A Near-Minimal Axiomatisation of ZX-Calculus for Pure Qubit Quantum Mechanics

Recent developments in the ZX-Calculus have resulted in complete axiomatisations first for an approximately universal restriction of the language, and then for the whole language. The main drawbacks were that the axioms that were added to achieve completeness were numerous, tedious to manipulate and lacking a physical interpretation. We present in this paper two complete axiomatisations for the general ZX-Calculus, that we believe are optimal, in that all their equations are necessary and moreover have a nice physical interpretation. To do so, we introduce the singular-value decomposition of a ZX-diagram, and use it to show that all the rules of the former axiomatisation are provable with the new one.