Relative Entropy and Moment Problems

We consider the von Neumann entropy I(ρ:=-trace(ρlogρ) and the Kullback-Leibler-Umegaki distance S(ρ∥σ):=trace(ρlogρ-ρlogσ) as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. We show how to obtain extrema for such functionals via a suitable homotopy and how to characterize all the solutions to moment problems. The range of possible applications includes the inverse problem of describing power spectra which are consistent with second-order statistics, measurement in classical thermodynamics as well as a quantum mechanics, as well as analytic interpolation encountered in modern robust control (cf.[6], [14], [15], [16], [17]).