The best you could hope for is a machine that reads the exact location of every atom in your body, as well as it’s chemical relationship to every nearby atom, then sends that blue print to another machine that builds a new body one atom at a time. Not only is every step of this a horrifying technical problem, but for “Uncertainty Principle” reasons is almost certainly impossible. Also, it doesn’t seem to be what they do on Star Trek.
Back in the 90′s it was shown that if two people share a pair of entangled particles, then they can use them to send 1 qbit instead of 2 bits, or they can send 2 bits instead of 1 qbit. The former is called “superdense coding” and the latter is called “quantum teleportation“. The guy who named it “quantum teleportation” and not the “2 bits = 1 qbit theorem” is a jerk.
The discovery, and more importantly the subsequent naming, of quantum teleportation lead to a new (false) hope that entire objects might be teleported. In fact the only thing being “teleported” is information about the particle involved (not the particle itself).
What follows is answer gravy (more complex):
For example: the polarization of a photon is a combination of both vertical and horizontal polarization. If you were to measure the polarization you would get a result of “vertical” or “horizontal”, but never the true combination of both. So in the process of measurement you lose some (quantum) information. Quantum teleportation (stupid name) allows you to get around this problem.
The exact technique is a little confusing, so if you intend to read the wikipedia article it might help to understand “classical teleportation” first. This is an experiment that also doubles as a party trick (nerd parties). The following technique will teleport the (classical) state of Alice’s coin A to Bob’s coin C.
1) Get 3 coins, A, B, and C.
2) Get B and C to be the same (heads or tails) without knowing which they both are. Maybe paper-clip them together, then flip them both without looking, or just get some one else to do this set up. B and C are now entangled.
3) Alice keeps coins A and B, and Bob takes coin C as far away as he likes.
4) Flip A and B together (like in step 2) and look at them. There is no way of telling what the original states of either A or B were, but you can tell if they were the same or different.
5) If A and B were not the same, then Alice tells Bob to flip C over. If A and B were the same, then Alice tells Bob to leave his coin alone. The idea is, since B=C: if A=B then A=C, and if A≠B, then A≠C (so C should be turned over to match A).
Without ever determining the exact state of any coin, but only comparing two of them, Alice and Bob have teleported the state of A to C. If A has a 77% chance of being heads and 23% chance of being tails (weird coin), then C will now also have a 77% chance of being heads and 23% chance of being tails. The information about A, including the probabilies on A, have been transfered to C. You’ll notice that the actual coin was never teleported, distance is irrelevant, the original state of A is destroyed, and the entire process is not even a little mysterious.