I will present several developments that collectively give us a fruitful new perspective on a fundamental result in quantum information theory, the strong sub-additivity of quantum entropy. I
will start by discussing quantum entropy and other information measures, and explain why they are emphatically more interesting and delicate than their commutative analogues. While exploring some of their properties we will encounter the Golden-Thompson inequality which illustrates the mathematical challenges faced when dealing with functions of matrices that do not commute. We will then see how the Golden-Thompson and other norm inequalities can be generalized from
two to arbitrarily many matrices using complex interpolation theory. Closing the circle, we will see how the resulting multivariate trace inequality can be used to strengthen strong sub-additivity of quantum entropy, leading us to strong bounds on the recoverability of quantum information.

This talk is based on work in arXiv:1512.02615, arXiv:1604.03023, and arXiv:1609.01999.