Explicit construction of a 2-design of ${\rm U}(2)$ from the theory of angular momentum
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911167874596864 |
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| author | Maggi, Rocco Lupo, Cosmo Pascazio, Saverio |
| author_facet | Maggi, Rocco Lupo, Cosmo Pascazio, Saverio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17127 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit construction of a 2-design of ${\rm U}(2)$ from the theory of angular momentum Maggi, Rocco Lupo, Cosmo Pascazio, Saverio Quantum Physics Mathematical Physics 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. |
| title | Explicit construction of a 2-design of ${\rm U}(2)$ from the theory of angular momentum |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2509.17127 |