0) dξ = ω µ (0) = ω µ 0 = 0, respectively, for any 1 ≤ µ ≤ n with n being the dimensionality of the underlying manifold. For additional remarks concerning the JLC equation, we suggest Refs. [18,21,22].
We point out that it would be intriguing to understand the behavior of the Jacobi vector fields within the geometry of the Kaniadakis statistical mechanics emerging from a one deformation parameter κ [13]. We leave this fascinating line of study to future scientific inquiry. For a schematic description of the behavior of the IGE and the Jacobi field for two-dimensional surfaces with distinct (Gaussian) curvatures, we refer to Table I and Fig. 1.
In the next section, making use of the complexity quantifiers introduced in Eqs. (3), (17), and (30), we present numerous illustrative examples within the IGAC framework.

III. APPLICATIONS
In this section, with the help of the three complexity quantifiers introduced above, we report the results of several applications of the IGAC in which the complexity of geodesic trajectories on statistical manifolds are quantified. We present these illustrative examples in a chronological order, from the first one to the last one. For brevity, we omit technical details and confine the presentation to our own information geometric approach to complexity. Early notions and applications of the IGAC originally appeared in Refs. [23][24][25]. For a recent review of the IGAC framework, we refer to Refs. [4,8,26,27] and Ref. [10], respectively.

A. Uncorrelated Gaussian statistical models
In [20,28], the IGAC framework was employed to study the information geometric features of a system of arbitrary nature, characterized by l degrees of freedom. Each of these degrees of freedom is described by two relevant pieces of information, namely its mean and variance. The infinitesimal line element for this model is given by [28], with µ k and σ k denoting the expectation value and the square root of the variance of the microvariable x k , respectively. It was found that the family of statistical models associated to such a system is Gaussian in form. Specifically, it was determined that this set of Gaussian distributions yields a non-maximally symmetric 2l-dimensional statistical manifold M s whose scalar curvature R Ms assumes a constant negative value that is proportional to the number of degrees of freedom of the system, It was determined that the system explores volume elements on M s at an exponential rate. In particular, the IGE S Ms was found to increase in a linear fashion in the asymptotic temporal limit (more precisely, in asymptotic limit of the statistical affine parameter τ ) and is proportional to the number of degrees of freedom l, The quantity λ in Eq. (35) denotes the maximal positive Lyapunov exponent that specifies the statistical model. Geodesic trajectories on M s were found to be hyperbolic curves. Finally, it was determined that in the asymptotic limit, the Jacobi vector field intensity J Ms is exponentially divergent and is proportional to the number of degrees of freedom l, Given that the exponential divergence of the Jacobi vector field intensity J Ms is an established classical feature of chaos, based on the results displayed in Eqs. (34), (35) and (36), the authors suggest that R Ms , S Ms and J Ms each behave as legitimate measures of chaoticity, with each indicator being proportional to the number of Gaussiandistributed microstates of the system. Although this result was verified in the context of this special scenario, the proportionality among R Ms , S Ms and J Ms constitutes the first known example appearing in the literature of a possible connection among information geometric indicators of chaoticity obtained from probabilistic modeling of dynamical systems. In this first example, we have compared all three measures R Ms , S Ms and J Ms . Although we have not performed such a comparative analysis in all applications, we shall attempt to mention curvature and/or Jacobi vector field intensity behaviors whenever possible. Our emphasis here is especially on our entropic measure of complexity. For more details on the other types of complexity indicators, we refer to our original works cited in this manuscript.

B. Correlated Gaussian statistical models
In [29], the IGAC framework was used to analyze the information constrained dynamics of a system comprised of two correlated, Gaussian-distributed microscopic degrees of freedom each having the same variance. The infinitesimal line element for this model is given by [29], with µ x and µ y denoting the expectation values of the microvariables x and y. The quantity σ 2 , instead, is the variance while r is the usual correlation coefficient between x and y. The scalar curvature R Ms of the manifold with line element in Eq. (37) is R Ms = −3/2. The inclusion of microscopic correlations give rise to asymptotic compression of the statistical macrostates explored by the system at a faster rate than that observed in the absence of microscopic correlations. Specifically, it was determined that in the asymptotic limit where the function F (r) in Eq. (38) with 0 ≤ F (r) ≤ 1 is defined as [29], The function F (r) is a monotone decreasing compression envelope ∀r ∈ (0, 1). This result provides an explicit link between correlations at the microscopic level and complexity at the macroscopic level. It also furnishes a transparent and concise description of the functional change of the macroscopic complexity of the underlying statistical manifold caused by the occurrence of microscopic correlations.

