In quantum mechanics, the pure state of a physical system is represented by rays in the Hilbert space. All the questions one can ask of the physical system do then depend on the state: e.g. the expectation value of given observables, the entanglement of a state, its coherence, whether the expectation values of a given observable will show typical values in their time evolution, how much certain observables are correlated etc.
One can then ask what are the average properties in a Hilbert space: what is the average entanglement of a state? What is the average coherence generated by a given quantum map? What is the average behavior of expectation values during time evolution? Moreover, one can also ask whether these properties are typical, i.e. if it is very unlikely that they differ from the average.
In this talk, I will show how to argue about typical properties of the Hilbert space using the Haar measure and Lévy lemma and some recent results when locality is imposed.