Homomorphic Aggregation of Continuous-Variable GKP States

Abstract

Aggregating logical information in continuous-variable quantum networks is essential for distributed quantum architecture. However, direct passive linear optics degrade non-Gaussian Gottesman-Kitaev-Preskill (GKP) grid states via symplectic lattice compression and entanglement-induced decoherence when auxiliary modes are discarded. We present an active, measurement-based continuous-variable network primitive for combining spatially distributed computational-basis payloads. Utilizing GKP Bell states, homodyne measurements, and conditional feed-forward phase-space displacements, we construct a completely positive trace-preserving (CPTP) map that evaluates a logical XOR operation on computational-basis inputs. We bound the Heisenberg action of the physical finite-squeezing channel on logical Pauli generators, demonstrate measurement-specific homodyne-outcome hiding for continuous-variable quantum one-time pads (CV-OTP) with an explicit prefactor bound $D_{\mathrm{TV}}\le 1.60e^{-r}$, and evaluate logical success probabilities under physical optical loss and network scaling constraints.

v2: Major revision. 15 pages, 2 figures. Clarified protocol scope to computational-basis payloads (logical XOR), added explicit prefactor bound for cryptographic homodyne-hiding, and included the full phase-space symplectic derivation of the physical router