What is the Difference Between Quantum and Classical Interference I was reading about Quantum decoherence and I came across this quote, "decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities)." It appears that this quote is distinguishing the phenomenon of quantum interference to classical interference, but what exactly is meant by the distinction of "additive probability amplitudes" and "additive probabilities?"
 A: When people talk about "classically adding probabilities", they're talking about finding the probability that either of two mutually exclusive events occurring. You just add the probabilities! $P(A \cup B) = P(A) + P(B)$ when $A$ and $B$ are mutually exclusive. Think about throwing darts at a dart board, and calculating the probability that you get a bull's eye or a triple 20.
When people talk about "quantum interference", they're talking about how quantum states combine when you carefully combine them. In this case, you get a new state that is obtained by summing the two input states and re-normalizing. Of course, you measure probabilities of future measurements using this new state. 
A: 
decoherence has irreversibly converted quantum behaviour (additive probability amplitudes) to classical behaviour (additive probabilities)."

Classical QM, in the Schrodinger picture, evolves the state of the system; this can be conceptualised as a wave, but properly is a probability amplitude which 'adds', that is in the same way classical waves 'add' ie we get a super-position of waves.
The state, in this picture, is simply a wave.
Though we call it a probability amplitude, it is not a probability; to obtain a probability - one must take the square of the amplitude; the general idea here is called the Born Rule.
Decoherence, takes into account the countless interactions that a system has with its environment to produce a macroscopic observation that is 'irreversible' - at least that would be my guess; and this ties in with 'conversion' from non-classical probability amplitudes, to probabilities themselves.
A: Decoherence means that we don't know about most of the total wavefunction of the combined system under test plus the measurement device (or the environment). This lack of knowledge can eliminate quantum behavior and lead to a classical response. 
This, however, is not universally so. In macroscopic quantum systems like solid matter, superconductors and permanent magnets (to pick the best known examples), decoherence does not eliminate the crucial quantum effects, at all. Solid matter does not become unstable as predicted by classical theory because it is subjected to decoherence. Superconducting magnets do not demagnetize when someone looks at them. Permanent magnets exhibit the collective behavior of their spins even under very significant thermal excitation (up to hundreds of K) and one can find "quantum interference effects" represented even in the detailed behavior of their seemingly classical response (this is also true for details of many chemical reactions, by the way).   
Therefor I would not paint decoherence and measurements that eliminate quantum mechanical interference with too broad a brush. They can and they do this, but in quantum systems with e.g. significant energy gaps the resulting behavior is all but trivial and it needs to be analyzed in detail, rather than characterized with a single sentence. That quantum measurement theory tends to look at these effects as fickle and completely covered by e.g. density matrix theory is somewhat of a selection effect: the optical and atomic systems that are easy to understand and treat with theory do, indeed, react the way as described by the source... that's just not all the systems there are. 
