  Quantum operations (QO) describe any state change allowed in quantum
mechanics, such as the evolution of an open system or the state change due to a
measurement. We address the problem of which unitary transformations and which
observables can be used to achieve a QO with generally different input and
output Hilbert spaces. We classify all unitary extensions of a QO, and give
explicit realizations in terms of free-evolution direct-sum dilations and
interacting tensor-product dilations. In terms of Hilbert space dimensionality
the free-evolution dilations minimize the physical resources needed to realize
the QO, and for this case we provide bounds for the dimension of the ancilla
space versus the rank of the QO. The interacting dilations, on the other hand,
correspond to the customary ancilla-system interaction realization, and for
these we derive a majorization relation which selects the allowed unitary
interactions between system and ancilla.
