Bell states and quantum teleportation play important roles in the study of quantum information and computation. But a comprehensive theoretical research on both of them remains to be performed. This work aims to investigate important algebraic properties of generalized Bell states as well as explore topological features of quantum teleportation. First, the basis theorem and basis group are introduced to explain that the extension of a generalized Bell basis by a unitary matrix is still an orthonormal basis. Then a twist operator is defined to make a connection between a generalized multiple qubit Bell state and a tensor product of two qubit Bell state. Besides them, the Temperley--Lieb algebra, the braid group relation and the Yang--Baxter equation are used to provide a topological diagrammatic description of generalized Bell states and quantum teleportation. It turns out that our approach is able to present a clear illustration of relevant quantum information protocols and exhibit a topological nature of quantum entanglement and quantum teleportation.
Bell states and quantum teleportation play important roles in the study of quantum information and computation. But a comprehensive theoretical research on both of them remains to be performed. This work aims to investigate important algebraic properties of generalized Bell states as well as explore topological features of quantum teleportation. First, the basis theorem and basis group are introduced to explain that the extension of a generalized Bell basis by a unitary matrix is still an orthonormal basis. Then a twist operator is defined to make a connection between a generalized multiple qubit Bell state and a tensor product of two qubit Bell state. Besides them, the Temperley--Lieb algebra, the braid group relation and the Yang--Baxter equation are used to provide a topological diagrammatic description of generalized Bell states and quantum teleportation. It turns out that our approach is able to present a clear illustration of relevant quantum information protocols and exhibit a topological nature of quantum entanglement and quantum teleportation.
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.
In this paper, we explore algebraic structures and low dimensional topology underlying quantum information and computation. We revisit quantum teleportation from the perspective of the braid group, the symmetric group and the virtual braid group, and propose the braid teleportation, the teleportation swapping and the virtual braid teleportation, respectively. Besides, we present a physical interpretation for the braid teleportation and explain it as a sort of crossed measurement. On the other hand, we propose the extended Temperley--Lieb diagrammatical approach to various topics including quantum teleportation, entanglement swapping, universal quantum computation, quantum information flow, and etc. The extended Temperley--Lieb diagrammatical rules are devised to present a diagrammatical representation for the extended Temperley--Lieb category which is the collection of all the Temperley--Lieb algebras with local unitary transformations. In this approach, various descriptions of quantum teleportation are unified in a diagrammatical sense, universal quantum computation is performed with the help of topological-like features, and quantum information flow is recast in a correct formulation. In other words, we propose the extended Temperley--Lieb category as a mathematical framework to describe quantum information and computation involving maximally entangled states and local unitary transformations.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly , from separable basis states, important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
Important developments in fault-tolerant quantum computation using the braiding of anyons have placed the theory of braid groups at the very foundation of topological quantum computing. Furthermore, the realization by Kauffman and Lomonaco that a specific braiding operator from the solution of the Yang-Baxter equation, namely the Bell matrix, is universal implies that in principle all quantum gates can be constructed from braiding operators together with single qubit gates. In this paper we present a new class of braiding operators from the Temperley-Lieb algebra that generalizes the Bell matrix to multi-qubit systems, thus unifying the Hadamard and Bell matrices within the same framework. Unlike previous braiding operators, these new operators generate directly from separable basis states important entangled states such as the generalized Greenberger-Horne-Zeilinger states, cluster-like states, and other states with varying degrees of entanglement.
We introduce the concept of the Bell transform to represent known quantum gates in the literature, which are a unitary basis transformation from the product basis to the Bell states or the Greenberger-Horne-Zeilinger (GHZ) states. The algebraic structure of the four dimensional Bell transform has been studied systematically in this paper and we point out that it may be not a Clifford gate. The representative examples of the four dimensional Bell transform are verified as maximally entangling Clifford gates and some of them are recognized as parity-preserving gates or matchgates or Yang--Baxter gates. We define the teleportation operator in terms of the four dimensional Bell transform and apply it to the reformulation of the fault-tolerant construction of single-qubit gates and two-qubit gates in teleportation-based quantum computation. The algebraic structure of the higher dimensional Bell transform is also included and representative examples for it are verified as multi-qubit Clifford gates. Our research suggests that the Bell transform may play important roles in quantum information and computation as a new type of quantum transform.
It is well-known that maximally entangled states such as the Greenberger-Horne-Zeilinger (GHZ) states, with the Bell states as the simplest examples, are widely exploited in quantum information and computation. We study the application of such maximally entangled states from the viewpoint of the GHZ transform, which is a unitary basis transformation from the product states to the GHZ states. The algebraic structure of the GHZ transform is made clear and representative examples for it are verified as multi-qubit Clifford gates. In this paper, we focus on the Bell transform as the simplest example of the GHZ transform and apply it to the reformulation of quantum circuit model of teleportation and the reformulation of the fault-tolerant construction of single-qubit gates and two-qubit gates in teleportation-based quantum computation. We clearly show that there exists a natural algebraic structure called the teleportation operator in terms of the Bell transform to catch essential points of quantum teleportation, and hence we expect that there would also exist interesting algebraic structures in terms of the GHZ transform to play important roles in quantum information and computation.