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

The main aim of this work is to present an explicit construction of a 2-design of $\mathrm{U}(2)$, 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 $\mathrm{U}(2)$ 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 $\text{unicode}x2014$ the prototypical unitary 1-design $\text{unicode}x2014$ one $\text{unicode}x201C$naturally$\text{unicode}x201D$ obtains the 2-design originally employed by Bennett and coauthors in $\mathit{\text{Mixed}}\mathit{\text{State}}\mathit{\text{Entanglement}}\mathit{\text{and}}\mathit{\text{Quantum}}\mathit{\text{Error}}\mathit{\text{Correction}}$. The present work also serves as a gentle and largely self-contained introduction to the subject.