Abstract

Relative entropies (or divergences) play a crucial role in both Classical and Quantum Information Theory. Their defining property, the data processing inequality, states that a relative entropy of two probability distributions/quantum states cannot increase when applying certain transformations to the system. In the quantum setting, one of the most important examples of a relative entropy satisfying the data processing inequality is the quantum relative entropy, which was first defined by Umegaki and proved to be the ‘correct’ quantum generalization of the well-known Kullback-Leibler divergence in the classical theory. The transformations considered in the quantum setting are completely positive, trace-preserving linear maps (also called quantum channels).

In this talk, we will review a seminal result by Petz that gives a necessary and sufficient condition for equality in the data processing inequality for the quantum relative entropy. More precisely, we derive a necessary and sufficient condition on the quantum channel and two states such that their quantum relative entropy remains unchanged after application of the quantum channel. Time permitting, we will also review this result in the light of strong subadditivity, an important and highly non-trivial property of the von Neumann entropy. In this case, the equality condition identifies states satisfying a quantum Markov chain condition.

Note that no prior knowledge of quantum mechanics is needed to follow the talk, as we will quickly review the mathematical foundation of quantum mechanics from the viewpoint of C*-algebras in the finite-dimensional setting.