Okay, let's start this last lecture in this semester.

And the topic will be quantum key distribution. That will be the last topic in the semester.

And the reason why I want to tell you about the quantum key distribution, because it's the most

mature application of quantum optics. So quantum optics is not just cool, it has some applications,

and among them quantum key distribution is something that is now even produced commercially.

There are companies who do this. So what is the idea of quantum key distribution? Suppose I want

to pass to Gaetano the tasks for the exam. And I don't want you to know the tasks.

The secret key or the secret transmission of messages is an old story, of course. Even

Romans did it. There were different tricks. The main idea is every symbol is substituted by some

other symbol. So the question is to pass the secret key. So the key to transfer, to transform

the message into some sequence of symbols and then to decrypt it. So to encrypt it into some

sequence of symbols and then to decrypt it. And the mathematical way to look at it is to consider

a message, for instance, in a binary code. For instance, I want to pass to Gaetano, very simple,

there will be, let's say, eight tasks. Actually, there will be five tasks. But suppose that there

will be eight tasks and the information I want to send to Gaetano is difficult, easy, difficult,

easy, zero and one, zero and one. So the easy task is zero and the difficult task is one.

And then I want to tell him that, for instance, the first two, four tasks will be easy and the

last four tasks will be difficult. How can I encrypt? If I, so this is the message.

Now I want to compose some key that is random and that is also binary code. So I have to write a

random sequence of zeros and ones. Well, I'll try to be random.

I'll try, I try to be random. So this is the secret key.

And encryption is just a summation modulo two. What it means that, it means that if I sum

zero and one, I get a one. If I sum zero and zero, I get a zero. And if I add one and one,

modulo two, I get zero again. So this is summation modulo two. So I add.

Here I get one, here I get zero, here I get zero, here one, one, zero, zero and zero. And the way to

decrypt it is again to add to this sequence again the secret key modulo two. So if I want to

obtain the previous, the original message, I just add these two numbers modulo two and I get here

zero, here zero, here zero, here zero, here one and so on. So this restores the message.

So the idea is that we have to share with Gaetano this secret key and then I send him this sequence.

He adds the secret key and he gets the message. And there is a theorem, it's called Vernam's theorem,

which is much older than the quantum key distribution. And the theorem is that if the

secret key has the same length as the message and if it is used just once, then the message is

unbreakable. So the key should be used just once and should have the same sequence as the message.

That's all. And then one cannot break the secret message in any possible way. So the whole

story about quantum key distribution is to transmit, to distribute, one says, distribute

this secret key between two users, which in our case are Gaetano and me. And then, well,

there are different ways to do it. For instance, I write this message on a paper and send him.

But someone can intercept it. In all classical ways, someone can intercept. For instance, I can

give it to someone and ask this someone to pass it to Gaetano, but then you can catch this someone,

bribe him, and take the message. The idea of quantum key distribution, quantum key distribution,

appeared first in 1984. And it was first introduced by Bennett and Brassard.

And that's why this protocol, well, a protocol is just some sequence of operations. And this

protocol is called for this reason BB84 for the year because of the year. And the idea was to

encode this secret key into the polarization state of a single photon. So the idea is that I'm sending

to the end user, to the other person. Actually, the terminology here is that the end user is

the one who is sending the message to the end user. So the end user is the one who is sending

terminology here is that the one who sends the message is called Alice A. And the one who

receives the message is called Bob B. Is it someone knocking? Hello. So Alice, Alice sends to Bob

the key in the form, in the shape of single photons. And the encoding is through polarization.

Why is it a cool idea? The reason why this idea is really cool, because you know from our lecture