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arXiv:2608.15916

Information Geometry of the Geodesic Quantum $f$-Divergences

Ángela Capel, Pablo Costa Rico

quant-phmath-phSubmitted Aug 16, 2026• Announced Aug 18, 2026• 0 citations
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Abstract

We study the differential, statistical, and geometrical consequences generated by the geodesic quantum $f$-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter $t\in [0,1]$. For an invertible state $ρ$ and an operator convex function $f$, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric $g_{ρ,t}^{(f)}$. Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter $t$ in this new geometry. We next show that the interpolation of relative modular operators $Γ_t$ in the reference purification of a state $σ$ defines a canonical finite binary experiment $(p_t,q_t)$, and introduce a log-likelihood cumulant function $Ψ_{ρ,σ}(t,s)$, recovering the Nussbaum-Szkoła distributions at $t=0$ and the Matsumoto construction at $t=1$. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.

Related papers

Geodesic Quantum fff-Divergences

Ángela Capel, Pablo Costa Rico

We introduce the geodesic quantum $f$-divergences $D_f^t$, $0\leq t\leq1$, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical $f$-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield $(t,α)$-Rényi divergences joining the Petz and geometric families. Our first main result is data processing of $D_f^t$ for every finite operator-convex generator $f$ and every $t\in[0,1]$. In particular, this gives DPI for the geodesic relative entropies and for the $(t,α)$-Rényi divergences when $0<α<1$ or $1<α\leq2$. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter $0\leq t<1$, exactly the Petz sufficiency class; at $t=1$, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every $t$. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the $t$-quantum Markov chains are precisely the quantum Markov chains for $0\leq t<1$, whereas at $t=1$ they are the invertible BS quantum Markov chains, a class that can be strictly larger. We also determine parameter-monotonicity regimes and the intersections with the $(α,z)$ family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.

quant-pharXiv:2608.15833•Aug 18, 2026

Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective

Sayantan Roy, Atin Gayen, Aditi Kar Gangopadhyay +1 more

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.

quant-pharXiv:2604.06908•Apr 9, 2026•0 citations

Relative Entropy and Modular Geometry in Algebraic Quantum Field Theory

Arşen Kaan

This preprint investigates structural relations between relative entropy, modular theory, and information geometry in algebraic quantum field theory (AQFT). In finite‑dimensional quantum systems, the Hessian of Umegaki relative entropy coincides with the Bogoliubov–Kubo–Mori (BKM) metric, and modular automorphisms act by isometries of this geometry; these standard results are reviewed as a mathematically controlled reference model. Motivated by this setting, the work formulates minimal structural conditions under which a perturbative modular information‑geometric programme for Type III1_11 von Neumann algebras could become mathematically meaningful. The analysis focuses on admissible perturbation sectors, Araki relative entropy, modular covariance, and the functional‑analytic obstructions caused by unbounded modular operators. A general consequence of positive semidefiniteness of the perturbative entropy Hessian is recorded: whenever the Hessian defines a well‑defined positive semidefinite bilinear form, its null directions annihilate the full form. In the Rindler wedge model, the corresponding regularity problem for candidate perturbations is formulated in terms of spectral properties and domain issues of the Lorentz boost generator. Throughout, explicit separation is maintained between rigorous results, conditional statements, heuristic motivations, and unresolved analytic problems. A conjecture concerning degenerate modular directions is stated together with a necessary falsifiable condition. This document is exploratory in nature and has not been peer reviewed.

•May 26, 2026•0 citations

Εφαρμογές της γεωμετρίας της πληροφορίας

Fotios D. Oikonomou, Φώτιος Οικονόμου

Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems. We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics. In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative. In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems. In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems. Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.

•Jan 1, 2019•0 citations

Constraining quantum fields using modular theory

Nima Lashkari

Abstract Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.

arXiv:1810.09306•Journal of High Energy Physics•Jan 1, 2019•44 citations

Information-Geometric Analysis of Quantum Collapse Dynamics: From State-Update Geometry to Dynamical Collapse Trajectories

Angelo Plastino

We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.

