{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T21:29:21Z","timestamp":1774387761469,"version":"3.50.1"},"reference-count":81,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T00:00:00Z","timestamp":1618444800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T00:00:00Z","timestamp":1618444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-18-CE47-0011 (ACOM)"],"award-info":[{"award-number":["ANR-18-CE47-0011 (ACOM)"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2022,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathrm {Tr}}\\big [H(D\\otimes E)\\big ]$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>Tr<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>[<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mi>H<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>D<\/mml:mi>\n                    <mml:mo>\u2297<\/mml:mo>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>]<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, maximized with respect to semidefinite constraints on <jats:italic>D<\/jats:italic> and <jats:italic>E<\/jats:italic>. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo\u2019s asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brand\u00e3o and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC\u201912, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from <jats:inline-formula><jats:alternatives><jats:tex-math>$$O(d^{k\/2})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>O<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathrm {poly}}(d,k)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>poly<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>d<\/jats:italic> denotes local dimension and <jats:italic>k<\/jats:italic> the number of copies.<\/jats:p>","DOI":"10.1007\/s10107-021-01650-1","type":"journal-article","created":{"date-parts":[[2021,4,15]],"date-time":"2021-04-15T12:26:08Z","timestamp":1618489568000},"page":"781-829","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Semidefinite programming hierarchies for constrained bilinear optimization"],"prefix":"10.1007","volume":"194","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0428-3429","authenticated-orcid":false,"given":"Mario","family":"Berta","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4067-6617","authenticated-orcid":false,"given":"Francesco","family":"Borderi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8491-0359","authenticated-orcid":false,"given":"Omar","family":"Fawzi","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3235-022X","authenticated-orcid":false,"given":"Volkher B.","family":"Scholz","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,15]]},"reference":[{"issue":"2","key":"1650_CR1","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1287\/moor.8.2.273","volume":"8","author":"FA Al-Khayyal","year":"1983","unstructured":"Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8(2), 273 (1983)","journal-title":"Math. Oper. Res."},{"key":"1650_CR2","unstructured":"ApS, M.: The MOSEK optimization toolbox for MATLAB manual. Version 8.1 (2017)"},{"issue":"5","key":"1650_CR3","doi-asserted-by":"publisher","first-page":"052203","DOI":"10.1063\/1.4921341","volume":"56","author":"R Arnon-Friedman","year":"2015","unstructured":"Arnon-Friedman, R., Renner, R.: De Finetti reductions for correlations. J. Math. Phys. 56(5), 052203 (2015)","journal-title":"J. Math. Phys."},{"key":"1650_CR4","doi-asserted-by":"crossref","unstructured":"Barak, B., Brandao, F.G.S.L., Harrow, A.W., Kelner, J., Steurer, D., Zhou, Y.: Hypercontractivity, sum-of-squares proofs, and their applications. In: Proceedings of the Forty-fourth Annual ACM Symposium on Theory of Computing, STOC\u201912, p. 307 (2012)","DOI":"10.1145\/2213977.2214006"},{"issue":"2","key":"1650_CR5","doi-asserted-by":"publisher","first-page":"1038","DOI":"10.1109\/TIT.2017.2696963","volume":"64","author":"S Barman","year":"2018","unstructured":"Barman, S., Fawzi, O.: Algorithmic aspects of optimal channel coding. IEEE Trans. Inf. Theory 64(2), 1038 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"12","key":"1650_CR6","doi-asserted-by":"publisher","first-page":"122202","DOI":"10.1063\/1.4838855","volume":"54","author":"S Beigi","year":"2013","unstructured":"Beigi, S.: Sandwiched R\u00e9nyi divergence satisfies data processing inequality. J. Math. Phys. 54(12), 122202 (2013)","journal-title":"J. Math. Phys."