Explicit construction of 2-designs of U(2) from the theory of angular momentum

Abstract The main aim of this work is to present an explicit construction of 2-designs of U(2), relying only on a tool that belongs to every physicist's 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 properties. Then we derive all 1-designs of 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 -the prototypical unitary 1-design -one "naturally" obtains the 2-design originally employed by Bennett and coauthors in Bennett and coauthors in Mixed-state entanglement and quantum error correction. The present work also serves as a gentle and largely self-contained introduction to the subject.