BTW; if we were theoretically able to entangle our bodies so we could "teleport", it might be very different from Trek teleporters.
Since you can't entangle at any distance, it might work by us stepping into the "entanglement machine" black box, and then stepping out. Inside that block box, there are entangled particles. That box will be transported to whereever we want to go, and when it's there, the parts can be "released". So basically, instead of sitting inside the plane, w would simply send two briefcases - one containing our stuff, and the other containing our quantum entangled particles that become us later.
But I think the part where we step out of the machine is where the whole process is failing... If I understand QE correctly, if you collapse the wave function of one partner in an entanglement, you do so with the other. But I bet that if you step out f the box, you will under go a lot of wave function collapses, so it's not actually possibe to work that way.