Core Concepts

The author presents a novel approach to unify different methods of Bregman relative entropies, extending them to various state spaces in quantum theory.

Abstract

The content introduces the theory of generalized Bregman relative entropies over non-reflexive Banach spaces. By utilizing nonlinear embeddings and Euler-Legendre functions, two former approaches are unified. This construction extends Bregman relative entropies to state spaces in probability, quantum, and postquantum theory. The Norden-Sen geometry is defined on C3 manifolds with specific properties. The global geometric properties of D can be analyzed using torsion-free Norden-Sen differential geometry. Two approaches to constructing functional encoding are discussed: one based on Brègman's method and another based on Brunk-Ewing-Utz method. The passage from probabilistic to quantum setting involves replacing L1(X, µ) by the Banach predual N* of a W*-algebra N. A fundamental feature of DΨ is characterized as a nonlinear generalization of the Pythagorean theorem. The dually flat geometry is characterized by the flatness of connections and existence of specific coordinate systems. A generalization called Dℓ,Ψ applicable to non-reflexive Banach spaces is introduced by pulling back properties exhibited by DΨ with Euler-Legendre Ψ acting on reflexive Banach spaces into properties exhibited by Dℓ,Ψ.

Stats

D(x, y) = 0 ⇐⇒ x = y ∀x, y ∈ Z.
g(t∇c(·), te∇c(·)) = g.
trH(Dξ(log ξ - log ζ) - ξ - ζ) ∀(ξ, ζ) ∈ K+ × K+0.
trH(γ|ξ|^(1/γ) + (1 - γ)ζ^(1/γ) - ξζ^(1/γ-1)) ∀(ξ, ζ) ∈ K × K+0.
Dα(ξ, ζ) := trH(ζ^α - (1/(1-α))ξ^α + (α/(1-α))ζ^α-1 * ξ) ∀(ξ, ζ, α) ∈ K+ × K+0 × ]0, 1[.

Quotes

"Duality between conjugate convex functions" - Fenchel W., 1949
"Convergence of Bregman projection methods for solving consistent convex feasibility problems" - Al'ber Ya.I., Butnariu D., 1997

Deeper Inquiries

The introduction of generalized Bregman relative entropies has a significant impact on current research in quantum theory. By extending the concept to non-reﬂexive Banach spaces, this approach allows for a more comprehensive analysis of state spaces in probability, quantum, and postquantum theory. This extension opens up new possibilities for studying complex systems where traditional methods may fall short. Researchers can now apply these generalizations to explore a wider range of scenarios and delve deeper into the intricacies of quantum phenomena.

Unifying different approaches to Bregman relative entropies brings about several implications. Firstly, it creates a cohesive framework that bridges previously disparate methodologies based on reﬂexive Banach spaces and differential geometry. This unification enables researchers to leverage insights from both domains and apply them synergistically to solve complex problems efficiently. Additionally, by extending Brègman relative entropies across various geometric and operator structures, this unified approach enhances the versatility and applicability of these concepts in diverse mathematical contexts.

The concept of dually flat geometry can be applied in various mathematical contexts beyond its original formulation within statistical manifolds or information geometry. For instance:
In optimization problems: Dually flat structures can provide efficient algorithms for optimization tasks by leveraging the properties of geodesics or projections onto convex sets.
In machine learning: The principles behind dually flat geometries can enhance models' interpretability and performance by incorporating curvature information into learning algorithms.
In signal processing: Utilizing dually flat geometries can lead to improved data representation techniques that capture underlying structures more effectively.
By applying dually flat geometry creatively across different mathematical disciplines, researchers can unlock novel insights and solutions tailored to specific problem domains.

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