# Teleportation and boosted observer, special relativity

According to relativistic quantum information, entanglement, purity of states are not Lorentz invariant.

Here I check such a problem. If we have two coordinate systems $R_A,R_B$ with relative velocity $v$ and two observers Alice/Bob fixed with $R_A,R_B$ respectively, then we can carry out two different teleportation experiments.

1. We carry out the experiment between Alice and Bob. This means we send a pair of MES (maximally entangled state) particles to Alice and Bob and teleport a single qubit from Alice to Bob. It's known that due to the boost between Alice and Bob, the entanglement between their shared entangled particle will not be a MES so the teleportation will be achieved with error. See "Relativistic quantum information and time machines"

2. Alice will carry out the experiment in a lab fixed w.r.t. $R_A$. Now we know for Alice the teleportation is perfect. According to the equivalent principle, Bob as an observer, he should also see a perfect teleportation in Alice's lab.

My questions are:

(1) Will the two entangled particles used by Alice also be MES for Bob? (Seems so since otherwise the teleportation can not be achieved perfectly). But why the boost of Bob does not affect the entanglement? You can say that since a boost is generated by something like $I_AX_B-X_AI_B$, so this is something like two SL operations on the two entangled qubits and this operation does not affect a MES like $00+11$. But this is strange for me since you can easily see the boost seems only work on A PAIR OF QUBITs. How will it work on a single qubit or 3 qubits?

(2) Generally even both Alice and Bob admit a perfect teleportation achieved, they will not agree on the STATE of the teleported qubit since a pure state for Alice can be a mixed state for Bob. So if the state of a single qubit is affected by Bob's boost, why the MES is not affected by Bob's boost?

(3) If the 2 qubit MES is not affected by Bob's boost, how about a 3-qubit GHZ or a W state? Or a pair of maximally entangled qutrits?

(4) Boost will not change vacuum since vacuum can be regarded as full of entangled qubits. Maybe boost works on fermions well since fermions appear in pairs. But how about Bosons? Even fermions are paired, but why boost works on 'this pair' instead of 'that pair'?