# Asymmetric Measurement-Device-Independent Quantum Key Distribution through Advantage Distillation
## 2 Methods  
### Asymmetric MDI-QKD  
Here, we mainly describe the process of the asymmetric seven-intensity MDI-QKD protocol, which develops from the four-intensity symmetric protocol, as follows:  
1. _State preparation_. [PERSON] ([PERSON]) randomly prepares the signal state only in \(Z\) basis with \(s_{A}\) (\(s_{B}\)), and prepares the decoy states only in \(X\) basis with intensities of \(w_{A}\),\(v_{A}\) (\(w_{B}\),\(v_{B}\)), satisfying the formula \(w_{A}<v_{A}\) (\(w_{B}<v_{B}\)). When preparing the vacuum state of intensity \(o\), [PERSON] ([PERSON]) does not choose any base. The prepared states will be sent to [PERSON] to perform measurement;
2. _Measurement_. [PERSON] performs the Bell state measurement (BSM) after receiving the quantum states sent from [PERSON] and [PERSON];
3. _Announcement_. After [PERSON] and [PERSON] repeat the above steps and enough counting events are recorded, [PERSON] publicly announces the BSM results. Subsequently, they announce the selected bases and intensities;
4. _Parameter estimation_. After finishing the quantum transmission phase, [PERSON] and [PERSON] can estimate the lower bound of single-photon yield \(Y_{11}^{Z,L}\) and the upper bound of single-photon error rate (QBER) \(e_{11}^{X,U}\) using the decoy-state technology;
5. _Post-processing_. [PERSON] and [PERSON] perform key reconciliation and privacy amplification on the raw key data to obtain the final secret key. The decoupled bases are used in the asymmetric seven-intensity MDI-QKD protocol, thus the protocol can perform decoy states in the \(X\) basis only to estimate \(Y_{11}^{X,L}\) and can use \(Y_{11}^{Z,L}=Y_{11}^{X,L}\) to obtain the single-photon yield in \(Z\) basis [11]. Then, the secret key rate can be calculated by the following formula [10; 11; 19]:  
\[R=P_{s_{A}}P_{s_{B}}\Big{\{}(s_{A}e^{-s_{A}})(s_{B}e^{-s_{B}})Y_{11}^{Z,L}[1- h(e_{11}^{X,U})]-f_{e}Q_{s_{A}s_{B}}^{Z}h(E_{s_{A}s_{B}}^{Z})\Big{\}}, \tag{1}\]  
where \(P_{s_{A}}\) and \(P_{s_{B}}\) each correspond to the probability that [PERSON] or [PERSON] emits the signal states of \(s_{A}\) or \(s_{B}\), respectively. \(Q_{s_{A}s_{B}}^{Z}\) and \(E_{s_{A}s_{B}}^{Z}\) are the gain and QBER in the \(Z\) basis, \(Y_{11}^{X,L}(e_{11}^{X,U})\) is the lower (upper) bound of single-photon yield (QBER), which can be estimated from the decoy-state technology, \(h(x)\) is the binary entropy function, and \(f_{e}\) is the error correction efficiency. Based on the asymmetric seven-intensity MDI-QKD protocol above, the performance can be further improved by optimization techniques such as joint estimations and collective constraints [11]. Referring to the joint estimations method, the common part \(\mathbb{H}\) is extracted from the following two parameters \(Y_{11}^{X,L}\), \(e_{11}^{X,U}\) to optimize the key rate. \(e_{11}^{Z,U}\)is used in the following subsection. \(Y_{11}^{X,L}\) is a piecewise function where \(P_{v_{A}}^{1}P_{v_{A}}^{2}P_{w_{B}}^{1}P_{v_{B}}^{2}<P_{w_{A}}^{1}P_{v_{A}}^{2}P_{v_ {B}}^{1}P_{w_{B}}^{2}P_{w_{B}}^{2}\)[19; 24]. These parameters \(Y_{11}^{X,L},e_{11}^{X,U}\) and \(e_{11}^{Z,U}\) can be estimated accurately by the decoy-state technology in the original MDI-QKD protocol [10; 11]. The following formulas can estimate these parameters which lead to a much higher rate in distilling the secure final key:  
\[