  Applications of quantum technology often require fidelities to quantify
performance. These provide a fundamental yardstick for the comparison of two
quantum states. While this is straightforward in the case of pure states, it is
much more subtle for the more general case of mixed quantum states often found
in practice. A large number of different proposals exist. In this review, we
summarize the required properties of a quantum fidelity measure, and compare
them, to determine which properties each of the different measures has. We show
that there are large classes of measures that satisfy all the required
properties of a fidelity measure, just as there are many norms of Hilbert space
operators, and many measures of entropy. We compare these fidelities, with
detailed proofs of their properties. We also summarize briefly the applications
of these measures in teleportation, quantum memories, quantum computers,
quantum communications, and quantum phase-space simulations.