},{"issue":"5","key":"1650_CR7","doi-asserted-by":"publisher","first-page":"3824","DOI":"10.1103\/PhysRevA.54.3824","volume":"54","author":"CH Bennett","year":"1996","unstructured":"Bennett, C.H., DiVincenzo, D.P., Smolin, J.A., Wootters, W.K.: Mixed-state entanglement and quantum error correction. Phys. Rev. A 54(5), 3824 (1996)","journal-title":"Phys. Rev. A"},{"issue":"3","key":"1650_CR8","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1007\/s00220-011-1309-7","volume":"306","author":"M Berta","year":"2011","unstructured":"Berta, M., Christandl, M., Renner, R.: The quantum reverse Shannon theorem based on one-shot information theory. Commun. Math. Phys. 306(3), 579 (2011)","journal-title":"Commun. Math. Phys."},{"issue":"3","key":"1650_CR9","doi-asserted-by":"publisher","first-page":"1529","DOI":"10.1137\/15M1037731","volume":"26","author":"M Berta","year":"2016","unstructured":"Berta, M., Fawzi, O., Scholz, V.B.: Quantum bilinear optimization. SIAM J. Optim. 26(3), 1529 (2016)","journal-title":"SIAM J. Optim."},{"key":"1650_CR10","volume-title":"Matrix Analysis. Graduate Texts in Mathematics","author":"R Bhatia","year":"1997","unstructured":"Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics. Springer, Berlin (1997)"},{"issue":"3","key":"1650_CR11","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1007\/s00220-011-1302-1","volume":"306","author":"FGSL Brandao","year":"2011","unstructured":"Brandao, F.G.S.L., Christandl, M., Yard, J.: Faithful squashed entanglement. Commun. Math. Phys. 306(3), 80 (2011)","journal-title":"Commun. Math. Phys."},{"issue":"1","key":"1650_CR12","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s00220-016-2575-1","volume":"342","author":"FGSL Brandao","year":"2016","unstructured":"Brandao, F.G.S.L., Harrow, A.W.: Product-state approximations to quantum ground states. Commun. Math. Phys. 342(1), 47 (2016)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"1650_CR13","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1007\/s00220-017-2880-3","volume":"353","author":"FGSL Brandao","year":"2017","unstructured":"Brandao, F.G.S.L., Harrow, A.W.: Quantum de Finetti theorems under local measurements with applications. Commun. Math. Phys. 353(2), 469 (2017)","journal-title":"Commun. Math. Phys."},{"issue":"5","key":"1650_CR14","doi-asserted-by":"publisher","first-page":"050501","DOI":"10.1103\/PhysRevLett.115.050501","volume":"115","author":"FGSL Brandao","year":"2015","unstructured":"Brandao, F.G.S.L., Harrow, A.W., Oppenheim, J., Strelchuk, S.: Quantum conditional mutual information, reconstructed states, and state redistribution. Phys. Rev. Lett. 115(5), 050501 (2015)","journal-title":"Phys. Rev. Lett."},{"issue":"9","key":"1650_CR15","doi-asserted-by":"publisher","first-page":"4537","DOI":"10.1063\/1.1494475","volume":"43","author":"CM Caves","year":"2002","unstructured":"Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43(9), 4537 (2002)","journal-title":"J. Math. Phys."},{"issue":"1","key":"1650_CR16","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s00220-014-1953-9","volume":"328","author":"E Chitambar","year":"2014","unstructured":"Chitambar, E., Leung, D., Man\u010dinska, L., Ozols, M., Winter, A.: Everything you always wanted to know about locc (but were afraid to ask). Commun. Math. Phys. 328(1), 303\u2013326 (2014)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"1650_CR17","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s00220-007-0189-3","volume":"273","author":"M Christandl","year":"2007","unstructured":"Christandl, M., K\u00f6nig, R., Mitchison, G., Renner, R.: One-and-a-half quantum de Finetti theorems. Commun. Math. Phys. 273(2), 473 (2007)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"1650_CR18","doi-asserted-by":"publisher","first-page":"020504","DOI":"10.1103\/PhysRevLett.102.020504","volume":"102","author":"M Christandl","year":"2009","unstructured":"Christandl, M., K\u00f6nig, R., Renner, R.: Postselection technique for quantum channels with applications to quantum cryptography. Phys. Rev. Lett. 102(2), 020504 (2009)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"1650_CR19","doi-asserted-by":"publisher","first-page":"042104","DOI":"10.1063\/1.3114986","volume":"50","author":"M Christandl","year":"2009","unstructured":"Christandl, M., Toner, B.: Finite de Finetti theorem for conditional probability distributions describing physical theories. J. Math. Phys. 50(4), 042104 (2009)","journal-title":"J. Math. Phys."},{"key":"1650_CR20","doi-asserted-by":"publisher","first-page":"052306","DOI":"10.1103\/PhysRevA.61.052306","volume":"61","author":"V Coffman","year":"2000","unstructured":"Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61, 052306 (2000)","journal-title":"Phys. Rev. A"},{"issue":"1","key":"1650_CR21","first-page":"1","volume":"7","author":"B de Finetti","year":"1937","unstructured":"de Finetti, B.: La pr\u00e9vision?: ses lois logiques, ses sources subjectives. Ann. Inst. Henri Poincar\u00e9 7(1), 1 (1937)","journal-title":"Ann. Inst. Henri Poincar\u00e9"},{"issue":"4","key":"1650_CR22","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1214\/aop\/1176994663","volume":"8","author":"P Diaconis","year":"1980","unstructured":"Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab. 8(4), 745 (1980)","journal-title":"Ann. Probab."},{"issue":"2","key":"1650_CR23","doi-asserted-by":"publisher","first-page":"830","DOI":"10.1103\/PhysRevA.57.830","volume":"57","author":"D DiVincenzo","year":"1998","unstructured":"DiVincenzo, D., Shor, P., Smolin, J.: Quantum-channel capacity of very noisy channels. Phys. Rev. A 57(2), 830 (1998)","journal-title":"Phys. Rev. A"},{"issue":"18","key":"1650_CR24","doi-asserted-by":"publisher","first-page":"187904","DOI":"10.1103\/PhysRevLett.88.187904","volume":"88","author":"AC Doherty","year":"2002","unstructured":"Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Distinguishing separable and entangled states. Phys. Rev. Lett. 88(18), 187904 (2002)","journal-title":"Phys. Rev. Lett."},{"issue":"2","key":"1650_CR25","doi-asserted-by":"publisher","first-page":"022308","DOI":"10.1103\/PhysRevA.69.022308","volume":"69","author":"AC Doherty","year":"2004","unstructured":"Doherty, A.C., Parrilo, P.A., Spedalieri, F.M.: Complete family of separability criteria. Phys. Rev. A 69(2), 022308 (2004)","journal-title":"Phys. Rev. A"},{"issue":"2","key":"1650_CR26","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1109\/TIT.2015.2507979","volume":"62","author":"R Duan","year":"2016","unstructured":"Duan, R., Winter, A.: No-signalling-assisted zero-error capacity of quantum channels and an information theoretic interpretation of the lov\u00e1sz number. IEEE Trans. Inf. Theory 62(2), 891 (2016)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1650_CR27","first-page":"1","volume":"2020","author":"K Fang","year":"2020","unstructured":"Fang, K., Fawzi, H.: The sum-of-squares hierarchy on the sphere and applications in quantum information theory. Math. Program. 2020, 1\u201330 (2020)","journal-title":"Math. Program."},{"issue":"3","key":"1650_CR28","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1007\/BF00398595","volume":"15","author":"M Fannes","year":"1988","unstructured":"Fannes, M., Lewis, J.T., Verbeure, A.: Symmetric states of composite systems. Lett. Math. Phys. 15(3), 255 (1988)","journal-title":"Lett. Math. Phys."},{"issue":"2","key":"1650_CR29","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1007\/s00220-015-2466-x","volume":"340","author":"O Fawzi","year":"2015","unstructured":"Fawzi, O., Renner, R.: Quantum conditional mutual information and approximate Markov chains. Commun. Math. Phys. 340(2), 575 (2015)","journal-title":"Commun. Math. Phys."