**3. 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 [23–25]. For a recent review of the IGAC framework, we refer to [4,8,26,27] and [10], respectively.

### *3.1. 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],

$$ds^2 \stackrel{\text{def}}{=} \sum\_{k=1}^{l} \frac{1}{\sigma\_k^2} d\mu\_k^2 + \frac{2}{\sigma\_k^2} d\sigma\_{k'}^2 \tag{33}$$

with *μk* and *σk* denoting the expectation value and the square root of the variance of the microvariable *xk*, 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 2*l*-dimensional statistical manifold M*s* whose scalar curvature R M*s* assumes a constant negative value that is proportional to the number of degrees of freedom of the system,

$$\mathcal{R}\_{\mathcal{M}\_s} = -l.\tag{34}$$

It was determined that the system explores volume elements on M*s* at an exponential rate. In particular, the IGE S M*s* 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*,

$$\mathcal{S}\_{\mathcal{M}\_\*} (\tau) \stackrel{\tau \to \infty}{\sim} l\lambda\tau. \tag{35}$$

The quantity *λ* in Equation (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*<sup>M</sup>*s* is exponentially divergent and is proportional to the number of degrees of freedom *l*,

$$J\_{\mathcal{M}\_s}(\tau) \stackrel{\tau \to \infty}{\sim} l \exp(\lambda \tau). \tag{36}$$

Given that the exponential divergence of the Jacobi vector field intensity *J*<sup>M</sup>*s* is an established classical feature of chaos, based on the results displayed in Equations (34)–(36), the authors sugges<sup>t</sup> that R M*s*, S M*s* and *J*<sup>M</sup>*s* each behave as legitimate measures of chaoticity, with each indicator being proportional to the number of Gaussian-distributed microstates of the system. Although this result was verified in the context of this special scenario, the proportionality among R M*s*, S M*s* and *J*<sup>M</sup>*s* 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 M*s*, S M*s* and *J*<sup>M</sup>*s*. 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.

### *3.2. 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]

$$ds^2 \stackrel{\text{def}}{=} \frac{1}{\sigma^2} \left[ \frac{1}{1 - r^2} d\mu\_x^2 + \frac{1}{1 - r^2} d\mu\_y^2 - \frac{2r}{1 - r^2} d\mu\_x d\mu\_y + 4d\sigma^2 \right],\tag{37}$$

with *μx* and *μy* 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 <sup>M</sup>*s*of the manifold with line element in Equation (37) is R M*s* = <sup>−</sup>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

$$\left[\exp(\mathcal{S}\_{\mathcal{M}\_{\rm s}}(\tau))\right]\_{\text{corrected}} \stackrel{\tau \to \infty}{\sim} \mathcal{F}(r) \cdot \left[\exp(\mathcal{S}\_{\mathcal{M}\_{\rm s}}(\tau))\right]\_{\text{uncorrelated}'} \tag{38}$$

where the function F(*r*) in Equation (38) with 0 ≤ F(*r*) ≤ 1 is defined as [29]

$$\mathcal{F}(r) \stackrel{\text{def}}{=} \frac{1}{2^{\frac{r}{2}}} \left[ \sqrt{\frac{4(4-r^2)}{(2-2r^2)^2}} \left( \frac{2+r}{4(1-r^2)} \right)^{-\frac{3}{2}} \right]. \tag{39}$$

The function F(*r*) is a monotone decreasing compression envelope ∀*r* ∈ (0, <sup>1</sup>). 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.

### *3.3. 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 spacetime evolves in response to the presence of mass–energy distributions; secondly, one seeks to investigate how configurations of mass–energy move in dynamical spacetime 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 spacetime 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 [31,32], Zurek and Paz explored the effects of decoherence in quantum chaos by analyzing a single unstable harmonic oscillator with frequency Ω and potential *<sup>V</sup>*(*x*),

$$V(\mathbf{x}) \stackrel{\text{def}}{=} -\frac{\Omega^2 \mathbf{x}^2}{2},\tag{40}$$

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

$$\mathcal{S}^{\text{(chaotic)}}\_{\text{quantum}}(\tau) \stackrel{\tau \to \infty}{\sim} \Omega \tau. \tag{41}$$

