bibkey: deboer2025wignermajorization
authors: Jan de Boer, Giuseppe Di Giulio, Esko Keski-Vakkuri, Erik Tonni
year: 2025
title: "Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states"
doi: 10.1103/w561-h3z5
url: https://arxiv.org/abs/2412.19698v3
claim: "The vacuum/one-photon mixtures form a Wigner-majorization chain increasing on [1/2,1] and decreasing on [0,1/2] under Proposal 1."
strata_touched:
  - D5/S3/Analytic/Convexity/DerivativeHingeMajorization
  - D5/S3/Quantum/QuantumChannels/FockAttenuator/VacuumOnePhotonWignerMajorization
license: citation-only
triage: anchor

Wigner-majorization chains for vacuum and one-photon mixtures

Verified locator

DOI: 10.1103/w561-h3z5

Source: https://arxiv.org/abs/2412.19698v3

The journal publication is Physical Review A 112, 032405. The cited arXiv version is v3. Its PDF numbers the level functional, threshold comparison, Fock Wigner function and mixture as Eqs. (97), (98), (99) and (100), respectively.

Source statements

Page 16, Eq. (99): “W_{|n⟩}(x,p) = ((−1)ⁿ/π) e^{−(x²+p²)} Lₙ(2(x²+p²))”. Eq. (100): “W̃_u(r) = (1 − u) W_{|0⟩}(r) + u W_{|1⟩}(r)”, with “u ∈ [0, 1]”. Eq. (97): “I_t[|W_ρ̂|] ≡ ∫ [|W_ρ̂(r)| − t]₊ dr”.

Page 16, Result 8: “We define that ρ̂₁ ≻_W ρ̂₂ if and only if |W₁| ≻_w |W₂|”. Eq. (98) is the comparison of the level functionals for every t ≥ 0.

Page 17, Section V.A.1: “By trying for several instances, we may conjecture that there is a majorization order along the family W̃_u only as long as u > 1/2.” “It would be interesting to understand this observation better analytically. We leave this question for future works.”

Mathematical scope

The proposition in issue #18571 specifies increasing order on [1/2,1] and decreasing order on [0,1/2], including both endpoints. These are assertions about Proposal 1. The Laguerre polynomial is the existing genLaguerre 0 n; the level functional is the real Bochner integral over product Lebesgue measure on ℝ × ℝ.

The general cancellation theorem assumes a continuous profile with an everywhere derivative, an integrable derivative, a nonnegative derivative before a nonnegative peak location, a nonpositive derivative afterwards, and zero endpoint limits. The convex minimum requires the profile to be nonnegative and both profile and derivative to be integrable. The concrete profile is h(s)=s exp(−s). Strictness, complete cross-branch classification and cancellation for general non-unimodal absolutely continuous profiles are outside these statements.

The experiment is in the-omega-institute/trureturing-experiments at commit fec5519e820aa7ae835e09692fb27fdccb2f49b7, path docs/reports/deboer-digiulio-keskivakkuri-tonni-2025-vacuum-one-photon-wigner-majorization/check.py, SHA-256 26165b657205c4f6d509c5a79fbaa3a07a13ecab3c2a79a61cbe378b33b5cfd8. Its reported exit code is 0, with 8610 comparisons on each branch and no violations. Numerical comparisons are distinct from the universal Lean proposition.

The literature screen in #18571 includes the source versions, the Wigner-positive vacuum result of Van Herstraeten–Jabbour–Cerf and the different preorder of Upadhyaya et al. Its completeness is ASSUMED-UNVERIFIED.