The fact black holes carry statistical entropy proportional to their horizon area implies that quantum information concepts are geometrized in gravity. This idea obtains a particular manifestation in the AdS/CFT correspondence, where it is believed that the quantum information content in the dual field theory state can be used to reconstruct the bulk space-time geometry. The calculation of entanglement entropy from geodesics in the bulk space-time have clarified this idea to some extend.

In this talk, I will consider two aspects of quantum information theory in relation to holography: First, I will discuss a refinement of entanglement entropy for systems with conserved charges, the so-called symmetry resolved entanglement. It measures the entanglement in a sector of fixed charge. I will present how to calculate the symmetry-resolved entanglement entanglement in two-dimensional conformal field theories with Kac-Moody symmetry, and also within W_3 higher spin theory. I will also discuss the geometric realization in the dual AdS space-time, and how the independent calculation there leads to a new test of the AdS3/CFT2 correspondence.

Second, I will discuss the large N limit of Nielsen's operator complexity on the SU(N) manifold, with a particular choice of cost function based on the Laplacian on the Lie algebra, which leads to polynomial (instead of exponential) penalty factors. I will first present numerical results that hint to the existence of chaotic and hence ergodic geodesic motion on the group manifold, as well show the existence of conjugate points. I will then discuss a mapping between the Euler-Arnold equation which governs the geodesic evolution, to the Euler equation of two-dimensional idea hydrodynamics, in the strict large N limit.