Energy-Constrained Commutator Variance for Weyl Pairs
Abstract
Let W be a representation of the Weyl relations on a separable Hilbert space and write ℛW(K)={W(f):f∈ K}''. If the symplectic pairing of K1 and K2 is nonzero, there is a self-adjoint unitary B∈ℛW(K2) such that, for every normal state ρ, one can choose a self-adjoint unitary Aρ∈ℛW(K1) with Varρ(−i[Aρ,B])=4. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness h, an exact identity for 4−h2 reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy GM=1/2 RTMR−1/2√(det M), with M>0 and uTΩv=π, we prove 4−γGM,E(h)=(uTMu+vTMv+2π√(det M))/4E+O(E−2). The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For G=dΓ(H1) and witness directions u,v∈domH11/2, the fixed-witness deficit is Θ(E−1) whenever at least one direction lies outside ker H1; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.
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BibTeX
@misc{nasreddine2026energyconstrained,
title = {{Energy-Constrained Commutator Variance for Weyl Pairs}},
author = {Hassan Nasreddine},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19163},
eprint = {2609.19163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19163},
}
RIS
TY - UNPB
TI - Energy-Constrained Commutator Variance for Weyl Pairs
AU - Hassan Nasreddine
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.19163
UR - https://arxiv.org/abs/2609.19163
AB - Let W be a representation of the Weyl relations on a separable Hilbert space and write ℛ_W(K)={W(f):f∈ K}''. If the symplectic pairing of K_1 and K_2 is nonzero, there is a self-adjoint unitary B∈ℛ_W(K_2) such that, for every normal state ρ, one can choose a self-adjoint unitary A_ρ∈ℛ_W(K_1) with Var_ρ(−i[A_ρ,B])=4. Hence the optimized mean-input-energy-constrained coefficient equals $4$ at every admissible energy threshold. For the fixed trigonometric Weyl witness h, an exact identity for 4−h^2 reduces the deficit to a phase-fixed problem for commuting squared Weyl translations. For the one-mode quadratic energy G_M=1/2 R^TMR−1/2√(det M), with M>0 and u^TΩv=π, we prove 4−γ_G_M,E(h)=(u^TMu+v^TMv+2π√(det M))/4E+O(E^−2). The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For G=dΓ(H_1) and witness directions u,v∈domH_1^1/2, the fixed-witness deficit is Θ(E^−1) whenever at least one direction lies outside ker H_1; if both directions are zero modes, the constrained supremum equals the endpoint at every positive threshold.
ER -