Weyl 对的能量约束交换子方差
Energy-Constrained Commutator Variance for Weyl Pairs
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中文总结 AI 辅助
本文研究 Weyl 对在能量约束下的交换子方差，证明优化系数恒为4，并给出二次能量与自由场情形下亏量的精确渐近估计。
中文摘要 AI 辅助
设 $W$ 为可分希尔伯特空间上 Weyl 关系的一个表示，并记 $\mathcal R_W(K)=\{W(f):f\in K\}''$。若 $K_1$ 与 $K_2$ 的辛配对非零，则存在自伴酉算子 $B\in\mathcal R_W(K_2)$，使得对每个正常态 $\rho$，可选择自伴酉算子 $A_\rho\in\mathcal R_W(K_1)$ 满足 $\operatorname{Var}_\rho(-i[A_\rho,B])=4$。因此，在任意容许能量阈值下，优化的平均输入能量约束系数等于 $4$。对于固定的三角 Weyl 见证 $h$，恒等式 $4-h^2$ 将亏量归结为对易的平方 Weyl 平移的相位固定问题。对于单模二次能量 $G_M=\frac12 R^TMR-\frac12\sqrt{\det M}$，其中 $M>0$ 且 $u^T\Omega v=\pi$，我们证明 $4-\gamma_{G_M,E}(h)=\frac{u^TMu+v^TMv+2\pi\sqrt{\det M}}{4E}+O(E^{-2})$。下界对所有满足平均能量约束的正常态成立，且局部 Zak 构造达到相同系数。对于 $G=d\Gamma(H_1)$ 及见证方向 $u,v\in\operatorname{dom}H_1^{1/2}$，若至少一个方向位于 $\ker H_1$ 之外，则固定见证亏量为 $\Theta(E^{-1})$；若两个方向均为零模，则约束上确界在每个正阈值处等于端点值。
英文摘要
Let $W$ be a representation of the Weyl relations on a separable Hilbert space and write $\mathcal R_W(K)=\{W(f):f\in K\}''$. 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 $ρ$, one can choose a self-adjoint unitary $A_ρ\in\mathcal R_W(K_1)$ with $\operatorname{Var}_ρ(-i[A_ρ,B])=4$. 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 $G_M=\frac12 R^TMR-\frac12\sqrt{\det M}$, with $M>0$ and $u^TΩv=π$, we prove $4-γ_{G_M,E}(h)=\frac{u^TMu+v^TMv+2π\sqrt{\det M}}{4E}+O(E^{-2})$. The lower bound holds over all normal states satisfying the mean-energy constraint, and a localized Zak construction attains the same coefficient. For $G=dΓ(H_1)$ and witness directions $u,v\in\operatorname{dom}H_1^{1/2}$, 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.
