Quantum Information for Quantum Cats (17)

Quantum Teleport : Instanteneous Port of State

In what state Bob would receive the arrow B from Eve? If you have followed the preceding arguments, you have the right answer: That is, there is no answer to such ill-posed question. Since the arrow he has at his hand is entangled with the rest of the system, i.e. with the arrows A and E which is at Alice's hand, Bob's arrow B alone cannot be assigned a unique state until Alice makes the measurement on arrows A and E. So Bob asks Alice to do the measurement, which, we assume, Alice choose to do with ultra high-tech "concurrent two arrow detector" which utilizes Bell basis. (Note that if Alice simply measures A alone, it is of course oin | \theta >A, and E and B are still in entangled state.) The result is that Alice gets one of the states | X+ >, | X- >, | Y+ >, | Y- > , and correspondingly, Bob has his arrow in the states

respectively. The close examination of the first case reveals

which means that if Alice finds A and E in the state | X+ >AE, Bob's arrow is determined to be in the state | \theta >B ! At the instant of the measurement, Alice's | \theta >A exists no more and | \theta >B apears in front of Bob. This is quantum teleportation.

An crucial remark is due now. The transport of the state | \theta > is done at the instase of Alice's observation, but this happens only in 25 % of the time as indicated by the factor 1/2 in the above formula. For Bob to be sure of his state being exactly the copy of Alice's he has to know that Alice hase observed the state AE<X+| in her measurement. Do something important with the state at hand without that assurence would be too risky! So for the teleportation in true sense of the word is achived only with that information, which has to be communicated through classical means. Since classical communication is limited by the speed of light, the utility of the quantum teleportation is also limited by it.

A further remark is that even when Alice finds the arrows A and E in other state than AE<X+| , namely in states AE<X- | 、AE<Y+| 、AE<Y- |, Bob can transform the state of B at his hand into | \theta >B by a single unitary transformation. So the probability of the teleportation is razed from 1/4 to 1. Of course this is achieved only with the information from Alice on her observation. And the teleport is limited by the limit of classical communication as before.

The teleportation has been experimentally tested just some five years ago. The key has been the creation and detection of Bell states.

On light-speed limit of quantum telepotation

As explained above, state teleportation with 100% success rate requires a suppliment by classical communication, and we sometimes encounter such statement like "quantum teleport itself is instanteneous, but sadly, it needs to be accompanied by classical communication whose spped is limited by spped of light, which therefore becomes the speed limit of teleportation itself - what a shame!" But, on calm reflection, this is obviously a wrong view. The speed limit of classical communication is a direct result of the fundamental assumption of special relativity. This requirement has to be imposed both on classical communication AND on quantum communication. The instanteneous quantum state teleportation is an artifact of our non-relativistic treetment of quantum mechanics. So above impression is simply a result of our inconsistent treetment of classical and quantum part of the process. This suggests that the full theory of quantum teleportation needs to be based on the relativistic quantum mevhanics, which has to be based rather on Dirac equation than Schoedinger equation, and possibly on full relativistic quantum field theory. At this point, there is no relativistic quantum infromation theory, and this should be one of the chalenge to young aspitrant of this field.