{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,24]],"date-time":"2026-10-24T00:47:07Z","timestamp":1792802827678,"version":"4.5.0"},"reference-count":53,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T00:00:00Z","timestamp":1684886400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FWF","award":["P 35509-N"],"award-info":[{"award-number":["P 35509-N"]}]},{"name":"GAMU","award":["MUNI\/G\/1596\/2019"],"award-info":[{"award-number":["MUNI\/G\/1596\/2019"]}]},{"DOI":"10.13039\/501100006109","name":"VEGA","doi-asserted-by":"crossref","award":["2\/0136\/19"],"award-info":[{"award-number":["2\/0136\/19"]}],"id":[{"id":"10.13039\/501100006109","id-type":"DOI","asserted-by":"crossref"}]},{"name":"European Research Council","award":["Consolidator grant \u2018Cocoquest\u2019 101043705"],"award-info":[{"award-number":["Consolidator grant \u2018Cocoquest\u2019 101043705"]}],"id":[{"id":"https:\/\/ror.org\/0472cxd90","id-type":"ROR","asserted-by":"crossref"}]},{"name":"JST PRESTO","award":["JPMJPR201A"],"award-info":[{"award-number":["JPMJPR201A"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"Computing the key rate in quantum key distribution (QKD) protocols is a long standing challenge. Analytical methods are limited to a handful of protocols with highly symmetric measurement bases. Numerical methods can handle arbitrary measurement bases, but either use the min-entropy, which gives a loose lower bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms. Based on a recently discovered semidefinite programming (SDP) hierarchy converging to the conditional von Neumann entropy, used for computing the asymptotic key rates in the device independent case, we introduce an SDP hierarchy that converges to the asymptotic secret key rate in the case of characterised devices. The resulting algorithm is efficient, easy to implement and easy to use. We illustrate its performance by recovering known bounds on the key rate and extending high-dimensional QKD protocols to previously intractable cases. We also use it to reanalyse experimental data to demonstrate how higher key rates can be achieved when the full statistics are taken into account.<\/jats:p>","DOI":"10.22331\/q-2023-05-24-1019","type":"journal-article","created":{"date-parts":[[2023,5,24]],"date-time":"2023-05-24T07:14:14Z","timestamp":1684912454000},"page":"1019","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":25,"title":["Quantum key distribution rates from semidefinite programming"],"prefix":"10.22331","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0155-354X","authenticated-orcid":false,"given":"Mateus","family":"Ara\u00fajo","sequence":"first","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marcus","family":"Huber","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"},{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Miguel","family":"Navascu\u00e9s","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matej","family":"Pivoluska","sequence":"additional","affiliation":[{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"},{"name":"Institute of Computer Science, Masaryk University, 602 00 Brno, Czech Republic"},{"name":"Institute of Physics, Slovak Academy of Sciences, 845 11 Bratislava, Slovakia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Armin","family":"Tavakoli","sequence":"additional","affiliation":[{"name":"Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria"},{"name":"Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2023,5,24]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Nicolas Gisin, Gr\u00e9goire Ribordy, Wolfgang Tittel, and Hugo Zbinden, ``Quantum cryptography'' Reviews of Modern Physics 74, 145-195 (2002).","DOI":"10.1103\/RevModPhys.74.145"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J. Cerf, Miloslav Du\u0161ek, Norbert L\u00fctkenhaus, and Momtchil Peev, ``The security of practical quantum key distribution'' Reviews of Modern Physics 81, 1301\u20131350 (2009).","DOI":"10.1103\/RevModPhys.81.1301"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Feihu Xu, Xiongfeng Ma, Qiang Zhang, Hoi-Kwong Lo, and Jian-Wei Pan, ``Secure quantum key distribution with realistic devices'' Reviews of Modern Physics 92, 025002 (2020).","DOI":"10.1103\/RevModPhys.92.025002"},{"key":"3","doi-asserted-by":"publisher","unstructured":"S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, ``Advances in quantum cryptography'' Advances