Explicit construction of a 2-design of ${\rm U}(2)$ from the theory of angular momentum

Abstract:The main aim of this work is to present an explicit construction of a 2-design of ${\rm U}(2)$, relying only on a tool that belongs to every physicists toolbox: the theory of angular momentum. Unitary designs are a rich and fundamental mathematical topic, with numerous fruitful applications in quantum information science and technology. In this work we take a peek under the hood. We begin with a minimal set of definitions and characterizations. Then we derive all 1-designs of ${\rm U}(2)$ of minimum size. Finally, we set out, step by step, a completion procedure extending such 1-designs to 2-designs. In particular, starting from the Pauli basis $\unicode{x2014}$ the prototypical unitary 1-design $\unicode{x2014}$ one $\unicode{x201C}$naturally$\unicode{x201D}$ obtains the 2-design originally employed by Bennett and coauthors in $\textit{Mixed State Entanglement and Quantum Error Correction}$. The present work also serves as a gentle and largely self-contained introduction to the subject.

pdfTeX-1.40.27, 42 pages, 5 figures, 1 table

arXiv:2509.17127 [quant-ph]

(or arXiv:2509.17127v2 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2509.17127 Focus to learn more arXiv-issued DOI via DataCite

Phys. Scr. 101, 305201 (2026)

https://doi.org/10.1088/1402-4896/ae59bf Focus to learn more DOI(s) linking to related resources

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