How quantum teleportation actually works

EDIT: Philip Ball has updated his article on Nature News, correcting the most serious of its errors. While everyone makes mistakes, few actually admit to them, so I think this action is rather praiseworthy. Correspondingly, I’m removing criticism of that mistake in my post.

Recently I have read an excellent essay by Philip Ball on the measurement problem: clear, precise, non-technical, free of bullshit and mysticism. I was impressed: a journalist managed to dispel confusion about a theme that even physicists themselves are confused about. It might be worth checking out what this guy writes in the future.

I was not so impressed, however, when I saw his article about quantum teleportation, reporting on Jian-Wei Pan’s group amazing feat of teleporting a quantum state from a ground station to a satellite. While Philip was careful to note that nothing observable is going on faster than light, he still claims that something unobservable is going on faster than light, and that there is some kind of conspiracy by Nature to cover that up. This is not only absurd on its face, but also needs the discredited notion of wavefunction collapse to make sense, which Philip himself noted was replaced by decoherence as a model of how measurements happen. For these reasons, very few physicists still take this description of the teleportation protocol seriously. It would be nice if the media would report on the current understanding of the community instead of repeating misconceptions from the 90s.

But enough ranting. I think the best way to counter the spreading of misinformation about quantum mechanics is not to just criticize people who get it wrong, but instead to give the correct explanation about the phenomena. I’m going to explain it twice, first in a non-technical way in the hope of helping interested laypeople, and then in a technical way, for people who do know quantum mechanics. So, without further ado, here’s how quantum teleportation actually works (this is essentially Deutsch and Hayden‘s description):

Alice has a quantum bit, which she wants to transmit to Bob. Quantum bits are a bit like classical bits as they can be in the states 0 or 1 (and therefore used to store information like blogs or photos1), and entirely unlike classical bits as they can also be in a superposition of 0 and 1. Now if Alice had a classical bit, it would be trivial to transmit it to Bob: she would just use the internet. But the internet cannot handle superpositions between 0 and 1: if you tried to send a qubit via the internet you would lose this superposition information (the Dutch are working on this, though). To preserve this superposition information Alice would need an expensive direct optical fibre connection to Bob’s place, that we assume she doesn’t have.

What she do? She can try to measure this superposition information, record it in classical bits, and transmit those via the internet. But superposition information is incredibly finicky: if Alice has only one copy of the qubit, she cannot obtain it. She can only get a good approximation to it if she measures several copies of the qubit. Which she might not have, or even if she does, it will be only an approximation to her qubit, not the real deal.

So again, what can she do? That’s where quantum teleportation comes in. If Alice and Bob share a Bell state (a kind of entangled state), they can use it to transmit this fragile superposition information perfectly. Alice needs to do a special kind of measurement — called Bell basis measurement — in the qubit she wants to transmit together with her part of the Bell state. Now, this is where everyone’s brains melt and all the faster-than-light nonsense comes from. It appears that after Alice does her measurement the part of the Bell state that belongs to Bob instantaneously becomes the qubit Alice wanted to send, just with some error that depends on her measurement result. In order to correct the error, Bob then needs to know Alice’s measurement result, which he can only find out after a light signal has had time to propagate from her lab to his. So it is as if Nature did send the qubit faster than light, but cleverly concealed this fact with this error, just so that we wouldn’t see any violation of relativity. Come on. Trying to put ourselves back in the centre of the universe, are we?

Anyway, this narrative only makes sense if you believe in some thoroughly discredit interpretations of quantum mechanics2. If you haven’t kept your head buried in the sand in the last decades, you know that measurements work through decoherence: Alice’s measurement is not changing the state of Bob in any way. She is just entangling her qubit with the Bell state and herself and anything else that comes in the way. And this entanglement spreads just through normal interactions: photons going around, molecules colliding with each other. Everything very decent and proper, nothing faster than light.

Now, in this precious moment after she has done her measurement and before this cloud of decoherence has had time to spread to Bob’s place, we can compare the silly story told in the previous paragraph with reality. We can compute the information about Alice’s qubit that is available in Bob’s place, and see that it is precisely zero. Nature is not trying to conceal anything from us, it is just a physical fact that the real quantum state that describes Alice and Bob’s systems is a complicated entangled state that contains no information about Alice’s qubit in Bob’s end. But the cool thing about quantum teleportation is that if Bob knows the measurement result he is able to sculpt Alice’s qubit out of this complicated entangled state. But he doesn’t, because the measurement result cannot get to him faster than light.

Now, if we wait a couple of nanoseconds more, the cloud of decoherence hits Bob, and then we are actually in the situation where Bob’s part of the Bell state has become Alice’s qubit, modulo some easily correctable error. But now there is no mystery to it: the information got there via decoherence, no faster than light.

