Law of Cosines and Shannon-Pythagorean theorem for quantum information

Belavkin, Roman V. ORCID: https://orcid.org/0000-0002-2356-1447 (2013) Law of Cosines and Shannon-Pythagorean theorem for quantum information. In: Geometric Science of Information : First International Conference, GSI 2013, Paris, France, August 28-30, 2013. Proceedings. Nielsen, Frank and Barbaresco, Frédéric, eds. Lecture Notes in Computer Science, Part X (8085) . Springer Verlag, Berlin, pp. 369-376. ISBN 9783642400193. [Book Section] (doi:10.1007/978-3-642-40020-9_40)

Abstract

The concept of information distance in non-commutative setting is re-considered. Additive information, such as Kullback-Leibler divergence, is defined using convex functional with gradient having the property of homomorphism between multiplicative and additive subgroups. We review several geometric properties, such as the logarithmic law of cosines, Pythagorean theorem and a lower bound given by squared Euclidean distance. We also prove a special case of Pythagorean theorem for Shannon information, which finds applications in informationtheoretic variational problems.

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