Superdense Coding with Gottesman-Kitaev-Preskill Codes over Lossy Bosonic Channels: Square and Hexagonal Lattices Compared

We analyze superdense coding with Gottesman-Kitaev-Preskill (GKP)-encoded bosonic modesover a lossy channel of transmissivity η, for square and hexagonal lattices. Combining loss, aquantum-limited pre-amplifier and the finite squeezing of both modes of the shared pair gives a twomode noise variance σ2eff = κ∆2 + (1 − η) for Bob’s destructive Bell measurement, with κ = 2 forindependent per-mode noise. Both lattices are treated in the same periodic model: Bob’s outcomeis a lattice point plus Gaussian noise, and the logical symbol is the class of that point modulo thestabilizer lattice. We report the capacity of the measured channel (mutual information with the rawoutcome) and the rate of hard-decision nearest-lattice-point decoding, the protocol as described.Both are computed by deterministic quadrature and checked against Monte Carlo. The hexagonaladvantage is modest and depends on the operating point. For the capacity, the hexagonal latticetolerates about 0.09 dB more noise variance at 1.5 bits and 0.17 dB more at 90% of the two-bitceiling (1.8 bits), approaching but staying below its 0.625 dB geometric coding gain as the errorrate falls, and it is slightly worse below about 1 bit; for hard decisions the crossover is near 1.4bits. The larger minimum distance is partly offset by six nearest neighbours (four for the squarelattice) and, for hard decisions, by a larger error entropy. For a 1.8-bit capacity the square latticeneeds 8.3, 9.7 and 12.4 dB of squeezing at η = 0.99, 0.95 and 0.90, of which the hexagonal latticesaves 0.19, 0.25 and 0.46 dB. Under thermal loss the saving grows only through the rising noisefloor, for which we give an exact expression. Evaluated at the pre-amplifier GKP resource energyNS (smaller than the energy actually entering the channel, which the pre-amplifier raises), thecapacity of GKP superdense coding stays below both the entanglement-assisted and the unassisted(Holevo) capacities of the channel throughout our sweep, never exceeding 0.65 of the latter in theexplored thermal-loss range. Using NS rather than the larger physical transmitted energy makesthis comparison favourable to GKP, while not charging the retained mode or the cost of distributingthe entangled pair makes it favourable to superdense coding; within this convention we find noadvantage over unassisted classical communication.