Okay…I'm going to go line by line on this abortion of a page.
You've probably read about relativity theory and quantum mechanics in one of those old books you dug up
There's nothing actually wrong with this sentence, but prefacing anything by mentioning a specific scientific field is the same as holding up a banner saying that you're about to spout some contrived bullshit.
The deeper truth is that everything boils down to entropy
This is also mostly correct, but becomes very complicated when you start looking at things on the Planck scale, as the aforementioned field of quantum mechanics does.
And entropy can be violated on local scales
The sentence itself is correct, and it's the reason why quantum mechanics is so bizarre, but the use of the word "local" in this sense is usually reserved for objects or areas that make protons look enormous. I suspect that Jeph is using the wrong "local", referring instead to one's personal space.
He also started the sentence with And, which is a literary no-no. If he was an actually competent writer, I might think that he's just simulating imperfect natural speech patterns, but he's not so it looks bad.
Drastically violated, if you have a way to make up the difference
This is where things start to go sideways. The violation of entropy that occurs at the Planck level is not well understood, and tends toward normal entropy when viewed across a large area (large being individual atoms) or a long time (such as a few seconds). The idea that one could "make up the difference" is only applicable at the smallest of scales, since the act of operating a device capable of doing so would result in normal entropy overall across the area the device occupies, because modifying the energy state of a system using outside energy requires you to use energy to insert energy into the system whose entropy you are attempting to modify.
If you entangle a system with an area of space containing the maximum possible entropy, you have an entropy sink
This is where we go full retard. First of all, quantum entanglement is when two particles are functionally identical to each other in every measurable way and each will simultaneously change in response to things happening to the other regardless of distance. If one particle in the pair has a particular energy state, they both do. One doesn't "pass" energy on to the other, they are identical at all times. It's possible to create a heat sink by creating many such pairs, placing them separately in two areas, and introducing nonpaired particles with different energy levels into the respective areas, but that just results in normal entropy across the system.
Which brings me to the second problem. Entropy is not a thing. It's a process. The idea of "maximum entropy" only exists when you take a snapshot of a system's energy state and try to predict the systemic change that has the highest probability of occurring.