{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T11:18:06Z","timestamp":1775819886309,"version":"3.50.1"},"reference-count":52,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T00:00:00Z","timestamp":1611100800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000925","name":"John Templeton Foundation","doi-asserted-by":"crossref","award":["61466"],"award-info":[{"award-number":["61466"]}],"id":[{"id":"10.13039\/100000925","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>We develop an extension of the process matrix (PM) framework for correlations between quantum operations with no causal order that allows multiple rounds of information exchange for each party compatibly with the assumption of well-defined causal order of events locally. We characterise the higher-order process describing such correlations, which we name the multi-round process matrix (MPM), and formulate a notion of causal nonseparability for it that extends the one for standard PMs. We show that in the multi-round case there are novel manifestations of causal nonseparability that are not captured by a naive application of the standard PM formalism: we exhibit an instance of an operator that is both a valid PM and a valid MPM, but is causally separable in the first case and can violate causal inequalities in the second case due to the possibility of using a side channel.<\/jats:p>","DOI":"10.22331\/q-2021-01-20-384","type":"journal-article","created":{"date-parts":[[2021,1,20]],"date-time":"2021-01-20T17:39:21Z","timestamp":1611164361000},"page":"384","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["The Multi-round Process Matrix"],"prefix":"10.22331","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7014-1958","authenticated-orcid":false,"given":"Timoth\u00e9e","family":"Hoffreumon","sequence":"first","affiliation":[{"name":"Centre for Quantum Information and Communication (QuIC), \u00c9cole polytechnique de Bruxelles, CP 165, Universit\u00e9 libre de Bruxelles, 1050 Brussels, Belgium."}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9390-1064","authenticated-orcid":false,"given":"Ognyan","family":"Oreshkov","sequence":"additional","affiliation":[{"name":"Centre for Quantum Information and Communication (QuIC), \u00c9cole polytechnique de Bruxelles, CP 165, Universit\u00e9 libre de Bruxelles, 1050 Brussels, Belgium."}]}],"member":"9598","published-online":{"date-parts":[[2021,1,20]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Alastair A. Abbott, Christina Giarmatzi, Fabio Costa, and Cyril Branciard. Multipartite causal correlations: Polytopes and inequalities. Phys. Rev. A, 94: 032131, 2016. 10.1103\/physreva.94.032131.","DOI":"10.1103\/physreva.94.032131"},{"key":"1","doi-asserted-by":"publisher","unstructured":"Alastair A. Abbott, Julian Wechs, Fabio Costa, and Cyril Branciard. Genuinely multipartite noncausality. Quantum, 1: 39, 2017. 10.22331\/q-2017-12-14-39.","DOI":"10.22331\/q-2017-12-14-39"},{"key":"2","doi-asserted-by":"publisher","unstructured":"John-Mark A. Allen, Jonathan Barrett, Dominic C. Horsman, Ciar\u00e1n M. Lee, and Robert W. Spekkens. Quantum common causes and quantum causal models. Phys. Rev. X, 7: 031021, 2017. 10.1103\/PhysRevX.7.031021.","DOI":"10.1103\/PhysRevX.7.031021"},{"key":"3","doi-asserted-by":"publisher","unstructured":"Mateus Ara\u00fajo, Philippe Allard Gu\u00e9rin, and \u00c4min Baumeler. Quantum computation with indefinite causal structures. Phys. Rev. A, 96: 052315, 2017. 