The notion of entropy plays a primary role in classic and quantum information theory. Due to the universal ultraviolet divergences of Quantum Field Theories (QFTs), its generalisation to the QFT is only possible in a relative (rather than absolute) form. Relative entropy can then be used to quantify the difference between pairs of states in a QFT, where it has found various applications.

In the first part of the talk, I will present the current state of the art on this topic. In particular, I will introduce and explain the techniques and mathematical tools necessary to deal with this subject, including the algebraic approach to QFT and the theory of Tomita-Takesaki. In the second part, I will discuss some open problems related to the computation of the relative entropy and briefly present some recent progress made together with collaborators.