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Observables in a quantum system, represented by a Hilbert space, are given bythe orthogonal bases of the aforementioned Hilbert space. Categorical QuantumMechanics provides further abstraction of such observables, allowing for adiagrammatic representation of measurements that extends to quantum processes.Our research studies this abstraction of observables, which has been dubbed asclassical structures, in a subtheory of quantum mechanics which focuses onqubit systems (or 2-dimensional quantum system and its composites). We haveconstructed a procedure that takes the complementary classical structures of asingle qubit system and compose them separably via the Kronecker product or'entangle' them via Bell states to obtain complementary classical structures in$n$-qubit systems. In this present work, we apply our procedure to two qubitand three qubit systems as examples. Then, using rewriting rules of ZX-calculusand tools in graph theory, we searched for maximal complete sets of mutuallycomplementary classical structures (the categorical counterpart of mutuallyunbiased bases) among our constructed composite classical structures. For twoqubits, we found 13 maximal complete sets of mutually complementary classicalstructures, and for three qubits, we found 32,448 maximal complete sets ofmutually complementary classical structures.

The Role of Compositionality in Constructing Complementarity Classical Structures Within Qubit Systems