Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The best known are the von Neumann entropy I(ρ): = -trace(ρ log ρ) and a generalization of the Kullback-Leibler distance S(ρ ∥ σ): = trace(ρlog ρ-ρ log σ), refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore I and S as regularizing functionals in seeking solutions to multivariable and multidimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application, we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.
Bibliographical noteFunding Information:
Manuscript received June 7, 2004; revised October 14, 2005. This work was supported in part by the National Science Foundation and the Air Force Office of Scientific Research. The author is with the Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: tryphon@ umn.edu). Communicated by V. V. Vaishampayan, Associate Editor At Large. Digital Object Identifier 10.1109/TIT.2005.864422
- Covariance realization
- Moment problem
- Quantum entropy
- Spectral analysis