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, 02138, U.S.A. Simeng WangLaboratoire de Mathematiques, Universite de Franche -Comte, Besancon Cedex, 25030, France and Institute of Mathematics, Polish Academy of Sciences, ul. Sniadeckin 8, 00-956 Warszawa, PolandJinsong WuSchool of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, P.R. China
Spectral Theory and Operator Algebramathscidoc:2206.32001
Journal of Operator Theory, 77, (1), 109-131, 2017.12
In this paper, we generalize Young's inequality for locally compact quantum groups and obtain some results for extremal pairs of Young's inequality and extremal functions of Hausdorff-Young inequality.
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).
The main purpose of this paper is to classify exchange relation planar algebras with 4 dimensional 2-boxes. Besides its skein theory, we emphasize the positivity of subfactor planar algebras based on the Schur product theorem. We will discuss the lattice of projections of 2-boxes, specifically the rank of the projections. From this point, several results about biprojections are obtained. The key break of the classification is to show the existence of a biprojection. By this method, we also classify another two families of subfactor planar algebras: subfactor planar algebras generated by 2-boxes with 4 dimensional 2-boxes and at most 23 dimensional 3-boxes; subfactor planar algebras generated by 2-boxes, such that the quotient of 3-boxes by the basic construction ideal is abelian. They extend the classification of singly generated planar algebras obtained by Bisch, Jones and the author.