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Hello, everyone. We're already a quarter
of the way through this class. So this

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might be a good time to take stock of the
situation and see where we are and

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re-orient ourselves. So the first thing
I'd like to do is say a little bit about

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the organizing principles of this course
just to make sure that we're on the same

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page. So, you may have noticed that this
class takes a certain approach to math

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formulas which, which I'll call the Kanban
approach. I don't know if you, well may,

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maybe you weren't even, some of you were
not even born then but you know, back in

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about 1990. There was, there was a lot of
talk about how Japanese, the Japanese were

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going to take over every aspect of, you
know they were taking over the

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semiconductor industry and they just did
everything well. And one, one thing that

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people talked about was that they used
this Kanban approach to manufacturing

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which, which was translated as just in
time. So with small inventories and they

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were extremely efficient about it. Okay so
what do we mean by Kanban approach to math

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formalism? So what I mean is that, you
know when we've in, in this class, I've

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tried to keep the mathematical formalism
down to the minimum. So, for example when

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we, when talked about two qubits, you know
I said well, if you have the first qubit

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in this state a0(0) + a1(1) and the second
qubit in b0(0) + b1(1) then the state of

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the two qubits system together is a0b0(00)
+ a0b1(01) + so on. Well, if you want to

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do this formally what, what you are doing
here is computing the tensor product of

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these two qubit states, you know but
rather than load you with all this

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mathematical formalism, what I want to do
is put off this formalism as far as

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possible. The same is the case with
measurements where, later, we'll describe

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quantum measurements in terms of, in terms
of the measurement operator. And the

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measurement outcomes in terms of
eigenvalues and eigenvectors of this

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operator but once, once again. It's, it's
actually useful to talk about as much of

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this as possible, without introducing a
[laugh] the math, all the mathematical

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notation. And that's what we are going to
do in this class. Now, as we go along as

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you get comfortable with these concepts,
okay? We will actually talk about this

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proper mathematical formalism with which
to discuss all these concepts. And so, I

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hope that this is actually going to be
better for many of you, it's a friendlier

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way of introducing the subject. And for
those of you who, who already know the

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formalism and more prefer to see it, well
you'll see it as we go along. Now, one of

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the reasons that I'm trying to introduce
the subject in this way, is that by

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keeping it light on formalism, it also
highlights the counter-intuitive aspects.

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The intuitive aspects of the subject and
once you look at things intuitively, you

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actually realize how counter-intuitive
quantum mechanics is. I think this is

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extremely important when you're doing
quantum computing, because quantum

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computing actually as a subject. It
exploits some of the most counter

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intuitive aspects of quantum mechanics.
And, so if you're going to understand

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quantum computing at a deep level, you've
got to grapple with these aspects of

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quantum mechanics. And you've got to
understand them at an intuitive level. And

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so, it's only by confronting your
intuition with how strange quantum

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mechanics is, how, how strange it's
behavior is. That you can start forming

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this new level of intuition, okay. So I
hope that this, this, you know may be some

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of you had already figured this out
implicitly but I hope this helps you

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understand how this course is being
treated and organized. Okay, so the second

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thing about, about taking stock is, well
at the beginning of the course we, we

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conducted a survey to try to understand
your background and so this is probably a

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good time to, to actually conduct another
survey to get a sense of your reaction to

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the lectures, the homework, you know how
easy, how difficult you find it, whether

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you actually attempt the optional
assignments and would you like to see more

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of those. How do you feel about your
linear algebra preparation for this

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course? Would you like to see a review of,
of, of the material and if so, which

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topics would you like to see a review? And
also would you like to see some discussion

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of, of current research topics related to
the lecture what's, what's being done in

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the lectures. So for example in the last
lecture I talked about the Bell experiment

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and I said actually, this Bell experiment
is , you know it's a subject of current

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research current research in, in
cryptography , and what's called device

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independent cryptography and in random
number generation. So there are, there are

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several, you know it, it is actually an
active subject of, of research. And so

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maybe I'll give you a little bit of a
hint, you know a little bit of a flavor of

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this, of this current research. And then
maybe you can, in the survey, you can

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comment about whether you'd like to see
more such minuet or whether you prefer to

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see just the basic lecture topics, okay.
So let's, let me just remind you what the

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Bell experiment setup is. The, you know
the apparatus was divided into two

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spatially separated parts which we think
of as two boxes. Each of those got a

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random bit as input x and y. Each of those
output a bit in b and what we wanted was.

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That if, if x and y were both one then we
wanted the alphabets to be different. In

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all of the three cases we wanted them to
be the same and what we said is that, if

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the boxes are described by local hidden
variable theory so if, if, if there is

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some sort of classical description of
what's going on in these boxes, then you

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can succeed in this task with no more than
75 percent probability. On the other hand,

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if the two boxes are allowed to share a
Bell State, if they share entanglement and

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if they measure these qubits in suitable
basis. Then they can succeed with

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probability as high as .85 cosine square
phi by eight and in fact, in each of the

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four cases, they succeed with probability
exactly .85 in that, in that particular

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protocol. Okay so, so now here's a
problem. Here's a, here's a question t hat

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people have been trying to think about for
a long time which is how do you construct

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a physical source of randomness? How do
you construct a device which outputs truly

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random bits. So Intel last year announced
a chip on which it digitally produces

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random bits for use in cryptography but
now you could ask the question. How do you

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know that the chip is working correctly?
So, in other words, you could ask suppose

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you have, you're given a black box which
produces as output what it claims is a

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random string. How you can test it? If it,
if it outputs let's say a thousand bits,

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how would you test that every one of the
two to the 1000 different 1000 bit strings

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are equally likely, or whether it's close
to that. Well, classically it seem

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impossible to do this. But it turns out
that you can use the Bell experiment to

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certify that you have got real randomness.
And the way this works is, if you run the

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Bell experiment and if, if the experiment
succeeds in it's goal with 85 percent

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probability or very close to 85 percent
probability, then the only way it can

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happen is if the output bits a and b, are
random bits. So for example you cloud, you

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could choose to pick b and you could
output it and it would be guaranteed to be

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a random bit if you could certify that you
succeeded with probability close to 85%.

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So now the, you know okay, so starting
from this observation, there are schemes

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that, that actually show how to, how to
create a random number generator so that

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you can certify that the particular output
that you've been you, you've got must have

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been sampled from a distribution which is
close to a uniform distribution on let's

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say, 1000 bit strings. Okay so, if you
want, if you want to read more about how

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this comes about, you know I've given you
pointers, references, to two different

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papers which are, which have been posted
on the archive. The archive is actually a

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good place for you to look for interesting
papers on the on, on, on quantum

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computing. Okay so, you know that's an
example of, you know active research, very

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close to somethi ng we've already talked
about in, in lecture.