{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:44:30Z","timestamp":1760237070905,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,16]],"date-time":"2020-02-16T00:00:00Z","timestamp":1581811200000},"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>Based on Arimoto\u2019s work in 1972, we propose an iterative algorithm for computing the capacity of a discrete memoryless classical-quantum channel with a finite input alphabet and a finite dimensional output, which we call the Blahut\u2013Arimoto algorithm for classical-quantum channel, and an input cost constraint is considered. We show that, to reach    \u03b5    accuracy, the iteration complexity of the algorithm is upper bounded by      log n log \u03b5  \u03b5     where n is the size of the input alphabet. In particular, when the output state      {  \u03c1 x  }   x \u2208 X      is linearly independent in complex matrix space, the algorithm has a geometric convergence. We also show that the algorithm reaches an    \u03b5    accurate solution with a complexity of     O (    m 3  log n log \u03b5  \u03b5  )    , and     O (  m 3  log \u03b5  log  ( 1 \u2212 \u03b4 )    \u03b5  D (  p *  | |  p  N 0   )   )     in the special case, where m is the output dimension,     D (  p *  | |  p  N 0   )     is the relative entropy of two distributions, and    \u03b4    is a positive number. Numerical experiments were performed and an approximate solution for the binary two-dimensional case was analysed.<\/jats:p>","DOI":"10.3390\/e22020222","type":"journal-article","created":{"date-parts":[[2020,2,18]],"date-time":"2020-02-18T10:10:25Z","timestamp":1582020625000},"page":"222","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Computing Classical-Quantum Channel Capacity Using Blahut\u2013Arimoto Type Algorithm: A Theoretical and Numerical Analysis"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4388-629X","authenticated-orcid":false,"given":"Haobo","family":"Li","sequence":"first","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"},{"name":"Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 201210, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ning","family":"Cai","sequence":"additional","affiliation":[{"name":"School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/TIT.1972.1054753","article-title":"An algorithm for computing the capacity of arbitrary discrete memoryless channels","volume":"18","author":"Arimoto","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1109\/TIT.1972.1054855","article-title":"Computation of channel capacity and rate-distortion functions","volume":"18","author":"Blahut","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/0034-4877(77)90010-6","article-title":"Problems in the mathematical theory of quantum communication channels","volume":"12","author":"Holevo","year":"1977","journal-title":"Rep. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1109\/18.651037","article-title":"The capacity of the quantum channel with general signal states","volume":"44","author":"Holevo","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","unstructured":"Wilde, M. (2016). From Classical to Quantum Shannon Theory, Cambridge University Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1109\/TIT.2015.2503755","article-title":"Efficient Approximation of Quantum Channel Capacities","volume":"62","author":"Sutter","year":"2016","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1007\/s10107-004-0552-5","article-title":"Smooth minimization of non-smooth functions","volume":"103","author":"Nesterov","year":"2005","journal-title":"Math. Program."},{"key":"ref_8","unstructured":"Yeung, R.W. (2008). Information Theory and Network Coding, Springer."},{"key":"ref_9","unstructured":"Nagaoka, H. (1998, January 16\u201321). Algorithms of Arimoto-Blahut type for computing quantum channel capacity. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Cambridge, MA, USA."},{"key":"ref_10","unstructured":"Beigi, S., and Shor, P.W. (2007). 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GEOMETRY OF QUANTUM STATES: An Introduction to Quantum Entanglement, Cambridge University Press.","DOI":"10.1017\/CBO9780511535048"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hiai, F., and Petz, D. (2014). Introduction to Matrix Analysis and Applications, Springer.","DOI":"10.1007\/978-3-319-04150-6"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1137\/S00361445024180","article-title":"Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later","volume":"45","author":"Moler","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"409601","DOI":"10.1088\/1751-8121\/aadb70","article-title":"Efficient optimization of the quantum relative entropy","volume":"51","year":"2018","journal-title":"J. Phys. A Math. Theor."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/222\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:58:18Z","timestamp":1760173098000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/2\/222"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,16]]},"references-count":17,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["e22020222"],"URL":"https:\/\/doi.org\/10.3390\/e22020222","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2020,2,16]]}}}