},{"key":"1650_CR30","unstructured":"Fazel, M., Hindi, H., Boyd, S.P.: Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices. In: Proceedings of the 2003 American Control Conference, 2003, vol.\u00a03, pp. 2156\u20132162 (2003)"},{"key":"1650_CR31","doi-asserted-by":"crossref","unstructured":"Fletcher, A.S.: Channel-Adapted Quantum Error Correction. PhD thesis, Massachusetts Institute of Technology (2007)","DOI":"10.1103\/PhysRevA.77.012320"},{"issue":"1","key":"1650_CR32","doi-asserted-by":"publisher","first-page":"012338","DOI":"10.1103\/PhysRevA.75.012338","volume":"75","author":"AS Fletcher","year":"2007","unstructured":"Fletcher, A.S., Shor, P.W., Win, M.Z.: Optimum quantum error recovery using semidefinite programming. Phys. Rev. A 75(1), 012338 (2007)","journal-title":"Phys. Rev. A"},{"key":"1650_CR33","first-page":"147","volume-title":"Unknown Quantum States and Operations, a Bayesian View","author":"CA Fuchs","year":"2004","unstructured":"Fuchs, C.A., Schack, R.: Unknown Quantum States and Operations, a Bayesian View, p. 147. Springer Berlin Heidelberg, Berlin (2004)"},{"issue":"6","key":"1650_CR34","doi-asserted-by":"publisher","first-page":"062305","DOI":"10.1103\/PhysRevA.69.062305","volume":"69","author":"CA Fuchs","year":"2004","unstructured":"Fuchs, C.A., Schack, R., Scudo, P.F.: De Finetti representation theorem for quantum-process tomography. Phys. Rev. A 69(6), 062305 (2004)","journal-title":"Phys. Rev. A"},{"key":"1650_CR35","unstructured":"Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming (2008)"},{"key":"1650_CR36","doi-asserted-by":"crossref","unstructured":"Harrow, A.W., Natarajan, A., Wu, X.: Limitations of semidefinite programs for separable states and entangled games. Commun. Math. Phys. 366, 423\u2013468 (2019)","DOI":"10.1007\/s00220-019-03382-y"},{"issue":"11","key":"1650_CR37","doi-asserted-by":"publisher","first-page":"4947","DOI":"10.1109\/TIT.2009.2030478","volume":"55","author":"M Hayashi","year":"2009","unstructured":"Hayashi, M.: Information spectrum approach to second-order coding rate in channel coding. IEEE Trans. Inf. Theory 55(11), 4947 (2009)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"10","key":"1650_CR38","doi-asserted-by":"publisher","first-page":"102201","DOI":"10.1063\/1.4964755","volume":"57","author":"M Hayashi","year":"2016","unstructured":"Hayashi, M., Tomamichel, M.: Correlation detection and an operational interpretation of the R\u00e9nyi mutual information. J. Math. Phys. 57(10), 102201 (2016)","journal-title":"J. Math. Phys."},{"issue":"3","key":"1650_CR39","doi-asserted-by":"publisher","first-page":"032310","DOI":"10.1103\/PhysRevA.62.032310","volume":"62","author":"P Horodecki","year":"2000","unstructured":"Horodecki, P., Lewenstein, M., Vidal, G., Cirac, I.: Operational criterion and constructive checks for the separability of low-rank density matrices. Phys. Rev. A 62(3), 032310 (2000)","journal-title":"Phys. Rev. A"},{"key":"1650_CR40","doi-asserted-by":"publisher","first-page":"865","DOI":"10.1103\/RevModPhys.81.865","volume":"81","author":"R Horodecki","year":"2009","unstructured":"Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865\u2013942 (2009)","journal-title":"Rev. Mod. Phys."},{"key":"1650_CR41","doi-asserted-by":"crossref","unstructured":"Huber, S., Koenig, R., Tomamichel, M.: Jointly constrained semidefinite bilinear programming with an application to Dobrushin curves. IEEE Trans. Inf. Theory Early Access (2019)","DOI":"10.1109\/TIT.2019.2939474"},{"issue":"4","key":"1650_CR42","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1007\/BF00534784","volume":"33","author":"RL Hudson","year":"1976","unstructured":"Hudson, R.L., Moody, G.R.: Locally normal symmetric states and an analogue of de Finetti\u2019s theorem. Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und Verwandte Gebiete 33(4), 343 (1976)","journal-title":"Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und Verwandte Gebiete"},{"key":"1650_CR43","unstructured":"Johnson, P.D., Romero, J., Olson, J., Cao, Y., Aspuru-Guzik, A.: QVECTOR: an algorithm for device-tailored quantum error correction. arXiv:1711.02249 (2017)"},{"key":"1650_CR44","unstructured":"Johnston, N.: Qetlab: A matlab toolbox for quantum entanglement, version 0.9 (2016)"},{"key":"1650_CR45","doi-asserted-by":"publisher","first-page":"070502","DOI":"10.1103\/PhysRevLett.123.070502","volume":"123","author":"E Kaur","year":"2019","unstructured":"Kaur, E., Das, S., Wilde, M.M., Winter, A.: Extendibility limits the performance of quantum processors. Phys. Rev. Lett. 123, 070502 (2019)","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"1650_CR46","doi-asserted-by":"publisher","first-page":"012105","DOI":"10.1063\/1.3049751","volume":"50","author":"R Koenig","year":"2009","unstructured":"Koenig, R., Mitchison, G.: A most compendious and facile quantum de Finetti theorem. J. Math. Phys. 50(1), 012105 (2009)","journal-title":"J. Math. Phys."},{"issue":"12","key":"1650_CR47","doi-asserted-by":"publisher","first-page":"122108","DOI":"10.1063\/1.2146188","volume":"46","author":"R Koenig","year":"2005","unstructured":"Koenig, R., Renner, R.: A de Finetti representation for finite symmetric quantum states. J. Math. Phys. 46(12), 122108 (2005)","journal-title":"J. Math. Phys."},{"issue":"1","key":"1650_CR48","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/BF01580367","volume":"11","author":"H Konno","year":"1976","unstructured":"Konno, H.: A cutting plane algorithm for solving bilinear programs. Math. Program. 11(1), 14 (1976)","journal-title":"Math. Program."},{"issue":"5","key":"1650_CR49","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s11128-009-0120-2","volume":"8","author":"RL Kosut","year":"2009","unstructured":"Kosut, R.L., Lidar, D.A.: Quantum error correction via convex optimization. Quantum Inf. Process. 8(5), 443 (2009)","journal-title":"Quantum Inf. Process."},{"issue":"1","key":"1650_CR50","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1088\/1367-2630\/6\/1\/026","volume":"6","author":"D Kretschmann","year":"2004","unstructured":"Kretschmann, D., Werner, R.F.: Tema con variazioni: quantum channel capacity. New J. Phys. 6(1), 26 (2004)","journal-title":"New J. Phys."},{"issue":"2","key":"1650_CR51","doi-asserted-by":"publisher","first-page":"555","DOI":"10.1007\/s00220-013-1779-x","volume":"323","author":"C Lancien","year":"2013","unstructured":"Lancien, C., Winter, A.: Distinguishing multi-partite states by local measurements. Commun. Math. Phys. 323(2), 555 (2013)","journal-title":"Commun. Math. Phys."},{"issue":"3","key":"1650_CR52","doi-asserted-by":"publisher","first-page":"796","DOI":"10.1137\/S1052623400366802","volume":"11","author":"JB Lasserre","year":"2000","unstructured":"Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796 (2000)","journal-title":"SIAM J. Optim."},{"issue":"8","key":"1650_CR53","doi-asserted-by":"publisher","first-page":"4486","DOI":"10.1109\/TIT.2015.2439953","volume":"61","author":"D Leung","year":"2015","unstructured":"Leung, D., Matthews, W.: On the power of PPT-preserving and non-signalling codes. IEEE Trans. Inf. Theory 61(8), 4486 (2015)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"1650_CR54","doi-asserted-by":"publisher","first-page":"2567","DOI":"10.1103\/PhysRevA.56.2567","volume":"56","author":"DW Leung","year":"1997","unstructured":"Leung, D.W., Nielsen, M.A., Chuang, I.L., Yamamoto, Y.: Approximate quantum error correction can lead to better codes. Phys. Rev. A 56(4), 2567 (1997)","journal-title":"Phys. Rev. A"},{"issue":"12","key":"1650_CR55","doi-asserted-by":"publisher","first-page":"7036","DOI":"10.1109\/TIT.2012.2210695","volume":"58","author":"W Matthews","year":"2012","unstructured":"Matthews, W.: A linear program for the finite block length converse of Polyanskiy-Poor-Verd\u00fa via nonsignaling codes. IEEE Trans. Inf. Theory 58(12), 7036 (2012)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"5","key":"1650_CR56","doi-asserted-by":"publisher","first-page":"052306","DOI":"10.1103\/PhysRevA.80.052306","volume":"80","author":"M Navascu\u00e9s","year":"2009","unstructured":"Navascu\u00e9s, M., Owari, M., Plenio, M.B.: Power of symmetric extensions for entanglement detection. Phys. Rev. A 80(5), 052306 (2009)","journal-title":"Phys. Rev. A"},{"key":"1650_CR57","unstructured":"Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information (2000)"},{"issue":"1","key":"1650_CR58","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1006\/jfan.1998.3342","volume":"161","author":"R Olkiewicz","year":"1999","unstructured":"Olkiewicz, R., Zegarlinski, B.: Hypercontractivity in noncommutative lp-spaces. J. Funct. Anal. 161(1), 246 (1999)","journal-title":"J. Funct. Anal."