2015/01/28 by Amar Hadzihasanovic, Hadzihasanovic, Amar · 4 citations
Computer Science · Physics and Astronomy · #16T05 (Secondary) #81P45 (Primary) 18D10 #Category Theory (math.CT) #Computability, Logic, AI Algorithms #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Logic in Computer Science (cs.LO) #Quantum Computing Algorithms and Architecture #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1501.07082
openalex publication_date 2015/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and Kissinger suggested that two 3-qubit states, GHZ and W, may be used as the building blocks of a new graphical calculus, aimed at a diagrammatic classification of multipartite qubit entanglement that would highlight the communicational properties of quantum states, and their potential uses in cryptographic schemes. In this paper, we present a full graphical axiomatisation of the relations between GHZ and W: the ZW calculus. This refines a version of the preexisting ZX calculus, while keeping its most desirable characteristics: undirectedness, a large degree of symmetry, and an algebraic underpinning. We prove that the ZW calculus is complete for the category of free abelian groups on a power of two generators - "qubits with integer coefficients" - and provide an explicit normalisation procedure.