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According to Heisenberg's Uncertainty Principle, it is impossible to measure the values of conjugate variables simultaneously as measuring one accurately would make the other equally corrupt.

So in QKD, there's a thing that states that Eve can't eavesdrop as the very act would change it's quantum state(Heisenberg's principle). I would like to know the measurement of which conjugate pair is being referred to here. Do answer fast as I'll need to do a seminar on this topic tomorrow :)

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    $\begingroup$ Measurement in the wrong (Hadamard) basis is not exactly the same as product of uncertainites in both such variables. So apart from searching for BB84 you might want to also search for squeezed states QKD and continuous variables model. Or maybe try again explaining what you want. $\endgroup$ – Vadym Fedyukovych Sep 17 '18 at 15:56
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In the simple BB84 protocol, Alice prepares a state in either the Hadamard or the Standard basis. In order to measure Eve need to choose a basis to measure in. The problem for Eve is these two are two orthogonal basis or in a sense "Conjugate". This however is not the same as the conjugate variables you mention in your question.

Measuring in the one of the bases, collapses the state onto a vector in that basis. Measuring in an orthogonal basis would result in a truly random collapse onto a vector in this basis. Hence, no new information is gained about the original state. Thus Eve's choice of basis and the basis which Alice used to encode her bit in must match if Eve hopes to reliably learn Alice's bit.

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  • $\begingroup$ okay thanks....but where does Heisenberg's principle apply in QKD then? $\endgroup$ – Allen Sep 16 '18 at 18:41
  • $\begingroup$ another doubt is what if the random outcome after Eve's examination is the same as the original state Alice sent? Will that mean "eavesdropping" was not detected by both Alice and Bob after cross-checking the polarizers used by them? $\endgroup$ – Allen Sep 16 '18 at 18:47
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    $\begingroup$ In short, the Heisenberg's uncertainty principle is not directly used in proving the security of QKD. Yes, you are correct, eavesdropping will not be detected in that case. $\endgroup$ – Guru Vamsi Policharla Sep 16 '18 at 18:53
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    $\begingroup$ Not quite. Alice chooses her basis uniformly at random. This would mean Eve can only measure in the right basis 50% of the time. Alice and Bob measure the amount of error in the bits Bob received. Alice and Bob can gauge how much information Eve has fro this. They abort if security of the key is compromised i.e Eve has too much information about the key. $\endgroup$ – Guru Vamsi Policharla Sep 16 '18 at 19:21
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    $\begingroup$ Nope. They're the bits for which the basis match. Note that all bits are not revealed. Only some of them are. The rest of them are used to arrive at the final key. Perhaps a better source would be lecture notes from a course such as the one on EdX as compared to Wikipedia. $\endgroup$ – Guru Vamsi Policharla Sep 18 '18 at 13:53

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