>Mathematicians are prone to taking words from elsewhere, either twisting their meaning or inventing wholly new meaning out of thin air, all according to their whimsy for their own particular needs.
True but one benefit of those guys is that they actually define what they mean in a formal way. "Programmers" generally don't. There is in fact some benefit in having consistent names for things, or if not at least a culture in which concepts have unambiguous definitions which are mandated.
>what this routing mechanism is (heating a substrate, maybe?)
You can engineer a waveguide if you understand the nonlinear theory they propose. There's no heat exchange involved, which is easy to get confused on because the writing in the article does not really understand "optical thermodynamics".
>if the routing is dynamically changeable
At this point probably not, it requires a finely engineered waveguide which has a well-defined "ground state"
>it works in reverse, eg light coming in can be routed to one of several output ports
In theory it works in reverse, as everything in this system is time-reversible (i.e., the "optical thermodynamics" is just an analogy and not real thermodynamics, which would break time reversibility). This is demonstrated via a simulation in the SI, but experimentally they did not achieve this (it may be difficult, I am not an experimentalist so cannot comment).
I really like the second part of the blogpost but starting with Gaussian elimination is a little "mysterious" for lack of a better word. It seems more logical to start with a problem ("how to solve linear equations?" "how to find intersections of lines?"), show its solution graphically, and then present the computational method or algorithm that provides this solution. Doing it backwards is a little like teaching the chain rule in calculus before drawing the geometric pictures of how derivatives are like slopes.
>My goal is to develop a practical, working understanding I can apply directly.
Apply directly... to what? IMO it is weird to learn theory (like linear algebra) expressly for practical reasons: surely one could just pick up a book on those practical applications and learn the theory along the way? And if in this process, you end up really needing the theory then certainly there is no substitute for learning the theory no matter how dense it is.
For example, linear algebra is very important to learning quantum mechanics. But if someone wanted to learn linear algebra for this reason they should read quantum mechanics textbooks, not linear algebra textbooks.
Although I am not an expert in quantum information, I think the problem you pose is resolved by the fact that the no-signalling theorem is about measurements of a quantum state, which is a microscopic state, and heat transfer is a measurement of a thermodynamic quantity, which is macroscopic. In much the same way that measuring the temperature of a classical gas doesn't give information on the location or momenta of the constituent particles, a thermodynamic probe of entanglement doesn't necessarily furnish precise information on how a state is entangled (e.g., Eq. 2 in https://arxiv.org/pdf/quant-ph/0406040).
True but one benefit of those guys is that they actually define what they mean in a formal way. "Programmers" generally don't. There is in fact some benefit in having consistent names for things, or if not at least a culture in which concepts have unambiguous definitions which are mandated.