Keshab Chandra BakshiInstitute of Mathematical Science, Homi Bhabha National Institute, Chennai, IndiaSayan DasDepartment of Mathematics, University of Iowa, Iowa City IA 52242Zhengwei LiuDepartment of Mathematics, Harvard University, Cambridge, MA 02138, USAYunxiang RenDepartment of Mathematics, University of Tennessee, Knoxville TN 37996
Transactions of the American Mathematical Society, 371, 5973-5991, 2018.12
We introduce a new notion of an angle between intermediate subfactors and prove various interesting properties of the angle and relate it to the Jones index. We prove a uniform 60 to 90 degree bound for the angle between minimal intermediate subfactors of a finite index irreducible subfactor. From this rigidity we can bound the number of minimal (or maximal) intermediate subfactors by the kissing number in geometry. As a consequence, the number of intermediate subfactors of an irreducible subfactor has at most exponential growth with respect to the Jones index. This answers a question of Longo’s published in 2003.
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.
Zhengwei LiuDepartment of Mathematics and Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USAJinsong WuSchool of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China
Quantum AlgebraSpectral Theory and Operator Algebramathscidoc:2206.29005
Journal of Mathematical Physics, 58, 052102, 2017.5
In this paper, we introduce the notation of bi-shift of biprojections in subfactor theory to unimodular Kac algebras. We characterize the minimizers of the Hirschman-Beckner uncertainty principle and the Donoho-Stark uncertainty principle for unimodular Kac algebras with biprojections and prove Hardy’s uncertainty principle in terms of the minimizers.
The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for , the Fuss-Catalan planar algebra, and the group/subgroup planar algebra for Z_2 ⊂ Z_5 \rtimes Z_2.. In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all Birman-Murakami-Wenzl algebras. Precisely it contains a depth 3 one from quantum O(3), and a one-parameter family from quantum S_p(4).