Abstract
This paper reports three almost trivial theorems that nevertheless appear to
have significant import for quantum foundations studies. 1) A Gleason-like
derivation of the quantum probability law, but based on the positive
operator-valued measures as the basic notion of measurement (see also Busch,
<a href="/abs/quant-ph/9909073">quant-ph/9909073</a>). Of note, this theorem also works for 2-dimensional vector
spaces and for vector spaces over the rational numbers, where the standard
Gleason theorem fails. 2) A way of rewriting the quantum collapse rule so that
it looks almost precisely identical to Bayes rule for updating probabilities in
classical probability theory. And 3) a derivation of the tensor-product rule
for combining quantum systems (and with it the very notion of quantum
entanglement) from Gleason-like considerations for local measurements on
bipartite systems along with classical communication.
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