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Okay. So now we are ready to talk about
entanglement which plays a central role in

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quantum computation. So, let's come back
to a two qubits system. And, you know,

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which let's say again, consists of,
consists of atomic qubits where our first

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qubit happens to be in the state
superposition α0|0> + α1|1>. And the

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second qubit happens to be in the
superposition, β0|0> + β1|1>. So now we

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could ask, what's the state of the two
qubit system? Well, the answer is very

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simple. What you do is, you just take a
product of these, according to this rule.

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It's sort of the obvious rules. So you,
the amplitude of |00> is, is just

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α0β0|00>. The amplitude of |01> is just
α0β1|01>. The amplitude of |10> is

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α1β0|10>. And the amplitude of |11> is
just α1β1|11>. So for example, suppose we,

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you know the, the, the states of the two
qubits where, the first one was in the, in

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the plus state, and the second one
happened to be in the state -3/√2,

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√3/2, sorry. Then the composite state,
this is, the composite system is in the

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state where 1/2√2|00> - √3/2√2|01> +
1/2√2|10> - √3/2√2|11>. Okay. So

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that's, that's quite simple. Okay. So now,
we could ask the opposite question. So

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suppose you have given the state of the
composite system, determine the state of

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each of the two qubits. So, we, we'll just
continue with our previous example.Suppose

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you are given that the state of the
composite system is, is this. Okay, so

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given the state of this system, now you
want to determine, what's the state of

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each qubit. How would you do that? Well
the answer is clear, what you would do is

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you take the state and you would factor it
in the form (α0|0> + α1|1>) (β0|0> +

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β1|1>). And this factorization may not be
unique but we don't care about the overall

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phases. And so for example, in this
previous case, what, what, we could do is

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we could factorize it as (1/√2|0> +
1/√2|1>) (1/2|0> - √3/2|1>. And

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we could say, that's the state of the
first system. Now of course, we could also

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you know, this, this product will remain
the same if we, if we apply minus sign to

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this and apply minus sign to that. So in
other words, we could, we could make this

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state be minus this, minus that, and we
could make this state be minus this, plus

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that. And the product of these two would
still be the same. In fact, we could apply

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any phase here and opposite phase here and
the product would still be the same with

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the phases, as long as the phase is
cancelled. But of course, the phase, the

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overall phase of a qubit doesn't matter.
So, we don't really care about that phase

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information. What we care about is,
understanding what's the, what's the state

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of each qubit up to a phase. So now we
could ask, given an arbitrary such state

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of the composite system, can you always
factorize it in this way? And the answer

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is no. There are states which you can not
factorize in this way and this is, these

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are the so-called entangled states of two
qubits. This is quite surprising. The

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reason is, that, a state that you could
not factorize, for such a state, each of

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the two qubits. The first qubit and the
second qubit, by itself does not have is

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not in a definite state, it does not have
a quantum state. All we can say is that

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the composite system is in some quantum
state, it's in some superposition. But if

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you look at just the first qubit by
itself, it's not in any given quantum

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state. The way we describe it is, it's
entangled with other qubit. The two qubits

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are, are completely entangled with each
other to the point where neither of them

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has an identity of its own. So let's see
an example. So a canonical example of

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entanglement of two qubits entanglement is
the Bell State. So it's this very simple

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state. It's equal superposition of |00> and 
|11>. So the claim is that,

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if the two qubits are in this joint state,
they are completely entangled with each

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other to the point where you cannot really
talk about what the state of each qubit

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individually is. So, why is that? Well,
let's try to factorize this. Let's try to

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write this as (α0|0> + α1|1>), so let's
try to factorize it, (β0|0> + β1|1>).

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And let's show that no such α is such
complex amplitudes α0, α1,

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β0, β1 exists. So why is
that? Well we know that the amplitude of

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|00> is the product of this and
this. So, α0β0 = 1/√2. We also know

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that |11>'s amplitude is α1β1 =
1/√2. What these two things facts imply,

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that α0, β0, α1, β1, they are all
non-zero. Okay. But now lets go back and

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look at the fact that what's the amplitude
of |01>. Well, that's zero and so what that

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means is that, α0β1 must be equal to
zero. But that's not possible because what

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we said is that all four amplitudes have
to be non-zero so that's a contradiction.

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So what this shows is that, we just cannot
write this state, we cannot factor this

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state as a state of the first qubit and a
state of the second qubit. Now, we are

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going to study the properties of this Bell
state, this entangled state over the

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course of this, the rest of this lecture
and the next lecture. But for now, we are

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going to study what happens if you measure
this Bell state. So, imagine that you

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create this Bell state and now you take
these two qubits, these two particles and

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you separate them by a great distance from
each other. Well of course, none of these

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affects the state of the particles. So the
state of the particle is still |00> + |11>.

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What happens when you measure the
first qubit? Well, when you measure the

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first qubit, you see zero with
probability, well the square of the

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magnitude of this which is a half. But
what happens when you, if you, if you

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actually happen to see a zero, measure
zero? Well, remember the rules of

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measurement? What they tell us is, that if
you see a zero, the new state is going to

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be determined like this. You just cross
off the parts of the superposition that

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are inconsistent with the measurement
outcome, and you re-normalize. So the new

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state is |00>. On the other hand, if
you see a one, you see that with

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probability half, and the new state would
be |11>. Now what happens if after

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this, you measure the second qubit. Well,
the answer is clear. If you measure,

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second qubit, well in this case, you would
see zero with probability one. And in this

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case, you would see one with probability
one. So it's as though, you know, there,

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there is this, there is this strange
coincidence between, between the two

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measurements of these two different
particles even though might be arbitrarily

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far apart. Well, so far this not, this is
something that you should be able to

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reconcile with your intuition because you
could say, well, wouldn't you get exactly

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the same result if, when these two
particles were initially entangled, they

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flipped a coin to determine whether were
they going to act as, as they were in

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state |00> or they were going act as they
were in state |11>. And then once they

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are separated, they just, happen to be in
that state. It's as though you flipped a

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coin when you created this state. Well, it
turns out that it's not so simple. And, in

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the next video we'll talk about the EPR
Paradox or the Einstein Podolsky Rosen

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Paradox which shows you just how strange
entanglement is.