# 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.