Quantum null energy condition, loop groups and modular nuclearity

When dealing with Quantum Information objects in Quantum Field Theory (QFT), the algebraic approach seems a good choice, since the tools regarding quantum entropy can be formulated very precisely in terms of operator algebras. Quantum information aspects of QFT naturally take place in the framework of quantum black holes thermodynamics, as for example the Bekenstein Bound and the Landauer's principle show. However, more unexpected and interesting connections between the relative entropy and the stress energy tensor have arisen, and in particular it is of interest to provide and prove an axiomatic formulation of the Quantum Null Energy Condition (QNEC). In this Ph.D. thesis we prove the QNEC for some solitonic states of physical interest on a generic conformal net, a particular class of algebraic models used to study 1+1 dimensional Conformal Field Theory. We also prove that such states verify the Bekenstein Bound. We then focus on conformal nets induced by vacuum Positive Energy Representations of a Loop Group LG with G a compact, simple and simply connected Lie group. More specifically, on these models we construct and study a family of solitonic states and we then show that these states verify the QNEC and the Bekenstein Bound. In order to do so, we study Sobolev extensions of Positive Energy Representations of a Loop Group LG. Finally, we study how entropy-like functionals can be used to measure the entanglement of a quantum system. In particular, we provide an explicit estimate for the mutual information on any quantum system satisfying some nuclearity condition, a mathematical requirement ensuring the system to exhibit the thermodynamical behaviour of a theory of particles.