Pythagorean theorem for Shannon capacity of a quantum channel

Pythagorean theorem is one of the most famous results in geometry, and there exist its information-theoretic counterpart, including a more general law of cosines. Here we shall consider a particular case where the vertexes of a triangle are defined by a compound (or ‘joint’) state on a product algebra. It is shown that such a triangle is always right, and the Pythagorean theorem gives a new, geometric interpretation of Shannon capacity of a quantum channel, defined by the compound state. In particular, the catheti of the triangle represent Shannon mutual information and divergence between the reduced (or marginal) states. We discuss applications of this Shannon-Pythagorean theorem to optimisation of dynamical systems with constraints on information divergence between states and channel capacity of the operation transforming the states.