Editing Quantum Conference Key Agreement (section)

==Properties==
<!-- All properties that should be satisfied by any protocol achieving the concerned functionality and other common terminologies used in all the protocols.-->
An ideal CKA protocol, with <math>N</math> users, Alice, Bob<math>_1</math>, Bob<math>_2</math>, ..., Bob<math>_{N-1}</math> should have the following properties:

* '''Correctness:''' A CKA protocol is said to be ''correct'' if all parties receive the same key at the end of the protocol with high probability. 
: Formally, a CKA protocol is <math>\epsilon_{corr}</math>-correct if: 
:: <math>p(K_A = K_{B_1} =  ... = K_{B_{N-1}}) \geq 1 - \epsilon_{corr}</math>
: where <math>K_A, K_{B_i}</math> are the final keys held by Alice and Bob<math>_i</math>, and <math>p(K_A = K_{B_1} =  ... = K_{B_{N-1}})</math> is the probability that all final keys are identical.
* '''Secrecy:''' A CKA protocol is said to be ''secret'' if an eavesdropper Eve cannot differentiate between the key established and a random bitstring.
: Formally, a CKA protocol is <math>\epsilon_{sec}</math>-secret if, for <math>\Omega</math> being the event that the protocol does not abort,
:: <math>p(\Omega)\frac{1}{2}||\rho_{K_AE|\Omega} - \tau_{K_A}\otimes\rho_{E|\Omega}|| \leq \epsilon_{sec}</math>,
: where <math>p(\Omega)</math> is the probability of the event <math>\Omega</math>, <math>\rho_{K_AE|\Omega}</math> is the state shared by Alice and Eve at the end of the protocol given the event <math>\Omega</math>, <math>\tau_{K_A} = \frac{1}{|S|}\sum_{s_i \in S} |s_i\rangle \langle s_i|</math> is the maximally mixed state over all possible values that the key <math>K_A</math> can assume, and <math>S = \{0,1\}^{\times l}</math> where <math>l</math> is the length of the key <math>K_A</math>.
* '''Completeness:''' A quantum CKA protocol is <math>\epsilon_c</math>-complete if there exists an honest implementation of the protocols, such that the probability of not aborting is greater than <math>1-\epsilon_c</math>.