Geometry from divergence functions and complex structures

• URL: http://arxiv.org/abs/2002.02891v1
• Date: Fri, 7 Feb 2020 16:47:18 GMT
• Title: Geometry from divergence functions and complex structures
• Authors: Florio M. Ciaglia, Fabio Di Cosmo, Armando Figueroa, Giuseppe Marmo, Luca Schiavone
• Abstract summary: We introduce an almost-complex structure $J$ on the product $Mtimes M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $Mtimes M$ from a divergence function.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Motivated by the geometrical structures of quantum mechanics, we introduce an almost-complex structure $J$ on the product $M\times M$ of any parallelizable statistical manifold $M$. Then, we use $J$ to extract a pre-symplectic form and a metric-like tensor on $M\times M$ from a divergence function. These tensors may be pulled back to $M$, and we compute them in the case of an N-dimensional symplex with respect to the Kullback-Leibler relative entropy, and in the case of (a suitable unfolding space of) the manifold of faithful density operators with respect to the von Neumann-Umegaki relative entropy.

Related papers

• Tensor network approximation of Koopman operators [0.0]
  We propose a framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)n$.
  arXiv  Detail & Related papers  (2024-07-09T21:40:14Z)
• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Equivariant Manifold Neural ODEs and Differential Invariants [1.6073704837297416]
  We develop a manifestly geometric framework for equivariant manifold neural ordinary differential equations (NODEs) We use it to analyse their modelling capabilities for symmetric data.
  arXiv  Detail & Related papers  (2024-01-25T12:23:22Z)
• Remarks on effects of projective phase on eigenstate thermalization hypothesis [0.0]
  We consider $mathbbZ_NtimesmathbbZ_N$ symmetries with nontrivial projective phases. We also perform numerical analyses for $ (1+1)$-dimensional spin chains and the $ (2+1)$-dimensional lattice gauge theory.
  arXiv  Detail & Related papers  (2023-10-17T17:36:37Z)
• Effects of detuning on $\mathcal{PT}$-symmetric, tridiagonal, tight-binding models [0.0]
  Non-Hermitian, tight-binding $mathcalPT$-symmetric models are extensively studied in the literature. Here, we investigate two forms of non-Hermitian Hamiltonians to study the $mathcalPT$-symmetry breaking thresholds and features of corresponding surfaces of exceptional points (EPs) Taken together, our results provide a detailed understanding of detuned tight-binding models with a pair of gain-loss potentials.
  arXiv  Detail & Related papers  (2023-02-26T01:36:59Z)
• Deep Learning Symmetries and Their Lie Groups, Algebras, and Subalgebras from First Principles [55.41644538483948]
  We design a deep-learning algorithm for the discovery and identification of the continuous group of symmetries present in a labeled dataset. We use fully connected neural networks to model the transformations symmetry and the corresponding generators. Our study also opens the door for using a machine learning approach in the mathematical study of Lie groups and their properties.
  arXiv  Detail & Related papers  (2023-01-13T16:25:25Z)
• $O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks [107.86545461433616]
  We propose permutation-equivariant architectures, on which a determinant Slater is applied to induce antisymmetry. FermiNet is proved to have universal approximation capability with a single determinant, namely, it suffices to represent any antisymmetric function. We substitute the Slater with a pairwise antisymmetry construction, which is easy to implement and can reduce the computational cost to $O(N2)$.
  arXiv  Detail & Related papers  (2022-05-26T07:44:54Z)
• Conditions for realizing one-point interactions from a multi-layer structure model [77.34726150561087]
  A heterostructure composed of $N$ parallel homogeneous layers is studied in the limit as their widths shrink to zero. The problem is investigated in one dimension and the piecewise constant potential in the Schr"odinger equation is given.
  arXiv  Detail & Related papers  (2021-12-15T22:30:39Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Modular Nuclearity and Entanglement Entropy [0.0]
  In this work we show that the Longo's canonical entanglement entropy is finite in any local QFT verifying a modular $p$-nuclearity condition. As application, in $1+1$-dimensional integrable models with factorizing S-matrices we study the behavior of the canonical entanglement entropy as the distance between two causally disjoint wedges diverges.
  arXiv  Detail & Related papers  (2021-08-20T09:01:59Z)
• A Unifying and Canonical Description of Measure-Preserving Diffusions [60.59592461429012]
  A complete recipe of measure-preserving diffusions in Euclidean space was recently derived unifying several MCMC algorithms into a single framework. We develop a geometric theory that improves and generalises this construction to any manifold.
  arXiv  Detail & Related papers  (2021-05-06T17:36:55Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.