  The classical de Finetti theorem provides an operational definition of the
concept of an unknown probability in Bayesian probability theory, where
probabilities are taken to be degrees of belief instead of objective states of
nature. In this paper, we motivate and review two results that generalize de
Finetti's theorem to the quantum mechanical setting: Namely a de Finetti
theorem for quantum states and a de Finetti theorem for quantum operations. The
quantum-state theorem, in a closely analogous fashion to the original de
Finetti theorem, deals with exchangeable density-operator assignments and
provides an operational definition of the concept of an "unknown quantum state"
in quantum-state tomography. Similarly, the quantum-operation theorem gives an
operational definition of an "unknown quantum operation" in quantum-process
tomography. These results are especially important for a Bayesian
interpretation of quantum mechanics, where quantum states and (at least some)
quantum operations are taken to be states of belief rather than states of
nature.
