Welcome to the forum!
It’s interesting to consider how quantum information – or quantum mechanics in general – could be related to BCI and brain processes.
I read your post on losttv-forum.com. I’m not a physicist or a neuroscientist. I am somewhat familiar with some of the theoretical efforts that relate QM to neurology and consciousness (eg Penrose/Hammeroff ORCH OR theory, Quantum Brain Dynamics, Scott Aaronson’s ‘Ghost in the Quantum Turing Machine’, Henry Stapp, Wigner, etc.). That said, I’m sure a real physicist could give you a much more technically sound response.
Although it’s interesting, in my view the quantum BCI technique you’ve proposed is probably not workable in principle (as well as facing some fairly serious practical challenges!), for two reasons.
The first reason is that EEG is not really suitable for capturing quantum information. EEG measures voltages, which are very, very far from the underlying quantum states of the brain’s microscopic components.
Consider: a wavefunction is the entire state of the system, in complete detail. In the case of the brain, or even any meaningfully-sized portion of it, the wavefunction (if such a thing exists – see below) includes the probability distribution for every feature of every single particle. No amount of gaussian quadrature would extract that level of detail from any number of voltage-measuring electrodes. Even with an electrode for every neuron, you still wouldn’t have enough information to reconstruct the wavefunction of even a single cell, because the 1-dimensional information that an electrode collects doesn’t let you know much about the billions or trillions of particles the neuron is made up of.
The second reason is that the brain is a large, noisy system that is in continuous interaction with its environment – not the sort of carefully isolated systems that quantum teleportation experiments are concerned with. This means that it probably does not (even if it would otherwise) have a wavefunction all its own but is entangled with its environment and therefore “cannot be described independently”, to quote the Wikipedia article on quantum entanglement. I think this environmental entanglement (and resulting decoherence) would probably a priori prohibit quantum teleportation of brain states, as currently understood.
(As a side note, depending on which interpretation of quantum mechanics you look at the assumption would be that the brain as a whole is not quantum mechanical at all – that the QM activity only applies at smaller scales. Just don’t ask why this is the case :))
All that said, I’m not an expert on the topic, only an enthusiastic spectator to the ongoing quantum-mechanics-meets-neuroscience circus.
Thanks for posting your hypothesis to the forum! Perhaps a more knowledgeable member may have a more encouraging reply.