Quantum Reports•Oct 28, 2026•0 citations
Information Geometry of the Geodesic Quantum fff-Divergences
We study the differential, statistical, and geometrical consequences generated by the geodesic quantum fff-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter t∈[0,1]t\in [0,1]t∈[0,1]. For an invertible state ρρρ and an operator convex function fff, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric gρ,t(f)g_{ρ,t}^{(f)}gρ,t(f)​. Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter ttt in this new geometry. We next show that the interpolation of relative modular operators ΓtΓ_tΓt​ in the reference purification of a state σσσ defines a canonical finite binary experiment (pt,qt)(p_t,q_t)(pt​,qt​), and introduce a log-likelihood cumulant function Ψρ,σ(t,s)Ψ_{ρ,σ}(t,s)Ψρ,σ​(t,s), recovering the Nussbaum-Szkoła distributions at t=0t=0t=0 and the Matsumoto construction at t=1t=1t=1. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.
We introduce the geodesic quantum fff-divergences DftD_f^tDft​, 0≤t≤10\leq t\leq10≤t≤1, obtained from the affine-invariant geodesic between the standard and maximal relative modular operators. They reduce to the classical fff-divergence for commuting states. The logarithmic generator yields geodesic relative entropies joining the Umegaki and Belavkin-Staszewski entropies, while the power generators yield (t,α)(t,α)(t,α)-Rényi divergences joining the Petz and geometric families. Our first main result is data processing of DftD_f^tDft​ for every finite operator-convex generator fff and every t∈[0,1]t\in[0,1]t∈[0,1]. In particular, this gives DPI for the geodesic relative entropies and for the (t,α)(t,α)(t,α)-Rényi divergences when 0<α<10<α<10<α<1 or 1<α≤21<α\leq21<α≤2. Our second main result identifies equality in the DPI: for invertible states, every equality-determining operator-convex generator has, at each nonmaximal parameter 0≤t<10\leq t<10≤t<1, exactly the Petz sufficiency class; at t=1t=1t=1, this changes to the generally larger maximal, or BS, class, which also coincides with the equality class of the divergences associated with the quadratic generator for every ttt. Our third main result is the corresponding collapse of invertible geodesic quantum Markov chains: for each of the three ordered conditional-mutual-information constructions, the ttt-quantum Markov chains are precisely the quantum Markov chains for 0≤t<10\leq t<10≤t<1, whereas at t=1t=1t=1 they are the invertible BS quantum Markov chains, a class that can be strictly larger. We also determine parameter-monotonicity regimes and the intersections with the (α,z)(α,z)(α,z) family. We prove strengthened data-processing and reconstruction estimates; we compare the three conditional orientations; we establish continuity bounds under positive lower-eigenvalue assumptions together with complementary discontinuity results; and we give a capacity-per-unit-cost interpretation.
Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.
This preprint investigates structural relations between relative entropy, modular theory, and information geometry in algebraic quantum field theory (AQFT). In finite‑dimensional quantum systems, the Hessian of Umegaki relative entropy coincides with the Bogoliubov–Kubo–Mori (BKM) metric, and modular automorphisms act by isometries of this geometry; these standard results are reviewed as a mathematically controlled reference model. Motivated by this setting, the work formulates minimal structural conditions under which a perturbative modular information‑geometric programme for Type III1_11 von Neumann algebras could become mathematically meaningful. The analysis focuses on admissible perturbation sectors, Araki relative entropy, modular covariance, and the functional‑analytic obstructions caused by unbounded modular operators. A general consequence of positive semidefiniteness of the perturbative entropy Hessian is recorded: whenever the Hessian defines a well‑defined positive semidefinite bilinear form, its null directions annihilate the full form. In the Rindler wedge model, the corresponding regularity problem for candidate perturbations is formulated in terms of spectral properties and domain issues of the Lorentz boost generator. Throughout, explicit separation is maintained between rigorous results, conditional statements, heuristic motivations, and unresolved analytic problems. A conjecture concerning degenerate modular directions is stated together with a necessary falsifiable condition. This document is exploratory in nature and has not been peer reviewed.
Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems. We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics. In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative. In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems. In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems. Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.
Abstract Tomita-Takesaki modular theory provides a set of algebraic tools in quantum field theory that is suitable for the study of the information-theoretic properties of states. For every open set in spacetime and choice of two states, the modular theory defines a positive operator known as the relative modular operator that decreases monotonically under restriction to subregions. We study the consequences of this operator monotonicity inequality for correlation functions in quantum field theory. We do so by constructing a one-parameter Rényi family of information-theoretic measures from the relative modular operator that inherit monotonicity by construction and reduce to correlation functions in special cases. In the case of finite quantum systems, this Rényi family is the sandwiched Rényi divergence and we obtain a simple proof of its monotonicity. Its monotonicity implies a class of constraints on correlation functions in quantum field theory, only a small set of which were known to us. We explore these inequalities for free fields and conformal field theory. We conjecture that the second null derivative of Rényi divergence is non-negative which is a generalization of the quantum null energy condition to the Rényi family.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.