Now, for the technical version: Alice has a qubit $\ket{\Gamma} = \alpha\ket{0} + \beta\ket{1}$, which she wishes to transmit to Bob, but she does not have a good noiseless quantum transmission channel that she can use, just a classical one (aka the Internet). So what can they do? Luckily they have maximally entangled state $\ket{\phi^+} = \frac1{\sqrt2}(\ket{00}+\ket{11})$ saved from the time when they did have a good quantum channel, so they can just teleport $\ket{\Gamma}$.

To do that, note that initial state they have, written in the order Alice’s state, Alice’s part of $\ket{\phi^+}$, and Bob’s part of $\ket{\phi^+}$, is
\[ \ket{\Gamma}\ket{\phi^+} = \frac{1}{\sqrt2}( \alpha\ket{000}+\alpha\ket{011} + \beta\ket{100} + \beta{111}), \] and if we rewrite the first two subsystems in the Bell basis we obtain
\[ \ket{\Gamma}\ket{\phi^+} = \frac{1}{2}( \ket{\phi^+}\ket{\Gamma} + \ket{\phi^-}Z\ket{\Gamma} + \ket{\psi^+}X\ket{\Gamma} + \ket{\psi^-}XZ\ket{\Gamma}),\] so we see that conditioned on Alice’s state being a Bell state, Bob’s state is just a simple function of $\ket{\Gamma}$. Note that at this point nothing was done to the quantum system, so Bob’s state did not change in any way. If we calculate the reduce density matrix at his lab, we see that it is the maximally mixed state, which contains no information about $\ket{\Gamma}$ whatsoever.

Now, clearly we want Alice to measure her subsystems in the Bell basis to make progress. She does that, first applying an entangling operation to map the Bell states to the computational basis, and then she makes the measurement in the computational basis.3 After the entangling operation, the state is
\[ \frac{1}{2}( \ket{00}\ket{\Gamma} + \ket{01}Z\ket{\Gamma} + \ket{10}X\ket{\Gamma} + \ket{11}XZ\ket{\Gamma}),\] and making a measurement in the computational basis — for now modelled in a coherent way — and storing the result in two extra qubits results in the state
\[ \frac{1}{2}( \ket{00}\ket{00}\ket{\Gamma} + \ket{01}\ket{01}Z\ket{\Gamma} + \ket{10}\ket{10}X\ket{\Gamma} + \ket{11}\ket{11}XZ\ket{\Gamma}).\] Now something was done to this state, but still there is no information at Bob’s: his reduced density matrix is still the maximally mixed state. Looking at this entangled state, though, we see that if Bob applies the operations $\mathbb{I}$, $X$, $Z$, or $ZX$ to his qubit conditioned on the measurement result he will extract $\ket{\Gamma}$ from it. So Alice simply sends the qubits with the measurement result to Bob, who uses it to get $\ket{\Gamma}$ in his side, the teleportation protocol is over, and Alice and Bob lived happily ever after. Nothing faster than light happened, and the information from Alice to Bob clearly travelled through the qubits with the measurement results. The interesting thing we saw was that by expending one $\ket{\phi^+}$ and by sending two classical bits we can transmit one quantum bit. Everything ok?

No, no, no, no, no!, you complain. What was this deal about modelling a measurement coherently? This makes no sense, measurements must by definition cause lots of decoherence! Indeed, we’re getting there. Now with decoherence, the state after the measurement in the computational basis is \[ \frac{1}{2}( \ket{E_{00}}\ket{00}\ket{00}\ket{\Gamma} + \ket{E_{01}}\ket{01}\ket{01}Z\ket{\Gamma} + \ket{E_{10}}\ket{10}\ket{10}X\ket{\Gamma} + \ket{E_{11}}\ket{11}\ket{11}XZ\ket{\Gamma}),\] where $\ket{E_{ij}}$ is the state of the environment, labelled according to the result of the measurement. You see that there is no collapse of the wavefunction4: in particular Bob’s state is in the same entangled superposition as before, and his reduced density matrix is still the maximally mixed state. Moreover, as any physical process, decoherence spreads at most as fast as the speed of light, so even after Alice has been engulfed by the decoherence and has obtained a definite measurement result, Bob will still for some time remain unaffected by it, with the state still being adequately described by the above superposition. Only after a relativity-respecting time interval he will become engulfed as well, coherence will be killed, and the state relative to him and Alice will be adequately described by (e.g.) \[ \ket{E_{10}}\ket{10}\ket{10}X\ket{\Gamma}.\] Now we are in the situation people usually describe: his qubit is in a definite state, and he merely does not know which is it. Alice then sends him the measurement result — 10 — via the Internet, from which he deduces that he needs to apply operation $X$ to recover $\ket{\Gamma}$, and now the teleportation protocol is truly over.

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