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 [33,34]. 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

$$ds^2 \stackrel{\text{def}}{=} [1 - \Phi(\theta)] \delta\_{\mu\nu}(\theta) d\theta^{\mu} d\theta^{\nu},\tag{42}$$

where Φ(*θ*) is defined as

$$\Phi(\theta) = \sum\_{k=1}^{l} \mu\_k(\theta),\tag{43}$$

with *uk*(*θ*) def = <sup>−</sup>(1/2)*ω*2*k <sup>θ</sup>*2*k*and *<sup>ω</sup>k*being the frequency of the *k*-th inverted harmonic oscillator. Neglecting mathematical details, it was demonstrated in [33,34] that the asymptotic temporal behavior of the IGE for such a system becomes

$$\mathcal{S}\_{\mathcal{M}^{(l)}\_{\text{HIO}}}(\boldsymbol{\pi};\omega\_{1},\ldots,\omega\_{l}) \stackrel{\boldsymbol{\tau}\to\infty}{\sim} \Omega \boldsymbol{\pi},\tag{44}$$

where,

$$
\Omega \stackrel{\text{def}}{=} \sum\_{i=1}^{l} \omega\_{i\prime} \tag{45}
$$

and *ωi* 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 [31,32] to an ensemble of anisotropic, uncoupled, inverted harmonic oscillators in the context of the IGAC. We remark that Equation (44) was proposed as the classical IG analogue of Equation (41) in [33,34].

### *3.4. 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

$$ds\_{\rm integrable}^2 \stackrel{\rm def}{=} ds\_{\rm Poisson}^2 + ds\_{\rm Exponential}^2 = \frac{1}{\mu\_A^2} d\mu\_A^2 + \frac{1}{\mu\_B^2} d\mu\_{B'}^2 \tag{46}$$

and,

$$ds\_{\rm chaotic}^2 \stackrel{\rm def}{=} ds\_{\rm Wigner-Dyson}^2 + ds\_{\rm Gaussian}^2 = \frac{4}{\mu\_A'^2} d\mu\_A'^2 + \frac{1}{\sigma\_B'^2} d\mu\_B'^2 + \frac{2}{\sigma\_B'^2} d\sigma\_B'^2,\tag{47}$$

respectively. In Equation (46), *μA* and *μB*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 Equation (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,

$$\mathcal{S}^{\text{(integrable)}}\_{\mathcal{M}\_s}(\tau) \stackrel{\tau \to \infty}{\sim} c \log(\tau) + \tilde{c},\tag{48}$$

whereas in the latter case, the IGE shows asymptotic linear growth,

$$\mathcal{S}^{\text{(chatotic)}}\_{\mathcal{M}\_s}(\tau) \stackrel{\tau \to \infty}{\sim} \mathcal{K}\tau. \tag{49}$$

We emphasize that the quantities *c* and *c*˜ in Equation (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 Equation (49) denotes a model parameter describing the asymptotic temporal rate of change of the IGE. The findings described above sugges<sup>t</sup> 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].

### *3.5. Statistical Embedding and Complexity Reduction*

Expanding upon the analysis presented in [39], Cafaro and Mancini utilized the IGAC framework in [40] to study the 2*l*-dimensional Gaussian statistical model M*s* induced by an appropriate embedding within a larger 4*l*-dimensional Gaussian manifold. The geometry of the 4*l*-dimensional Gaussian manifold is defined by a Fisher–Rao information metric *gμν* with non-vanishing off-diagonal elements. It should be noted that these non-vanishing off-diagonal terms arise due to the occurrence of macroscopic correlation coefficients *ρk* 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]

$$ds^2 \stackrel{\text{def}}{=} \sum\_{k=1}^{l} \frac{1}{\sigma\_{2k-1}^2} \left[ d\mu\_{2k-1}^2 + 2\rho\_{2k-1} d\mu\_{2k-1} d\sigma\_{2k-1} + 2d\sigma\_{2k-1}^2 \right],\tag{50}$$

with *ρ*2*k*−1defined as

$$\rho\_{2k-1} \stackrel{\text{def}}{=} \frac{\frac{\partial \mu\_{2k}}{\partial \mu\_{2k-1}} \frac{\partial \mu\_{2k}}{\partial x\_{2k-1}}}{\left[1 + \left(\frac{\partial \mu\_{2k}}{\partial \mu\_{2k-1}}\right)^2\right]^{1/2} \left[2 + \frac{1}{2} \left(\frac{\partial \mu\_{2k}}{\partial \sigma\_{2k-1}}\right)^2\right]^{1/2}} \tag{51}$$

