If a pure quantum state is drawn at random, this state will almost surely be almost maximally entangled. This is a well-known example for the "concentration of measure" phenomenon, which has proved to be tremendously helpful in recent years in quantum information theory. It was also used as a new method to justify some foundational aspects of statistical mechanics.
In this talk, I discuss recent work with David Gross and Jens Eisert on concentration in the set of pure quantum states with fixed mean energy: We show typicality in this manifold of quantum states, and give a method to evaluate expectation values explicitly. This involves some interesting mathematics beyond Levy's Lemma, and suggests potential applications such as finding stronger counterexamples to the additivity conjecture.