Okay. So we are, let's move on to the third video of this lecture, which is on
Tensor products. Let me just say a word before we go on. Which is, first, for many
of you, this might be too much all at once. So you know, take it in whatever
doses you, you, you can digest it and let's see. So, some of this stuff, you
know, some of the things that I'm speaking about, of course. It's just a
formalization of stuff you've already seen in the, in the previous lectures.
Especially these, tensor products. You know, we've been implicitly using them for
the last couple of weeks. And, hopefully this only, you know, this only help, helps
clarify to you what the, what the underlying math behind it, behind it all
is. Okay, so. So, with that, with that small warning let's, let's just go on and
let's talk about, tensor products. So, let's, let's go back and remember, what a
qubit was. So, so you remember we, you know, our picture of it, was, was like
this. We have a hydrogen atom. Its electron, we think of it as being either
in the ground state, which we think of as a, as representing zero, and an excited
state, which we think of as a, as representing one. And now, of course, the
state of this, this qubit, is, is a unit vector in. So it's, it's some unit vector,
phi in a two dimensional complex vector space, which we call a Hilbert space.
Okay. We call it a Hilbert space mainly because it's a, it's a complex vector
space with a defined inner product. And then. You know this is additional property
that we want from Hilbert's spaces when, when they are, you know which is more
complicated, when they are infinite, and infinite dimensional Hilbert spaces so
when we talked about continuous quantum states. But, in this course, we are not
going to deal with all those subtleties about their completeness, you know limits
and so on. So formerly we'll only think about finite eventual Hilbert spaces. And
then when we talk about continuous quantum states, we'll just do it intuitively.
Okay, so, so , back here, okay, so we have, we have a cubit, it's a unit vector
in this two dimensional complex vector space, which we're calling a Hilbert
space, and. And now, let's say we have another such electron in another hydrogen
atom. And so, of course, it's state also is a unit vector in a, in a two
dimensional Hilbert space. Okay, but, but now, of course. In general, we know that
we cannot really talk about if we are given the two hydrogen atoms together, we
have, we are given these two qubits together, we cannot in general talk about
the state of the first qubit and the state of the second qubit separately because of
the phenomenon of entanglement. So, in general, we cannot talk about, any more
about the state of the first qubit and the state of the second qubit, but we still
have these two Hilbert spaces associated with the first qubit and the second qubit.
So now, what happens when we bring these two qubits together? Okay, so what you
already saw is that. When we think of this as a composite system, there is actually a
four dimensional complex vector space associated with it. And the basis for this
state is all the classical possibilities which is 0-0, 0-1, 1-0, and 1-1. But
still, you can ask what was the operation by which we, we took two such complex
vector spaces, two dimensional complex vector spaces. And we glued them together
to get a four dimensional complex vector space. What is this operation? So if you
call this Hilbert space H1, and we call this Hilbert space H2, then what's the
operation that they perform on these two Hilbert spaces to get this new Hilbert
space H, which is a four dimensional complex vector space. So, this operation
is called a tensor product. And when we perform this tensor product, what we do
is, we look, we take a basis vector from the first Hilbert space, like zero and a
basis vector from the second vector space, like zero. And we take a tensor product
between them. Okay. And, well it might be. You know, just to, just for clarity, it
might be, it might be a little, easier for you t o understand if I write them a
little closer together. So, I take. Okay. So, let's, let's, let's erase all this,
and let's call this, let's call this, this one, H1. H2, and we've taken a tensor
product between them. So, we are taking a basis vector from the first Hilbert space,
one from the second Hilbert space, and we take the tensor product of them. And this
tensor product we also write as so, when you write 0x0 we also you know this is
what we have been calling 00. Right? And sometime we will also write just as cat
zero, cat zero. So, this is what we mean by saying well the, the first electron was
in the state zero and the second electron was in the state zero. Similarly, if you
take the tensor product of these two, we, we can also abuse notation and write it
like, like this or like that, and so on. For the, for the, for the, for each of the
four cases. Okay and then, okay so this is how we glue these two Hilbert spaces
together and got a new Hilbert space, new complex vector space, which is now four
dimensional. Okay, we can also ask, you know, of course we can, we can glue
together any vector here and any vector here so if you have, if you have a state
phi one in the first Hilbert space, and phi two in the second Hilbert space, we
could, we could glue them together by putting this, this symbol in between them.