10.1103\/physreva.96.052315.","DOI":"10.1103\/physreva.96.052315"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Mateus Ara\u00fajo, Fabio Costa, and \u010caslav Brukner. Computational advantage from quantum-controlled ordering of gates. Phys. Rev. Lett., 113: 250402, 2014. 10.1103\/PhysRevLett.113.250402.","DOI":"10.1103\/PhysRevLett.113.250402"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Mateus Ara\u00fajo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and \u010caslav Brukner. Witnessing causal nonseparability. New Journal of Physics, 17 (10): 1\u201328, 2015. 10.1088\/1367-2630\/17\/10\/102001.","DOI":"10.1088\/1367-2630\/17\/10\/102001"},{"key":"6","unstructured":"Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. Quantum causal models. arXiv:quant-ph\/1906.10726, 2019."},{"key":"7","unstructured":"Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov. Cyclic quantum causal models. arXiv:quant-ph\/2002.12157, 2020."},{"key":"8","doi-asserted-by":"publisher","unstructured":"\u00c4min Baumeler and Stefan Wolf. The space of logically consistent classical processes without causal order. New Journal of Physics, 18 (1): 013036, 2016. 10.1088\/1367-2630\/18\/1\/013036.","DOI":"10.1088\/1367-2630\/18\/1\/013036"},{"key":"9","doi-asserted-by":"publisher","unstructured":"\u00c4min Baumeler, Fabio Costa, Timothy C. Ralph, Stefan Wolf, and Magdalena Zych. Reversible time travel with freedom of choice. Classical and Quantum Gravity, 36 (22): 224002, 2019. 10.1088\/1361-6382\/ab4973.","DOI":"10.1088\/1361-6382\/ab4973"},{"key":"10","doi-asserted-by":"publisher","unstructured":"Alessandro Bisio and Paolo Perinotti. Theoretical framework for higher-order quantum theory. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475 (2225): 20180706, 2019. 10.1098\/rspa.2018.0706.","DOI":"10.1098\/rspa.2018.0706"},{"key":"11","doi-asserted-by":"publisher","unstructured":"\u010caslav Brukner. Quantum causality. Nature Physics, 10 (4): 259\u2013263, 2014. 10.1038\/nphys2930.","DOI":"10.1038\/nphys2930"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Esteban Castro-Ruiz, Flaminia Giacomini, and \u010caslav Brukner. Dynamics of Quantum Causal Structures. Phys. Rev. X, 8: 011047, 2018. 10.1103\/PhysRevX.8.011047.","DOI":"10.1103\/PhysRevX.8.011047"},{"key":"13","doi-asserted-by":"publisher","unstructured":"Giulio Chiribella. Perfect discrimination of no-signalling channels via quantum superposition of causal structures. Phys. Rev. A, 86: 040301, 2011. 10.1103\/PhysRevA.86.040301.","DOI":"10.1103\/PhysRevA.86.040301"},{"key":"14","doi-asserted-by":"publisher","unstructured":"Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. Theoretical framework for quantum networks. Phys. Rev. A, 80 (2), 2009. 10.1103\/PhysRevA.80.022339.","DOI":"10.1103\/PhysRevA.80.022339"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Giulio Chiribella, Giacomo Mauro D\u2019Ariano, and Paolo Perinotti. Informational derivation of quantum theory. Phys. Rev. A, 84 (1), 2011. 10.1103\/physreva.84.012311.","DOI":"10.1103\/physreva.84.012311"},{"key":"16","doi-asserted-by":"publisher","unstructured":"Giulio Chiribella, Giacomo Mauro D'Ariano, Paolo Perinotti, and Benoit Valiron. Quantum computations without definite causal structure. Phys. Rev. A, 88: 022318, 2013. 10.1103\/PhysRevA.88.022318.","DOI":"10.1103\/PhysRevA.88.022318"},{"key":"17","doi-asserted-by":"publisher","unstructured":"Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications, 10 (3): 285\u2013290, 1975. 