Can anyone working in this field comment on how it's applied? For example, what is computed? Homology groups, just the ranks of these groups, or something else? Given these, what does one learn about the data set under analysis? https://en.wikipedia.org/wiki/Topological_data_analysis#Appl... is pretty scant on detail.
Not sure exactly what you're linking the Wikipedia article on quantum states, but if you like it as a reference, check out https://en.wikipedia.org/wiki/Quantum_state#Mixed_states. The operation of "projecting out" the E&M field in my previous comment would be realized on the density matrix as a partial trace over the electromagnetic field. You can also take the partial trace of this field of the Hamiltonian operator to see the effective dynamics of the atom when "ignoring" the E&M field. This is a fully quantum description of the state, so I stand by the statement that your claim "An atom can be described by its quantum state only if it's isolated and in that case its energy is constant." is incorrect.
"An atom can be described by its quantum state only if it's isolated and in that case its energy is constant."
How do you figure? As a contradiction, take your atom+electromagnetic field system, describe the transition from excited atom to unexcited atom + photon state, and project out the E&M field. Voila, now you have a quantum description of an atom transitioning between different energy states. Its dynamics may look funny, i.e. they may appear nonlocal, they may not conserve energy, etc. but that's different from saying "there is not a quantum description of these dynamics" which is what you're claiming.
"Basically any fundamentally correct buffer encoded as message A will decode successfully as message B for any B."
This is incorrect. I suspect you're overextending proto3's treatment of unknown fields to include discarding incorrectly typed fields too. If A has field 1 types as an int, and B has field 1 typed as a string, an A message with field 1 set will not parse as a B message. However, if the A message has no fields set, or sets a field number unknown to B, that could parse successfully with "leftover" unknown fields.