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Bob has two detectors + and x. You can see it in the picture I sent. Alice sends a photon with vertical polarization and Bob uses the + detector and the detector is detecting Alice's photon.

Bob's + detector detects when the polarization of the photon from Alice is either vertical or horizontal. (because it is + detector)

Bob can only tell if his detector is detecting a photon or not, if it is detecting, how does he know that the photon that causes his detector to give an output is vertical or horizontal?

enter image description here

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  • $\begingroup$ It looks like you mean quantum-key-distribution. It is usually not considered part of quantum-cryptography. $\endgroup$
    – Maarten Bodewes
    Mar 11, 2023 at 16:13
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    $\begingroup$ @MaartenBodewes: actually, I believe that QKD is considered the prime example of Quantum Cryptography (that is, using the properties of Quantum Mechanics to achieve a cryptographical goal) $\endgroup$
    – poncho
    Mar 11, 2023 at 20:14
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    $\begingroup$ On the other hand, if the question is "how do photon polarization detectors work?", perhaps this be better answered on the physics stack exchange... $\endgroup$
    – poncho
    Mar 11, 2023 at 20:46
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    $\begingroup$ @poncho In e.g. the Wikipedia article it mentions that it is not quantum cryptography. I kind-of agree that it doesn't have much to do with cryptographic algorithms. That said, key establishment is very much cryptography in my opinion and it definitely uses quantum effects. So yeah, OK, it's quantum-crypto. In any case, I'll tag it QKD in addition (even if it is a subset and usually no other tags are required, I guess it won't hurt, and it's not like there is a lot of discussion running specific algorithms on a QC anyway). $\endgroup$
    – Maarten Bodewes
    Mar 12, 2023 at 2:57

1 Answer 1

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Bob randomly chooses to measure each photon in either one of two bases: the horizontal-vertical basis or the $45^o$ basis. The former can distinguish a horizontal from a vertical photon, while the latter can distinguish a $+45^{o}$ photon from a $-45^o$ photon.

What Bob doesn't know, is whether he measured the photon via the same basis it is aligned by. For example, Bob may measure a $-45^o$ photon via the horizontal-vertical detector, which will give him an unreliable result. That is why, as you notice in the line before last of your diagram there is a "compatibility" property. After sending all the necessary photons to Bob, Alice uses a classical channel to communicate to Bob the bases she used when sending each photon, and Bob tells her for each one whether he used the compatible detector or not.

Since the bases themselves don't reveal the orientation of the photon (For example: knowing it is the horizontal-vertical basis doesn't tell you whether the photon was sent with the horizontal or vertical polarization) there's in theory no security problem in doing that. So, Alice and Bob throw away all the photon values that weren't sent and measured with the same basis (hence, such photons are incompatible) so they are left only with the photon values (bits) that are identical (have the same $1/0$ value) which they can then use as a key.

Of course, if very few photons are sent, an attacker listening on this channel might get lucky and orient his detector exactly like Bob did, and hence recover the key also. So in general such a protocol becomes more secure the more photons are sent. But there are probably many other attacks which are more complicated, involving weak measurements and so on. This is a separate topic.

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  • $\begingroup$ Re. last para. So the attacker's advantage would be through pure guesswork? And so a 256 bit key would be guessed as in other protocols at $P=\frac{1}{{2}^{256}}$? $\endgroup$
    – Paul Uszak
    Mar 13, 2023 at 13:17
  • $\begingroup$ No, since recall the classical channel communication is, AFAIK also assumed to be unencrypted (otherwise why do we need to agree on a key) so an attacker verifies whether he guessed the correct basis by intercepting this also. If he did, he measures the same bit as Bob did and he knows that. $\endgroup$
    – Amit
    Mar 13, 2023 at 13:34
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    $\begingroup$ BTW this protocol is called BB84 in case you or anyone else want to learn more about it, in particular the security analysis... $\endgroup$
    – Amit
    Mar 13, 2023 at 13:48
  • $\begingroup$ Although Alice and Bob use same + basis, it can be either vertical or horisontal, right? How can all the photons that they have left is all identical? $\endgroup$
    – Tatsuya
    Mar 13, 2023 at 14:48
  • $\begingroup$ @Tatsuya No, Alice and Bob randomly select either the x or + basis without knowing whether they used the same one or not. But the basis itself does not determine the value sent under this basis. + basis can distinguish $0^o$ from $90^o$ photons, and x can distinguish $45^o$ from $-45^o$ photons. The values are encoded in the angles (polarizations) of the photons, not in the basis. Alice is the one that selects the secret polarization for each photon she sends. For example if she selects a $+45^o$ it can only be correctly measured via the x basis. $\endgroup$
    – Amit
    Mar 13, 2023 at 14:57

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