A Structure Theorem for Phase-Space Representations of Continuous-Variable Quantum Error-Correcting Codes

In this paper we connect the structure theorem for quasiprobability representation of generalised probabilistic theories to bosonic quantum error correction codes, giving both a general phase-space representation for continuous-variable error-correcting codes, and showing as specific examples the phase-space representations obtained through this method for Gottesman-Knill-Preskill codes, cat codes, and binomial codes. This representation allows us to define both generally and for each of these codes the mathematical structure in phase space that errors can take, which we show both abstractly and for the specific example of single photon loss errors.