Comprehension questions on quantum cryptography especially BB84

Question

I have recently read a lot about the BB84 protocol, I have used three primary sources, the original work, a QK book, and a diploma thesis.

My questions refer to the photons sent by Alice, the base of Bob and a possible attack on the protocol (MITM attack).

My first question is about sending the photons from Alice to Bob. The original paper states that Alice sends a single photon in one out of four polarization directions, my question. How does Alice know the direction of the photon? I imagine that I look at a single unpolarized photon, this is sent through a random filter. But there are cases in which a photon does not come through a filter (photon has 90° and the filter is 0°). So how does Alice know if a photon is ever sent? How can she even produce a single say we make a vertically polarized photon?

My second question is based on Bob, in the original paper, only Bob is measured in one of two possible bases (0°, 90°) or (45° and -45°). In my second source (the book), however, it is stated that Bob simply uses a filter from the respective base. I'll explain it a bit more carefully: The statement of the book is that Bob measures (always) at 90° or 45°. But now the diploma thesis, which I use as a third source, says that Bob uses 0°, 90° 45° -45° as a filter for the detection at random. I understand both possibilities, because assuming a photon comes in 90° and I measure in 0° then I can indeed from the non-detection conclude that the photon has been polarized in 90° direction. So I suspect the statements are equivalent, both in the book and in the thesis. Is it correct that way? and what does measuring in a base mean?

My third question relates to a possible attack. I have read a paper in which a MITM (man-in-the-middle) attack is carried out. My book source also lists this attack. But in what way is that an attack scenario, if only I have to authenticate the connection? Then the actual attack is still witless?

I hope my questions are understandable. It is important for me to understand that. I am looking forward to your answers. If I should explain one or the other in more detail, then, of course, I would correct my question and specify. Thanks so far!

Answer 1

How does Alice know the direction of the photon?

You're talking about the photons as if we've just magically plucked them out of the air with no idea what polarisation they're in. In practice, we're producing the photons. A source of individual photons that can be produced 'on demand' is still an experimental challenge (not that I'm completely up to date with the latest experimental literature). But the point is that we have systems that produce photons for us, and they can produce polarised photons. For example, some designs of laser produce polarised outputs.

Another way that could help you think of this (although I don't think people really do this), is imagine you have a photon going along a path. If you put a polarising beamsplitter in the way, then you direct one output to a detector. If that detector doesn't click then (if it's a perfect detector), the photon is travelling along the other path in the opposite polarisation.

I suspect the statements are equivalent

Yes, they are. The overall protocol is to make a random choice of which basis you want (0/90 or $\pm$45) and then perform that basis measurement. This is basically the same as the original statement. Those individual measurements could be made using a polarising beamsplitter with detectors on both outputs of the beamsplitter, or you could just put the detector on one output and assume that if you don't get a hit, it was the other outcome. At that point, you can use a filter instead of the beamsplitter because, either way, you just lose the other polarisation. So, the random basis choice is making a random choice between whether you're using a 0 or a 45 degree filter. This is the same as the book's statement.

Equally, using the combination of 0,90,$\pm$45 will achieve just the same, it's just more complicated than necessary (because 90 will always give the opposite result to 0, so why not just use 0?).

what does measuring in a base mean?

That's a much bigger question which deserves a much fuller answer than I can reasonably give as a sub-answer, and would preferably require some mathematical formalism. Still, to give you a sense: in quantum, when you want to ask a system what its state is, you can only ask limited questions. You cannot ask "what is the polarisation angle of the photon?" but, instead, you can ask if it's in one of two perpendicular directions, such as 0 and 90, and get the answer with varying probabilities.

But in what way is that an attack scenario, if only I have to authenticate the connection? Then the actual attack is still witless?

I don't really understand what you're asking here. Of course, the point of the protocol is to make sure that a man in the middle attack cannot work. But you have to prove that, and this helps give insight as to why the quantum protocol is a good one. The sort of thing that might go on is that Eve, sat in the middle between Alice and Bob, can take an (approximate) copy of the photons as they zoom past, and later use her copies to get as much information about the key shared between Alice and Bob as possible. The point is that Eve trying to do this has a knock-on effect on what Bob measures, and so Eve's meddling is detectable. In principle, they can even estimate how much she might know, and perform a protocol called privacy amplification to reduce that knowledge arbitrarily.