C. Inverted harmonic oscillators
Generally speaking, the fundamental issues addressed by the General Theory of Relativity are twofold: firstly, one wishes to understand how the geometry of space-time evolves in response to the presence of mass-energy distributions; secondly, one seeks to investigate how configurations of mass-energy move in dynamical space-time geometry. By contrast, within the IGAC framework, one is concerned only with the manner in which systems move within a given statistical geometry, while the evolution of the statistical manifold itself is neglected. The recognition that there exist two separate and distinct characteristics to consider regarding the interplay between mass-energy and space-time geometry served as a catalyst in the development of the IGAC framework, ultimately leading to a rather interesting finding. The first result obtained in this novel research direction was proposed by Caticha and Cafaro in [30]. In that article, the possibility of utilizing well established principles of inference to obtain Newtonian dynamics from relevant prior information encoded in a suitable statistical manifold was investigated. The primary working assumption in that derivation was the assumed existence of an irreducible uncertainty in the location of particles. This uncertainty requires the state of a particle to be described by a probability distribution. The resulting configuration space is therefore a statistical manifold whose Riemannian geometry is specified by the Fisher-Rao information metric. The expected trajectory is a consequence of the MrE method, with the latter being regarded as a principle of inference. An unexpected consequence of this approach is that no additional physical postulates such as an equation of motion, principle of least action, nor the concept of momentum, mass, phase space or external time are required. Newton's mechanics involving any number of self-interacting particles as well as particles interacting with external fields is entirely recovered by the resulting entropic dynamics. Indeed, a powerful result of this approach is the fact that interactions among particles as well as particle masses are all justified in terms of the underlying statistical manifold.
Our next example will be of a more applied nature. In [33,34], Zurek and Paz explored the effects of decoherence in quantum chaos by analyzing a single unstable harmonic oscillator with frequency Ω and potential V (x), coupled to an external environment. They determined that in the reversible classical limit, the von Neumann entropy of such a system increases linearly at a rate determined by the Lyapunov exponent Ω according to, Building upon the results obtained in [30], an information geometric analogue of the Zurek-Paz quantum chaos criterion in the classical reversible limit was proposed in [31,32]. In these works, the IGAC framework was employed to study a set of l, three-dimensional, anisotropic, uncoupled, inverted harmonic oscillators (IHO) with an Ohmic distributed frequency spectrum. In this example, the infinitesimal line element is given by where Φ (θ) is defined as, with u k (θ) def = −(1/2)ω 2 k θ 2 k and ω k being the frequency of the k-th inverted harmonic oscillator. Neglecting mathematical details, it was demonstrated in Refs. [31,32] that the asymptotic temporal behavior of the IGE for such a system becomes, where, and ω i with 1 ≤ i ≤ l is the frequency of the i th IHO. Equation (44) indicates an asymptotic, linear IGE growth for the set of IHOs and can be regarded as an extension of the result of Zurek and Paz appearing in Refs. [33,34] to an ensemble of anisotropic, uncoupled, inverted harmonic oscillators in the context of the IGAC. We remark that Eq. (44) was proposed as the classical IG analogue of Eq. (41) in Refs. [31,32].

D. Quantum spin chains
In [35,36], the IGAC was used to study the ED on statistical manifolds whose elements are classical probability distribution functions routinely employed in the study of regular and chaotic quantum energy level statistics. Specifically, an IG description of the chaotic (integrable) energy level statistics of a quantum antiferromagnetic Ising spin chain immersed in a tilted (transverse) external magnetic field was presented. The IGAC of a Poisson distribution coupled to an Exponential bath (that specifies a spin chain in a transverse magnetic field and corresponds to the integrable case) along with that of a Wigner-Dyson distribution coupled to a Gaussian bath (that specifies a spin chain in a tilted magnetic field and corresponds to the chaotic case) were investigated. The line elements in the integrable and chaotic cases are given by, and, respectively. In Eq. (46), µ A and µ B are the average spacing of the energy levels and the average intensity of the magnetic field, respectively. A similar notation is employed for the second scenario described in Eq. (47) where, clearly, σ 2 B denotes the variance of the intensity of the magnetic field. Remarkably, it was determined that in the former case, the IGE shows asymptotic logarithmic growth, whereas in the latter case, the IGE shows asymptotic linear growth, We emphasize that the quantities c andc in Eq. (48) are integration constants that depend upon the dimensionality of the statistical manifold and the boundary constraint conditions on the statistical variables, respectively. The quantity K appearing in Eq. (49) denotes a model parameter describing the asymptotic temporal rate of change of the IGE. The findings described above suggest that the IGAC framework may prove useful in the analysis of applications involving quantum energy level statistics. It is worth noting that in such cases, the IGE effectively serves the role of the standard entanglement entropy used in quantum information science [37,38].