in Optics and Photonics 12, 1012 (2020).","DOI":"10.1364\/AOP.361502"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Charles H. Bennettand Gilles Brassard ``Quantum cryptography: Public key distribution and coin tossing'' Theoretical Computer Science 560, 7\u201311 (1984) (reprint).","DOI":"10.1016\/j.tcs.2014.05.025"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Dagmar Bru\u00df ``Optimal Eavesdropping in Quantum Cryptography with Six States'' Physical Review Letters 81, 3018\u20133021 (1998).","DOI":"10.1103\/PhysRevLett.81.3018"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Nicolas J. Cerf, Mohamed Bourennane, Anders Karlsson, and Nicolas Gisin, ``Security of Quantum Key Distribution Using $d$-Level Systems'' Physical Review Letters 88, 127902 (2002).","DOI":"10.1103\/PhysRevLett.88.127902"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Lana Sheridanand Valerio Scarani ``Security proof for quantum key distribution using qudit systems'' Physical Review A 82, 030301(R) (2010).","DOI":"10.1103\/physreva.82.030301"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Robert K\u00c3\u00b6nig, Renato Renner, and Christian Schaffner, ``The operational meaning of min- and max-entropy'' IEEE Transactions on Information Theory 55, 4337\u20134347 (2009).","DOI":"10.1109\/TIT.2009.2025545"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Jean-Daniel Bancal, Lana Sheridan, and Valerio Scarani, ``More randomness from the same data'' New Journal of Physics 16, 033011 (2014).","DOI":"10.1088\/1367-2630\/16\/3\/033011"},{"key":"10","doi-asserted-by":"publisher","unstructured":"O. Nieto-Silleras, S. Pironio, and J. Silman, ``Using complete measurement statistics for optimal device-independent randomness evaluation'' New Journal of Physics 16, 013035 (2014).","DOI":"10.1088\/1367-2630\/16\/1\/013035"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Mirdit Doda, Marcus Huber, Gl\u00c3\u00a1ucia Murta, Matej Pivoluska, Martin Plesch, and Chrysoula Vlachou, ``Quantum Key Distribution Overcoming Extreme Noise: Simultaneous Subspace Coding Using High-Dimensional Entanglement'' Physical Review Applied 15, 034003 (2021).","DOI":"10.1103\/physrevapplied.15.034003"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Yukun Wang, Ignatius William Primaatmaja, Emilien Lavie, Antonios Varvitsiotis, and Charles Ci Wen Lim, ``Characterising the correlations of prepare-and-measure quantum networks'' npj Quantum Information 5, 17 (2019).","DOI":"10.1038\/s41534-019-0133-3"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Ernest Y. Z. Tan, Ren\u00e9 Schwonnek, Koon Tong Goh, Ignatius William Primaatmaja, and Charles C. W. Lim, ``Computing secure key rates for quantum cryptography with untrusted devices'' npj Quantum Information 7, 158 (2021).","DOI":"10.1038\/s41534-021-00494-z"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Adam Winick, Norbert L\u00c3\u00bctkenhaus, and Patrick J. Coles, ``Reliable numerical key rates for quantum key distribution'' Quantum 2, 77 (2018).","DOI":"10.22331\/q-2018-07-26-77"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Hao Hu, Jiyoung Im, Jie Lin, Norbert L\u00c3\u00bctkenhaus, and Henry Wolkowicz, ``Robust Interior Point Method for Quantum Key Distribution Rate Computation'' Quantum 6, 792 (2022).","DOI":"10.22331\/q-2022-09-08-792"},{"key":"16","unstructured":"Peter Brown, Hamza Fawzi, and Omar Fawzi, ``Device-independent lower bounds on the conditional von Neumann entropy'' (2021)."},{"key":"17","doi-asserted-by":"publisher","unstructured":"Miguel Navascu\u00c3\u00a9s, Stefano Pironio, and Antonio Ac\u00c3\u00adn, ``A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations'' New Journal of Physics 10, 073013 (2008).","DOI":"10.1088\/1367-2630\/10\/7\/073013"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Hoi-Kwong Lo, H. F. Chau, and M. Ardehali, ``Efficient Quantum Key Distribution Scheme and a Proof of Its Unconditional Security'' Journal of Cryptology 18, 113\u2013165 (2005).","DOI":"10.1007\/s00145-004-0142-y"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Igor Devetakand Andreas Winter ``Distillation of secret key and entanglement from quantum states'' Proceedings of the Royal Society of London Series A 461, 207\u2013235 (2005).","DOI":"10.1098\/rspa.2004.1372"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Gene H. Golub ``Some Modified Matrix Eigenvalue Problems'' SIAM Review 