To start with the double slit experiment gives interference even when the beam is composed by one photon at a time. The spot on the screen a photon/particle the statistical accumulation the interference seen as expected classically too.
The joint comes because the photon as a quantum mechanical entity has a wavefunction that is the solutions of Maxwell's equation, treated as operators operating on the wave function. The E=h*nu identity the photon carries is the same as the frequency of the classical solution of Maxwell's equation and that , together with the phase attached to the wavefunction allow the continuity of classical down to quantum single photon level For a formal treatment how classical emerges from an ensemble of photons have a look at this blog entry.
There exists a fundamental difference between classical and quantum equations and their solutions, but also a continuity at the interface.
Edit after question edit
How is the theory of partial coherent light related to quantum-mechanics?
This needs somebody familiar with the formalism of both, but I believe the connection should follow the method in the link, how classical electromagnetic beams emerge from an ensemble of photons.
]>... the amplitude of a wave function ... But is this really a fundamental difference, or just a difference in the common practices of the respective theories?
the square of the wavefunction is the connection with predictions and experiments in quantum mechanics, it is the probabilistic nature that makes the difference with the classical framework, as far as I know.
How much of the strange phenomena of quantum-mechanics can be explained by the theory of partial coherent light alone, without any reference to particles or measurement processes?
Phenomenon is " an observable" , observing something implies a measurement process, measurement implies interaction, picking up a point that will contribute to the quantum mechanical probability distribution (or building up the distribution itself by continuous observations) so there is an inherent contradiction in this part of the question.
I assume in your figure the red lines represent the phase profile of the wave (the points where the phase is constant), and not the intensity. And that the wave propagates from top to bottom.
Is the following wave spatially coherent? Yes.
If those lines were straight, the wave would be a plane wave, perfectly coherent in time and space.
I consider then your plot as a propagating wave with a periodically undulated phase profile. Again, it's perfectly coherent in time and space.
The definition of coherence which I consider accepted says that a field is coherent where (when) there is a fixed phase relation between the electric field in different locations (times).
Mathematically this is described by the cross correlation of the field at two points in space or time.
Experimentally, we measure the intensity $|E|^2$, not the field $E$, so coherence is tested using interference as follows:
Spatial coherence: you measure the intensity fringe contrast (max-min) in the double slit experiment, as a function of the distance between the slits. The maximum distance between the slits at which there is still some interference is the spatial coherence length of the wavefront (the contrast depends on the phase correlation between two different points in space).
Temporal coherence: in an interferometer with 2 different arm lengths (e.g. Michelson) each point, or beam, of the wavefront interferes with a delayed copy of itself. The maximum delay (difference between the lengths of the 2 arms) at which there is still some interference is the temporal coherence length of the field.
Applying the above 2 criteria to your wave, you can see that it's both time and space coherent. This will hold true if you distort the red lines even more (you can make them random), IF they remain one identical to the others and equispaced (like the figure below, left). To decrease the temporal coherence, you can make the distance between the lines, i.e. the wavelength, more randomly changing (fig. below, middle). In this way, the phase difference $\Delta \phi$ between the field $E(x_o,t)=P1$ and the field $E(x_o,t+\tau)=P'1$ is not constant anymore, and the spectrum is not a delta but a broader peak. To decrease the spatial coherence, while keeping the temporal one, you can change the wavelength across the wavefront (fig. below right).
Probably a useful way to see if a wave is coherent is to consider the local wavelength $\lambda(x,t)$, and asking if it constant in $t$ (vertical direction in the figure) and / or in $x$ (horizontal direction in the figure), etc.
Best Answer
Coherence means waves which maintain the same phase difference between them as you mentioned. So the two waves in the first picture are coherent if they maintain the same phase difference. Note that this implies that they must have the same frequency.
Polarized waves are just waves oscillating about one plane. They need not be monochromatic, coherent or in phase.
In phase waves are those which maintain no phase difference between them, so saying in phase implies the waves are coherent.
The 2nd picture says that the waves are in phase so they are also coherent.