Related papers: Clifford groups are not always 2-designs

Clifford groups are not always 2-designs

URL: http://arxiv.org/abs/2108.04200v1
Date: Mon, 9 Aug 2021 17:37:59 GMT
Title: Clifford groups are not always 2-designs
Authors: Matthew A. Graydon and Joshua Skanes-Norman and Joel J. Wallman
Abstract summary: We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. We also clarify the structure of projective group unitary $2$-designs.
Score: 0.0
License: http://creativecommons.org/licenses/by/4.0/
Abstract: The Clifford group is the quotient of the normalizer of the Weyl-Heisenberg group in dimension $d$ by its centre. We prove that when $d$ is not prime the Clifford group is not a group unitary $2$-design. Furthermore, we prove that the multipartite Clifford group is not a group unitary 2-design except for the known cases wherein the local Hilbert space dimensions are a constant prime number. We also clarify the structure of projective group unitary $2$-designs. We show that the adjoint action induced by a group unitary $2$-design decomposes into exactly two irreducible components; moreover, a group is a unitary 2-design if and only if the character of its so-called $U\overline{U}$ representation is $\sqrt{2}$.

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