{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T01:12:07Z","timestamp":1728177127474},"reference-count":49,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T00:00:00Z","timestamp":1653350400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001870","name":"Foundation for Polish Science","doi-asserted-by":"crossref","award":["2018\/MAB\/5"],"award-info":[{"award-number":["2018\/MAB\/5"]}],"id":[{"id":"10.13039\/501100001870","id-type":"DOI","asserted-by":"crossref"}]},{"name":"QuantERA, an ERA-Net cofund in Quantum Technologies","award":["eDICT"],"award-info":[{"award-number":["eDICT"]}]},{"name":"NCN through grant SHENG","award":["2018\/30\/Q\/ST2\/00625"],"award-info":[{"award-number":["2018\/30\/Q\/ST2\/00625"]}]},{"name":"Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u2019s Excellence Strategy \u2013 Cluster of Excellence Matter and Light for Quantum Computing","award":["EXC 2004\/1 \u2013 390534769"],"award-info":[{"award-number":["EXC 2004\/1 \u2013 390534769"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.<\/jats:p>","DOI":"10.22331\/q-2022-05-24-717","type":"journal-article","created":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T11:27:15Z","timestamp":1653391635000},"page":"717","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Quantum Bell inequalities from Information Causality \u2013 tight for Macroscopic Locality"],"prefix":"10.22331","volume":"6","author":[{"given":"Mariami","family":"Gachechiladze","sequence":"first","affiliation":[{"name":"Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bart\u0142omiej","family":"B\u0105k","sequence":"additional","affiliation":[{"name":"Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcin","family":"Paw\u0142owski","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nikolai","family":"Miklin","sequence":"additional","affiliation":[{"name":"International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gda\u0144sk, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2022,5,24]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics Physique Fizika, 1: 195\u2013200, Nov 1964. 10.1103\/PhysicsPhysiqueFizika.1.195. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysicsPhysiqueFizika.1.195.","DOI":"10.1103\/PhysicsPhysiqueFizika.1.195"},{"key":"1","doi-asserted-by":"publisher","unstructured":"A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47: 777\u2013780, May 1935. 10.1103\/PhysRev.47.777. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRev.47.777.","DOI":"10.1103\/PhysRev.47.777"},{"key":"2","doi-asserted-by":"publisher","unstructured":"Antonio Ac\u00edn, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett., 98: 230501, Jun 2007. 10.1103\/PhysRevLett.98.230501. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.98.230501.","DOI":"10.1103\/PhysRevLett.98.230501"},{"key":"3","doi-asserted-by":"publisher","unstructured":"S. Pironio, A. Ac\u00edn, S. Massar, A. Boyer de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe. Random numbers certified by Bell&apos;s theorem. Nature, 464 (7291): 1021\u20131024, Apr 2010. ISSN 1476-4687. 10.1038\/nature09008. URL https:\/\/doi.org\/10.1038\/nature09008.","DOI":"10.1038\/nature09008"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Sergey Bravyi, David Gosset, and Robert K\u00f6nig. Quantum advantage with shallow circuits. Science, 362 (6412): 308\u2013311, 2018. ISSN 0036-8075. 10.1126\/science.aar3106.","DOI":"10.1126\/science.aar3106"},{"key":"5","doi-asserted-by":"publisher","unstructured":"John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett., 23: 880\u2013884, Oct 1969. 10.1103\/PhysRevLett.23.880. