Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.
Arthur JaffeDepartments of Mathematics and Physics, Harvard University, Cambridge, MA, 02138, USAZhengwei LiuDepartments of Mathematics and Physics, Harvard University, Cambridge, MA, 02138, USAAlex WozniakowskiPresent address: Current address: School of Physical and Mathematical Sciences and Complexity Institute, Nanyang Technological University, Singapore, 637723, Singapore; Departments of Mathematics and Physics, Harvard University, Cambridge, MA, 02138, USA
Mathematical PhysicsQuantum AlgebraSpectral Theory and Operator AlgebraarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.22002
We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the n-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information.
Chi-Ming ChangJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 USAYing-Hsuan LinJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 USAYifan WangCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 USAXi YinJefferson Physical Laboratory, Harvard University, Cambridge, MA 02138 USA
Mathematical PhysicsarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.22001
We study deformations of maximally supersymmetric gauge theories by higher dimensional operators in various spacetime dimensions. We classify infinitesimal deformations that preserve all 16 supersymmetries, while allowing the possibility of breaking either Lorentz or R-symmetry, using an on-shell algebraic method developed by Movshev and Schwarz. We also consider the problem of extending the deformation beyond the first order.
Arthur JaffeHarvard University, Cambridge, MA 02138, United States of America; Max Planck Institute for Mathematics, Bonn, GermanyZhengwei LiuHarvard University, Cambridge, MA 02138, United States of AmericaAlex WozniakowskiHarvard University, Cambridge, MA 02138, United States of America
We give a topological simulation for tensor networks that we call the two-string model. In this approach we give a new way to design protocols, and we discover a new multipartite quantum communication protocol. We introduce the notion of topologically compressed transformations. Our new protocol can implement multiple, non-local compressed transformations among multi-parties using one multipartite resource state.
Habib AmmariDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, FranceEmmanuel BossyInstitut Langevin, ESPCI ParisTech, CNRS UMR 7587, 10 rue Vauquelin, 75231 Paris Cedex 05, FranceJosselin GarnierLaboratoire de Probabilités et Modèles Aléatoires & Laboratoire Jacques-Louis Lions, Université Paris VII, 75205 Paris Cedex 13, FranceWenjia JingDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, FranceLaurent SeppecherDepartment of Mathematics and Applications, Ecole Normale Supérieure, 45 Rue d’Ulm, 75005 Paris, France
Analysis of PDEsMathematical Physicsmathscidoc:2206.03007
Journal of Mathematical Physics, 54, (2), 021501, 2013.2
The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.