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.