For a physical layer message authentication procedure based on the comparison of channel estimates obtained from the received messages, we focus on an outer bound on the type I/II error probability region. Channel estimates are modeled as multivariate Gaussian vectors, and we assume that the attacker has only some side information on the channel estimate, which he does not know directly. We derive the attacking strategy that provides the tightest bound on the error region, given the statistics of the side information. This turns out to be a zero mean, circularly symmetric Gaussian density whose covariance matrices can be obtained by solving a constrained optimization problem. We propose an iterative algorithm for its solution: starting from the closed-form solution of a relaxed problem, we obtain, by projection, an initial feasible solution; then, by an iterative procedure, we look for the fixed-point solution of the problem. Numerical results show that for cases of interest the iterative approach converges, and perturbation analysis shows that the found solution is a local minimum.

We propose a general framework for solving quantum state estimation problems using the minimum relative entropy criterion. A convex optimization approach allows us to decide the feasibility of the problem given the data, find the maximal common kernel of all admissible states and, whenever necessary, to relax the constraints in order to allow for a physically admissible solution. Building on these results, the variational analysis can be completed ensuring existence and uniqueness of the optimum. The latter can then be computed by efficient standard algorithms for convex optimization, without resorting to approximate methods or restrictive assumptions on its rank.

Dempster's covariance selection method is extended first to general nonsingular matrices and then to full rank rectangular matrices. Dempster observed that his completion solved a maximum entropy problem. We show that our generalized completions are also solutions of a suitable entropy-like variational problem.