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Unitary designs and codes. (English) Zbl 1172.05310
Summary: A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary \(t\)-design in \(U(d)\), for any \(d\) and \(t\). We also introduce the notion of a unitary code — a subset of \(U(d)\) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values — and give an upper bound for the size of a code with \(s\) inner product values in \(U(d)\), for any \(d\) and \(s\). These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary \(t\)-design in \(U(d)\), and we catalogue some \(t\)-designs that arise from finite groups.

MSC:
05B30 Other designs, configurations
41A55 Approximate quadratures
81P15 Quantum measurement theory, state operations, state preparations
94A20 Sampling theory in information and communication theory
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