This is a bit earlier in the sequence than I'd intended but I wanted to rant about it.
What is so-called quantum teleportation?
Imagine you have a small particle. If this were a classical world, you could measure everything about it (it's speed, it's spin, etc), and then use a bunch of fiddly experiments to recreate one (or more) new copies of it that had all those same properties. Of course, it's *practically* impossible, to scan the state of millions of particles so this actually only happens to single particles (or we mass-manufacture consumer goods, but we don't try and make sure they all have corresponding atoms in the same place in each).
As we live in a quantum world, you can't "measure everything about it". Electrons don't exist at a particular point, they exist as a wave of possibility in a sphere round an atom, and only when another particle interacts with them, does it interact with them at one particular place on that sphere. Each photon isn't "in a particular place", even if you have a single photon you have a very very very faint beam of light and if you repeat the experiment, you find "places a photon hits" and "places the beam of light would cover" are the same thing. If you have a qubit made up of a single atom, you can measure its value as 0 or 1, or send it through a quantum logic gate, and find out about the parts of its state you can't measure directly *instead* but you can't do both.
Hence, in a quantum world, even in theory, it's weirder to construct a new particle the same as an existing particle, because you can't "measure everything, and then move the new particle so it has all those values".
So you *can't* make multiple copies.
What can you do
However, it turns out, there *is* a way of making an exact copy of a particle's property. You create two other objects (photons?) with opposite values for polarisation or something, even though you can't measure what that value is. (aka "an entangled pair", although all "entangled" means is "they have the opposite polarisation even if you don't know what it is"). You interact the original with that one and measure some values. Those values don't tell you what the property is (because if it WAS one particular thing, you'd have destroyed the information you were trying to copy). But you can apply them to a new particle via the second entangled particle. And you don't know what the state *is*, the original particle no longer has it, but the new one does.
That is, "You might imagine that you could copy a quantum electron the same way you could copy a classical particle by measuring the values and applying them to a new electron. But you can't, that's actually a meaningless concept. Knowing that, you might give up. But there's a way to do sort-of do that."
Specifically, "quantum teleportation" means "there's a special and fiddly way you can construct a new particle exactly the same as an old particle, but only EXACTLY ONE, and it destroys the original state". As in, you can do SOME of what you'd expect to be able to do to a classical particle, but not all of it.
What doesn't it mean?
What doesn't it mean? Firstly, it means "teleportation of quantum", not "teleportation by means of quantum". It doesn't give you some magic way of scanning macroscopic objects or reconstructing them elsewhere. It just means that, if you happened to already have one, you might be able to copy quantum states too.
Secondly, nothing anyone cares about day-to-day is encoded in quantum states. It might matter for quantum computers. Maybe for quantum cryptography. Certain scientific experiments. That sort of thing.
If you actually cared about quantum states, this might be exciting. Suppose brains encoded what they did in something like a quantum computer. Then startrek teleportation would only be normally impossible because you can't scan a human like that, not logically impossible. However, brains don't do anything of the sort.
If you care about startrek-teleporting a human, you probably want to end up with the same DNA molecule. But you probably don't need each atom to have the same quantum state. So it doesn't really matter.
A: Startrek is awesome, right?
A: But teleporting people is impossible right?
B: Pretty much. I suppose there might be some way discovered, but it doesn't seem very practical.
A: But, doesn't quantum say something about this?
B: Oh right. Yes, it says if you care about replicating all the quantum states in the transportee, you can only have one source (which is destroyed) and one copy.
A: That seems fair. That's how it works in startrek.
B: Well, it rules out "lets keep a backup of our most valuable engineers and seconds in command". Which did happen in startrek but only by accident.
A: Oh yeah, I guess.
A: So, *do* I care about replicating the quantum states in the transportee?
B: No, not really.
A: So quantum doesn't really change the answer?
A: What about "quantum teleportation". Doesn't that let you... teleport people?
B: No. It just means, you CAN do the up-to-one perfect-quantum-states copy (assuming you have a way of teleporting people at all).
A: So why do people keep writing news articles about it?
B: Because it sounds startrek-y.
B: And to be fair, is relevant for how QM works.
How do I know that? Well, I might be wrong. But firstly, maintaining atoms in a particular quantum state which can encoded a qubit used for quantum computing needs a whole bunch of vacuums and stuff. MAYBE brains could do that, but it seems unlikely. Sorry Penrose, I know you're a genius and I'm not, but I don't believe you.
Secondly, quantum computers have certain distinguishing features. They're about the same as classical computers for most problems. Notably, most every-day stuff. Also, NP-complete problems they're not significantly better. However, they ARE better than normal calculations for some specific things, like factoring numbers with thousands of digits, and other maths problems which share some features in common with that. If you look at a human brain, do you think, "boy, that's optimised for simple but powerful heuristics used for catching balls, recognising objects, and social interaction, but is mediocre at factorising incredibly large numbers"? Or the reverse?
Thirdly, there's no reason to think brains DO have quantum information used in any particular way. If they did... it wouldn't change anything significant. It wouldn't make the free will argument any different. It wouldn't give them a magical insight into parallel universes (as awesome as Anathem makes it sound). So why would you think that?