Fidelity of quantum operations

We present a derivation and numerous applications of a compact explicit formula for the average fidelity of a quantum operation on a finite dimensional quantum system. The formula can be applied to averages over particularly relevant subspaces; it is easily generalized to multi-component systems, and as a special result, we show that when the same completely positive trace-preserving map is applied to a large number of qubits with one-bit fidelity F close to unity, the average fidelity of the operation on the full K-bit register scales as $F^{3K/2}$.