> So Anne takes her bit, A, and puts it through her handy entanglement-decoding machine, which spits out an answer: either B or C.
(Note: I have a Ph.D. in Math and half a major in Physics that include a few quantum mechanics curses (for example, with the Sakurai book))
What is a entanglement-decoding machine? You cant use entanglement to send information. Just repeat after me: You cant use entanglement to send information. This is one of the most common misunderstandings about entanglement.
When you make the measurement and get the collapse, you only get a random value (with a weighted probability). It's not possible to use that to get information about B or C.