Minimality in Finite-Dimensional ZW-Calculi

The ZW-calculus is a graphical language capable of representing 2-dimensional quantum systems (qubit) through its diagrams, and manipulating them through its equational theory. We extend the formalism to accommodate finite dimensional Hilbert spaces beyond qubit systems. First we define a qu$d$it version of the language, where all systems have the same arbitrary finite dimension $d$, and show that the provided equational theory is both complete -- i.e. semantical equivalence is entirely captured by the equations -- and minimal -- i.e. none of the equations are consequences of the others. We then extend the graphical language further to allow for mixed-dimensional systems. We again show the completeness and minimality of the provided equational theory.