> Shanon & Von Neumann were hacking a new theory of communication, not doing real physics
Maybe I’m misunderstanding the reference to von Neumann but his work on entropy was about physics, not about communication.
Maybe I’m misunderstanding the reference to von Neumann but his work on entropy was about physics, not about communication.
OTOH, old Von Neumann was wealthy, hobnobbing with politicians and glitterati musing about life, biology, econ and anything else that would amuse his social circles. "Entropy", as he's alleged to have told Shannon, was his ace in the pocket to win arguments.
Formal similarity with Shannon's entropy is superfluous and conveys no new information about any system, quantum or otherwise. But it does make for lot's PhD dissertations, for exactly the same reason Von Nuemann stated.
We agree then! John von Neumann’s work on entropy was about physics, not about communication theory. S(p)=Tr(p ln(p)) is physics. If you still claim that he “was extending Shannon's information theoretic entropy to quantum channels” at some point could you maybe give a reference?
> Formal similarity with Shannon's entropy is superfluous and conveys no new information about any system, quantum or otherwise
What I still don’t understand is your fixation with that.
“Entropy can't be a measure of uncertainty, because all the uncertainty is in the probability distribution p(x)” makes zero sense given that the entropy is a property of the probability distribution. (Any measure of “all the uncertainty” which is “in the probability distribution p(x)” will be a property of p(x). The entropy checks that box so why can’t it be a measure of uncertainty?)
It is a measure of the uncertainty in the probability distribution that describes a physical system in statistical mechanics. It is a measure of the lack of knowledge about the system. For a quantum system, von Neumann’s entropy becomes zero when the density matrix corresponds to a pure state and there is nothing left to know.