Annals of Mathematical Sciences and Applications

Volume 4 (2019)

Number 2

Geometric measures of information for quantum state characterization

Pages: 395 – 409



Warner A. Miller (Department of Physics, Florida Atlantic University, Boca Raton, Florida, U.S.A.)

Shahabeddin M. Aslmarand (Department of Physics, Florida Atlantic University, Boca Raton, Florida, U.S.A.)

Paul M. Alsing (Air Force Research Laboratory, Information Directorate, Rome, New York, U.S.A.)

Verinder S. Rana (Naval Information Warfare Center Pacific (NIWC PAC), San Diego, California, U.S.A.)


We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon’s entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski.We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.

We dedicate this paper in honor of the 70th birthday of S.-T. Yau for his pioneering research and leadership on the subject of geometry.

PMA and WAM would like thank support from the Air Force Office of Scientific Research (AFOSR). WAM research was supported under AFOSR/ AOARD grant #FA2386-17-1-4070. WAM wished to thank the Griffiss Institute and AFRL/RI for support under the Visiting Faculty Research Program. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of AFRL.

Corresponding author: Warner A. Miller. ORCID: 0000-0002-5883-3596.

Received 9 July 2019

Accepted 15 August 2019

Published 2 October 2019