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.23.880.","DOI":"10.1103\/PhysRevLett.23.880"},{"key":"6","doi-asserted-by":"publisher","unstructured":"Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88: 040404, Jan 2002. 10.1103\/PhysRevLett.88.040404. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.88.040404.","DOI":"10.1103\/PhysRevLett.88.040404"},{"key":"7","doi-asserted-by":"publisher","unstructured":"Daniel Collins and Nicolas Gisin. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. Journal of Physics A: Mathematical and General, 37 (5): 1775\u20131787, jan 2004. 10.1088\/0305-4470\/37\/5\/021. URL https:\/\/doi.org\/10.1088\/0305-4470\/37\/5\/021.","DOI":"10.1088\/0305-4470\/37\/5\/021"},{"key":"8","doi-asserted-by":"publisher","unstructured":"Stefano Pironio. Lifting Bell inequalities. Journal of Mathematical Physics, 46 (6): 062112, 2005. 10.1063\/1.1928727. URL https:\/\/doi.org\/10.1063\/1.1928727.","DOI":"10.1063\/1.1928727"},{"key":"9","doi-asserted-by":"publisher","unstructured":"Denis Rosset, Jean-Daniel Bancal, and Nicolas Gisin. Classifying 50 years of Bell inequalities. Journal of Physics A: Mathematical and Theoretical, 47 (42): 424022, oct 2014. 10.1088\/1751-8113\/47\/42\/424022. URL https:\/\/doi.org\/10.1088\/1751-8113\/47\/42\/424022.","DOI":"10.1088\/1751-8113\/47\/42\/424022"},{"key":"10","doi-asserted-by":"publisher","unstructured":"E. Zambrini Cruzeiro and N. Gisin. Complete list of tight Bell inequalities for two parties with four binary settings. Phys. Rev. A, 99: 022104, Feb 2019. 10.1103\/PhysRevA.99.022104. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevA.99.022104.","DOI":"10.1103\/PhysRevA.99.022104"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Mariami Gachechiladze and Otfried G\u00fchne. Completing the proof of ``Generic quantum nonlocality&apos;&apos;. Physics Letters A, 381 (15): 1281\u20131285, 2017. ISSN 0375-9601. https:\/\/doi.org\/10.1016\/j.physleta.2016.10.001. URL https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0375960116311355.","DOI":"10.1016\/j.physleta.2016.10.001"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Lynden K. Shalm, Evan Meyer-Scott, Bradley G. Christensen, Peter Bierhorst, Michael A. Wayne, Martin J. Stevens, Thomas Gerrits, Scott Glancy, Deny R. Hamel, Michael S. Allman, Kevin J. Coakley, Shellee D. Dyer, Carson Hodge, Adriana E. Lita, Varun B. Verma, Camilla Lambrocco, Edward Tortorici, Alan L. Migdall, Yanbao Zhang, Daniel R. Kumor, William H. Farr, Francesco Marsili, Matthew D. Shaw, Jeffrey A. Stern, Carlos Abell\u00e1n, Waldimar Amaya, Valerio Pruneri, Thomas Jennewein, Morgan W. Mitchell, Paul G. Kwiat, Joshua C. Bienfang, Richard P. Mirin, Emanuel Knill, and Sae Woo Nam. Strong loophole-free test of local realism. Phys. Rev. Lett., 115: 250402, Dec 2015. 10.1103\/PhysRevLett.115.250402. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.115.250402.","DOI":"10.1103\/PhysRevLett.115.250402"},{"key":"13","doi-asserted-by":"publisher","unstructured":"B. Hensen, H. Bernien, A. E. Dr\u00e9au, A. Reiserer, N. Kalb, M. S. Blok, J. Ruitenberg, R. F. L. Vermeulen, R. N. Schouten, C. Abell\u00e1n, W. Amaya, V. Pruneri, M. W. Mitchell, M. Markham, D. J. Twitchen, D. Elkouss, S. Wehner, T. H. Taminiau, and R. Hanson. Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. Nature, 526 (7575): 682\u2013686, Oct 2015. ISSN 1476-4687. 