Information relationship to Special Relativity

How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?

Answer

How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity.

Since you are looking for an equation (you say "mathematically"), I would undoubtedly choose this: $$\left[\hat O (x),\, \hat O' (y)\right]=0, \, \mbox{if}\; x-y \; \mbox{is spacelike}$$ where $\hat O$ and $\hat O'$ are the (linear self-adjoint) operators corresponding to two physical observables —in particular, both may be the same observable and therefore the same operator ($\hat O=\hat O'$)—, and $x, y$ are two points in space-time. This equation summarizes the fact that information cannot travel faster than light because it says that the results of two experiments separated by a space-like interval cannot be correlated. And this is what "information" means since one codes information with physical effects. Please, see this Definitions: 'locality' vs 'causality' if you are interested in the different usages of the terms "causality" and "locality", which are physically more relevant or why entanglement do not imply faster than light propagation.

The previous formula assumes that the physical laws obey the principles of quantum mechanics and special relativity, and are thus quantum field theories. This is the case for the electromagnetic, the weak and the strong interactions and also likely for the case of the gravitational interaction in the weak field limit and in the sense of an effective field theory; which are the fundamental interactions that we know.

Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity)

Sometimes defining the speed of a wave is tricky. The signal or information velocity is often the group velocity (which is the velocity of a wave packet), even though in some media (see http://en.wikipedia.org/wiki/Signal_velocity) it is not. But the phase velocity (the rate at which the phase of the wave propagates) cannot carry information (see http://en.wikipedia.org/wiki/Phase_velocity) and may be faster than $c$.

Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?

You may take any inertial frame provided its speed be lower than $c$. Note that according to special relativity one needs an infinite amount of energy to cross the speed of light $c$ threshold.

Edit: SMeznaric points out —and I agree with him— that space-like separated measurements may give correlated results. What is not possible is to send information one has control over, such as the choice of measurement operators.