Okay. So in this video, we are going to look at some remarkable properties of the
Bell state. So, the property that we'll be interested in is what's called the
rotational invariance of the Bell State. So remember what the, what the Bell State
is. It's a, it's a state of two qubits where they are in a equal superposition of
being in the state 00 or eleven. So now you remember that if we measure the first
qubit, if we see a zero, then if we measure the second qubit, its also going
to be, we, we are going to see it in the state zero with probability one. Now what
the rotational invariance of the Bell State says is the, is the following day
remarkable property. What it says is, suppose that we were to measure this Bell
State so that's a, base is 01. Instead of measuring it in this 01 basis, let's
measure it in some basis uu perp. The u and u perp are just arbitrary rotations.
They're, this is just some arbitrary angle here with the zero state in the real
plane. And we do the same thing for the other Bell State. We measure this one the
second, sorry, the other qubit in the Bell State. We measured also In you. You built.
Okay. The remarkable fact is that if we, if you are to carry out this measurement,
well first of all, we'll see u or u perp with equal probability with 50, 50
probability. Okay, so the probability that the outcome is u is one-half. Probability
of u perp is also one-half but if the probability we see the first outcome is u
if that's what happens, then when we make a measurement on the second qubit, we'll
also see u with probability one. So in other words, we'll always see matching
outcomes in the two cases even though if you look at any one case it's 50, 50
whether we get u or u perp. So this is exactly the same situation we had, when we
measured in the 01 basis except it holds no matter what the basis is, okay. This
remarkable property is called the Rotational Invariance of the Bell State.
So now, let's try to understand this a little more. You know let's, let's see how
well we understand this so let's say that we make a measu rement on this first qubit
in some arbitrary basis uu perp and now they pick a different basis. So we pick,
pick some, some different bases v, v perp for the second qubit and we measure it in
this basis. So let's say, this was u and this angle is theta. So what we want to
know is, suppose we do these measurements, we measure the first qubit of the Bell
State in this uu perp basis. We measure the second cubit in the vv perp where
these two bases make an angle of theta with each other. Now, what we want to know
is. What is the probability of matching outcomes? Where we will see, we see that
the outcomes are matching if we get this outcome on the first qubit and that
outcome on the second qubit or if we get this outcome on the first cubit and that
one on the second cubit right? So we see this matches with that and u perp matches
with v perp. So, whats the chance that we will get matching outcomes? Well I claim
that we already, from what we saw in the last slide, we already know how to answer
this question. Why? Because when we measure the first qubit, we either get
back in u or u perp with 50, 50 probability. So it doesn't matter which
one we get. Let's say we get the outcome u. Now, what's the state of the second
qubit? Well, I claim the second qubit is in the state u, right? That's what we saw
in the previous slide but now, this is the state of the second qubit and we are
measuring it in the v perp, vv perp basis. What's the chance we get outcome v? Well,
the answers exactly cosine squared theta. Okay, so the same thing, the same
reasoning holds if you happen to get u perp in the first as, as the outcome of
our first measurement. Well, then the state of the second qubit must be u perp
as well because they completely match. But now, if you measure the second qubit in
the vv perp basis, what's the chance we get output v perp? Well, it's still cosine
squared theta so the, so the, the answer is if you measure the two qubits in basis
which make an angle of theta with each other, the chance that we get matching
outcomes is exactly cosine squared theta, okay. So how do we prove this? Well the
way you prove this is by saying that if you were to write your state psi it's,
it's an equal superposition of 00 + eleven. In the last lecture we saw that we
could also write it as an equal super position of plus, plus and minus, minus.
What we are claiming is in fact if you have an arbitrary uu perp, then you can
write 00 + eleven as uu perp + vv perp. Okay so, so just think of u as being some
arbitrary vector is a(0) + b(1) where a and b are real numbers. And then u perp,
you can write as b(0) - a(1). Actually, the way I've drawn it here, it's - b(0) +
a(1). And now what you would do is you would substitute this into, into this
formula and you'd say well, what's one / two^2 uu perp. It's just a(0) + b(1) a(0)
+ b(1) + one / two^2 x u perp u perp which is - b(0) + a(1) - b(0) + a(1). And if you
expand this out and you simplify you'll get one / two^2 x 00 + one / two^2 x
eleven. Let's see, should I leave this as an exercise? Well, here's the other
condition we have. Of course u and u perp unit vectors, they have a normalization
condition which isa^2 + b^2 = one. Okay, so let's just, let's just eyeball it and
lets see, what's the amplitude of 00? Well, there are two ways of achieving 00.
The, the amplitude here is a squared. The amplitude from here is b^2 x one-half^2.
So it's one / two^2 x a^2 + b^2 but a^2 + b^2 = one so you get total amplitude is
one / two^2. The same thing holds for the amplitude of eleven. It's a^2 + b^2 and so
that's, that's one / two^2. What about the amplitude of the 01? Will you achieve 01
from here with amplitude ab one / two^2? And from here with amplitude -b so b
cancel, and so you get amplitude zero, okay. The same is true for ten and so, so
you get, that the Bell State can be written in any basis as uu perp. Uu + u
perp u perp. Now, there's one other thing that I should say that here I use the
restriction that, that this uu perp basis has to be a real, valued basis, okay. So
the, the a and b, the, the coefficients here are not allowed to be complex value.
If you do want to use complex coefficients, then you can use a different
kind of Bell State. This, this Bell State is called psi minus. And it's of the form
one / two^2 01 - one / two^2 ten. And this Bell State is actually invariant on the,
any change of base. You know if you, if you measure it in uu perp. Where u is now
of the form a(0) + b(1) where the alpha and beta are complex numbers. And u perp
is -b0 + a1. These are complex conjugates. You know of course
they satisfy this condition a^2 + b^2 = one. And now you can, you can actually
verify if you like that, you know the same sort of rotational invariance holds except
now its not just real rotations but also complex rotations on the complex plane but
this is not, you know if you don't follow this, its okay because we are not going to
be dealing with this. For the purposes of this lecture we are only going to be
dealing with real rotations and what I said so far is all that you need to know.