where *σ*2*k* = *σ*2*k*−1 and *μ*2*k* = *μ*2*<sup>k</sup>*(*μ*2*k*−1, *<sup>σ</sup>*2*k*−<sup>1</sup>)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 *ρk* was observed

$$\mathcal{S}\_{\mathcal{M}\_s}(\tau; l, \lambda\_{k'}\rho\_k) \stackrel{\tau \to \infty}{\sim} \log \left[\Lambda(\rho\_k) + \frac{\bar{\Lambda}(\rho\_{k'}\lambda\_k)}{\tau}\right]^l,\tag{52}$$

with *ρk* = *ρs* ∀*k* and *s* = 1, . . . , *l*, where

$$\Lambda(\rho\_k) \stackrel{\text{def}}{=} \frac{2\rho\_k \sqrt{2-\rho\_k^2}}{1+\sqrt{\Delta(\rho\_k)}},\\\bar{\Lambda}(\rho\_{k'}, \lambda\_k) \stackrel{\text{def}}{=} \frac{\sqrt{\Delta(\rho\_k)\left(2-\rho\_k^2\right)}\log[\Sigma(\rho\_{k'}\lambda\_k\mathfrak{a}\pm)]}{\rho\_k\lambda\_k},\\\text{and } a\_\pm(\rho\_k) \stackrel{\text{def}}{=} \frac{1}{2}\left(3\pm\sqrt{\Delta(\rho\_k)}\right). \tag{53}$$

The quantity <sup>Σ</sup>(*ρk*, *λk*, *<sup>α</sup>*±) is a strictly positive function of its arguments for 0 ≤ *ρk* < 1 and is given by [40]

$$
\Sigma(\rho\_k, \lambda\_k, \alpha\_\pm) \stackrel{\text{def}}{=} -\frac{\Xi\_k}{4\lambda\_k} \frac{1 + \sqrt{\Delta(\rho\_k)}}{1 - \sqrt{\Delta(\rho\_k)}} \sqrt{\frac{2\alpha\_-(\rho\_k)}{\alpha\_+(\rho\_k)}},\tag{54}
$$

where Ξ*k* and *λk* are real, positive constants of integration, and

$$
\Delta(\rho\_k) \stackrel{\text{def}}{=} 1 + 4\rho\_k^2. \tag{55}
$$

Equation (52) represents the first main finding reported in [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 4*l*-dimensional manifold results in an attenuation of the asymptotic exponential divergence of the Jacobi field intensity on the embedded 2*l*-dimensional manifold. Neglecting mathematical details, it was determined in [40] that in the asymptotic limit *τ* ! 1,

$$0 \le \frac{J\_{\mathcal{M}\_s}^{2l\text{-embedded}}(\tau)}{J\_{\mathcal{M}\_s}^{4l\text{-larger}}(\tau)} < 1.\tag{56}$$

Equation (56) constitutes the second main finding reported in [40]. The observed attenuation of the asymptotic exponential divergence of the Jacobi vector field associated with the larger 4*l*-manifold, suggests that the occurrence of such embedding constraint relations results in an asymptotic compression of the macrostates explored on the statistical manifold M*<sup>s</sup>*. 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 modeling of complex systems.

### *3.6. 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 [42,43]. Within the IGAC framework, the pre and post quantum scattering scenarios associated with elastic, headon 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

$$ds\_{\rm no\text{-}correlation}^2 = \frac{1}{\sigma^2} \left[ d\mu\_x^2 + d\mu\_y^2 + 4d\sigma^2 \right],\tag{57}$$

and,

$$ds\_{\text{correlations}}^2 \stackrel{\text{def}}{=} \frac{1}{\sigma^2} \left[ \frac{1}{1-r^2} d\mu\_x^2 + \frac{1}{1-r^2} d\mu\_y^2 - \frac{2r}{1-r^2} d\mu\_x d\mu\_y + 4d\sigma^2 \right],\tag{58}$$

respectively. The scalar curvature RM*s*of the manifolds with line elements in Equations (57) and (58) is RM*s* = −3/2. 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,

