Device-Independent Quantum Key Distribution: Difference between revisions

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Either Protocol (see [[Device Independent Quantum Key Distribution#Pseudocode|Pseudocode]]) abort with probability higher than <math>1-(\epsilon_{EA}+\epsilon_{EC})</math>, or it generates a</br>
Either Protocol (see [[Device Independent Quantum Key Distribution#Pseudocode|Pseudocode]]) abort with probability higher than <math>1-(\epsilon_{EA}+\epsilon_{EC})</math>, or it generates a</br>
<math>(2\epsilon_{EC}+\epsilon_{PA}+\epsilon_s)</math>-correct-and-secret key  of length</br>
<math>(2\epsilon_{EC}+\epsilon_{PA}+\epsilon_s)</math>-correct-and-secret key  of length</br>
<math>l\geq \frac{{n}}{\bar{s}}\eta_{opt} -\frac{{n}}{\bar{s}}h(\omega_{exp}-\delta_{est}) -\sqrt{\frac{{n}}{\bar{s}}}\nu_1  -\mbox{leak}_{EC}</math></br>
<math>
<math>-3\log\Bigg(1-\sqrt{1-\Bigg(\frac{\epsilon_s}{4(\epsilon_{EA} + \epsilon_{EC})}\Bigg)^2}\Bigg)+2\log\Bigg(\frac{1}{2\epsilon_{PA}}\Bigg)</math>,</br>
\begin{align}
where <math>\mbox{leak}_{EC}</math> is the leakage due to error correction step and the functions <math>\bar{s}</math>, <math>\eta_{opt}</math>, <math>\nu_1</math> and <math>\nu_2</math> are specified in below.
l\geq &\frac{{n}}{\bar{s}}\eta_{opt} -\frac{{n}}{\bar{s}}h(\omega_{exp}-\delta_{est}) -\sqrt{\frac{{n}}{\bar{s}}}\nu_1  -\mbox{leak}_{EC} \\
The security parameters of the error correction protocol, <math>\epsilon_{EC}</math> and <math>\epsilon'_{EC}</math>, mean that if the error correction step in Protocol 1 does not abort, then <math>K_A=K_B</math> with probability at least <math>1-\epsilon_{EC}</math>, and for an honest implementation, the error correction protocol aborts with probability at most <math>\epsilon'_{EC}+\epsilon_{EC}</math>.
& -3\log\Bigg(1-\sqrt{1-\Bigg(\frac{\epsilon_s}{4(\epsilon_{EA} + \epsilon_{EC})}\Bigg)^2}\Bigg)+2\log\Bigg(\frac{1}{2\epsilon_{PA}}\Bigg)
\end{align}
</math>,</br>
where <math>\mbox{leak}_{EC}</math> is the leakage due to error correction step and the functions <math>\bar{s}</math>, <math>\eta_{opt}</math>, <math>\nu_1</math> and <math>\nu_2</math> are specified in below.
The security parameters of the error correction protocol, <math>\epsilon_{EC}</math> and <math>\epsilon'_{EC}</math>, mean that if the error correction step of the protocol (see below) does not abort, then <math>K_A=K_B</math> with probability at least <math>1-\epsilon_{EC}</math>, and for an honest implementation, the error correction protocol aborts with probability at most <math>\epsilon'_{EC}+\epsilon_{EC}</math>.
*<math>\bar{s}=\frac{1-(1-\gamma)^{\left\lceil \frac{1}{\gamma} \right\rceil}}{\gamma}</math>
*<math>\bar{s}=\frac{1-(1-\gamma)^{\left\lceil \frac{1}{\gamma} \right\rceil}}{\gamma}</math>
*<math>\eta_{opt}=\max_{\frac{3}{4}<\frac{{p}_t(1)}{1-(1-\gamma)^{s_{max}}}<\frac{2+\sqrt{2}}{4}} \Bigg(F_{\min}(\vec{p},\vec{p}_t)-\frac{1}{\sqrt{m}}\nu_2\Bigg)</math>
*<math>\eta_{opt}=\max_{\frac{3}{4}<\frac{{p}_t(1)}{1-(1-\gamma)^{s_{max}}}<\frac{2+\sqrt{2}}{4}} \Bigg(F_{\min}(\vec{p},\vec{p}_t)-\frac{1}{\sqrt{m}}\nu_2\Bigg)</math>
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