AI 中文总结
研究构建三点连续变量量子麦克威廉姆斯恒等式，给出积分核。利用该恒等式推导量子纠错码维度界，证明三点装置未改进两点界，确定GKP晶格码三点最优值，还对一般玻色子码相关情况进行分析，验证了部分坍缩。
AI 中文摘要
我们构建了三点连续变量（CV）量子麦克威廉姆斯恒等式，扩展了布尔查德的两点框架，并给出了其封闭形式的积分核。其配置空间带有一个没有经典对应物的辛不变量，它编码了GKP量子化条件和一个三点符号相位。利用该恒等式，我们推导出它对CV量子纠错码维度所支持的半定规划界，并在两个坍缩定理中证明，三点装置并未改进两点界。对于GKP晶格码，三点最优值恒等于布尔查德的两点线性规划最优值。这是对晶格三点最优值的精确确定，所以$E_8$和李奇特魔法函数使其饱和而非超越它。对于一般玻色子码，一个完全正定的重新表述绕过了排除自然因式形式构造的正性障碍；相位符号条件与崔正性随后迫使三点项消失。我们在单模的前八个拉盖尔能级上对径向崔形式验证了这种坍缩，并将全迹类锥问题留待解决。这两种坍缩都有一个没有经典类似物的单一原因，即码投影器：它正确地确定了界的方向，但也消除了推动经典三点改进的完全正性。
英文摘要
We construct the three-point continuous-variable (CV) quantum MacWilliams identity, extending the two-point framework of Burchards, and give its closed-form integral kernel. Its configuration space carries a symplectic invariant with no classical counterpart, which encodes the GKP quantization condition and a three-point sign phase. Using the identity, we derive the semidefinite-programming bounds it supports on the dimension of CV quantum error-correcting codes, and we prove, in two collapse theorems, that the three-point apparatus does not improve on the two-point bound. For GKP lattice codes the three-point optimum equals the Burchards two-point linear-programming optimum identically. This is an exact determination of the lattice three-point optimum, so the $E_8$ and Leech magic functions saturate it rather than beat it. For general bosonic codes a completely-positive reformulation bypasses the positivity obstruction that rules out the natural factored-form constructions; the phase-sign condition together with Choi positivity then force the three-point term to vanish. We certify this collapse for radial Choi forms on the first eight Laguerre levels at one mode, and leave the full trace-class cone open. Both collapses have a single cause with no classical analogue, the code projector: it orients the bound correctly but also removes the full positivity that powers the classical three-point improvement.
Comments35 pages. Ancillary files: positive-definite certificates for the CP collapse theorem (JSON, N=1 exact-rational and N=2 floating-point) and a stdlib-only exact verification script (anc/)
