Okay, so now in this video we are going to finally complete the Quantum Teleportation
Protocol. So in the last video, we saw how to, how Alice could teleport a arbitrary
qubit α0 + β1 to Bob. If she could do this unrealistic thing of performing a
CNOT with her qubit as a control bit and Bob's qubit as a target bit. So, if she
does that then she can entangle the two qubits into this rectangle state, α0(0) +
β1(1). And then, Alice measures her qubit in the sign basis, and if she gets out
from plus then Bob's qubit is in the, in the desired state alpha zero + beta one.
But if she gets out to minus, then Bob's qubit is in a slightly related cube state
alpha zero - beta one. So, well, Alice calls up Bob and tells him which outcome
she got and if the outcome was plus Bob, Bob just uses his qubit as is, as is. If
he got a minus, then he performs a phase-flip on his, on his qubit using the
Z gate, the phase-flip gate. Okay. So, now, what, what this does is, it reduces
our task of teleportation to somehow creating this entangled state alpha zero,
zero plus beta one but without the benefit of a CNOT gate. And that's what we are
going to do and once we do that, we'll have our teleportation protocol completed.
Okay. So, so what's our setup now? Our setup is, that Alice has a qubit, alpha
zero plus beta one, which we, let's, let's think of you know, since we are going to
be drawing a circuit, let's think of why that can, that holds this qubit. But now,
we are also going to assume that Alice and Bob cannot, you know, they are too far
apart to perform a, to perform a gate or anything else. But we are going to assume
that sometime in the past, they managed to talk to each other enough to create a bell
state, so they share a bell state. So Alice has this qubit, and Bob has that
qubit and these two qubits are entangled with each other and they are in the states
zero, zero plus one, one. So we think of time as going from left to right so there
are these three wires now that are carrying these three qubits. Two of them
entangled with eac h other and the third one which is the one that Alice is started
with, is the qubit that she wants to eventually teleport to Bob. Okay. So what
we want to do is, we want to use this bell state to effectively apply a CNOT gate
remotely. So how do we do this? Well, let's start by well, Alice performs a CNOT
from her qubit out you know, the unknown qubit, the one she wants to teleport to
her half of the bell state. Okay. So now, let's try to understand what the state of,
of the three qubits is after she does this CNOT. Now so first, what was the state
before she, she did this? Well, before she did this, it was clearly, α0 + β1
1/√2(00) + 1/√2(11). α/√2(000) + α/√2(011) + β/√2(100) +
β/√2(111). So that's the state of her qubits, of the, of the three qubits,
initially before the CNOT gate. Now what happens after the CNOT gate is applied? So
after we apply the CNOT, the CNOT is from the first qubit to the second. So, in
these first two cases, whether, whether first qubit is zero, everything remains
the same so you get α/√2(000) + α/√2(011). And then in these two cases
where the control bit is a one, the target bit flips, so you get β/√2(110) +
β/√2(101). Okay. So that's the state that her, of, of the three qubits after
the CNOT. Okay, so now let's, let's try to understand what happens if we measure this
middle qubit, what happens to the first two, first and the third qubits? Do we
magically end up in the desired state, alpha zero, zero plus beta one, one, the
entangled state that we want to get to. Okay, so lets, lets try to see what
happens. So, this was, this was the state of our, of our three qubits, I have just
rearranged the state a little bit. And, now, what, what we, what we wish to do is,
we wish to measure the middle qubit. So, when we measure the middle qubit, the
outcome is either, is either zero or one. Now if it happens to be zero, which is,
which is these red possibilities here, then the remaining state of the other two
qubits would be, zero, zero plus one, one with amplitud e alpha and beta. So, you
would get alpha zero, zero plus beta one, one. On the other hand, if the middle
qubit turns out to be one, if it's measured to be one, then the, then the
state of the remaining qubits, would be one zero, be, let's see. Alpha times zero,
one + beta times one, zero. Okay. So, at this point, Alice picks up her phone,
calls Bob and says, you know, I did the measurement. The result was either zero or
one. If the result is zero, then Bob can, Bob needs to do nothing because, because,
the, the two now share this, this entangled state that we wanted, Alpha
zero, zero plus beta one, one. On the other hand, if the result is one, then,
what Bob can do is, but he, he, you know, they share the state alpha zero one plus
beta one, zero. All he has to do is perform a bit flip on his bit, and this
will get transformed to alpha zero, zero plus beta one, one. So, meaning, Alice
makes a measurement on the second bit. She calls up Bob. If the, if the bit is zero,
Bob does nothing. If it's a one, then he performs a bit flip or an x gate on his
qubit. So now we have, we have reduced to the case that we knew how to solve. So we,
we are ready to see what the full quantum teleportation protocol is. So here's the
protocol. Alice starts with this qubit in the unknown state, Psi. Alice and Bob
share this bell state. So this is Alice's qubit, that's Bob's qubit and they're
entangled with each other. Now what Alice does is, she performs a CNOT from her
qubit to, to sorry, her unknown qubit to her part of the bell state. She measures,
she, she calls up, sends the results to Bob. If it's a one, he performs a bit
flip. Okay. So now, to deal with the first and third qubits, Alice wants to measure
her qubit in the, in the sign basis of the Hadamard basis so what she does is, she
first does a Hadamard transform, and then she measures. So she is measuring in the
plus-minus basis, she calls up Bob, tells him what the result of that is, if it's,
if the result is zero, Bob does nothing. If it's one, then he performs a phas
e-flip. And low and behold, his qubit is now transformed into Alice's original
qubit. In the meantime, Alice measured both her cubits. And so this quantum state
on these two qubits is completely destroyed. And so, Alice's qubit got
destroyed and later, after the phone call, the, the and, and, and these two, the x,
the potential x and Z gates, the, the, the quantum state magically materializes at