Editing Device-Independent Quantum Key Distribution
Jump to navigation
Jump to search
The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 55: | Line 55: | ||
</math></br> | </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 below. | 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 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>. | 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> | ||
*<math>F_{\min}(\vec{p},\vec{p}_t) = \frac{d}{d {p}(1)}g(\vec{p}) \Big|_{\vec{p}_t}\cdot {p}(1)+\Bigg( g(\vec{p}_t)- \frac{d}{d{p}(1)}g(\vec{p})|_{\vec{p}_t}\cdot {p}_t(1) \Bigg)</math> | *<math>F_{\min}(\vec{p},\vec{p}_t) = \frac{d}{d {p}(1)}g(\vec{p}) \Big|_{\vec{p}_t}\cdot {p}(1)+\Bigg( g(\vec{p}_t)- \frac{d}{d{p}(1)}g(\vec{p})|_{\vec{p}_t}\cdot {p}_t(1) \Bigg)</math> | ||
*<math>g({\vec{p}}) = {s}\Bigg(1-h\ | *<math>g({\vec{p}}) = {s}\Bigg(1-h\Big(\frac{1}{2}+\frac{1}{2}\sqrt{16\frac{{p}(1)}{1-(1-\gamma)^{s_{max}}}\Bigg(\frac{{p}(1)}{1-(1-\gamma)^{s_{max}}} -1}+3 )\Bigg)</math> | ||
*<math>\nu_2 =2 \ | *<math>\nu_2 =2 \Big(\log\Bigg(1+2\cdot 2^{s_{\max}}3}+\left\lceil \frac{d}{d{p}(1)}g(\vec{p})\big|_{\vec{p}_t}\right\rceil\Bigg)\sqrt{1-2\log \epsilon_s)</math> | ||
*<math>\nu_1=2 \Big(\log 7 +\left\lceil\frac{|h'(\omega_{exp}+\delta_{est})|}{1-(1-\gamma)^{s_{\max}}}\right\rceil\Big)\sqrt{1-2\log\epsilon_s}</math> | *<math>\nu_1=2 \Big(\log 7 +\left\lceil\frac{|h'(\omega_{exp}+\delta_{est})|}{1-(1-\gamma)^{s_{\max}}}\right\rceil\Big)\sqrt{1-2\log\epsilon_s}</math> | ||