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

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 et al (1996 Phys. Rev. A 54 3824–51). The present work also serves as a gentle and largely self-contained introduction to the subject.