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I have questions regarding quantum teleportation, which keep confusing me.

  1. Suppose Alice and Bob are in the same inertial frame K, and at time t (in K) Alice teleports a quantum state to Bob. What I always hear is that this means that at time t, Bob has then got one of four states, although he does not yet know exactly which one of the four. Is this true?

  2. Now, what if Alice and Bob are both moving along the x-axis of K, in the same direction, both with the same speed v? If Alice does her part of the protocol at time t (again, as seen in K), then if Bob is behind Alice (w.r.t. their common direction of movement in K), he must get the quantum state before t in K, due to special relativity (as calculated by the Lorentz transformation, assuming his quantum state "arrives" at the same time as Alice sends it, in the inertial frame where both of them are at rest). This sounds weird, as if the cause had happened after the effect.

  3. And what if Alice and Bob are not in the same inertial frame? Then the point in time Alice executes her part in her inertial frame does not correspond to any single point in time in Bob’s inertial frame. So what can we say about the arrival time of the quantum state to Bob?

Note: Cross-posted in quantumcomputing.SE.

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The relativity of simultaneity ensures that for non-causal events, the order in which they happen can depend on the reference frame you choose.

However, in the case of teleportation, if Bob chooses the correct unitary transformation, then he can only do so having received classical information from Alice regarding the outcome of Alice's measurement. These events are well ordered in all reference frames. Every frame sees Alice perform the measurement, send a signal to Bob and then Bob performs his unitary in that order.

The problem with your way of looking at this is because the state's "arrival time" to Bob is not a well defined event. Somebody (either Alice or Bob) has to actually do a measurement/unitary transformation and then you have a well defined event that can be analyzed using any reference frame.

See also this related answer in the case of entanglement

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  • $\begingroup$ Thanks for the answer, I have to think about it more. The related answer makes perfect sense. $\endgroup$
    – Tamás V
    Commented Feb 10, 2019 at 21:01
  • $\begingroup$ I think the "arrival time" is well defined, via the following experiment: in K, let Alice teleport |0> at time t, and let Bob measure at time t+dt in the computational basis. Let them repeat this 100 times, where dt>0 is a small constant. Then they run another 100 times using -dt (i.e. Bob will measure before t). Using Alices's records, let them later select those cases when Bob really got |0>: they will see that during the first 100 runs Bob always got |0> as measurement result in those cases, but not during the second 100 runs. So they conclude that the "arrival time" must have been at dt=0. $\endgroup$
    – Tamás V
    Commented Feb 10, 2019 at 21:08
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    $\begingroup$ No, the experiment for the "arrival time" would not work. Answer accepted, thanks. $\endgroup$
    – Tamás V
    Commented Feb 10, 2019 at 21:39
  • $\begingroup$ My only concern is that the (quantum computing) books I read so far talk about immediate effect on Bob's qubit. But why if there is no such thing as "arrival time"? $\endgroup$
    – Tamás V
    Commented Feb 10, 2019 at 21:49
  • $\begingroup$ I agree, the notion that the effect is immediate can be misleading. Hence the link to the related answer where things are a bit easier to understand. Although you might say Alice performs a measurement and Bob's qubit then "immediately collapses" to a correlated state, this way of thinking is not ideal as it implies causation which is not really present. $\endgroup$
    – Navneeth Ramakrishnan
    Commented Feb 10, 2019 at 22:39

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