Convergence of Density Operators and Security of Discrete Modulated CVQKD Protocols

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.