15, 318\u2013334 (1973).","DOI":"10.1137\/1015032"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Miguel Navascu\u00c3\u00a9s, Gonzalo de la Torre, and Tam\u00c3\u00a1s V\u00c3\u00a9rtesi, ``Characterization of Quantum Correlations with Local Dimension Constraints and Its Device-Independent Applications'' Physical Review X 4, 011011 (2014).","DOI":"10.1103\/PhysRevX.4.011011"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Dmitriy Drusvyatskiyand Henry Wolkowicz ``The many faces of degeneracy in conic optimization'' Foundations and Trends in Optimization 3, 77\u2013170 (2017).","DOI":"10.1561\/2400000011"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Karin Gatermannand Pablo A. Parrilo ``Symmetry groups, semidefinite programs, and sums of squares'' Journal of Pure and Applied Algebra 192, 95\u2013128 (2004).","DOI":"10.1016\/j.jpaa.2003.12.011"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Jos F. Sturm ``Using SeDuMi 1.02, A MATLAB toolbox for optimization over symmetric cones'' Optimization Methods and Software 11, 625\u2013653 (1999).","DOI":"10.1080\/10556789908805766"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Chris Coey, Lea Kapelevich, and Juan Pablo Vielma, ``Solving natural conic formulations with Hypatia.jl'' INFORMS Journal on Computing 34, 2686\u20132699 (2022) https:\/\/github.com\/chriscoey\/Hypatia.jl.","DOI":"10.1287\/ijoc.2022.1202"},{"key":"26","unstructured":"MOSEK ApS ``The MOSEK Optimization Suite 10.0.40'' manual (2023) https:\/\/docs.mosek.com\/latest\/intro\/index.html."},{"key":"27","doi-asserted-by":"publisher","unstructured":"J. L\u00c3\u00b6fberg ``YALMIP : a toolbox for modeling and optimization in MATLAB'' Proceedings of the CACSD Conference 284\u2013289 (2004).","DOI":"10.1109\/CACSD.2004.1393890"},{"key":"28","doi-asserted-by":"publisher","unstructured":"William K Woottersand Brian D Fields ``Optimal state-determination by mutually unbiased measurements'' Annals of Physics 191, 363\u2013381 (1989).","DOI":"10.1016\/0003-4916(89)90322-9"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Ingemar Bengtsson, Wojciech Bruzda, \u00c5sa Ericsson, Jan-\u00c5ke Larsson, Wojciech Tadej, and Karol \u00c5\u00bbyczkowski, ``Mutually unbiased bases and Hadamard matrices of order six'' Journal of Mathematical Physics 48, 052106 (2007).","DOI":"10.1063\/1.2716990"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Ingemar Bengtsson ``Three Ways to Look at Mutually Unbiased Bases'' AIP Conference Proceedings 889, 40\u201351 (2007).","DOI":"10.1063\/1.2713445"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Jessica Bavaresco, Natalia Herrera Valencia, Claude Kl\u00f6ckl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik, and Marcus Huber, ``Measurements in two bases are sufficient for certifying high-dimensional entanglement'' Nature Physics 14, 1032\u20131037 (2018).","DOI":"10.1038\/s41567-018-0203-z"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Yeong Cherng Liang, Dagomir Kaszlikowski, Berthold-Georg Englert, Leong Chuan Kwek, and C. H. Oh, ``Tomographic quantum cryptography'' Physical Review A 68, 022324 (2003).","DOI":"10.1103\/PhysRevA.68.022324"},{"key":"33","unstructured":"Yongtao Zhanand Hoi-Kwong Lo ``Tomography-based Quantum Key Distribution'' (2020)."},{"key":"34","doi-asserted-by":"publisher","unstructured":"Alexey Tiranov, S\u00e9bastien Designolle, Emmanuel Zambrini Cruzeiro, Jonathan Lavoie, Nicolas Brunner, Mikael Afzelius, Marcus Huber, and Nicolas Gisin, ``Quantification of multidimensional entanglement stored in a crystal'' Physical Review A 96, 040303 (2017).","DOI":"10.1103\/PhysRevA.96.040303"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Sebastian Ecker, Fr\u00e9d\u00e9ric Bouchard, Lukas Bulla, Florian Brandt, Oskar Kohout, Fabian Steinlechner, Robert Fickler, Mehul Malik, Yelena Guryanova, Rupert Ursin, and Marcus Huber, ``Overcoming Noise in Entanglement Distribution'' Physical Review X 9, 041042 (2019).","DOI":"10.1103\/PhysRevX.9.041042"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Lukas Bulla, Matej Pivoluska, Kristian Hjorth, Oskar Kohout, Jan Lang, Sebastian Ecker, Sebastian P. Neumann, Julius Bittermann, Robert Kindler, Marcus Huber, Martin Bohmann, and Rupert Ursin, ``Nonlocal Temporal Interferometry for Highly Resilient Free-Space Quantum Communication'' Physical Review X 13, 021001 (2023).","DOI":"10.1103\/physrevx.13.021001"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Zdenek