Pivoting makes the ZX-calculus complete for real stabilizers

TL;DR

This paper demonstrates that adding pivoting to the ZX-calculus does not derive from existing axioms and introduces an angle-free version that is complete for real stabilizer quantum mechanics.

Contribution

It shows the limitations of pivoting within the ZX-calculus and introduces a new angle-free calculus that achieves completeness for real stabilizers.

Findings

Pivoting cannot be derived from ZX-calculus axioms.

Pivoting does not imply local complementation.

Angle-free ZX-calculus is complete for real stabilizer quantum mechanics.

Abstract

We show that pivoting property of graph states cannot be derived from the axioms of the ZX-calculus, and that pivoting does not imply local complementation of graph states. Therefore the ZX-calculus augmented with pivoting is strictly weaker than the calculus augmented with the Euler decomposition of the Hadamard gate. We derive an angle-free version of the ZX-calculus and show that it is complete for real stabilizer quantum mechanics.