bibkey: brodygraefemelanathuru2026phasespace authors: Dorje C. Brody; Eva-Maria Graefe; Rishindra Melanathuru year: 2026 title: “Phase-space measurements and decoherence for angular momentum systems” doi: 10.48550/arXiv.2605.02696 url: https://arxiv.org/abs/2605.02696v1 claim: “Equation (53) conjectures that coefficientwise domination by the Husimi kernel restores positivity for J > 1/2.” strata_touched:
- D5/S3/Quantum/Decoherence/DampedSpinKernelPositivityRefutation license: citation-only triage: anchor
Phase-space measurements and decoherence for angular momentum systems
Page numbers refer to arXiv:2605.02696v1.
Page 2, equation (12):
the matrix elements of the irreducible tensors in the standard Ĵz-bases are given by ⟨J, m′|T̂^J_{L,k}|J, m⟩ = √((2L + 1)/(2J + 1)) C^{Jm′}{Jm Lk}, where the C^{JM}{j1 m1 j2 m2} denote the Clebsch-Gordan coefficients.
Page 2, equation (13):
ρ̂ = Σ_{L=0}^{2J} Σ_{k=−L}^{L} ρ_{Lk} T̂^J_{L,k}, ρ_{L,k} = tr((T̂^J_{L,k})† ρ̂).
Page 3, footnote 1:
We use the convention Y^0_0 = 1, so that the spherical harmonics are orthonormal with respect to the uniform probability measure dµ^0_{θ,ϕ} = (4π)^{−1} sin θ dθ dϕ. This differs from the Condon–Shortley convention by a factor of √4π: Y^m_L = √4π Y^{m,CS}_L, or equivalently Y^m_L = √(2L+1) Y^{m,Racah}_L, where Y^{m,Racah}_L = (−1)^m √((L−m)!/(L+m)!) P^m_L(cos θ) e^{imϕ} for m ≥ 0.
Equations (35)–(36) fix . Page 5, equation (44):
F^σ(θ, ϕ, t) = a_J Σ_{L,k} e^{−γ L(L+1) t/2} ( C(2J, L) / C(2J+L+1, L) )^{−σ/2} ρ_{Lk}(0) overline(Y_{Lk}(θ, ϕ)).
Page 8, Section VI, the conjecture after equation (52) and equation (53):
For J > 1/2 we do not have an exact result, but it seems reasonable to conjecture that positivity is ensured provided that the damped σ kernel in (44), due to decoherence, becomes no sharper than the Husimi (σ = −1) kernel. That is, if e^{−½γL(L+1)} ( C(2J,L)/C(2J+L+1,L) )^{−σ/2} ≤ ( C(2J,L)/C(2J+L+1,L) )^{1/2} (53) for all L, then we restore positivity.
The Lean encoding uses , doubled integer arguments for Racah’s
Condon–Shortley Clebsch–Gordan formula, the standard spin basis ordered
, and the scaled associated-Legendre harmonics of footnote 1.
The general kernel sums all and .
Density matrices reuse the positive-semidefinite unit-trace predicate in
CoPRelativeQuantumnessRefutation.IsDensity.
Equation (53) omits in its printed exponent; the witness has , so
that reading and the time-dependent reading agree.
The refutation uses , , , the
lowest-weight pure state and the north pole. Its exact value is
while every coefficient inequality holds.
Verified locator
- DOI: https://doi.org/10.48550/arXiv.2605.02696
- URL: https://arxiv.org/abs/2605.02696v1