A New Diagrammatic Approach to Quantum Information

We introduce a new diagrammatic approach to quantum information, that we call holographic software [1]. Our new approach is a small modification of previous diagrammatic approaches, yet our small change leads to a great deal of new understanding and to a large number of new insights and motivation. Its application leads us to new protocols, designed in a “topological” way. (1) The fundamental idea to use a diagrammatic notation to describe tensor manipulation originated in the work of Penrose [2]. This work has been extensively developed by Abramsky, Coecke, and their coworkers, yielding many diagrammatic representations of tensors and other algebraic structures in tensor networks [3, 4]. (2) Many diagrams have topological meaning, and its importance in quantum information was recognized in the pioneering work of Kitaev, Freedman, Larsen, Wang, Kauffman, and Lomonaco [5, 6, 7, 8, 9]. In addition, the topological model of Kitaev, Levin, and Wen provide powerful tools in quantum information [5, 10]. Manin and Feynman introduced the concept of quantum simulation for quantum systems [11, 12, 13]. In our framework we simulate quantum networks using a topological model PAPPA constructed in [14]. This fits into the philosophy of (2). Our holographic software allows us to translate between the topological approach and the algebraic approach. Thus we obtain new diagrammatic interpretations of algebraic structures. This fits into the philosophy of (1).