{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:22:28Z","timestamp":1760239348196,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T00:00:00Z","timestamp":1605052800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We introduce a quantum key distribution protocol using mean multi-kings\u2019 problem. Using this protocol, a sender can share a bit sequence as a secret key with receivers. We consider a relation between information gain by an eavesdropper and disturbance contained in legitimate users\u2019 information. In BB84 protocol, such relation is known as the so-called information disturbance theorem. We focus on a setting that the sender and two receivers try to share bit sequences and the eavesdropper tries to extract information by interacting legitimate users\u2019 systems and an ancilla system. We derive trade-off inequalities between distinguishability of quantum states corresponding to the bit sequence for the eavesdropper and error probability of the bit sequence shared with the legitimate users. Our inequalities show that eavesdropper\u2019s extracting information regarding the secret keys inevitably induces disturbing the states and increasing the error probability.<\/jats:p>","DOI":"10.3390\/e22111275","type":"journal-article","created":{"date-parts":[[2020,11,11]],"date-time":"2020-11-11T19:08:28Z","timestamp":1605121708000},"page":"1275","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Distinguishability and Disturbance in the Quantum Key Distribution Protocol Using the Mean Multi-Kings\u2019 Problem"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2403-2093","authenticated-orcid":false,"given":"Masakazu","family":"Yoshida","sequence":"first","affiliation":[{"name":"Faculty of Design Technology, Osaka Sangyo University, 3-1-1 Daito-shi, Osaka 574-8530, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ayumu","family":"Nakayama","sequence":"additional","affiliation":[{"name":"Independent Researcher, Chiba 263-8522, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7771-2132","authenticated-orcid":false,"given":"Jun","family":"Cheng","sequence":"additional","affiliation":[{"name":"Faculty of Science and Engineering, Doshisha University, 1-3 Kyotanabe-shi, Kyoto 610-0394, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1080\/00107510010002599","article-title":"Quantum state discrimination","volume":"41","author":"Chefles","year":"2000","journal-title":"Contemp. 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