  The ZX-Calculus is a powerful graphical language for quantum mechanics and
quantum information processing. The completeness of the language -- i.e. the
ability to derive any true equation -- is a crucial question. In the quest of a
complete ZX-calculus, supplementarity has been recently proved to be necessary
for quantum diagram reasoning (MFCS 2016). Roughly speaking, supplementarity
consists in merging two subdiagrams when they are parameterized by antipodal
angles. We introduce a generalised supplementarity -- called cyclotomic
supplementarity -- which consists in merging n subdiagrams at once, when the n
angles divide the circle into equal parts. We show that when n is an odd prime
number, the cyclotomic supplementarity cannot be derived, leading to a
countable family of new axioms for diagrammatic quantum reasoning.We exhibit
another new simple axiom that cannot be derived from the existing rules of the
ZX-Calculus, implying in particular the incompleteness of the language for the
so-called Clifford+T quantum mechanics. We end up with a new axiomatisation of
an extended ZX-Calculus, including an axiom schema for the cyclotomic
supplementarity.