Hradil ``Quantum-state estimation'' Physical Review A 55, R1561\u2013R1564 (1997).","DOI":"10.1103\/PhysRevA.55.R1561"},{"key":"38","doi-asserted-by":"publisher","unstructured":"V. Bu\u00c5\u00beek, R. Derka, G. Adam, and P.L. Knight, ``Reconstruction of Quantum States of Spin Systems: From Quantum Bayesian Inference to Quantum Tomography'' Annals of Physics 266, 454\u2013496 (1998).","DOI":"10.1006\/aphy.1998.5802"},{"key":"39","doi-asserted-by":"publisher","unstructured":"R\u00c3\u00bcdiger Schack, Todd A. Brun, and Carlton M. Caves, ``Quantum Bayes rule'' Physical Review A 64, 014305 (2001).","DOI":"10.1103\/PhysRevA.64.014305"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Robin Blume-Kohout ``Optimal, reliable estimation of quantum states'' New Journal of Physics 12, 043034 (2010).","DOI":"10.1088\/1367-2630\/12\/4\/043034"},{"key":"41","unstructured":"Robin Blume-Kohout ``Robust error bars for quantum tomography'' (2012)."},{"key":"42","doi-asserted-by":"publisher","unstructured":"Jiangwei Shang, Hui Khoon Ng, Arun Sehrawat, Xikun Li, and Berthold-Georg Englert, ``Optimal error regions for quantum state estimation'' New Journal of Physics 15, 123026 (2013).","DOI":"10.1088\/1367-2630\/15\/12\/123026"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Christopher Ferrie ``High posterior density ellipsoids of quantum states'' New Journal of Physics 16, 023006 (2014).","DOI":"10.1088\/1367-2630\/16\/2\/023006"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Christopher Granade, Joshua Combes, and D G Cory, ``Practical Bayesian tomography'' New Journal of Physics 18, 033024 (2016).","DOI":"10.1088\/1367-2630\/18\/3\/033024"},{"key":"45","doi-asserted-by":"crossref","unstructured":"Lukas Bulla, Kristian Hjorth, Oskar Kohout, Jan Lang, Sebastian Ecker, Sebastian P. Neumann, Julius Bittermann, Robert Kindler, Marcus Huber, Martin Bohmann, Rupert Ursin, and Matej Pivoluska, ``Distribution of genuine high-dimensional entanglement over 10.2 km of noisy metropolitan atmosphere'' (2023).","DOI":"10.1103\/PhysRevA.107.L050402"},{"key":"46","doi-asserted-by":"publisher","unstructured":"Natalia Herrera Valencia, Vatshal Srivastav, Matej Pivoluska, Marcus Huber, Nicolai Friis, Will McCutcheon, and Mehul Malik, ``High-Dimensional Pixel Entanglement: Efficient Generation and Certification'' Quantum 4, 376 (2020).","DOI":"10.22331\/q-2020-12-24-376"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Jessica Bavaresco, Mio Murao, and Marco T\u00falio Quintino, ``Strict Hierarchy between Parallel, Sequential, and Indefinite-Causal-Order Strategies for Channel Discrimination'' Physical Review Letters 127, 200504 (2021).","DOI":"10.1103\/PhysRevLett.127.200504"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Hoi-Kwong Lo, Marcos Curty, and Bing Qi, ``Measurement-Device-Independent Quantum Key Distribution'' Physical Review Letters 108, 130503 (2012).","DOI":"10.1103\/PhysRevLett.108.130503"},{"key":"49","doi-asserted-by":"publisher","unstructured":"M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, ``Overcoming the rate-distance limit of quantum key distribution without quantum repeaters'' Nature 557, 400\u2013403 (2018).","DOI":"10.1038\/s41586-018-0066-6"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Won-Young Hwang ``Quantum Key Distribution with High Loss: Toward Global Secure Communication'' Physical Review Letters 91, 057901 (2003).","DOI":"10.1103\/PhysRevLett.91.057901"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Frederic Dupuis, Omar Fawzi, and Renato Renner, ``Entropy accumulation'' Communications in Mathematical Physics 379, 867\u2013913 (2020).","DOI":"10.1007\/s00220-020-03839-5"},{"key":"52","unstructured":"Ian George, Jie Lin, Thomas van Himbeeck, Kun Fang, and Norbert L\u00fctkenhaus, ``Finite-Key Analysis of Quantum Key Distribution with Characterized Devices Using Entropy Accumulation'' (2022)."}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-05-24-1019\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2023,12,13]],"date-time":"2023-12-13T17:13:37Z","timestamp":1702487617000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2023-05-24-1019\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,24]]},"references-count":53,"URL":"https:\/\/doi.org\/10.22331\/q-2023-05-24-1019","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,24]]},"article-number":"1019"}}