10.1016\/0024-3795(75)90075-0.","DOI":"10.1016\/0024-3795(75)90075-0"},{"key":"18","doi-asserted-by":"publisher","unstructured":"Timoteo Colnaghi, Giacomo Mauro D'Ariano, Stefano Facchini, and Paolo Perinotti. Quantum computation with programmable connections between gates. Physics Letters, Section A: General, Atomic and Solid State Physics, 376 (45): 2940\u20132943, 2012. 10.1016\/j.physleta.2012.08.028.","DOI":"10.1016\/j.physleta.2012.08.028"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Adrien Feix, Mateus Ara\u00fajo, and \u010caslav Brukner. Quantum superposition of the order of parties as a communication resource. Phys. Rev. A, 92: 052326, 2015. 10.1103\/PhysRevA.92.052326.","DOI":"10.1103\/PhysRevA.92.052326"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Adrien Feix, Mateus Ara\u00fajo, and \u010caslav Brukner. Causally nonseparable processes admitting a causal model. New Journal of Physics, 18 (8): 083040, 2016. 10.1088\/1367-2630\/18\/8\/083040.","DOI":"10.1088\/1367-2630\/18\/8\/083040"},{"key":"21","doi-asserted-by":"publisher","unstructured":"Kaumudibikash Goswami, Christina Giarmatzi, Michael Kewming, Fabio Giarmatzi, Cyril Branciard, Jacqueline Romero, and Andrew G. White. Indefinite Causal Order in a Quantum Switch. Phys. Rev. Lett., 121: 090503, 2018. 10.1103\/PhysRevLett.121.090503.","DOI":"10.1103\/PhysRevLett.121.090503"},{"key":"22","doi-asserted-by":"publisher","unstructured":"Philippe Allard Gu\u00e9rin, Adrien Feix, Mateus Ara\u00fajo, and \u010caslav Brukner. Exponential Communication Complexity Advantage from Quantum Superposition of the Direction of Communication. Phys. Rev. Lett., 117: 100502, 2016. 10.1103\/PhysRevLett.117.100502.","DOI":"10.1103\/PhysRevLett.117.100502"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Philippe Allard Gu\u00e9rin and \u010caslav Brukner. Observer-dependent locality of quantum events. New Journal of Physics, 20 (10): 103031, 2018. 10.1088\/1367-2630\/aae742.","DOI":"10.1088\/1367-2630\/aae742"},{"key":"24","doi-asserted-by":"publisher","unstructured":"Lucien Hardy. Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure. Journal of Physics A: Mathematical and Theoretical, 40 (12): 3081\u20133099, 2007. 10.1088\/1751-8113\/40\/12\/S12.","DOI":"10.1088\/1751-8113\/40\/12\/S12"},{"key":"25","unstructured":"Israel N. Herstein. Topics in algebra. Open University set book. Wiley, 1975. ISBN 978-0-471-00258-1. URL https:\/\/books.google.be\/books?id=LJRxPQAACAAJ."},{"key":"26","doi-asserted-by":"publisher","unstructured":"Fumio Hiai and D\u00e9nes Petz. Introduction to matrix analysis and applications. Springer Science & Business Media, 2014. ISBN 978-3-319-04149-0. 10.1007\/978-3-319-04150-6.","DOI":"10.1007\/978-3-319-04150-6"},{"key":"27","unstructured":"Timoth\u00e9e Hoffreumon. Processes with indefinite causal structure in quantum theory: the multi-round process matrix. Master's thesis, Universit\u00e9 Libre de Bruxelles, 2019. http:\/\/quic.ulb.ac.be\/_media\/publications\/2019_mfe_timothee_hoffreumon.pdf."},{"key":"28","doi-asserted-by":"publisher","unstructured":"Andrzej Jamio\u0142kowski. Linear transformations which preserve trace and positive semidefiniteness of operators. Reports on Mathematical Physics, 3 (4): 275\u2013278, 1972. 10.1016\/0034-4877(72)90011-0.","DOI":"10.1016\/0034-4877(72)90011-0"},{"key":"29","unstructured":"Ding Jia. Correlational quantum theory and correlation constraints. arXiv:quant-ph\/2001.03142, 2020."