E. Statistical embedding and complexity reduction
Expanding upon the analysis presented in Ref. [39], Cafaro and Mancini utilized the IGAC framework in Ref. [40] to study the 2l-dimensional Gaussian statistical model M s induced by an appropriate embedding within a larger 4l-dimensional Gaussian manifold. The geometry of the 4l-dimensional Gaussian manifold is defined by a Fisher-Rao information metric g µν with non-vanishing off-diagonal elements. It should be noted that these non-vanishing offdiagonal terms arise due to the occurrence of macroscopic correlation coefficients ρ k with 1 ≤ k ≤ l that specify the embedding constraints among the statistical variables on the larger manifold. The infinitesimal line element is given by [40], with ρ 2k−1 defined as, where σ 2k = σ 2k−1 and µ 2k = µ 2k (µ 2k−1 , σ 2k−1 ) for any 1 ≤ k ≤ l. Two significant results were obtained. First, a power law decay of the IGE at a rate determined by the correlation coefficients ρ k was observed with ρ k = ρ s ∀k and s = 1, . . . , l, where The quantity Σ (ρ k , λ k , α ± ) is a strictly positive function of its arguments for 0 ≤ ρ k < 1 and is given by [40], where Ξ k and λ k are real, positive constants of integration, and Equation (52) represents the first main finding reported in Ref. [40] and can be interpreted as a quantitative indication that the IGC of a system decreases in response to the emergence of correlational structures. Second, it was demonstrated that the presence of embedding constraints among the Gaussian macrovariables of the larger 4ldimensional manifold results in an attenuation of the asymptotic exponential divergence of the Jacobi field intensity on the embedded 2l-dimensional manifold. Neglecting mathematical details, it was determined in Ref. [40] that in the asymptotic limit τ 1, Equation (56) constitute the second main finding reported in Ref. [40]. The observed attenuation of the asymptotic exponential divergence of the Jacobi vector field associated with the larger 4l-manifold, suggests that the occurrence of such embedding constraint relations results in an asymptotic compression of the macrostates explored on the statistical manifold M s . These two findings serve to advance, in a non-trivial manner, the goal of developing a description of complexity of either macroscopically or microscopically correlated, multi-dimensional Gaussian statistical models relevant in the modelling of complex systems.

F. Entanglement induced via scattering
Guided by the original study appearing in [41], the IGAC framework was employed to furnish an IG viewpoint on the phenomena of quantum entanglement emerging via s-wave scattering between interacting Gaussian wave packets in Refs. [42,43]. Within the IGAC framework, the pre and post quantum scattering scenarios associated with elastic, head-on collision are hypothesized to be macroscopic manifestations arising from underlying microscopic statistical structures. By exploiting this working hypothesis, the pre and post quantum scattering scenarios were modeled by uncorrelated and correlated Gaussian statistical models, respectively. Using the standard notation used so far in this article, the infinitesimal line elements in the absence and presence of correlations are given by and, respectively. The scalar curvature R Ms of the manifolds with line elements in Eqs. (57) and (58) is Using such a hybrid modeling approach enabled the authors to express the entanglement strength in terms of the scattering potential and incident particle energy. Moreover, the manner in which the entanglement duration is related to the scattering potential and incident particle energy was furnished with a possible explanation. Finally, the link between complexity of informational geodesic paths and entanglement was discussed. In particular, it was demonstrated that in the asymptotic limit, where the function F (r) in Eq. (59) with 0 ≤ F (r) ≤ 1 is defined as, The function F (r) is a monotone decreasing compression factor with 0 < r < 1. The analysis proposed in Refs. [42,43] is a significant progress toward the understanding among the concepts of entanglement and statistical microcorrelations, as well as the impact of micro-correlations on the complexity of informational geodesic paths. The finding appearing in Eq. (59) suggest that the IGAC construct may prove useful in developing a sound IG perspective of the phenomenon of quantum entanglement.