$$\left[\exp(\mathcal{S}\_{\mathcal{M}\_s}(\tau))\right]\_{\text{corrected}} \stackrel{\tau \to \infty}{\sim} \mathcal{F}(r) \cdot \left[\exp(\mathcal{S}\_{\mathcal{M}\_s}(\tau))\right]\_{\text{uncorrelated}'} \tag{59}$$

where the function F(*r*) in Equation (59) with 0 ≤ F(*r*) ≤ 1 is defined as

$$\mathcal{F}(r) \stackrel{\text{def}}{=} \sqrt{\frac{1-r}{1+r}}.\tag{60}$$

The function F(*r*) is a monotone decreasing compression factor with 0 < *r* < 1. The analysis proposed in [42,43] is a significant progress toward the understanding among the concepts of entanglement and statistical micro-correlations, as well as the impact of microcorrelations on the complexity of informational geodesic paths. The finding appearing in

Equation (59) suggests that the IGAC construct may prove useful in developing a sound IG perspective of the phenomenon of quantum entanglement.

### *3.7. Softening of Classical Chaos by Quantization*

Expanding upon the original analysis presented in [44–46], the IGAC framework was utilized to investigate the entropic dynamics and information geometry of a threedimensional Gaussian statistical model as well as the two-dimensional Gaussian statistical model derived from the former model by introducing the following macroscopic information constraint,

$$
\sigma\_x \sigma\_y = \Sigma^2,
$$

where Σ<sup>2</sup> ∈ <sup>R</sup>+0 . The quantities *x* and *y* label the microscopic degrees of freedom of the system. The constraint given by Equation (61) resembles the standard minimum uncertainty relation encountered in quantum mechanics [47]. The infinitesimal line elementsin the 3*D*and 2*D*-Gaussian statistical models are given by

$$ds\_{3D}^2 \stackrel{\text{def}}{=} \frac{1}{\sigma\_\chi^2} d\mu\_\chi^2 + \frac{2}{\sigma\_\chi^2} d\sigma\_\chi^2 + \frac{2}{\sigma\_\chi^2} d\sigma\_{y\_\cdot}^2 \tag{62}$$

and,

$$ds\_{2D}^2 \stackrel{\text{def}}{=} \frac{1}{\sigma^2} d\mu\_x^2 + \frac{4}{\sigma^2} d\sigma^2,\tag{63}$$

respectively. Note that the expectation value *μy* of the microvariable *y* is set equal to zero in Equation (62), while *σx* = *σ* with *<sup>σ</sup>xσy* = Σ<sup>2</sup> in Equation (63). Furthermore, the scalar curvatures corresponding to the 3*D* and 2*D* cases are equal to R3*D* = −1 and R2*D* = −1/2, respectively. It was determined that the complexity of the 2*D*-Gaussian statistical model specified by the IGE is relaxed when compared with the complexity of the 3*D*-Gaussian statistical model,

$$\mathcal{S}^{(2D)}\_{\mathcal{M}\_s}(\tau) \stackrel{\tau \to \infty}{\sim} \left(\frac{\lambda\_{2D}}{\lambda\_{3D}}\right) \cdot \mathcal{S}^{(3D)}\_{\mathcal{M}\_s}(\tau),\tag{64}$$

with *λ*2*D* and *λ*3*D* being both positive model parameters (satisfying the condition *λ*2*D* ≤ *λ*3*D*) that express the asymptotic temporal rates of change of the IGE in the 2*D* and 3*D* 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 sugges<sup>t</sup> 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 [48] where the work in [47] was generalized to a scenario where—in conjunction with the macroscopic constraint in Equation (61)—the microscopic degrees of freedom *x* and *y* of the system are also correlated.

### *3.8. Topologically Distinct Correlational Structures*

In [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 is 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

$$
\sigma \left( \left[ ds^2 \right]\_{\text{trivariate}}^{\text{(mildly connected)}} \right)\_{\text{corrected}} \stackrel{\text{def}}{=} \frac{1}{\sigma^2} \frac{3-4r}{1-2r^2} d\mu^2 + \frac{6}{\sigma^2} d\sigma^2. \tag{65}
$$

Moreover, the infinitesimal line element in the uncorrelated trivariate case is given by

$$
\int \left( \left[ ds^2 \right]\_{\text{trivariate}} \right)\_{\text{unnormalized}} \stackrel{\text{def}}{=} \frac{3}{\sigma^2} d\mu^2 + \frac{6}{\sigma^2} d\sigma^2. \tag{66}
$$

