The research you posted doesn't say anything about communication at faster than light, it just tries to determine the speed of "spooky action at a distance". The question is: Is it instant or does it have a finite speed?
I'm preparing two envelopes into which I put two identical sheets of papers with a short text. One of them I give to you, the other gets shipped by Elon Musk with the first manned Mars mission. On a certain date and time, you both agree to open the envelopes.
As soon as you open your envelope, you will
instantanelousy know what text Elon Musk has in his hands
in this very moment - despite the fact that Mars may be half an hour lightspeed delay apart.
This deceptively looks like instantaneous information exchange, except the information hasn't been exchanged in this moment - it has laboriously been flown to Mars for six months. Only your
knowledge of the information has suddenly changed.
Likewise, if you quantum entangle two particles, they have to be close. You can separate them, at which point you move the information, but you can't separate the system faster than lightspeed. Once you measure the system, what changes is not so much the physics (no energy gets exchanged, no spin gets flipped), it's your knowledge of the state.
The key thing to notice is that information here is not what
you know, it's what's knowable. A newspaper has an information content even before you read it, and an entangled system transfers information even before you measure its state.
Hope that explains some of the paradoxies.
First of all light speed is undefined, and is only what we've observed so far dependent on the medium density (like any wave/particle), which does by no means, mean it's constant.
That's true but misleading - usually we refer to light speed in vacuum, and that is very well defined and one of the basic constants of nature. In a medium you get a split into phase and group velocity - but information is only carried by group velocity, and that's always slower than vacuum lightspeed.
Secondly We cannot even define gravity, which I suspect is way (instantaneous) faster than light - an ideal candidate for entanglement communication, but I reckon it's pot luck as thus far our reality is a quantum average, which is like a lottery ticket with two numbers.
In General Relativity, gravity waves propagate with vacuum lightspeed. So do gravitons in leading order Quantum Gravity theories. That's all rather important for causality to work, and causality is... sort of... really important, which is why theories with anything in them usually face great problems explaining a universe that's anywhere close to what we observe.
You can of course always hope for the next theory, but since causality is a cornerstone of our world, it's probably not going to happen.
Btw: Concluding that
because we cannot define gravity that
it is way faster than light is faulty reasoning - there are no valid conclusions you can draw from not understanding something about its nature.
