we can transmit information at 3 trillion meters per second, a LOT faster than the speed of light.
Unfortunately, no, we cannot - unless I grossly misunderstood the whole thing (always a possibility) - the universe, in a way, does and that was the spooky part Einstein did non like.
There is no way we can use the entanglement to transmit information of our choice, we can only measure that entangled particles are, in fact, entangled (they have one single state for the couple, not one each).
E.g.: we cannot force the spin of one particle to be up when we measure it, so forcing the distant one to be measured as down, effectively transmitting our piece of information faster than light.
Yep, these quantum effects are in a sense, "action at a distance", but it's non-causal and cannot transmit information.
And speaking of faster-than-light, in fact there are many interesting phenomenon that give such an illusion. The experiment result I'm most impressed is that some exotic non-linear materials can have a phase velocity and a group velocity,
both faster than c. This is known as anomalous dispersion. If you transmit a Gaussian impulse and measure "delay" between the rise edge / peak of the input and output impulses, the "delay" is "faster than light". It has been demonstrated that you can even run the experiment in both air/vacuum and the non-linear material simultaneously, and non-linear material is always "faster" than the air/vacuum, the results are as consistent as it can be. Better, this apparent delay can even be negative.
The simplest explanation is when the pulse shape is distorted by the medium: no magic here, the location of the edge or the peak of the impulse was shifted, giving a misleading propagation delay measurement
. But experiments showed that in some cases, the effect is still possible without wave shape distortion. I still can't wrap my head around it, but all we need to know is that if you define the "speed" as "the time interval between the moment the ON button of the transmitter is pushed, and the moment that the receiver detects a signal", this propagation delay is always slower than or equal to c. Here's an introduction to these phenomenon:
https://cpb-us-w2.wpmucdn.com/u.osu.edu/dist/7/38882/files/2016/09/PhotonicsSpectraFastLightSlowLight2007-1gb2tao.pdfI read that similar experiments exist using microwave waveguides below cut-off. Sometimes the phase shift of the evanescent wave can appear faster-than-light, but it's not even a traveling wave, just a stray field, and there's no signaling.
Regardless of how exotic the material, it's basically a variation of this idea: you can build linear circuit (an opamp active filter) with a negative group delay, but it's only because the signal is bandwidth limited, and the active filter is essentially an analog computer that predicts and recreates the expected output in advance.
https://www.dsprelated.com/showarticle/54.php