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Device-Independent Quantum Key Distribution: Difference between revisions
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</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\Bigg(\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\Bigg)+3} \Bigg)\Bigg)</math> | ||
*<math>\nu_2 =2 \ | *<math>\nu_2 =2 \Bigg(\log(1+6\cdot 2^{s_{\max}}})+\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> | ||