Law of Cosines and Shannon-Pythagorean Theorem for Quantum Information

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 information-theoretic variational problems.