In Equations (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,

$$\left(\exp\left[\mathcal{S}\_{\text{trivariate}}^{\text{(mildy connected)}}(\tau)\right]\right)\_{\text{correlated}} \stackrel{\tau \to \infty}{\sim} \mathcal{R}\_{\text{trivariate}}^{\text{(mildy connected)}}(\tau) \left(\exp\left[\mathcal{S}\_{\text{trivariate}}^{\text{(mildy connected)}}(\tau)\right]\right)\_{\text{uncorrelations}'} \tag{67}$$

where

$$
\bar{\mathcal{R}}\_{\text{trivariate}}^{\text{(mildly connected)}}(r) \stackrel{\text{def}}{=} \sqrt{\frac{3(1-2r^2)}{3-4r}}.\tag{68}
$$

In Equation (67), the quantity *r* is the micro-correlation coefficient. The function R ˜ (mildly connected) trivariate (*r*) shows non-monotone behavior in the correlation parameter *r* and assumes a value of zero at the extrema of the permitted range *r* ∈ −√2/2, <sup>√</sup>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 *μx* = *μy* = *μ* and *σx* = *<sup>σ</sup>y* = *σ*, the infinitesimal line element is

$$
\left( \left[ ds^2 \right]\_{\text{bivariant}}^{\text{(fully connected)}} \right)\_{\text{corrected}} \stackrel{\text{def}}{=} \frac{2}{\sigma^2} \frac{1}{1+r} d\mu^2 + \frac{4}{\sigma^2} d\sigma^2. \tag{69}
$$

It was found that expS(fully connected) bivariate (*τ*)correlated *τ*→∞∼ R˜ (fully connected) bivariate (*r*)expS(fully connected) bivariate (*τ*)uncorrelated, (70)

> where

$$\mathcal{R}^{\text{(fully connected)}}\_{\text{bivariant}}(r) \stackrel{\text{def}}{=} \sqrt{1+r}.\tag{71}$$

Finally, in the fully connected trivariate Gaussian case with trivariate models having all microscopic variables correlated with each other, the infinitesimal line element is

$$
\left( \left[ ds^2 \right]\_{\text{trivariate}}^{\text{(fully connected)}} \right)\_{\text{corrected}} \stackrel{\text{def}}{=} \frac{3}{\sigma^2} \frac{1}{1+2r} d\mu^2 + \frac{6}{\sigma^2} d\sigma^2. \tag{72}
$$

It was determined in this case that

$$\left(\exp\left[\mathcal{S}\_{\text{trivariate}}^{\text{(fully connected)}}(\tau)\right]\right)\_{\text{corrected}} \overset{\tau \to \infty}{\sim} \mathcal{R}\_{\text{trivariate}}^{\text{(fully connected)}}(\tau) \left(\exp\left[\mathcal{S}\_{\text{trivariate}}^{\text{(fully connected)}}(\tau)\right]\right)\_{\text{uncoridated}} \tag{73}$$

where

$$
\bar{\mathcal{R}}\_{\text{trivariate}}^{\text{(fully connected)}}(r) \stackrel{\text{def}}{=} \sqrt{1+2r}.\tag{74}
$$

These results imply that in the fully connected bivariate and trivariate configurations, the ratios R ˜ (fully connected) bivariate (*r*) and R ˜ (fully connected) trivariate (*r*) both present monotone behavior in *r* over the open intervals (−1, 1) and (−1/2, <sup>1</sup>), respectively. On the other hand, in the mildly connected trivariate scenario appearing in Equation (67), an extremum in the function R ˜ (mildly connected) trivariate (*r*) occurs at *<sup>r</sup>*pea<sup>k</sup> = 1/2 ≥ 0. Such a 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 [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 arises when loops are present [50].

### **4. 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 2. 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 [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 [53–78].

**Table 2.** Schematic description of existing mathematical, classical, and quantum investigations within the IGAC.


**Author Contributions:** The authors contributed equally to this work. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No new data were created or analyzed in this study. Data sharing is not applicable to this article.

**Acknowledgments:** C. C. acknowledges the hospitality of the *United States Air Force Research Laboratory* in Rome where part of his initial contribution to this work was completed.

**Conflicts of Interest:** The authors declare no conflict of interest.
