Energy-Constrained Commutator Variance for Weyl Pairs

Abstract

Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write ${\mathcal{R}}_W(K)=\{W(f):f\in K{\}}^{\prime \prime }$. If the symplectic pairing of $K_1$ and $K_2$ is nonzero, there is a self-adjoint unitary $B\in {\mathcal{R}}_W(K_2)$ such that, for every normal state $\rho$, one can choose a self-adjoint unitary $A_{\rho }\in {\mathcal{R}}_W(K_1)$ with ${\mathrm{Var}}_{\rho }(-i[A_{\rho },B])=4$. 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 $G_M=\frac{1}{2}{R}^{T}MR-\frac{1}{2}\sqrt{\det M}$, with $M>0$ and $u^T\Omega v=\pi$, we prove $4-{\gamma }_{G_M,E}(h)=\frac{u^{T}Mu+v^{T}Mv+2\pi \sqrt{\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=d\Gamma (H_1)$ and witness directions $u,v\in \mathrm{dom}{H}_1^{1/2}$, the fixed-witness deficit is $\Theta (E^{-1})$ 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.

arXiv comment23 pages, no figures