Editing Quantum Key Distribution (section)

==Properties==
A quantum key distribution protocol is secure if it is ''correct'' and ''secret''. Correctness is the statement that Alice and Bob share the same string of bits, namely the secret key, at the end of the protocol. Secrecy is the statement that the eavesdropper is (nearly) ignorant about the final key. 

*'''Correctness''' A QKD protocol is <math>\epsilon_{\rm corr}</math>-correct if the probability that the final key of Alice differs from the final key of Bob, is smaller than <math>\epsilon_{\rm corr}</math>

*'''Secrecy''' A QKD protocol is <math>\epsilon_{\rm sec}</math>-secret if for every input state it holds that
<math> \frac{1}{2}{\|{\rho_{K_AE}}-{\tau_{K_A}\otimes \rho_E}\|}_1\leq \epsilon_{\rm sec},</math>
where <math>\tau_{K_A}=\frac{1}{|K_A|}\sum_{k}|{k}\rangle\langle{k}|_A</math> is the maximally mixed state in the space of strings <math>K_A</math>, and <math>{\|\cdot \|}_1</math> is the trace norm.

*A protocol implements a <math>(n,\epsilon_{\rm corr},\epsilon_{\rm sec},\ell)</math>-QKD if with <math>n</math> rounds it generates an <math>\epsilon_{\rm corr}</math>-correct and <math>\epsilon_{\rm sec}</math>-secret key of size <math>\ell</math> bits.