bibkey: litan2024bargmann
authors: Mao-Sheng Li; Yi-Xi Tan
year: 2024
title: On the Bargmann invariants for quantum imaginarity
doi: 10.48550/arXiv.2412.08022
url: https://arxiv.org/abs/2412.08022v1
claim: Section III conjectures that the circulant formed from geometric means of moduli and arithmetic means of principal phases preserves positive semidefiniteness of correlation matrices in general dimension.
strata_touched:
  - D5/S3/Quantum/Matrix/LiTanCirculantPositivityRefutation
license: citation-only
triage: anchor

Li and Tan: circulant positivity

Verified locator

DOI: 10.48550/arXiv.2412.08022

Source: https://arxiv.org/abs/2412.08022v1

Source statements

Section II, page 2: “Let 𝐇_n denote the set of all n × n Hermitian matrices H that are positive semidefinite and have h_ii = 1 for all i = 1, …, n.”

Section II, page 2: “Let C_n = G_{(0,1,0,…,0)}”. Its displayed matrix has ones on the superdiagonal and in the bottom-left corner.

Section III, page 5: “For each H = (h_kl) ∈ Mat_{n×n}(ℂ), we can associate it with a circulant matrix C_H = Σ_{k=0}^{n−1} z_k C_nᵏ, where the coefficients are given by z_k = (Π_{l=1}^n r_{l(l⊕k)})^{1/n} e^{iθ_k}, with θ_k := (Σ_{l=1}^n θ_{l(l⊕k)})/n and h_kl = r_kl e^{iθ_kl}, where r_kl ≥ 0 and θ_kl ∈ (−π, π].”

Section III, page 5: “For the case when n = 3, we deduced that the positive semidefiniteness of C_H follows from the positive semidefiniteness of H. This leads us to the conjecture that the positive semidefiniteness of H implies the positive semidefiniteness of C_H (which may depend on n) for general n.”

Encoding and scope

Indices are Fin n, with addition modulo n. The phase is Complex.arg, the modulus is the complex norm, and the root is Real.rpow of the nonnegative product. The source’s shift is written directly in the sum defining CH. The conjecture is restricted to positive semidefinite matrices with unit diagonal; a counterexample in this smaller class refutes the assertion for unrestricted positive semidefinite matrices as well.

The repository refutation uses the real rational matrix with rows (1,5/8,1/9,5/8), (5/8,1,5/8,4/9), (1/9,5/8,1,5/8), and (5/8,4/9,5/8,1). A diagonal Gram factorization proves positivity. The transformed first row is (1,5/8,2/9,5/8), and the quadratic value at (1,−1,1,−1) is −1/9. The parameterized family and the source’s n = 3 assertion are outside this Lean theorem.

Literature completeness and agreement of the journal version, Phys. Rev. A 111 (2025) 022409, with this arXiv statement are ASSUMED-UNVERIFIED.