And this is what, you know, so for example, if we had if we had the state one
over square root to zero plus one over square root to one, and the first overt
space and one over square root to zero minus one over square root to one in the
second overt space. And if it took a product of these, a tensor product, this
is what we were, we were writing informally just by writing a product. And
now, when we take tensor products it's distributive, as you know, exactly the
rules that we followed. And we'd write down series. One half zero times tensor to
zero minus one half zero tensor to one plus one half one tensor of zero minus one
half, one tensor of one. Okay. That's what the that's what the tensor product of two
states would look like. We could also a sk, well what happens if you, if you look
at two set states? Phi one tensor and phi two and phi one tensor at phi two. What's
the inner product between them? And the answer is, it's just the inner product
between phi one and phi one. So, whatever the states are in the first, in the first
Hilbert space times the inner product between phi two and phi two. Okay, finally
lets look at the more general case. So, now suppose we have a, we have a K-state
particle. A K-state, sorry a K-state quantum system. So, if the first Hilbert
space is a K-dimensional Hilbert space. And let's say, let's, let's say the second
Hilbert space is, is made up of an L-state quantum system. So H2 is an L-dimensional
complex vector space. So this, has a basis zero through K - one, let's say. And let's
call these basis vectors, say, zero through, L - one. Okay, so now what
happens when we glue these two together, when we put them together and call this
one big system? So, we get some Hilbert space H. And H is a tensor product of H1
and H2. Okay, what's, what's another way of saying it? Well another, you know the
intuitive way of saying it is look, you had a quantum system here, the first one,
where you could be in one of these key distinguishable states, zero through K -
one, or in any super-position of these. In each two you can be in one of L
distinguished will states. Why need super positions of those? And so when you put
them together, what are the distinguishable states? Well, you could be
in 00, 01, 02 through 0L - one, or ten, eleven through 1L - one. So, they are
exactly K times L different distinguishable states. So H, its a K
times L dimensional space. And again. You know, the, the, the basis vectors in H are
going to be tensor, tensor products of basis vectors in H1 and H2. So they're
going to be of the form zero tensor zero. Zero tensor one. Zero tensor L-1. One
tensor zero one tensor L-1, all the way to K-1 tensor zero. K-1 tensor L-1. Okay, so
there are exactly K times L of these. And of course, your state can be any linear
combination of these. Again, remember by abuse of notation, w e, we, we can, we are
going to denote the state, also by. By this or also by. By that. Okay. So now,
there's something to think about here. There's something very interesting going
on here. Okay, what, what's this interesting thing? So, if you were to
write a state of your first system, how many parameters would you need for that?
Well since it's a K-dimensional system, you need. K. Complex numbers to describe
your state in of, of the first system. How many, how many parameters would you need
if you were just to put, describe a state in your second system? Well, you need
exactly L-complex numbers to describe a state in your second system. But now, if
you put them together, if you call your system the union of, of your first and
second quant, quantum system. Then you need K L complex numbers. Okay, so it's
worth thinking about this intuitively. What would have happened if this was a
classical system? Let's say that, You know, you had some system where you needed
a million parameters to describe, describe your system. So, for example, what's,
what's, a, what's a system where you need a million parameters to describe it? Well,
let's say it's the memory in your computer. So, let's say that you got a,
got a mem, you know, you know, you, you, you bought a mega, megabit, megabyte of,
memory. So, ten^6. You need ten^6 numbers to describe what the state of that memory
is. And now you are running short of memory. So, you go out and purchase
another, you know, some more rams, so its, lets say you purchase other ten^6
megabytes. How much memory do you have? Well, you have ten^6 + ten^6 which is two
ten^6. Alright? They add up, so the number of the parameters you need to describe the
state of the existence once you have, once you have, one you add these new memories
in, it's just the sum of the parameters you need for the first one and the
parameters you need for the second one. What happens if this was a complex system,
sorry, quantum system? So, you needed ten^6 parameter for your first system,
your first quantum memory , ten^6 for the second one. Right? If you use them
individually. But if you put them together, and call the whole thing one big
system. Now you need ten^6 ten^6. So, instead of a megabyte, you now end up with
a terabyte of memory. This sounds ridiculous. But this is, this is really,
you know, this is the basic fact about quantum mechanics or quantum systems which
make them so strange and therefore also so useful from a computational viewpoint.
Okay, one last thing to help you think about this. So, how could it be that you
needed only K-complex numbers to describe a state in H1. L to describe a state in
H2. But now, if you put, put the two systems together you need K L complex
numbers. How could this be? What, what's going on? Well the answer is,
entanglement. Okay, so, so why is the answer entanglement. So. So you, you had
H1 but you needed K-complex numbers, you had H2 but you needed L-complex numbers.
So if you had a state, phi one, you need only K-parameters to describe phi one. A
state of H, of, of the first quan, quantum system. If you have phi two, the state of
the second quantum system, you only need L-parameters, L-complex numbers to
describe it. If your have state of the two systems is of the form phi one tensor phi
two, you only need K + L parameters to describe it. But remember, the state of a
composite system does not need to be a product state. In general, it's going to
be an entangled state. And these states cannot be described using K-parameters on
the, on the first space and L-parameters on the second space. And that's the reason
why, in general most of the states of this, of this composite system are going
to be entangled. And this is why we need K L parameters to describe a general state
of this composite, quantum system, which we describe by the Hilbert space H, which
is H1 tensor H2. Okay? So, how much of this should you understand? Well, you
know, it depends upon, it depends upon how deeply you want to get into the subject.
But hopefully this gives you, some sense of the, you know, of sol id ground, in
terms of, what are all these symbols that we've been using so far? What do all these
operations we've been using so far mean? So use it, you know, use this, this
lecture. I, I, I guess What I should say is try to digest as much of this as you
find comfortable, and you know, I'll try to be as gentle as possible going forward,
and use, use this notation as sparingly as possible. Okay.