10.1038\/nature15759. URL https:\/\/doi.org\/10.1038\/nature15759.","DOI":"10.1038\/nature15759"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Marissa Giustina, Marijn A. M. Versteegh, S\u00f6ren Wengerowsky, Johannes Handsteiner, Armin Hochrainer, Kevin Phelan, Fabian Steinlechner, Johannes Kofler, Jan-\u00c5ke Larsson, Carlos Abell\u00e1n, Waldimar Amaya, Valerio Pruneri, Morgan W. Mitchell, J\u00f6rn Beyer, Thomas Gerrits, Adriana E. Lita, Lynden K. Shalm, Sae Woo Nam, Thomas Scheidl, Rupert Ursin, Bernhard Wittmann, and Anton Zeilinger. Significant-loophole-free test of Bell&apos;s theorem with entangled photons. Phys. Rev. Lett., 115: 250401, Dec 2015. 10.1103\/PhysRevLett.115.250401. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.115.250401.","DOI":"10.1103\/PhysRevLett.115.250401"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Itamar Pitowsky. Correlation polytopes: Their geometry and complexity. Mathematical Programming, 50 (1): 395\u2013414, Mar 1991. ISSN 1436-4646. 10.1007\/BF01594946. URL https:\/\/doi.org\/10.1007\/BF01594946.","DOI":"10.1007\/BF01594946"},{"key":"16","doi-asserted-by":"publisher","unstructured":"B. S. Tsirel&apos;son. Quantum analogues of the Bell inequalities. The case of two spatially separated domains. Journal of Soviet Mathematics, 36 (4): 557\u2013570, Feb 1987. ISSN 1573-8795. 10.1007\/BF01663472. URL https:\/\/doi.org\/10.1007\/BF01663472.","DOI":"10.1007\/BF01663472"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Lawrence J. Landau. Empirical two-point correlation functions. Foundations of Physics, 18 (4): 449\u2013460, Apr 1988. ISSN 1572-9516. 10.1007\/BF00732549. URL https:\/\/doi.org\/10.1007\/BF00732549.","DOI":"10.1007\/BF00732549"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Jos Uffink. Quadratic Bell inequalities as tests for multipartite entanglement. Phys. Rev. Lett., 88: 230406, May 2002. 10.1103\/PhysRevLett.88.230406. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.88.230406.","DOI":"10.1103\/PhysRevLett.88.230406"},{"key":"19","doi-asserted-by":"publisher","unstructured":"S. Zohren, P. Reska, R. D. Gill, and W. Westra. A tight Tsirelson inequality for infinitely many outcomes. EPL (Europhysics Letters), 90 (1): 10002, apr 2010. 10.1209\/0295-5075\/90\/10002. URL https:\/\/doi.org\/10.1209\/0295-5075\/90\/10002.","DOI":"10.1209\/0295-5075\/90\/10002"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Tzyh Haur Yang, Miguel Navascu\u00e9s, Lana Sheridan, and Valerio Scarani. Quantum Bell inequalities from macroscopic locality. Phys. Rev. A, 83: 022105, Feb 2011. 10.1103\/PhysRevA.83.022105. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevA.83.022105.","DOI":"10.1103\/PhysRevA.83.022105"},{"key":"21","doi-asserted-by":"publisher","unstructured":"B. S. Cirel&apos;son. Quantum generalizations of Bell&apos;s inequality. Letters in Mathematical Physics, 4 (2): 93\u2013100, Mar 1980. ISSN 1573-0530. 10.1007\/BF00417500. URL https:\/\/doi.org\/10.1007\/BF00417500.","DOI":"10.1007\/BF00417500"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP* = RE. Commun. ACM, 64 (11): 131\u2013138, oct 2021. ISSN 0001-0782. 10.1145\/3485628. URL https:\/\/doi.org\/10.1145\/3485628.","DOI":"10.1145\/3485628"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Dominic Mayers and Andrew Yao. Self testing quantum apparatus. arXiv preprint quant-ph\/0307205, 2003. https:\/\/doi.org\/10.48550\/arXiv.quant-ph\/0307205.","DOI":"10.48550\/arXiv.quant-ph\/0307205"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Andrea Coladangelo, Koon Tong Goh, and Valerio Scarani. All pure bipartite entangled states can be self-tested. Nature Communications, 8 (1): 15485, May 2017. ISSN 2041-1723. 