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

TL;DR

This paper explores how to construct and analyze mutually complementary classical structures in multi-qubit systems using categorical quantum mechanics, revealing a large number of such structures in three-qubit systems.

Contribution

It introduces a procedure to construct and identify mutually complementary classical structures in multi-qubit systems using categorical quantum mechanics and ZX-calculus.

Findings

Identified 13 maximal sets for two qubits

Found 32,448 maximal sets for three qubits

Applied diagrammatic and graph-theoretic methods

Abstract

Observables in a quantum system, represented by a Hilbert space, are given by the orthogonal bases of the aforementioned Hilbert space. Categorical Quantum Mechanics provides further abstraction of such observables, allowing for a diagrammatic representation of measurements that extends to quantum processes. Our research studies this abstraction of observables, which has been dubbed as classical structures, in a subtheory of quantum mechanics which focuses on qubit systems (or 2-dimensional quantum system and its composites). We have constructed a procedure that takes the complementary classical structures of a single 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 qubit and three qubit systems as examples. Then, using rewriting rules of ZX-calculus and tools in graph theory, we searched for maximal complete sets of mutually complementary classical structures (the categorical counterpart of mutually unbiased bases) among our constructed composite classical structures. For two qubits, we found 13 maximal complete sets of mutually complementary classical structures, and for three qubits, we found 32,448 maximal complete sets of mutually complementary classical structures.