Friday, February 7, 2014

Quantum Teleportation

One of the important results in quantum mechanics is the teleportation protocol. There are several misconceptions about teleportation, fueled in part by Star Trek episodes.

For example, you cannot teleport faster than the speed of light. Quantum mechanics exhibits correlations above what can be expected by a causal local theory and an experiment here can obtain data instantaneously correlated with an experiment located at the other side of Milky Way. To verify the correlation however, one needs to physically travel from here to there and this can only be done at speeds below the speed of light. Quantum mechanics is non-local, but it does not permit violation of special theory of relativity. Just because one measures something over here it does not carry any instantaneous signals across the galaxy (this is usually called no-signaling).

Then, unlike in Star Trek, the teleported object does nor vanish. Instead, all its information is extracted and in the process the object gets destroyed. From the extracted information, the quantum state is reconstructed at the destination using local materials. In a way, it is like breaking up an apply pie to extract the baking instructions, and then using those instructions re-baking the pie at the destination using local ingredients. If you are teleported, you will die on the teleporting pad, and hopefully your complete atom state information is transmitted and used to make up an exact replica of yourself at the destination. A few mistakes in measuring and reconstruction and you will end up like someone from a Picasso painting-not an appealing prospect.

So what is the big deal then with quantum teleportation? Is there any difference between a fax machine and a quantum teleportation device? In the quantum world there are two basic barriers. First, in quantum mechanics a state cannot be copied (cloned). The no cloning theorem means that a classical fax machine is impossible to be implemented in the quantum realm. The second roadblock is that measuring the quantum state destroys (or collapses) what it is measured and you cannot access all of the quantum state information. So it was a very exciting time when the teleportation protocol was discovered. This protocol can enable future quantum computers to transmit quantum information. But how does it work? We can start with the mathematical description (see here ) but it is much more entertaining to repeat a funny story originally told by Charles Bennett, one of the discoverers of teleportation. I could not find a good reference for this story, so I will tell it from memory attempting to be as close as possible with the original story (I originally blogged about this at FQXi and I had a link there to Bennett’s story but the link is now broken).

Suppose there are two brothers Romulus and Remus who don’t know much about anything. When asked about any question they answer randomly, but they both give the same answer:

Teacher: What color is the grass?
Romulus: Pink ma’am.
Another Teacher in another room: What color is the grass?
Remus: Pink sir.

Now as the story goes, a murder was committed in Boston and the FBI wants to talk with the sole witness. They do not trust the local cops, and since the witness is still in a state of shock, they cannot transport him to Washington DC to be interviewed by FBI experts because they risk tampering the brittle witness’ state of mind. Fortunately Romulus happens to be in Washington DC and Remus in Boston. Then they ask Remus to spend time with the witness talking about any topics they want: the weather last weekend, the best movie showing right now, the stock market, etc. So Remus spends an hour with the witness and at the end of the hour, the witness said he hates Remus because he dislikes every single thing Remus likes. Moreover, the stress of the meeting has completely erased his recollection of the crime. Can the FBI agents in Washington DC have any chance to find out about the crime? Surprisingly the answer is yes. They will ask Romulus about the crime (he did not witness anything; he was in Washington DC all the time) and armed with the information about the outcome of the meeting between Remus and the witness, they reverse every single the answer Romulus provides them when asked about the crime and solve the case.

Nice story? Unbelievable?

Let’s say it again using quantum mechanics mathematics this time.

Let’s meet the key people:
| ψ > = α |0> + β |1>  ------ the witness state to be teleported from Boston to DC
| Φ+> = 1/sqrt(2) [|0> |0> + |1> |1>] – maximal entangled state -------Romulus-Remus pair

The total state is:

| Φ+>|ψ> = 1/sqrt(2) (0> |0> + |1> |1>) (α |0> + β |1>)

Let’s introduce 4 completely entangled states:
| Φ+> = 1/sqrt(2) [|0> |0> + |1> |1>]
| Φ-> =  1/sqrt(2) [|0> |0> - |1> |1>]
| Ψ+> = 1/sqrt(2) [|0> |1> + |1> |0>]
| Ψ-> =  1/sqrt(2) [|0> |1> - |1> |0>]

|0> |0> = 1/sqrt(2) [|Φ+> + | Φ->]
|0> |1> = 1/sqrt(2) [|Ψ+> + | Ψ->]
|1> |0> = 1/sqrt(2) [|Ψ+> -  | Ψ->]
|1> |1> = 1/sqrt(2) [|Φ+> - | Φ ->]

and therefore

| Φ+>|ψ> = 1/sqrt(2) (0> |0> + |1> |1>) (α |0> + β |1>) =

½( α |0>r |0>r|0>w  + β |0>r |0>r|1>w  + α |1>r |1>r|0>w  + β |1>r |1>r|1>w)

where the indices r,r,w represent Romulus, Remus, and the witness. Now we will swap Romulus to the right with witness to the left and after a bit of elementary algebra we get the expression above written in this form:

½ (|Φ+>rw(α |0>r + β |1>r) + |Φ->rw(α |0>r - β |1>r) + |Ψ+>rw(β |0>r + α |1>r) + |Ψ->rw(β |0>r - α |1>r))

This rewrite is the clever idea of teleportation.

The Boston meeting between Remus and the witness corresponds to a measurement for the Remus-witness part which will collapse the state to one of the 4 possible outcomes:

Consequently the state of Romulus in Washington DC is one of the four possibilities:

α |0>r + β |1>r
α |0>r - β |1>r
β |0>r + α |1>r
β |0>r - α |1>r

Then the measurement outcome is sent by classical means to Washington DC. (OK, the story is a bit more complex than the Romulus-Remus story above, in the real protocol there are 4 outcomes)

All that is left to do is to apply a local transformation (like reversing the answers from Romulus) in Washington DC to transform Romulus state to:

α |0> + β |1>

thus achieving the teleportation of the unknown original state from one place to the other. In the process we did not copy the state (the original state got destroyed) and the no-clone theorem was obeyed. Also we did not extract the coefficients α and β like in a classical faxing process. The memory of crime got teleported from the witness into the head of Romulus who for the purposes of the FBI investigation has become the witness. This was achieved by what is called quantum steering: measuring on one entangled particle changes remotely the state of its pair. This is instantaneous and can be done from one end of the galaxy to the other if you like, but in the absence of the right key to unlock the information it is useless. The key still has to travel slower than the speed of light and relativity is ultimately obeyed.  

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