Okay, so in this video and the next video, we finally talk about quantum
teleportation. I'm sure you've heard about quantum teleportation. You've seen, seen
it in Star Trek. How to teleport an object from one place to another. This is, and
quantum teleportation is a rough analogy with that so you will see that there are
similarities. There are, there are very big differences but what, what is the
basic idea in quantum teleportation? So, you know that it's impossible to clone a
qubit but in quantum teleportation you want to, you want to transport a qubit
from one location to another. And well, one way to transport it is if you have a
quantum channel. If you could carry the quantum bit from one place to the other
but it turns out that there is other way of transporting it from one place to
another which is called teleportation. So lets see what that means. Well first,
here, here is the set up. Okay, so you have two, let's say there are two
physicists. Alice and Bob, experimentalists and Alice has created a
state which it took her a long time to create it. It's a, it's a very interesting
state. It, it's some superposition, a(0) + b(1). She doesn't quite know what, what
alpha and beta are but it's just the output of some process that she designed.
And now she wants to, she wants to run this through this quantum state through
some, some other apparatus which she doesn't have but Bob has in his lab. But
Bob is, Bob's lab is across campus at the other end of campus. How should she get a
qubit from her lab to his lab? Maybe it's, it's all the way across the city. How, how
should she transport this qubit from one location to another? Okay so, one thing
she could do is she could try to send it from her lab to his lab but let's say that
these two labs are very distant and it's very hard to transport it from one place
to the other. So what else could she do? Well of course she, she, she cannot really
make a copy of her qubit but what she might try to do is try to figure out.
Alpha and beta. And then give that inform ation to both but of course alpha and beta
are complex numbers. Not only do they require, you know she, what, what Alice
has to do is figure out to sufficient precision, what alpha and beta are. But
what you can do is measure the state and she doesn't, you know if she measures the
state, she does not really get alpha and beta. What she sees is zero with
probability a^2 and one with probability b^2. Moreover, as soon as she does the
measurement the new state is whatever she saw, and so she can't repeat the
experiment. So if she has only this one copy of the state, she really cannot
figure out what alpha and beta are. So what is she to do? Okay, so this is where
quantum teleportation comes in. So let's say that Alice and Bob were ready for this
eventuality. So once upon a time, they set up, you know they shared a Bell pair with
each other. So meaning that they have these two qubits. Alice has a qubit and
Bob has a qubit. And they happen to be in the state (00) + (eleven) in the Bell
State. Of course this, this Bell State has nothing to do with, with the qubit that
Alice wants to transport both. So, quantum teleportation is this, remarkable protocol
by which Alice can do something with her, with her two qubits. The one that she
wants to transport and the qubit out of the Bell pair. She does something to these
two, she measures these two. So what, what Alice does is she takes these two qubits
and she performs a measurement. Okay, what's the result of this measurement?
Well, you know she is measuring two qubits, the result is she gets two bits of
information. She gets two bits, Lets call them b1 and b2. These are classical bits.
Right its 00, 01, ten or eleven. So now she picks up the phone and calls up Bob
and she transmits these two bits across to Bob by phone. At this point you might
think, Alice what did you do? You've destroyed your qubit. Well, now what Bob
does is based on what, what Alice told him about b1 and b2. He preforms some single
qubit gates or he actually performs two gates on his qubit. Which gate? Really
depends o n what b1 and b2 are. And lo and behold, what happens is, after performing,
so, so he performs some unitary rotation, it's going to be a simple unitary rotation
depending upon b1 and b2 and what happens? Well, what happens is magically this qubit
gets transformed to being in the state a(0) + b(1). So this quantum state which
got destroyed at Alice's end suddenly reappears on Bob's end. Okay? So, in what
sense is it like teleportation? Well, it's like teleportation in the sense, you now
on Star Trek, in the sense that well, you had this entity here on Alice's end and
you perform a certain protocol. This entity disappear and then you do something
on Bob's end and it reappears on that endso it's like teleportation. In what
sense is it not like teleportation on Star Trek? Well it's not like that
teleportation because you have, it's not instantaneous. You have to, Alice does her
measurement at hand, she destroys her qubit, and she is left with just this
classical information. And now, she has to pick up the phone. She has to send this
information across to Bob and it's only after he gets this classical information
that he can reconstruct the quantum information that Alice started with. Okay,
so in this and the next video we are going to understand how this protocol works.
