Zhengwei Liu

Exchange relation planar algebras of small rank

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.

Noncommutative Uncertainty Principles

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the Hausdorff–Young inequality, Youngs inequality, the Hirschman–Beckner uncertainty principle, the Donoho–Stark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardys uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popas λ-lattices, modular tensor categories, etc.

Holographic Software for Quantum Networks

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.

Planar Para Algebras, Reflection Positivity

We define a planar para algebra, which arises naturally from combining planar algebras with the idea of

Quon 3D language for quantum information

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Uncertainty principles for Kac algebras

ZN para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects that are invariant under para isotopy. For each

Singly generated planar algebras of small dimension, Part III

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