Information Geometry (IG) is a thriving field of research that brings together, among others, Mathematics, Physics, and Mathematical Engineering.

Born originally in Classical Statistics and Classical Information theory, IG has been extended to various research fields: Quantum Physics, Quantum Information Theory, Machine Learning, and Artificial Intelligence.

The original idea behind IG is to investigate and exploit the differential geometric properties of suitable manifolds of classical probability distributions/quantum states/neural networks to address specific operational tasks and provide geometric understandings. For example, estimation problems (e.g., Cramer-Rao bound and its quantum generalizations), hypothesis testing (e.g., classical and quantum error probability), optimization problems (e.g., projection with respect to dual connections), and machine learning (e.g., natural gradient learning), just to name a few.

One of the recent topics in IG focuses on the analysis of manifolds that cannot be handled with the conventional approaches.

The main geometrical structures arising in IG are smooth manifolds endowed with a Riemannian metric tensor g (e.g., the Fisher-Rao metric tensor and its quantum generalizations), and a pair of g-dual affine connections, thus giving IG the mathematical flavor of a "generalized Riemannian/affine geometry".

Quite often, most of the work is usually done in the context of dually-flat manifolds (i.e., both connections are flat), and/or using torsion-free connections. However, since the theoretical premises of IG clearly allows more freedom in the choice of these geometric structures, it is not yet completely clear what we are missing by not fully exploiting this freedom.

Moreover, from a less mathematical point of view, an interesting interplay between IG and quantum field theory (and its neighborhoods) is recently developing and growing, thus attracting new avenues of cross-fertilization which is important to explore.

Therefore, we think it would be useful for both people actively working in IG and for people simply curious about IG to gather in one place some of the most well-established results and some of the more recent developments gravitating around non-standard aspects and applications of IG.

In this Research Topic we aim to gather Original Research articles as well as Review articles dealing with non-standard aspects and applications of IG.

For regular articles, the successful submission should develop new ideas and present original results in (but not limited to) the following topics :

1) non-dually-flat structures in IG;

2) connections with torsion in IG;

3) symplectic and Poisson aspects of IG;

4) quantumness and incompatibility using Uhlmann's curvature;

5) singular models of classical and quantum states;

6) Bayesian aspects of IG;

7) Tomita-Takesaki modular theory and IG;

8) theoretical particle physics, quantum field theory and IG;

9) applications of IG in condensed-matter systems and critical systems.

Information Geometry (IG) is a thriving field of research that brings together, among others, Mathematics, Physics, and Mathematical Engineering.

Born originally in Classical Statistics and Classical Information theory, IG has been extended to various research fields: Quantum Physics, Quantum Information Theory, Machine Learning, and Artificial Intelligence.

The original idea behind IG is to investigate and exploit the differential geometric properties of suitable manifolds of classical probability distributions/quantum states/neural networks to address specific operational tasks and provide geometric understandings. For example, estimation problems (e.g., Cramer-Rao bound and its quantum generalizations), hypothesis testing (e.g., classical and quantum error probability), optimization problems (e.g., projection with respect to dual connections), and machine learning (e.g., natural gradient learning), just to name a few.

One of the recent topics in IG focuses on the analysis of manifolds that cannot be handled with the conventional approaches.

The main geometrical structures arising in IG are smooth manifolds endowed with a Riemannian metric tensor g (e.g., the Fisher-Rao metric tensor and its quantum generalizations), and a pair of g-dual affine connections, thus giving IG the mathematical flavor of a "generalized Riemannian/affine geometry".

Quite often, most of the work is usually done in the context of dually-flat manifolds (i.e., both connections are flat), and/or using torsion-free connections. However, since the theoretical premises of IG clearly allows more freedom in the choice of these geometric structures, it is not yet completely clear what we are missing by not fully exploiting this freedom.

Moreover, from a less mathematical point of view, an interesting interplay between IG and quantum field theory (and its neighborhoods) is recently developing and growing, thus attracting new avenues of cross-fertilization which is important to explore.

Therefore, we think it would be useful for both people actively working in IG and for people simply curious about IG to gather in one place some of the most well-established results and some of the more recent developments gravitating around non-standard aspects and applications of IG.

In this Research Topic we aim to gather Original Research articles as well as Review articles dealing with non-standard aspects and applications of IG.

For regular articles, the successful submission should develop new ideas and present original results in (but not limited to) the following topics :

1) non-dually-flat structures in IG;

2) connections with torsion in IG;

3) symplectic and Poisson aspects of IG;

4) quantumness and incompatibility using Uhlmann's curvature;

5) singular models of classical and quantum states;

6) Bayesian aspects of IG;

7) Tomita-Takesaki modular theory and IG;

8) theoretical particle physics, quantum field theory and IG;

9) applications of IG in condensed-matter systems and critical systems.