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Symmetric informationally complete measurements: Can we make big ones out of small ones?

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For a quantum system with a d-dimensional Hilbert space, a symmetric informationally complete measurement (SIC) can be thought of as a set of d^2 pure states all having the same overlap. Constructions of SICs for composite systems usually do not make use of the composite structure but treat the system as a whole. Indeed for some cases, one can prove that a SIC cannot have the symmetry that one naturally associates with the composite structure.
In this talk I give one example showing how a SIC for three qubits can be constructed from SICs for the individual qubits. I ask whether the strategy used in this example might apply to other composite cases.