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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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