We give an introduction to de Finetti theorems for quantum states. Such a theorem asserts that, given a symmetric state on a system composed of n subsystems, the state obtained by tracing out n-k of the subsystems can be represented approximately as a convex sum of product states. Furthermore, the precision of the approximation increases as k/n decreases, i.e. as a larger fraction of the subsystems are traced out. This can be proved using simple representation-theoretic ideas, which can also be extended to prove a number of interesting variants of the de Finetti theorem. Finally, we consider states that are both symmetric and unitarily invariant (symmetric Werner states). These give a rich supply of examples and have a close connection to the de Finetti theorem for probability distributions. They can also be used to establish limits on the closeness of approximation by product states, and therefore provide information about the tightness of de Finetti theorems.

Joint work with Matthias Christandl and Renato Renner