  This communication deals with the problem of bounding the approximation error
on weak convergence of mixed coherent state towards a Gaussian thermal state.
In the context of CVQKD with discrete modulation, we develop expressions for
two specific cases. The first one is the distance between the Gaussian
equivalent bipartite state and a reference Gaussian modulated (GG02) and the
second one is for the trace distance between the constellation and a thermal
state with same photon number. Since, in the convex set of density operators,
weak convergence implies convergence in the trace norm, knowing how fast the
sequence gets close to the equivalent Gaussian state has implication on the
security of QKD Protocols. Here we derive two bounds on the $L_1$ distance, one
of them related with an energy test that can be used in the security proof.