10.1038\/ncomms15485. URL https:\/\/doi.org\/10.1038\/ncomms15485.","DOI":"10.1038\/ncomms15485"},{"key":"25","doi-asserted-by":"publisher","unstructured":"Ivan \u0160upi\u0107 and Joseph Bowles. Self-testing of quantum systems: a review. Quantum, 4: 337, September 2020. ISSN 2521-327X. 10.22331\/q-2020-09-30-337. URL https:\/\/doi.org\/10.22331\/q-2020-09-30-337.","DOI":"10.22331\/q-2020-09-30-337"},{"key":"26","doi-asserted-by":"publisher","unstructured":"Marcin Paw\u0142owski, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, Andreas Winter, and Marek \u017bukowski. Information causality as a physical principle. Nature, 461 (7267): 1101\u20131104, Oct 2009. ISSN 1476-4687. 10.1038\/nature08400. URL https:\/\/doi.org\/10.1038\/nature08400.","DOI":"10.1038\/nature08400"},{"key":"27","doi-asserted-by":"publisher","unstructured":"Miguel Navascu\u00e9s and Harald Wunderlich. A glance beyond the quantum model. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466 (2115): 881\u2013890, 2010. 10.1098\/rspa.2009.0453. URL https:\/\/royalsocietypublishing.org\/doi\/abs\/10.1098\/rspa.2009.0453.","DOI":"10.1098\/rspa.2009.0453"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Carlo Rovelli. Quantum Gravity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004. 10.1017\/CBO9780511755804.","DOI":"10.1017\/CBO9780511755804"},{"key":"29","doi-asserted-by":"publisher","unstructured":"Sandu Popescu and Daniel Rohrlich. Quantum nonlocality as an axiom. Foundations of Physics, 24 (3): 379\u2013385, Mar 1994. ISSN 1572-9516. 10.1007\/BF02058098. URL https:\/\/doi.org\/10.1007\/BF02058098.","DOI":"10.1007\/BF02058098"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Marcin Paw\u0142owski and Marek \u017bukowski. Entanglement-assisted random access codes. Phys. Rev. A, 81: 042326, Apr 2010. 10.1103\/PhysRevA.81.042326. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevA.81.042326.","DOI":"10.1103\/PhysRevA.81.042326"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Miguel Navascu\u00e9s, Stefano Pironio, and Antonio Ac\u00edn. Bounding the set of quantum correlations. Phys. Rev. Lett., 98: 010401, Jan 2007. 10.1103\/PhysRevLett.98.010401. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.98.010401.","DOI":"10.1103\/PhysRevLett.98.010401"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Daniel Cavalcanti, Alejo Salles, and Valerio Scarani. Macroscopically local correlations can violate information causality. Nature Communications, 1 (1): 136, Dec 2010. ISSN 2041-1723. 10.1038\/ncomms1138. URL https:\/\/doi.org\/10.1038\/ncomms1138.","DOI":"10.1038\/ncomms1138"},{"key":"33","unstructured":"Ronald A Howard. Dynamic programming and Markov processes. John Wiley, 1960."},{"key":"34","unstructured":"Alan V Oppenheim, John R Buck, and Ronald W Schafer. Discrete-time signal processing. Vol. 2. Upper Saddle River, NJ: Prentice Hall, 2001."},{"key":"35","doi-asserted-by":"publisher","unstructured":"Wim Van Dam. Implausible consequences of superstrong nonlocality. arXiv preprint quant-ph\/0501159, 2005. https:\/\/doi.org\/10.48550\/arXiv.quant-ph\/0501159.","DOI":"10.48550\/arXiv.quant-ph\/0501159"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Wim van Dam. Implausible consequences of superstrong nonlocality. Natural Computing, 12 (1): 9\u201312, Mar 2013. ISSN 1572-9796. 10.1007\/s11047-012-9353-6. URL https:\/\/doi.org\/10.1007\/s11047-012-9353-6.","DOI":"10.1007\/s11047-012-9353-6"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Marcin Paw\u0142owski and Valerio Scarani. Information Causality, pages 423\u2013438. Springer Netherlands, Dordrecht, 2016. ISBN 978-94-017-7303-4. 10.1007\/978-94-017-7303-4_12. URL https:\/\/doi.org\/10.1007\/978-94-017-7303-4_12.","DOI":"10.1007\/978-94-017-7303-4_12"},{"key":"38","unstructured":"Robert M Fano. Transmission of information: a statistical theory of communications. Mit Press, 1968."