We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2+1 Levin–Wen model to obtain 1+1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.
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.