},{"key":"30","doi-asserted-by":"publisher","unstructured":"Aleks Kissinger and Sander Uijlen. A categorical semantics for causal structure. In 2017 32nd Annual ACM\/IEEE Symposium on Logic in Computer Science (LICS), pages 1\u201312, 2017. 10.1109\/LICS.2017.8005095.","DOI":"10.1109\/LICS.2017.8005095"},{"key":"31","doi-asserted-by":"publisher","unstructured":"Hl\u00e9r Kristj\u00e1nsson, Giulio Chiribella, Sina Salek, Daniel Ebler, and Matthew Wilson. Resource theories of communication. New Journal of Physics, 22 (7): 073014, 2020. 10.1088\/1367-2630\/ab8ef7.","DOI":"10.1088\/1367-2630\/ab8ef7"},{"key":"32","doi-asserted-by":"publisher","unstructured":"Jean-Philippe W. Maclean, Katja Ried, Robert W. Spekkens, and Kevin J. Resch. Quantum-coherent mixtures of causal relations. Nature Communications, 8: 1\u201310, 2017. 10.1038\/ncomms15149.","DOI":"10.1038\/ncomms15149"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Michael A. Nielsen and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2009. 10.1017\/cbo9780511976667.","DOI":"10.1017\/cbo9780511976667"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Ognyan Oreshkov. Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics. Quantum, 3: 206, 2019. 10.22331\/q-2019-12-02-206.","DOI":"10.22331\/q-2019-12-02-206"},{"key":"35","doi-asserted-by":"publisher","unstructured":"Ognyan Oreshkov and Nicolas J. Cerf. Operational quantum theory without predefined time. New Journal of Physics, 18 (7): 073037, 2016. 10.1088\/1367-2630\/18\/7\/073037.","DOI":"10.1088\/1367-2630\/18\/7\/073037"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Ognyan Oreshkov and Christina Giarmatzi. Causal and causally separable processes. New Journal of Physics, 18 (9): 1\u201336, 2016. 10.1088\/1367-2630\/18\/9\/093020.","DOI":"10.1088\/1367-2630\/18\/9\/093020"},{"key":"37","doi-asserted-by":"publisher","unstructured":"Ognyan Oreshkov, Fabio Costa, and \u010caslav Brukner. Quantum correlations with no causal order. Nature Communications, 3: 1\u201313, 2012. 10.1038\/ncomms2076.","DOI":"10.1038\/ncomms2076"},{"key":"38","doi-asserted-by":"publisher","unstructured":"Paolo Perinotti. Causal structures and the classification of higher order quantum computations. Tutorials, Schools, and Workshops in the Mathematical Sciences, page 103\u2013127, 2017. 10.1007\/978-3-319-68655-4_7.","DOI":"10.1007\/978-3-319-68655-4_7"},{"key":"39","doi-asserted-by":"publisher","unstructured":"D\u00e9nes Petz. Complementarity in quantum systems. Reports on Mathematical Physics, 59 (2): 209\u2013224, 2007. 10.1016\/S0034-4877(07)00010-9.","DOI":"10.1016\/S0034-4877(07)00010-9"},{"key":"40","doi-asserted-by":"publisher","unstructured":"Robert Piziak, Patrick L. Odell, and R. Hahn. Constructing projections on sums and intersections. Computers and Mathematics with Applications, 37 (1): 67\u201374, 1999. 10.1016\/S0898-1221(98)00242-9.","DOI":"10.1016\/S0898-1221(98)00242-9"},{"key":"41","doi-asserted-by":"publisher","unstructured":"Christopher Portmann, Christian Matt, Ueli Maurer, Renato Renner, and Bjorn Tackmann. Causal boxes: Quantum information-processing systems closed under composition. IEEE Transactions on Information Theory, page 1\u20131, 2017. 10.1109\/tit.2017.2676805.","DOI":"10.1109\/tit.2017.2676805"},{"key":"42","doi-asserted-by":"publisher","unstructured":"Lorenzo M. Procopio, Amir Moqanaki, Mateus Ara\u00fajo, Fabio Costa, Irati Alonso Calafell, Emma G. Dowd, Deny R. Hamel, Lee A. Rozema, \u010caslav Brukner, and Philip Walther. Experimental superposition of orders of quantum gates. Nature Communications, 6: 1\u201310, 2015. 