},{"issue":"9\u201310","key":"1650_CR59","first-page":"751","volume":"13","author":"L Pankowski","year":"2013","unstructured":"Pankowski, L., Brandao, F.G.S.L., Horodecki, M., Smith, G.: Entanglement distillation by extendible maps. Quantum Inf. Comput. 13(9\u201310), 751\u2013770 (2013)","journal-title":"Quantum Inf. Comput."},{"issue":"2","key":"1650_CR60","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/s10107-003-0387-5","volume":"96","author":"PA Parrilo","year":"2003","unstructured":"Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293 (2003)","journal-title":"Math. Program."},{"issue":"1","key":"1650_CR61","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01377629","volume":"85","author":"D Petz","year":"1990","unstructured":"Petz, D.: A de Finetti-type theorem withm-dependent states. Probab. Theory Relat. Fields 85(1), 65 (1990)","journal-title":"Probab. Theory Relat. Fields"},{"key":"1650_CR62","doi-asserted-by":"publisher","first-page":"2157","DOI":"10.1137\/090760155","volume":"20","author":"S Pironio","year":"2010","unstructured":"Pironio, S., Navascu\u00e9s, M., Ac\u00edn, A.: Convergent relaxations of polynomial optimization problems with noncommuting variables. SIAM J. Optim. 20, 2157\u20132180 (2010)","journal-title":"SIAM J. Optim."},{"issue":"5","key":"1650_CR63","doi-asserted-by":"publisher","first-page":"2307","DOI":"10.1109\/TIT.2010.2043769","volume":"56","author":"Y Polyanskiy","year":"2010","unstructured":"Polyanskiy, Y., Poor, H.V., Verd\u00fa, S.: Channel coding rate in the finite blocklength regime. IEEE Trans. Inf. Theory 56(5), 2307 (2010)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1650_CR64","first-page":"980","volume":"62","author":"GA Raggio","year":"1989","unstructured":"Raggio, G.A., Werner, R.F.: Quantum statistical mechanics of general mean field systems. Helv. Phys. Acta 62, 980 (1989)","journal-title":"Helv. Phys. Acta"},{"key":"1650_CR65","unstructured":"Reimpell, M.: Quantum Information and Convex Optimization. PhD thesis, TU Braunschweig (2008)"},{"issue":"8","key":"1650_CR66","doi-asserted-by":"publisher","first-page":"080501","DOI":"10.1103\/PhysRevLett.94.080501","volume":"94","author":"M Reimpell","year":"2005","unstructured":"Reimpell, M., Werner, R.F.: Iterative optimization of quantum error correcting codes. Phys. Rev. Lett. 94(8), 080501 (2005)","journal-title":"Phys. Rev. Lett."},{"key":"1650_CR67","unstructured":"Rosset, D.: Symdpoly: symmetry-adapted moment relaxations for noncommutative polynomial optimization. arXiv preprint arXiv:1808.09598 (2018)"},{"issue":"5","key":"1650_CR68","doi-asserted-by":"publisher","first-page":"055308","DOI":"10.1088\/1751-8113\/41\/5\/055308","volume":"41","author":"AJ Scott","year":"2008","unstructured":"Scott, A.J.: Optimizing quantum process tomography with unitary 2-designs. J. Phys. A: Math. Theor. 41(5), 055308 (2008)","journal-title":"J. Phys. A: Math. Theor."},{"issue":"1","key":"1650_CR69","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/0022-1236(69)90050-0","volume":"3","author":"E St\u00f8rmer","year":"1969","unstructured":"St\u00f8rmer, E.: Symmetric states of infinite tensor products of C*-algebras. J. Funct. Anal. 3(1), 48 (1969)","journal-title":"J. Funct. Anal."},{"issue":"3","key":"1650_CR70","doi-asserted-by":"publisher","first-page":"1461","DOI":"10.1109\/TIT.2009.2039162","volume":"56","author":"S Taghavi","year":"2010","unstructured":"Taghavi, S., Kosut, R.L., Lidar, D.A.: Channel-optimized quantum error correction. IEEE Trans. Inf. Theory 56(3), 1461 (2010)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1650_CR71","doi-asserted-by":"crossref","unstructured":"Toh, K.