bibkey: stacey2025sicsmajorization
authors: Blake C. Stacey
year: 2025
title: "SICs, the Stabilizer Polytope and Majorization"
doi: null
url: https://arxiv.org/abs/2509.16819v1
claim: "For a pure state in prime dimension d, the matrix A(psi)_{mk} = sum_j p_{m,j} p_{m,j+k} built from the Weyl-Heisenberg MUB probabilities is asserted to be changed by Clifford unitaries only by permutations of its rows and columns, so that every unitarily invariant norm of A is Clifford-invariant and gives a candidate magic monotone."
strata_touched:
  - D5/S3/Quantum/Information/StabilizerAutocorrelationNormRefutation
license: citation-only
triage: anchor

SICs, the Stabilizer Polytope and Majorization

Blake C. Stacey, arXiv:2509.16819v1 (quant-ph), 2025-09-20. Quotations are from the v1 TeX source.

The matrix (Section III):

let us define the matrix $A(|\psi ⟩)$ by [A(\ket{\psi})]{mk} = \sum_j p{m,j} p_{m,j+k} , , where $p_{m,j}$ is the probability of obtaining outcome $j$ when measuring in the basis labeled by~$m$, given the pure state $|\psi ⟩$.

The invariance assertion and the proposal:

Acting upon $|\psi ⟩$ with a Clifford unitary can at most permute the rows and columns of the matrix $A$. … Consequently, Clifford unitaries will leave invariant the Frobenius norm of~$A$. Indeed, let $||\cdot ||$ be any unitarily invariant norm; because permutation matrices are a very special subset of the unitary matrices, left- and right-multiplying $A$ by any pair of permutation matrices will leave $||A||$ unchanged. It seems, then, that we have a large family of new candidate monotones that are all extremized by Weyl–Heisenberg SIC fiducials.

Verified locator

• URL: https://arxiv.org/abs/2509.16819v1 (v1, 2025-09-20, the only version; source schur-stabilizer.tex, md5 88608961707f4cd7368de90b5ad5eb3d): the matrix A (l. 378–386) and the invariance assertion and proposal (l. 385–397).