Why group speed can contain information but phase speed can't? I was watching sixty symbol on YouTube about reflective index and learnt that it is the sum of all the speed of light or phase speed that appears slower than speed of light in the medium, and also the phase speed can also move greater than the speed of light but cannot carry information. I thought this is very interesting and confusing because a pulse is a group speed which can contain information but can never travel faster than speed of light in a vacuum, what is the difference between phase speed and group speed? Is quantum physics needed to explain the phenomenon?
 A: No quantum mechanics is needed here; the following only requires a classical treatment of waves.
Any wave propagating in a medium is composed of a sum of simple sine and cosine waves. A single sine wave is described by three properties: a frequency $\omega$, a "wavenumber" $k$ (which is really just the inverse wavelength), and a phase $\delta$. But you can't just use any old combination of $\omega$ and $k$; the medium will only support particular kinds of oscillations. When we quantify this, we end up with a dispersion relation $\omega(k)$, which gives the frequency of a valid sine wave as a function of its wavenumber.
You may be familiar with the simplest expression for the speed of a sine wave: it's usually written as the wavelength times the frequency. (Intuitively, it's easy to see why this should be the speed; you're multiplying the number of times the wave oscillates per second by the amount of distance it covers per oscillation.) But this definition is only valid for a sine wave, not more complicated waves, and also fails if your medium responds differently to oscillations of different wavelengths (a property which is true of basically all realistic media). If we want to generalize the concept of wave propagation speed, there are several ways for us to take our dispersion relation $\omega(k)$ and get something that looks like a speed. Each different manipulation carries with it a different physical meaning, and tells us something slightly different about the physical propagation of the wave. The two most common speeds are the phase velocity
$$\frac{\omega(k)}{k}$$
and the group velocity
$$\frac{d\omega(k)}{dk}$$
The phase velocity at a wavenumber $k$ tells you how fast a sine wave of wavenumber $k$ will propagate. Generally waves are made up of many sine waves of different wavelengths (and therefore different wavenumbers), which means that if your dispersion relation $\omega(k)$ is nonlinear, your wave will spread out, as the faster wavenumbers gradually outpace the slower ones.
The group velocity at wavenumber $k$ is a bit more complicated, but in a nutshell, it tells you how fast a peak in your wave will move, if your wave is made up of components that are mostly around wavenumber $k$. Since we usually transmit information by way of pulses, we usually associate the speed of information propagation to the speed that those pulses travel. 
