Equivalence of approximate Gottesman-Kitaev-Preskill codes

The Gottesman-Kitaev-Preskill (GKP) quantum error correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error correcting routines and the high tolerance against Gaussian errors. Since the GKP code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate GKP codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-from expressions for the Wigner functions, the inner products, and the average photon numbers in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate GKP codes.