-C., Todd, M.J., T\u00fct\u00fcnc\u00fc, R.H.: On the Implementation and Usage of SDPT3\u2014A Matlab Software Package for Semidefinite-Quadratic-Linear Programming, Version 4.0, pp. 715\u2013754. Springer US, Boston, MA (2012)","DOI":"10.1007\/978-1-4614-0769-0_25"},{"key":"1650_CR72","doi-asserted-by":"publisher","first-page":"11419","DOI":"10.1038\/ncomms11419","volume":"7","author":"M Tomamichel","year":"2016","unstructured":"Tomamichel, M., Berta, M., Renes, J.M.: Quantum coding with finite resources. Nat. Commun. 7, 11419 (2016)","journal-title":"Nat. Commun."},{"issue":"10","key":"1650_CR73","doi-asserted-by":"publisher","first-page":"103032","DOI":"10.1088\/1367-2630\/16\/10\/103032","volume":"16","author":"JJ Wallman","year":"2014","unstructured":"Wallman, J.J., Flammia, S.T.: Randomized benchmarking with confidence. New J. Phys. 16(10), 103032 (2014)","journal-title":"New J. Phys."},{"key":"1650_CR74","doi-asserted-by":"crossref","unstructured":"Wang, X., Duan, R.: A semidefinite programming upper bound of quantum capacity. In: Proceedings IEEE ISIT 2016, p. 1690 (2016)","DOI":"10.1109\/ISIT.2016.7541587"},{"issue":"4","key":"1650_CR75","first-page":"2581","volume":"65","author":"X Wang","year":"2018","unstructured":"Wang, X., Fang, K., Duan, R.: Semidefinite programming converse bounds for quantum communication. IEEE Trans. Inf. Theory 65(4), 2581\u20132592 (2018)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1650_CR76","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1109\/TIT.2017.2741101","volume":"64","author":"X Wang","year":"2017","unstructured":"Wang, X., Xie, W., Duan, R.: Semidefinite programming strong converse bounds for classical capacity. IEEE Trans. Inf. Theory 64, 640\u2013653 (2017)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"11","key":"1650_CR77","doi-asserted-by":"publisher","first-page":"217","DOI":"10.4086\/toc.2009.v005a011","volume":"5","author":"J Watrous","year":"2009","unstructured":"Watrous, J.: Semidefinite programs for completely bounded norms. Theory Comput. 5(11), 217\u2013238 (2009)","journal-title":"Theory Comput."},{"key":"1650_CR78","doi-asserted-by":"publisher","DOI":"10.1017\/9781316848142","volume-title":"The Theory of Quantum Information","author":"J Watrous","year":"2018","unstructured":"Watrous, J.: The Theory of Quantum Information, 1st edn. Cambridge University Press, Cambridge (2018)","edition":"1"},{"issue":"3","key":"1650_CR79","first-page":"1","volume":"1","author":"RF Werner","year":"2001","unstructured":"Werner, R.F., Wolf, M.M.: Bell inequalities and entanglement. Quantum Inf. Comput. 1(3), 1 (2001)","journal-title":"Quantum Inf. Comput."},{"key":"1650_CR80","doi-asserted-by":"publisher","unstructured":"Wilde, M.M.: Quantum Information Theory. Cambridge University Press, Cambridge. https:\/\/doi.org\/10.1017\/CBO9781139525343","DOI":"10.1017\/CBO9781139525343"},{"key":"1650_CR81","unstructured":"Wolf, M.: Quantum channels and operations: Guided tour. Lecture notes https:\/\/wwwm5.ma.tum.de\/foswiki\/pub\/M5\/Allgemeines\/MichaelWolf\/QChannelLecture.pdf (July 2012)"}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-021-01650-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-021-01650-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-021-01650-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,31]],"date-time":"2023-01-31T13:10:04Z","timestamp":1675170604000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-021-01650-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,4,15]]},"references-count":81,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2022,7]]}},"alternative-id":["1650"],"URL":"https:\/\/doi.org\/10.1007\/s10107-021-01650-1","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,4,15]]},"assertion":[{"value":"26 March 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 March 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}