The representation theory of the Clifford group, with applications to resource theories

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I will report on an ongoing project to work out and exploit an analogue of Schur-Weyl duality for the Clifford group. Schur-Weyl establishes a one-one correspondence between irreps of the unitary group and those of the symmetric group. A similar program can be carried out for Cliffords.
The permutations are then replaced by certain discrete orthogonal maps.
As is the case for Schur-Weyl, this duality has many applications for problems in quantum information. It can be used, e.g., to derive quantum property tests for stabilizerness and Cliffordness, a new direct interpretation of the sum-negativity of Wigner functions, bounds on stabilizer rank, the construction of designs using few non-Clifford resources, etc.

[arXiv:1609.08172, arXiv:1712.08628, arXiv:1906.07230, arXiv:out.soon].