Okay, and to make this easy, I am going to break this protocol down into two parts.
First, we are going to start by making an unrealistic assumption. I am going to
assume, okay so, on top here is Alice's lab so she has a qubit in the state a(0) +
b(1). On bottom here is Bob's initial qubit which is initially in the state
zero. I Am going to assume that Alice was able to perform a gate between her qubit
and Bob's qubit. This is completely unrealistic but it'll help us understand
how to proceed with teleportation. Okay, so she performs a CNOT, what happens? Well
obviously the, the state of the two qubits after the CNOT becomes a(00) + b(11). Now
what we'd like is, we'd like somehow to do something on Alice's end so that Bob gets
left with a(0) + b(1). An d Alice gets left with just a measured qubit. Okay, so
what should Alice do? Well, she could try measuring her qubit. I guess if she
measures in the standard basis, what happens? Well, if she measures and the
outcome is zero, then the new state of the qubits is just 00. And whats the state of
Bob's qubit? Well its just, its just that, it's zero. On the other hand is, is she
sees, sees a one, the state of his qubit, its just that, its one, So alpha and beta
just disappear if you the do a measurement in the standard basis. So measuring the
first qubit in the standard basis was a terrible idea. Okay, so here is an idea,
why not use a different basis? So what basis should we use? So let's try the
plus, minus basis. So now, what we are doing is, what Alice does is she measures
the first qubit in the plus, minus basis and we want to know what's the result of
this measurement. So let's write out her state, let's write out the state a(00) +
b(11) and let's rewrite her part of the state in the plus, minus basis. So it's
a(0) is one / square root two(+) + one / square root two (-) and then of course you
have his qubit which is in the state (zero) + b(1), one is (one / square root
two (+),, - one / square root two (-)) and then his qubit is in the state one and so
now what we want to do is we want to collect them, so this is one / square root
two (+) . So plus comes from here and< /i> then you have a(0) for his qubit. And
then you have + and you have + b(1). I am just, I am just factoring things out and
then you have - here, one / square root two (-) a(0). And for this you have -b(1).
Okay, so that's the state of the two qubits. If you write the first qubit in
the plus, minus basis in the second qubit you leave alone, as zero. So now what
happens if you do a measurement? So, if Alice does a measurement well, how come
either plus or minus if the outcome is, is plus? Then the new state, well her qubit
is being in the state plus but also his qubit will be in the state a(0) + b(1).
That's great because thats exactly the qubit that Alice was trying to send to Bob
. And now he has got it, so thats great. But on the other hand if the outcome is
minus. Then the state of his qubit is a(0) - b(1). So now what should he do? Well
Alice can pick up the phone and she can call him and say, in this first case she
says, the outcome is plus, you've got a bit, qubit. In the second case, she calls
him up and says sorry, the outcome was minus but then Bob says you know that's
not a problem because I have a phase flip gate sitting around. Phase flip gate is
the z gate and what it does is it, it changes the freeze in front of the, of one
to minus. And so that's going to take this qubit and restore it to look exactly like
Alice's qubit. Okay so, so what did we, what did we discover? We discovered that
if Alice was able to perform a CNOT from her qubit to Bob's qubit then she can
create this special state a(00) + b(11). And then, if Alice does a measurement in
the Hadamard basis, in the plus, minus basis, and the sign basis then, and she
communicates that to Bob, he can apply a correction or not depending upon what the
outcome was and he can recover Alice's cubit. So, the challenge is, how to create
the state without the CNOT and that's what we'll see in the next video.