Teleportation, Braid Group and Temperley--Lieb Algebra

We explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and Temperley--Lieb algebra. We realize the braid teleportation configuration, teleportation swapping and virtual braid representation in the standard description of the teleportation. We devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation: the transfer operator, quantum measurements and characteristic equations, and further propose the Temperley--Lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation. We compare our diagrammatical approach with two known recipes to the quantum information flow: the teleportation topology and strongly compact closed category, in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation.Comment: 33 pages, 19 figures, latex. The present article is a short version of the preprint, quant-ph/0601050, which includes details of calculation, more topics such as topological diagrammatical operations and entanglement swapping, and calls the Temperley--Lieb category for the collection of all the Temperley--Lieb algebra with physical operations like local unitary transformation

Groupoid Semantics for Thermal Computing

A groupoid semantics is presented for systems with both logical and thermal degrees of freedom. We apply this to a syntactic model for encryption, and obtain an algebraic characterization of the heat produced by the encryption function, as predicted by Landauer's principle. Our model has a linear representation theory that reveals an underlying quantum semantics, giving for the first time a functorial classical model for quantum teleportation and other quantum phenomena.Comment: We describe a groupoid model for thermodynamic computation, and a quantization procedure that turns encrypted communication into quantum teleportation. Everything is done using higher category theor

Environment and classical channels in categorical quantum mechanics

We present a both simple and comprehensive graphical calculus for quantum computing. In particular, we axiomatize the notion of an environment, which together with the earlier introduced axiomatic notion of classical structure enables us to define classical channels, quantum measurements and classical control. If we moreover adjoin the earlier introduced axiomatic notion of complementarity, we obtain sufficient structural power for constructive representation and correctness derivation of typical quantum informatic protocols.Comment: 26 pages, many pics; this third version has substantially more explanations than previous ones; Journal reference is of short 14 page version; Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag (2010

Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)

We give a scheme for interpreting shaded tangles as quantum programs, with the property that isotopic tangles yield equivalent programs. We analyze many known quantum programs in this way -- including entanglement manipulation and error correction -- and in each case present a fully-topological formal verification, yielding in several cases substantial new insight into how the program works. We also use our methods to identify several new or generalized procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973

Teleportation-Based Quantum Computation, Extended Temperley-Lieb Diagrammatical Approach and Yang--Baxter Equation

This paper focuses on the study of topological features in teleportation-based quantum computation as well as aims at presenting a detailed review on teleportaiton-based quantum computation (Gottesman and Chuang, Nature 402, 390, 1999). In the extended Temperley-Lieb diagrammatical approach, we clearly show that such topological features bring about the fault-tolerant construction of both universal quantum gates and four-partite entangled states more intuitive and simpler. Furthermore, we describe the Yang--Baxter gate by its extended Temperley-Lieb configuration, and then study teleportation-based quantum circuit models using the Yang--Baxter gate. Moreover, we discuss the relationship between the extended Temperley-Lieb diagrammatical approach and the Yang-Baxter gate approach. With these research results, we propose a worthwhile subject, the extended Temperley-Lieb diagrammatical approach, for physicists in quantum information and quantum computation.Comment: Latex, 47 pages, many figure