# Quantum teleportation with moving Alice and Bob

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?

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.