G. Softening of classical chaos by quantization
Expanding upon the original analysis presented in Refs. [44][45][46], the IGAC framework was utilized to investigate the entropic dynamics and information geometry of a three-dimensional Gaussian statistical model as well as the two-dimensional Gaussian statistical model derived from the former model by introducing the following macroscopic information constraint, where Σ 2 ∈ R + 0 . The quantities x and y label the microscopic degrees of freedom of the system. The constraint equation (61) resembles the standard minimum uncertainty relation encountered in quantum mechanics [47]. The infinitesimal line elements in the 3D-and 2D-Gaussian statistical models are given by and, respectively. Note that the expectation value µ y of the microvariable y is set equal to zero in Eq. (62), while σ x = σ with σ x σ y = Σ 2 in Eq. (63). Furthermore, the scalar curvatures corresponding to the 3D and 2D cases are equal to R 3D = −1 and R 2D = −1/2, respectively. It was determined that the complexity of the 2D-Gaussian statistical model specified by the IGE is relaxed when compared with the complexity of the 3D-Gaussian statistical model, with λ 2D and λ 3D being both positive model parameters (satisfying the condition λ 2D ≤ λ 3D ) that express the asymptotic temporal rates of change of the IGE in the 2D and 3D cases, respectively. Motivated by the connection between the macroscopic information constraint (61) on the variances and the phase-space coarse-graining due to the Heisenberg uncertainty relations, the authors suggest their work may shed light on the phenomenon of classical chaos suppression arising from the process of quantization when expressed in an IG setting. It is worth noting that a similar analysis was implemented in Ref. [48] where the work in Ref. [47] was generalized to a scenario where -in conjunction with the macroscopic constraint in Eq. (61) -the microscopic degrees of freedom x and y of the system are also correlated.

H. Topologically distinct correlational structures
In Ref. [49], the asymptotic behavior of the IGE associated with either bivariate or trivariate Gaussian statistical models, with or without micro-correlations, was analyzed by Felice and coworkers. For correlated cases, several correlational configurations among the microscopic degrees of freedom of the system were taken into consideration. It was found that the complexity of macroscopic inferences is dependent on the quantity of accessible microscopic information, as well as on how such microscopic information are correlated. Specifically, in the mildly connected case defined by a trivariate statistical model with two correlations among the three degrees of freedom of the system, the infinitesimal line element is Moreover, the infinitesimal line element in the uncorrelated trivariate case is given by In Eqs. (65) and (66), µ, σ, and r denote the expectation value, the standard deviation, and the correlation coefficient, respectively. It was determined that in the asymptotic limit, whereR (mildly connected) trivariate In Eq. (67), the quantity r is the micro-correlation coefficient. The functionR (mildly connected) trivariate (r) shows nonmonotone behavior in the correlation parameter r and assumes a value of zero at the extrema of the permitted range r ∈ − √ 2/2, √ 2/2 . By contrast, for closed bivariate configurations where all microscopic variables are correlated with each other, the complexity ratio between correlated and uncorrelated cases presents monotone behavior in the correlation parameter r. For example, in the fully connected bivariate Gaussian case with µ x = µ y = µ and σ x = σ y = σ, the infinitesimal line element is whereR (fully connected) trivariate These results imply that in the fully connected bivariate and trivariate configurations, the ratiosR (fully connected) bivariate (r) andR (fully connected) trivariate (r) both present monotone behavior in r over the open intervals (−1, 1) and (−1/2, 1), respectively. On the other hand, in the mildly connected trivariate scenario appearing in Eq. (67), an extrema in the functioñ R (mildly connected) trivariate (r) occurs at r peak = 1/2 ≥ 0. Such distinctly different behavior between mildly and fully connected trivariate configurations can be attributed to the fact that when making statistical inferences subject to the hypothesis of three positively correlated Gaussian random variables, the system becomes frustrated because the maximum entropy favorable state -characterized by minimum complexity -is incompatible with the initial working hypothesis. Guided by these results, it was suggested in Ref. [49] that the impossibility of realizing the maximally favorable state for specific correlational configurations among microscopic degrees of freedom, viewed from an entropic inference perspective, yields an information geometric analogue of the statistical physics frustration effect that arise when loops are present [50].

IV. FINAL REMARKS
In this paper, we discussed the primary results obtained by the authors and colleagues over an extended period of work on the IGAC framework. A summary of the IGAC applications can be found in Table II. For ease of readability, we have chosen to omit technicalities in our discussion. We are aware of several unresolved issues within the IGAC framework, including a deep understanding of the foundational aspects of the IGE measure of complexity. Further developments of the framework are necessary, especially within a fully quantum mechanical setting. For a more detailed list on limitations and future directions of the IGAC approach, we refer the interested reader to Ref. [8]. In particular, we mentioned there that one of our main objectives in the near future is to extend our comprehension of the relationship between the IGE and the Kolmogorov-Sinai dynamical entropy [51], the coarse-grained Boltzmann entropy [51] and the von Neumann entropy [52], depending upon the peculiarity of the system being investigated. Despite its limitations, we are pleased that our theoretical modeling approach is steadily gaining interest in the community of researchers. Indeed, there appears to be an increasing number of scientists who either actively use, or who's work is linked to the theoretical framework described in the present brief feature review article .