},{"key":"39","doi-asserted-by":"publisher","unstructured":"A. J. Leggett and Anupam Garg. Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Phys. Rev. Lett., 54: 857\u2013860, Mar 1985. 10.1103\/PhysRevLett.54.857. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.54.857.","DOI":"10.1103\/PhysRevLett.54.857"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Johannes Kofler and \u010caslav Brukner. Classical world arising out of quantum physics under the restriction of coarse-grained measurements. Phys. Rev. Lett., 99: 180403, Nov 2007. 10.1103\/PhysRevLett.99.180403. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.99.180403.","DOI":"10.1103\/PhysRevLett.99.180403"},{"key":"41","doi-asserted-by":"publisher","unstructured":"George E Collins. Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In Automata theory and formal languages, pages 134\u2013183. Springer, 1975. https:\/\/doi.org\/10.1007\/3-540-07407-4_17.","DOI":"10.1007\/3-540-07407-4_17"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Karin Gatermann and Pablo A. Parrilo. Symmetry groups, semidefinite programs, and sums of squares. Journal of Pure and Applied Algebra, 192 (1): 95 \u2013 128, 2004. ISSN 0022-4049. https:\/\/doi.org\/10.1016\/j.jpaa.2003.12.011. URL http:\/\/www.sciencedirect.com\/science\/article\/pii\/S0022404904000131.","DOI":"10.1016\/j.jpaa.2003.12.011"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Denis Rosset. Characterization of correlations in quantum networks. 2015. https:\/\/doi.org\/10.13097\/archive-ouverte\/unige:77401.","DOI":"10.13097\/archive-ouverte\/unige:77401"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Nikolai Miklin and Marcin Paw\u0142owski. Information causality without concatenation. Phys. Rev. Lett., 126: 220403, Jun 2021. 10.1103\/PhysRevLett.126.220403. URL https:\/\/link.aps.org\/doi\/10.1103\/PhysRevLett.126.220403.","DOI":"10.1103\/PhysRevLett.126.220403"},{"key":"45","doi-asserted-by":"publisher","unstructured":"Miguel Navascu\u00e9s, Yelena Guryanova, Matty J Hoban, and Antonio Ac\u00edn. Almost quantum correlations. Nature Communications, 6 (1): 6288, Feb 2015. ISSN 2041-1723. 10.1038\/ncomms7288. URL https:\/\/doi.org\/10.1038\/ncomms7288.","DOI":"10.1038\/ncomms7288"},{"key":"46","doi-asserted-by":"crossref","unstructured":"G Lejeune Dirichlet. Sur la convergence des s\u00e9ries trigonom\u00e9triques qui servent \u00e0 repr\u00e9senter une fon... Journal f\u00fcr die reine und angewandte Mathematik, 4, 1829.","DOI":"10.1515\/crll.1829.4.157"},{"key":"47","doi-asserted-by":"publisher","unstructured":"Daniel Zwillinger, Victor Moll, I.S. Gradshteyn, and I.M. Ryzhik, editors. 1 - Elementary Functions. Academic Press, Boston, eighth edition edition, 2014. ISBN 978-0-12-384933-5. https:\/\/doi.org\/10.1016\/B978-0-12-384933-5.00001-1. URL https:\/\/www.sciencedirect.com\/science\/article\/pii\/B9780123849335000011.","DOI":"10.1016\/B978-0-12-384933-5.00001-1"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Thomas M Cover. Elements of information theory. 1999. 10.1002\/047174882X.","DOI":"10.1002\/047174882X"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-05-24-717\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2022,5,24]],"date-time":"2022-05-24T11:27:28Z","timestamp":1653391648000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2022-05-24-717\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,24]]},"references-count":49,"URL":"https:\/\/doi.org\/10.22331\/q-2022-05-24-717","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,24]]},"article-number":"717"}}