Basics of Group Representation Theory The Unitary Group and Random Quantum States

In fields like statistical dynamics or chaos theory, we use probabilistic models to come to conclusions about the state and the evolution of a system. If we want to apply quantum theory to these fields, it would be helpful to know whether a state whose classical analog is chaotic is effectively random [Woo90]. For the definition of a random quantum state we need at least a probability measure on the set of quantum states. It turns out that the Haar measure can be seen as a probability measure, and that some results of group representation theory allow us to calculate some results for random quantum states. In this seminar talk, we will start with some basics on the unitary group, followed by a bit of group representation theory. We will then introduce the Haar measure as a probability measure on the unitary group, and we will close with the calculation of expectation values of random pure states: E(|ψ〉) = |ψ 〉 d|ψ〉. (2.1