Optical coherence versus quantum coherence Optical coherence is usually defined to be the measure of statistical correlation (for example if the phase difference is constant) of a pair of waves.
Recently I came across the notion of quantum coherence which is the measure of how many off-diagonal terms in the density matrix are non-zero. One can think of it as a measure of superposition - if the state is in a superposition of the basis elements it is coherent. And for example, a diagonal density matrix is perfectly incoherent.
Now, as far as I can see, there seems to be no connection between the two kinds of coherence and it makes me wonder whether it is mere chance that both these properties have been given the same name. Is that true?
Note: The idea of a coherent state is again totally different. It is the state
$|\alpha \rangle = e^{-\alpha ^2/2} \sum _n \frac {\alpha ^n}{\sqrt {n!}} |n \rangle.$ 
and also has (as far as I can see) nothing to do with the idea of quantum coherence.
 A: Coherence is the ability to track the phase of a wave. In optical coherence, a light source is coherent in time/space if you can determine the phase difference between two points in time/space. For an incoherent light source, one cannot perform Young double slit experiment or just a simple Michelson interferometer, because these require a stable interference pattern, and when the phase difference is too noisy, the interference pattern fluctuates rapidly from constructive to destructive interference and on average it seems like there is no interference. 
Quantum coherence is the same principle - in quantum mechanics effects of superposition show how states interfere with one another, and losing track of the phase difference between the states - that's what quantum decoherence is all about, quantum effects average to the case of no interference because of the fluctuating phase.
And finally, the coherent state is just a special state that is useful in quantum optics to describe the state of light coming out of lasers - not with a certain amount of photons but a superposition that looks like a regular EM wave in time.
A: Optical and quantum coherence are intimately related. For example, in the case of a pair of polarized photons prepared in a net-zero polarization state (a superposition of s-p and p-s pairs), both optical and quantum coherence are degraded by scattering. To scatter light, a charged particle oscillates in the incident e-field direction, thereby "observing" the photon polarization state, causing the collapse of the superposition into a definite state. The scattered photons are also out of step with the incident radiation, yielding a degraded optical coherence.
