What is 'coherence' of a quantum system (with examples)? The term 'coherence' is frequently used in relation to certain quantum systems such as LASER, superconductors, etc. Please help me understand what we mean by the coherence of a quantum system. How do we describe it quantitatively and interpret physically? Also, why is it important/interesting?
 A: Coherence is a very general concept applied to many wave phenomena, and it would be hard to give a comprehensive coverage. Therefore I will give only a few examples.
Lasers
In the context of laser we are talking about coherence of light, which can be thought of as our ability predict the phase of an electromagnetic wave in a point of time and space, knowing its phase in a different point of time and/or space, and we can correspondingly talk about time coherence and space coherence. The coherence is notably absent when using thermal light sources, such as light bulbs, which emit many of photons with random phases and directions. On the other hand, lasers and other quantum light generators emit photons that are identical, i.e. they have exactly the same phase. This allows studying such phenomena as diffraction from objects (basic wave phenomena that is however hard to reproduce with conventional light sources) or creating holographic images (dependent on preserving the phase information).
Radio waves
Just to mention manifestly non-quantum applications.
For radio waves preserving coherence is less of an issue, as their wavelength is much bigger than that of light, which is why phased arrays are in common use in radio location and similar applications.
Another place where coherence is crucial is electrical grids, supplied by multiple power stations. Here one is obviously interested in adding the ac electric signal constructively, rather than losing most of it in destructive inetrference.
Quantum transport
In the context of solid state coherence is usually not preserved, due to multiple interaction processes - scattering from impurities, phonons, and Coulomb scattering. In this context one would often distinguish decoherence and dephasing, although the distinction is not well established one. Dephasing would usually imply randomization of phase by elastic scattering, while decoherence involves inelastic interactions. A good introduction to this domain is the Yoseph Imry's book.
Creating nanostructures with long coherence lengths allows observing a multitude of interesting quantum phenomena, including weak and strong localization, conductance quantization, Josephson tunneling, integer and quantum hall effect, etc.
In particular, implementation of two-slit experiments in solid-state Aharonov-Bohm interferometers has long been a subject of study: one can destroy this interference completely or partially by causing decoherence in one of the arms of such an interferometer in what is called "which path?" experiments.
And also quantum computation...
Quantum computation is certainly an important subject where coherence is crucial. I do not say about it much, because, as you can see, the subject is much broader.
