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However, the original proposal using polarization encoding is sensitive to polarization rotations stemming from birefringence in fibers or misalignment. To overcome this problem, here we propose a robust QKD protocol without detector vulnerabilities based on decoherence-free subspaces using polarization-entangled photon pairs. A logical Bell state analyzer is designed specifically for such encoding. The protocol exploits common parametric down-conversion sources, for which we develop a MDI-decoy-state method, and requires neither complex measurements nor a shared reference frame. We have analyzed the practical security in detail and presented a numerical simulation under various parameter regimes, showing the feasibility of the logical Bell state analyzer along with the potential that double communication distance can be achieved without a shared reference frame.<\/jats:p>","DOI":"10.3390\/e25060869","type":"journal-article","created":{"date-parts":[[2023,5,30]],"date-time":"2023-05-30T02:33:27Z","timestamp":1685414007000},"page":"869","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Measurement-Device-Independent Quantum Key Distribution Based on Decoherence-Free Subspaces with Logical Bell State Analyzer"],"prefix":"10.3390","volume":"25","author":[{"given":"Jun-Hao","family":"Wei","sequence":"first","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin-Yu","family":"Xu","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5895-825X","authenticated-orcid":false,"given":"Shu-Ming","family":"Hu","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5126-1938","authenticated-orcid":false,"given":"Qing","family":"Zhou","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nai-Le","family":"Liu","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9833-8197","authenticated-orcid":false,"given":"Kai","family":"Chen","sequence":"additional","affiliation":[{"name":"Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China, Hefei 230026, China"},{"name":"CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China"},{"name":"Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1103\/RevModPhys.74.145","article-title":"Quantum cryptography","volume":"74","author":"Gisin","year":"2002","journal-title":"Rev. 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