# Shared Quantum Key Distribution Based on Asymmetric Double Quantum Teleportation
## 5 A QKD Protocol Based on a Shared Collaboration among Three Parties
### QKD Protocol Description Based on a Shared Generation  
In this section, we will describe how to afford a QKD scheme with the previous procedure based on double teleportation. The process is partially based on the BB84 protocol [11], but there, any sensitive information is transmitted through a quantum channel directly. Instead, it is generated by post-measurement. BB84 protocol is based on the transmission of a series of unknown states by the receiver, to then compare the outcomes between two bases independently selected. Still, they should be communicated directly through a quantum channel, nevertheless, it is relatively secure. Other protocols, as the E91 [14], exploit the entangled properties of certain states to transmit no-communicated correlations to set the key. Similarly, as the BB84, the B92 protocol [12] uses the comparison between two non-orthogonal bases to shade part of the information to a possible eavesdropper, while previously agreed correlations allow us to set the key. Nevertheless, BB92 protocol still lets an eavesdropper gain more information [46] as compared with other protocols. Thus, in this subsection, we use the previous procedure to set an improved BB84-like QKD protocol. First, [PERSON] supported by [PERSON]\({}_{0}\) (which could be assumed to be part of the same system) sets the double teleportation algorithm to virtually transmit a state just known by her. It is previously configured to be generated as one element of two different bases selected at a time by [PERSON], \(B_{A}\). It is reached by selecting \(\theta=0\), \(\frac{\pi}{2}\) and \(\phi=0\) (it implies \(p=0\)), setting after the basis \(H=\{|0\rangle,|1\rangle\}\) or \(D=\{|-\rangle,|+\rangle\}\). \(K,p_{0}\) should be selected at this point to define the control state (if the strategy is to increase the probability of success \(P_{j}\) in the stochastic step to choose the correct control state \(|b_{j}\rangle\), otherwise it could be delayed after the processing of [PERSON]\({}_{1}\)). Thus, through a public key statement dictating which \(\omega_{1}\) is applied by [PERSON]\({}_{1}\) in a concrete time (see Figure 6a exhibiting a step pseudo-random function to generate \(\omega_{1}\) for instance), [PERSON] can improve the election of the basis and the outcome for the QKD protocol. At the same time, she has settled, in advance, the future state in possession of [PERSON]\({}_{1}\) upon the selection of the parameter \(\omega_{0}\) depending on \(K\) (then also \(\beta_{0}\), related with the further measurement on the control, by using the maximal prescription \(p_{0_{max}}\)). In fact, in the process, \(K\) could be selected randomly but always secretly by a classical procedure. Figure 6: (**a**) Timely generation of \(\omega_{1}\) agreed between [PERSON] and [PERSON]\({}_{1}\) through a classical public channel, (**b**) \(SWAP\) gates used by [PERSON] to stole and reinsert the state of [PERSON]\({}_{1}\), (**c**,**d**) alternating \(CNOT\) gates to infer the [PERSON]\({}_{1}\) state by [PERSON]. Then, through the controlled processing \(\mathcal{U}_{0_{2}}\) and \(\mathcal{U}_{14}\) by [PERSON]\({}_{0}\) and [PERSON]\({}_{1}\) respectively, we get the state (18), which is ready to begin the QKD protocol, still without transmitting any sensible information. Finally, all is decided by the control measurement performed by [PERSON]. It is reached by settling an apparatus of measurement in agreement with the \(\beta_{0}\) value (which indeed was already decided upon the selection of \(\omega_{1}\) and \(K\)), and the selection of \(j,m\) in agreement with \(j+m=1\) to get the maximum \(P_{j}\) (if that is the strategy being followed). Note that \(0.5P_{j}\) represents the efficiency in the generation of a useful key, compared with \(0.5\) for the traditional BB84 protocol. Despite possibly lower, additional advantages against eavesdropping are present as it will be discussed below.