A Three-Point Continuous-Variable Quantum MacWilliams Identity

Abstract

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.