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BB84 Quantum Key Distribution: Difference between revisions
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\epsilon_{\rm sec}= \epsilon_{\rm PA}+\epsilon_{\rm PE},</math> | \epsilon_{\rm sec}= \epsilon_{\rm PA}+\epsilon_{\rm PE},</math> | ||
and the amount of key <math>\ell</math> that is generated is given by</br> | and the amount of key <math>\ell</math> that is generated is given by</br> | ||
<math> | <math> \begin{align} | ||
\ell \geq & (1-\gamma)^2n (1-h(Q_X+\nu) -h(Q_Z)) \\ &-\sqrt{(1-\gamma)^2n}\big(4\log(2\sqrt{2}+1)(\sqrt{\log\frac{2}{\epsilon_{\rm PE}^2}}+ \sqrt{\log \frac{8}{{\epsilon'}_{\rm EC}^2}})) \\& -\log(\frac{8}{{\epsilon'}_{\rm EC}^2}+\frac{2}{2-\epsilon'_{\rm EC}})-\log (\frac{1}{\epsilon_{\rm EC}})- 2\log(\frac{1}{2\epsilon_{\rm PA}}) \end{align}</math> | \ell \geq & (1-\gamma)^2n (1-h(Q_X+\nu) -h(Q_Z)) \\ &-\sqrt{(1-\gamma)^2n}\big(4\log(2\sqrt{2}+1)(\sqrt{\log\frac{2}{\epsilon_{\rm PE}^2}}+ \sqrt{\log \frac{8}{{\epsilon'}_{\rm EC}^2}})) \\& -\log(\frac{8}{{\epsilon'}_{\rm EC}^2}+\frac{2}{2-\epsilon'_{\rm EC}})-\log (\frac{1}{\epsilon_{\rm EC}})- 2\log(\frac{1}{2\epsilon_{\rm PA}}) | ||
\end{align} | |||
</math> | |||
</br>where <math>\nu = \sqrt{ \frac{(1+\gamma^2n)((1-\gamma)^2+\gamma^2)}{(1-\gamma)^2\gamma^4n^2}\log(\frac{1}{\epsilon_{\rm PE}}})</math> | </br>where <math>\nu = \sqrt{ \frac{(1+\gamma^2n)((1-\gamma)^2+\gamma^2)}{(1-\gamma)^2\gamma^4n^2}\log(\frac{1}{\epsilon_{\rm PE}}})</math> | ||
and <math>h(\cdot)</math> is the [[binary entropy function]]. | and <math>h(\cdot)</math> is the [[binary entropy function]]. | ||