  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.
