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 (+) <i>. So plus comes from here and<
/i> then you have a(0) for his qubit. And</i>
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.