## 5 Conclusions  
We have examined the BBM92 protocol and shown that, in this case (and quite possibly others like it), autocompensation is either not possible (full perturbation) or not a very good solution even for simpler cases (phase). As an alternative, we have proposed another entanglement-based QKD protocol making use of the exchange parity of Bell states. Such a protocol has a simple, built-in autocompensation mechanism that allows transmission of the secure key over longer distances at a larger rate. Moreover, by construction, the BSEP protocol has MDI characteristics. This extends previous work [21] addressing purely MDI protocols, where entanglement occurred only at the final time upon measurement by postselection, or more conventional plug and play versions of the BB84 protocol [12], which are not robust in general against detection side-channel attacks. We believe that the BSEP protocol does not introduce new backdoors for an eavesdropper to take advantage of, apart from the proposed attack, for which we present a simple way to tackle. That said, the protocol we propose is not free from other attacks common to such classes of MDI-QKD protocols. In particular, the MDI protocol is still vulnerable to other attacks, except those on the measurement system, for example, the source [48], as the MDI protocol does not provide full device independence as DI-QKD does [49], which nonetheless remains experimentally challenging. At the present time, MDI protocols remain the most feasible QKD protocols that are able to avoid a general class of attacks; thus, the interest in working on such a direction, as we did, is high. To compare BBM92 with the new BSEP protocol, we have analyzed the relevant perturbation errors. We have conducted a simplified but clarifying security analysis by introducing these errors into a simple version of the key rate, taking into account real-life optical hardware imperfections such as fiber attenuation, detector inefficiency, and detector dark counts. In short, the proposed QKD protocol inherently removes coupling and phase perturbations in two-mode fiber-optic systems, overcoming prior problems associated with the direct use of entangled states for information encoding, such as in the BBM92 protocol.  
Figure 6: Secret key rate comparison between protocols for the realistic value \(\gamma_{\theta}=5\times 10^{-5}\) (units of km\({}^{-1}\)) and the hypothetical value \(\gamma_{\delta}=3\times 10^{-2}\). Four cases are considered: BBM92 with full perturbations (phase + coupling) and no autocompensation (blue dotted line); BBM92 when only phase perturbations are present with and without autocompensation (green and orange dotted lines, respectively); and the BSEP protocol (red solid line).  
**Author Contributions:** Conceptualization, G.M.C., J.L., X.P.-B. and E.F.M.; methodology, G.M.C., J.L. and X.P.-B.; software, G.M.C.; validation, G.M.C., J.L., X.P.-B. and E.F.M.; formal analysis, G.M.C., J.L. and X.P.-B.; writing--original draft preparation, G.M.C.; writing--review and editing, G.M.C., J.L., X.P.-B. and E.F.M.; supervision, J.L. and X.P.-B. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This work was supported in part by the MICIN, European Union Next Generation EU under Grant PRTR-C17.II, and in part by the Galician Regional Government through Planes Complementarios de I + D + i con las Comunidades Autonomas in quantum communication.  
**Data Availability Statement:** Data are contained within the article. The code used for the simulations is available upon reasonable request to the authors.  
**Acknowledgments:** We acknowledge financial support by the Galician Regional Government through a predoctoral grant (G.M.C-2020), co-financed by the European Social Fund.  
**Conflicts of Interest:** Author [PERSON] was employed by the company NEC-Corporation. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.