  Given a quantum system consisting of many parts, we show that symmetry of the
system's state, i.e., invariance under swappings of the subsystems, implies
that almost all of its parts are virtually identical and independent of each
other. This result generalises de Finetti's classical representation theorem
for infinitely exchangeable sequences of random variables as well as its
quantum-mechanical analogue. It has applications in various areas of physics as
well as information theory and cryptography. For example, in experimental
physics, one typically collects data by running a certain experiment many
times, assuming that the individual runs are mutually independent. Our result
can be used to justify this assumption.
