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I am working on a paper about Continuous Variable QKD. (https://arxiv.org/abs/1711.08500v2)

I read about direct and reverse reconciliation in this paper. I don't understand what exactly Reconciliation is? Reconciliation algorithms, reconciliation protocols and so on.

I am really confused. Can any one explain reconciliation to me?

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  • $\begingroup$ Please provide a link to the paper where you encountered this so that we can better help you. $\endgroup$
    – JSdJ
    Nov 17 '20 at 10:12
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    $\begingroup$ I believe the term "reconciliation" or "information reconciliation" refers to the error correction step that comes after Alice and Bob have generated their raw key. Their raw keys $K_A$ and $K_B$ will not be exactly equal so Alice and Bob have to communicate some classical information to allow them to error correct $K_A$ and $K_B$ so that after the procedure they have new keys $J_A$, $J_B$ such that $J_A = J_B$ (with high probability). $\endgroup$
    – Rammus
    Nov 17 '20 at 10:26
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    $\begingroup$ @Rammus Exactly what I thought, I was writing up an answer:) $\endgroup$
    – JSdJ
    Nov 17 '20 at 10:34
  • $\begingroup$ @JSdJ thanks for your comment. I added the paper link. $\endgroup$ Nov 19 '20 at 6:58
  • $\begingroup$ @Rammus as I understand, reverse reconciliation and post selection are two different method to reach significant transmission distances. Am I right? $\endgroup$ Nov 19 '20 at 7:12
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I assume the paper you are reading is referring to information reconciliation.

Information reconciliation is a vital part of post-processing in QKD, to limit (or erase in the best-case scenario) the amount of errors/differences between the key of Alice and Bob.

In that sense, it is a form of (classical) error-correction, and, broadly speaking, it works like this:

  • Alice and Bob run a QKD protocol together and both obtain a raw key $k_{a}$ and $k_{b}$ of whatever length.
  • In real world scenarios, $k_{a} \not = k_{b}$ due to measurement errors, imperfect channels etc. Alice and Bob want the exact same key, so they need to fix this.
  • However, $k_{a} \simeq k_{b}$; we can write $k_{b} = k_{a} + \epsilon$, where $\epsilon$ is the error/difference between Alice and Bob, and therefore $\epsilon \simeq \overrightarrow{0}$ (i.e. $\epsilon$ is a bitstring with almost all entries equal to $0$ because there are not that many errors).
  • Alice and Bob have (in advance) agreed to use a certain error code to perform error correction. This error code is some linear function $f$ that calculates the syndrome $s$ of a key. Note that the error code may or may not be secret to Alice and Bob.
  • It is hard to reverse this function in general. $s$ is of considerably shorter length than $k$. So, there are multiple $k$ with the same $s$.
  • Alice calculates $s_{a} = f(k_{a})$ and publicly communicates $s_{a}$. Any Eve cannot do anything with this because the function is in general hard to reverse.
  • Bob receives $s_{a}$ and calculates $s_{b}$. The function is linear, so $s_{b} = f(k_{b}) = f(k_{a} + \epsilon) = f(k_{a}) + f(\epsilon) = s_{a} + s_{\epsilon}$. Therefore, the 'error syndrome' $s_{\epsilon} = s_{a} - s_{b}$ encodes only information about $\epsilon$.
  • Only Bob can calculate $s_{\epsilon}$. Because the code was designed properly (and for this exact function), and because $\epsilon \simeq \overrightarrow{0}$, Bob can decode the error syndrome to estimate $\epsilon$: $\hat{\epsilon} = f^{-1}(s_{\epsilon})$.
  • Finally, Bob calculates $\hat{k_{a}} = k_{b} - \hat{\epsilon}$. If the code is any good, $\hat{k_{a}} - k_{a} \rightarrow 0$, i.e. Alices key and Bobs corrected key are very close/almost always the same.

There are a lot of important details I have grossed over, but this is the general idea. For a nice introduction of information reconciliation/error correction in the scope of QKD, check this paper.

Note that the above is error-correction for general QKD, not necessarily only for continuous variable QKD. I suspect that there are small discrepancies between the two, but the main goal should be the same.

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