Existence of centralizing bases for arbitrary group representations

Determine whether, for every compact group G with a unitary (possibly projective) representation R on a Hilbert space H, there exists a basis W of H such that the associated classical-shadows measurement channel M = E_{U∼G} Ad_{U†} ∘ A_W ∘ Ad_U acts as a scalar on each G-isotypic component of the visible operator space, i.e., M = Σ_λ a^H_λ P^V_λ for scalars a^H_λ and projectors P^V_λ onto the corresponding isotypic components. This asks whether a centralizing basis always exists, beyond the sufficient (but not necessary) case of a non-degenerate commuting subgroup eigenbasis (NDCSE).

Background

The paper develops a unified framework for classical shadows based on sampling from arbitrary group representations. A key notion is a centralizing basis: a measurement basis W for which the shadows measurement channel M acts as a scalar on each G-isotypic component of the visible operator space, making channel inversion trivial and enabling analytic variance bounds.

The authors prove that any non-degenerate commuting subgroup eigenbasis (NDCSE)—a Fourier basis that is simultaneously diagonal for an abelian subgroup with multiplicity-free weights—yields a centralizing channel with explicit coefficients a^H_λ = d^H_λ/d_λ. However, this condition is only sufficient and not necessary: for example, global Clifford shadows are centralizing even when a generic basis is not an NDCSE, while some S_n irreducible representations lack NDCSEs. This leaves unresolved whether a centralizing basis exists for every group representation.