Teleportation is cool isn’t it? Disappearing from one place and appearing in another is in fact a very powerful super power. We have seen this in movies like X-men and similar science fiction movies. As this heading says, it isn’t just a fantasy anymore. In fact scientists have actually conducted experiments and successfully did teleportation. Wait, there is a catch, it’s not people or objects we teleported, instead a quantum state. Yes, here we are looking at how we can actually teleport a quantum state. Before that let’s see what is quantum state and quantum mechanics.
Quantum mechanics is a branch of physics which studies how subatomic particles behave, which is completely different from what we have seen. Suppose an electron is trapped in a cavity and it doesn’t have that much energy to go out of it, but still quantum mechanics says that electrons still have a probability to escape from it. This phenomenon is called quantum tunneling. Similarly there are other cool things that subatomic particles can do such as quantum entanglement, superposition and much more. But this is all a topic for another discussion, Now let’s stick on to teleportation.
Like our computers, there are something called quantum computers, which works according to the law of quantum mechanics. Computers are systems which process information. Our computers (Classical computers) store information in binary bits. We can think of this as an on-off state of a switch. Switch in on condition representing information 1 and off means 0 (on -1 off -0 picture of switch).
Story is different when it comes to quantum computers. Here information is stored in qubit, unlike bits the value of qubit can also be somewhere between zero and one. Mathematically a qubit or a quantum state |> can be represented as
|> = a|0> + b|1>
This means before we make a measurement, we don’t know whether the qubit or quantum state is in the on or in the off state-it’s in a superposition of both. Once we measure, the state collapses to either on or off.
It is the information stored in this state we are teleporting. Before we go into explaining teleportation, we need to understand what happens when we add more qubits. In other words, we need to understand the multi qubit state. Consider a two qubit state, we can think of this as two switches. This has a total of 4 different combinations.
|> = a|00> + b|01> + c|10> + d|11>
If 0 represents off and 1 represents on, we can interpret the above equation as – off,off + off,on + on,off + on,on. Also a,b,c,d gives probability corresponds to qubit when we try to measure. Now lets substitute b=c=0 and a=b=1, in the above equation to obtain a bell state. This state will be studied when we discuss entanglement.
|> = 12|00> + |11>
Bell state
Now we are in a position to discuss quantum entanglement. This phenomenon is perhaps most controversial in the whole area of quantum mechanics, even Einstein called it spooky action at a distance. Quantum entanglement is a phenomenon where two or more states are correlated such that observing one will have an instant effect on others, even if it is light years away. We may doubt this would violate the fact that no information can travel more than the speed of light. Suppose Alice and Bob both picked a black opaque box from two. One box contains a gold coin, the other has silver. They both dont know what is inside their box, unless they open it. Now let’s send Alice in a spaceship about 10 light years away. Before opening they have no way to figure out which box contains the gold coin. Now if Bob opens the box, Bob will be able to know what is inside Alice’s box. At an instant Bob will be able to get that information and that is totally fine. ‘Spookiness’ comes only in the context of quantum mechanics. Instead of gold and silver coins, if those boxes contain an entangled quantum state with one state – spin up and other spin down, then the story is different. Quantum mechanics tells that before measuring there is no definite answer for spin up or spin down. In both boxes the state will be a combination of spin up and down. Now when Alice measures one box, at that instant the other state in the other box will also collapse. This is the spooky action at a distance that Einstein had mentioned. Now let’s discuss Bell state again.
|> = 12|00> + |11>
We know this is a 2 qubit state, so let’s assume first qubit taken by Alice and second by Bob. Since both qubits are entangled, they have an entangled state like the bell state. So, if Alice measured 0, then Alice is sure that the above state has collapsed to |00> also Alice would know that Bob’s state should be 0. Now if Alice measured 1 then state has collapsed to |11>. Bob’s state will be 1. So this is the entanglement, measurement of qubit will give us the information of others.
Finally we are in the position to talk about teleportation ! Quantum teleportation is the method of teleporting a quantum state using an entangled pair of qubits. Lets see this in action. For the sake of completeness we are going through all mathematical steps but it’s fine even if you didn’t follow maths.
Suppose Alice and Bob met somewhere in the universe and generated an entangled bell pair and took each with them to very far away. Alice has a third qubit |> which is to be teleported to Bob.
|> = |0> + |1>
Altogether three qubits are in the state
|>1200> + |11> = (|0> + |1>)12|00> + |11>
Above what we did was not multiplication, it’s called outer product or tensor product. We don’t want to worry about all the intricacies of mathematics, conceptually the above step means we are combining the qubit we want to teleport with the bell pair. Note that |0>|00> = |000> in tensor product. Therefor above expression can simplified to
12|000> + |011> |100> |111>
Just a reminder so that we are all on track – first two qubits are with Alice and third is with Bob. In Order to perform teleportation Alice should perform some operations to qubits she have(first and second). Precisely, Alice should perform the CNOT followed by Hardmand operator. These calculations are not done here, it can be a topic of another blog. After these operations the above state will be
12(|00>(|0> + |1>) + |01>(|1> + |0>) + |10>(|0> – |1>) + |11>(|1> – |0>))
Now is the crucial part, If Alice measures her two qubits. She gets 00,01,10,11 with 14 probability. If she obtained 00 then state with Bob will be |0> + |1> or if she obtained 01 Bob’s will be |1> + |0> , similarly the other two. Important part is Bob doesn’t know the final state to which it collapsed so Alice should send 2 bit information about her state. Then Bob can recreate the state.
In this way Alice successfully teleported a quantum state to Bob. It is important to note that
teleportation does not violate the maximum speed limit of light because in the process Alice has to send two bit information which travels at speed of light. Still this is very fascinating because we could in principle send quantum information or state just through quantum entanglement.
Even after I understood the whole concept about quantum teleportation, this idea still fascinates me. Not just this, science always fascinates me with its beauty and elegance. However quantum teleportation fascinates me a bit more, maybe because teleportation has been one of the wildest science fiction I ever encountered. Also, if we just sit back and go through what we have discussed. For the most part of our way, we did a lot of mathematical manipulations and got a result that we interpreted as teleportation and this scientists made practical in the lab. We actually performed this teleportation and varied. A very elegant example of how mathematics describes our reality.
Reference
- Wong, Thomas G. Introduction to Classical and Quantum Computing. Cambridge University Press, 2022.
- Offord, Wilf. Quantum Teleportation and Entanglement: How Do They Work? Springer, 2023.
- Zettili, Nouredine. Quantum Mechanics: Concepts and Applications. 2nd ed., Wiley, 2009.