10.1038\/ncomms8913.","DOI":"10.1038\/ncomms8913"},{"key":"43","doi-asserted-by":"publisher","unstructured":"Lorenzo M. Procopio, Francisco Delgado, Marco Enr\u00edquez, Nadia Belabas, and Juan Ariel Levenson. Communication Enhancement through Quantum Coherent Control of N Channels in an Indefinite Causal-Order Scenario. Entropy, 21 (10): 1012, 2019. 10.3390\/e21101012.","DOI":"10.3390\/e21101012"},{"key":"44","doi-asserted-by":"publisher","unstructured":"Giulia Rubino, Lee A. Rozema, Adrien Feix, Mateus Ara\u00fajo, Jonas M. Zeuner, Lorenzo M. Procopio, \u010caslav Brukner, and Philip Walther. Experimental verification of an indefinite causal order. Science Advances, 2017a. 10.1126\/sciadv.1602589.","DOI":"10.1126\/sciadv.1602589"},{"key":"45","unstructured":"Giulia Rubino, Lee A. Rozema, Francesco Massa, Mateus Ara\u00fajo, Magdalena Zych, \u010caslav Brukner, and Philip Walther. Experimental entanglement of temporal orders. arXiv:quant-ph\/1712.06884, 2017b."},{"key":"46","doi-asserted-by":"publisher","unstructured":"Ralph Silva, Yelena Guryanova, Anthony J. Short, Paul Skrzypczyk, Nicolas Brunner, and Sandu Popescu. Connecting processes with indefinite causal order and multi-time quantum states. New Journal of Physics, 19 (10): 103022, 2017. 10.1088\/1367-2630\/aa84fe.","DOI":"10.1088\/1367-2630\/aa84fe"},{"key":"47","doi-asserted-by":"publisher","unstructured":"M\u00e1rcio M. Taddei, Ranieri V. Nery, and Leandro Aolita. Quantum superpositions of causal orders as an operational resource. Phys. Rev. Research, 1: 033174, 2019. 10.1103\/PhysRevResearch.1.033174.","DOI":"10.1103\/PhysRevResearch.1.033174"},{"key":"48","doi-asserted-by":"publisher","unstructured":"Germain Tobar and Fabio Costa. Reversible dynamics with closed time-like curves and freedom of choice. Classical and Quantum Gravity, 37 (20): 205011, 2020. ISSN 1361-6382. 10.1088\/1361-6382\/aba4bc.","DOI":"10.1088\/1361-6382\/aba4bc"},{"key":"49","doi-asserted-by":"publisher","unstructured":"Julian Wechs, Alastair A. Abbott, and Cyril Branciard. On the definition and characterisation of multipartite causal (non)separability. New Journal of Physics, 21 (1): 013027, 2019. 10.1088\/1367-2630\/aaf352.","DOI":"10.1088\/1367-2630\/aaf352"},{"key":"50","doi-asserted-by":"publisher","unstructured":"Kejin Wei, Nora Tischler, Si-Ran Zhao, Yu-Huai Li, Juan Miguel Arrazola, Yang Liu, Weijun Zhang, Hao Li, Lixing You, Zhen Wang, Yu-Ao Chen, Barry C. Sanders, Qiang Zhang, Geoff J. Pryde, Feihu Xu, and Jian-Wei Pan. Phys. Rev. Lett., 122: 120504, Mar 2019. 10.1103\/PhysRevLett.122.120504.","DOI":"10.1103\/PhysRevLett.122.120504"},{"key":"51","doi-asserted-by":"publisher","unstructured":"Magdalena Zych, Fabio Costa, Igor Pikovski, and \u010caslav Brukner. Bell\u2019s theorem for temporal order. Nature Communications, 10 (1), 2019. ISSN 2041-1723. 10.1038\/s41467-019-11579-x.","DOI":"10.1038\/s41467-019-11579-x"}],"container-title":["Quantum"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/quantum-journal.org\/papers\/q-2021-01-20-384\/pdf\/","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T19:55:18Z","timestamp":1613073318000},"score":1,"resource":{"primary":{"URL":"https:\/\/quantum-journal.org\/papers\/q-2021-01-20-384\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,20]]},"references-count":52,"URL":"https:\/\/doi.org\/10.22331\/q-2021-01-20-384","archive":["CLOCKSS"],"relation":{},"ISSN":["2521-327X"],"issn-type":[{"value":"2521-327X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,20]]},"article-number":"384"}}