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arXiv · 2509.17127

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

Rocco Maggi·Cosmo Lupo·Saverio Pascazio
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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.

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Rocco Maggi, Cosmo Lupo, Saverio Pascazio. 2025-09-21. Explicit construction of a 2-design of ${\rm U}(2)$ from the theory of angular momentum. https://arxiv.org/abs/2509.17127

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