Depicting qudit quantum mechanics and mutually unbiased qudit theories
arXiv:1404.1288 · doi:10.4204/EPTCS.172.6
Abstract
We generalize the ZX calculus to quantum systems of dimension higher than two. The resulting calculus is sound and universal for quantum mechanics. We define the notion of a mutually unbiased qudit theory and study two particular instances of these theories in detail: qudit stabilizer quantum mechanics and Spekkens-Schreiber toy theory for dits. The calculus allows us to analyze the structure of qudit stabilizer quantum mechanics and provides a geometrical picture of qudit stabilizer theory using D-toruses, which generalizes the Bloch sphere picture for qubit stabilizer quantum mechanics. We also use our framework to describe generalizations of Spekkens toy theory to higher dimensional systems. This gives a novel proof that qudit stabilizer quantum mechanics and Spekkens-Schreiber toy theory for dits are operationally equivalent in three dimensions. The qudit pictorial calculus is a useful tool to study quantum foundations, understand the relationship between qubit and qudit quantum mechanics, and provide a novel, high level description of quantum information protocols.
arXiv commentIn Proceedings QPL 2014, arXiv:1412.8102
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- The Qupit Stabiliser ZX-travaganza: Simplified Axioms, Normal Forms and Graph-Theoretic Simplification
- Diagrammatic Analysis for Parameterized Quantum Circuits
- Building Qutrit Diagonal Gates from Phase Gadgets
- Scaling W state circuits in the qudit Clifford hierarchy
- Mermin Non-Locality in Abstract Process Theories
- ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
- Transversal AND in Quantum Codes