Okay. So now we are ready to talk about entanglement which plays a central role in
quantum computation. So, let's come back to a two qubits system. And, you know,
which let's say again, consists of, consists of atomic qubits where our first
qubit happens to be in the state superposition α0|0> + α1|1>. And the
second qubit happens to be in the superposition, β0|0> + β1|1>. So now we
could ask, what's the state of the two qubit system? Well, the answer is very
simple. What you do is, you just take a product of these, according to this rule.
It's sort of the obvious rules. So you, the amplitude of |00> is, is just
α0β0|00>. The amplitude of |01> is just α0β1|01>. The amplitude of |10> is
α1β0|10>. And the amplitude of |11> is just α1β1|11>. So for example, suppose we,
you know the, the, the states of the two qubits where, the first one was in the, in
the plus state, and the second one happened to be in the state -3/√2,
√3/2, sorry. Then the composite state, this is, the composite system is in the
state where 1/2√2|00> - √3/2√2|01> + 1/2√2|10> - √3/2√2|11>. Okay. So
that's, that's quite simple. Okay. So now, we could ask the opposite question. So
suppose you have given the state of the composite system, determine the state of
each of the two qubits. So, we, we'll just continue with our previous example.Suppose
you are given that the state of the composite system is, is this. Okay, so
given the state of this system, now you want to determine, what's the state of
each qubit. How would you do that? Well the answer is clear, what you would do is
you take the state and you would factor it in the form (α0|0> + α1|1>) (β0|0> +
β1|1>). And this factorization may not be unique but we don't care about the overall
phases. And so for example, in this previous case, what, what, we could do is
we could factorize it as (1/√2|0> + 1/√2|1>) (1/2|0> - √3/2|1>. And
we could say, that's the state of the first system. Now of course, we could also
you know, this, this product will remain the same if we, if we apply minus sign to
this and apply minus sign to that. So in other words, we could, we could make this
state be minus this, minus that, and we could make this state be minus this, plus
that. And the product of these two would still be the same. In fact, we could apply
any phase here and opposite phase here and the product would still be the same with
the phases, as long as the phase is cancelled. But of course, the phase, the
overall phase of a qubit doesn't matter. So, we don't really care about that phase
information. What we care about is, understanding what's the, what's the state
of each qubit up to a phase. So now we could ask, given an arbitrary such state
of the composite system, can you always factorize it in this way? And the answer
is no. There are states which you can not factorize in this way and this is, these
are the so-called entangled states of two qubits. This is quite surprising. The
reason is, that, a state that you could not factorize, for such a state, each of
the two qubits. The first qubit and the second qubit, by itself does not have is
not in a definite state, it does not have a quantum state. All we can say is that
the composite system is in some quantum state, it's in some superposition. But if
you look at just the first qubit by itself, it's not in any given quantum
state. The way we describe it is, it's entangled with other qubit. The two qubits
are, are completely entangled with each other to the point where neither of them
has an identity of its own. So let's see an example. So a canonical example of
entanglement of two qubits entanglement is the Bell State. So it's this very simple
state. It's equal superposition of |00> and |11>. So the claim is that,
if the two qubits are in this joint state, they are completely entangled with each
other to the point where you cannot really talk about what the state of each qubit
individually is. So, why is that? Well, let's try to factorize this. Let's try to
write this as (α0|0> + α1|1>), so let's try to factorize it, (β0|0> + β1|1>).
And let's show that no such α is such complex amplitudes α0, α1,
β0, β1 exists. So why is that? Well we know that the amplitude of
|00> is the product of this and this. So, α0β0 = 1/√2. We also know
that |11>'s amplitude is α1β1 = 1/√2. What these two things facts imply,
that α0, β0, α1, β1, they are all non-zero. Okay. But now lets go back and
look at the fact that what's the amplitude of |01>. Well, that's zero and so what that
means is that, α0β1 must be equal to zero. But that's not possible because what
we said is that all four amplitudes have to be non-zero so that's a contradiction.
So what this shows is that, we just cannot write this state, we cannot factor this
state as a state of the first qubit and a state of the second qubit. Now, we are
going to study the properties of this Bell state, this entangled state over the
course of this, the rest of this lecture and the next lecture. But for now, we are
going to study what happens if you measure this Bell state. So, imagine that you
create this Bell state and now you take these two qubits, these two particles and
you separate them by a great distance from each other. Well of course, none of these
affects the state of the particles. So the state of the particle is still |00> + |11>.
What happens when you measure the first qubit? Well, when you measure the
first qubit, you see zero with probability, well the square of the
magnitude of this which is a half. But what happens when you, if you, if you
actually happen to see a zero, measure zero? Well, remember the rules of
measurement? What they tell us is, that if you see a zero, the new state is going to
be determined like this. You just cross off the parts of the superposition that
are inconsistent with the measurement outcome, and you re-normalize. So the new
state is |00>. On the other hand, if you see a one, you see that with
probability half, and the new state would be |11>. Now what happens if after
this, you measure the second qubit. Well, the answer is clear. If you measure,
second qubit, well in this case, you would see zero with probability one. And in this
case, you would see one with probability one. So it's as though, you know, there,
there is this, there is this strange coincidence between, between the two
measurements of these two different particles even though might be arbitrarily
far apart. Well, so far this not, this is something that you should be able to
reconcile with your intuition because you could say, well, wouldn't you get exactly
the same result if, when these two particles were initially entangled, they
flipped a coin to determine whether were they going to act as, as they were in
state |00> or they were going act as they were in state |11>. And then once they
are separated, they just, happen to be in that state. It's as though you flipped a
coin when you created this state. Well, it turns out that it's not so simple. And, in
the next video we'll talk about the EPR Paradox or the Einstein Podolsky Rosen
Paradox which shows you just how strange entanglement is.