Friday, 13 June 2014

How to teleport a qubit

One of the fascinating things we can do with quantum entanglement is a scheme called quantum teleportation. In the original proposal by Charlie Bennett, Gilles Brassar, Claude Crepeau, Richard Jozsa, Asher Peres and Bill Wootters, it describes a way to transmit an arbitrary quantum state between two parties who may be far apart, using only a Bell state shared between the two parties, a few qubit operations that each party can perform independently, and two bits of information that can be communicated by one party to the other.

Suppose Alice and Bob are in separate locations but they share a pair of electrons that are in the entangled state

|E) = (|u,u) + |d,d)) / sqrt(2)

where as usual |u) denotes the state of an electron having its spin pointing in the up-direction, |d) denotes that with spin in the down-direction, and |u,u) refer to the state of the first and second electrons, respectively. Let's say that Bob has the first electron on his side and Alice has the second electron on her side. 

Alice also possesses a third electron in the state

|q) = a |u) + b |d)

and she wants Bob to obtain this state. If Alice does not know what the value of a and b precisely, she can not clone the state and send a copy to Bob. However, since Alice and Bob have shared entanglement, it is possible to transfer the state of this electron into Bob's electron using teleportation, which is shown in the figure below. 

A sketch of how to teleport a qubit from Alice to Bob using an entangled state. Alice applies CNOT to the 2 electrons on her side and then proceeds to measure one of them in the  |+),|-) basis while the other one in the |u),|d) basis. She tells Bob the 2 outcomes she gets. If Alice got a |d), Bob applies an up-down flipper to his electron. If Alice got a |-), Bob applies a plus-minus flipper to his electron. After performing the required operations, the state of Bob's electron will be |q).

We will now describe the method in detail.  First, we note that the combined state of the three electrons can be written as

|E, q) = a |u,u,u) + a |d,d,u) + b |u,u,d) + b |d,d,d).

Written this way, Bob holds the first qubit while Alice holds the second and third qubits. The first thing she does is perform a 2-qubit operation known as CNOT, or controlled NOT gate. The figure below illustrates how CNOT operates. 

The controlled-NOT gate. For each box shown above, the top qubit controls whether the spin of the bottom qubit is flipped or not. More specifically, spin up is flipped to down and vice-versa only when the top qubit is spin down, as seen in the two boxes on the right.

For the three electrons we have, it will flip the third qubit from |u) to |d), or vice-versa,  if the second qubit is in the |d) state, and does nothing to the third qubit if the second qubit is in the |u) state. This means that after applying CNOT(2,3) to the 2nd and 3rd electrons, the combined state of the electrons becomes

CNOT(2,3) |E,q) =  a |u,u,u) + a |d,d,d) + b |u,u,d) + b |d,d,u).

Now Alice is ready to perform the following steps: 

(a) She measures the spin of the 3rd electron in the up-down basis. Because she starts with

CNOT(2,3) |E,q) = [a |u,u) + b |d,d)] |u) +  [a |d,d) + b |u,u) ] |d),

we see that if she gets outcome |u), the state of the first 2 qubits will be a|u,u) + b|d,d).
On the other hand, if she gets outcome |d) the state of the first 2 qubits will be a|d,d) + b|u,u).

(b) She measures the spin of the 2nd electron in the plus-minus basis. In this case, it is useful to note that 

a |u,u) + b |d,d) = [ a |u,+) + b|d, +) + a |u,-) - b|d,-) ] / sqrt(2),
a |d,d) + b |u,u) = [ a |d,+) + b|u, +) + a |d,-) - b|u,-) ] / sqrt(2),

where we used |u) = [|+) + |-)]/sqrt(2) and |d) = [|+) - |-)]/sqrt(2). Thus, state the first qubit ends up is one of four possibilities: 

a|u) + b|d),      a|u) - b|d),       a|d) + b|u),      a|d) - b|u).

(c) The last thing Alce needs to do is tell Bob the 2 outcomes she obtained from the measurements. So we need her to have access to a phone or email or a fax machine. Bob will not need to say anything to Alice so her means of communication can be something that works just one way, from Alice to Bob.

All this time,Bob just waits to hear from Alice about her measurement results. The results tell him what sort of corrections he needs to perform so that his electron ends up in the state |q). In particular, there are 2 kinds of corrections he might need, which uses 2 different types of flippers:

(i) If Alice reports an outcome |d), Bob applies an up-down flipper to his qubit.

(ii) If Alice reports an outcome |-), Bob applies a plus-minus flipper to his qubit, that is, a flipper that flips state |+) to |-), and vice-versa.

If he gets both of those outcomes from Alice, he applies both flippers, the order of which is not important. If he gets neither of those outcomes, he does nothing.

Finally, after doing corrections, Bob's qubit will be in the state

a|u) + b|d) = |q),

which originally was the state of the third qubit.

A summary of what happens when teleporting a qubit (after CNOT is used) is summarized below:

There are a few important remarks that need to be said:

1. The state of the qubit is arbitrary since the operations Alice and Bob need to perform do not depend on the value of a and b.

2. Teleportation does not make a second copy of the state. Because Alice has to measure the third electron in the up-down basis, at the end of the scheme, the original qubit would now be in either state |u) or |d).

3. Teleportation does not allow us to transmit information faster than the speed of light. Note that before Alice tells Bob about the measurement outcomes, there are 4 possible states for Bob's electron. As far as he is concerned, these four states are equally likely, without the extra information provided by Alice. But Alice has to send this information by some other method whose speed is most certainly limited by laws of relativity.

4. Teleportation does not work in the way depicted in science fiction accounts like Star Trek, where actual material is transferred instantaneously from one location to another. Here, only the quantum state is transferred to Bob, the electron that originally carried that state stays on Alice side.

5. Still, quantum teleportation seems to suggest that it may be possible in principle to transfer the quantum states of all the particles that make up an object and "reconstruct" it in another location. This has been achieved with states of a few photons, or more recently, with electrons trapped in diamonds. The problem with bigger objects is the operations you need to perform become increasingly complex as more and more particles are involved, not to mention you need in complete detail the quantum state of all particles if you want an accurate "reproduction".

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