To study which are the most general causal structures which are compatible with local quantum mechanics, Oreshkov $et$ $al.$  introduced the notion of a process: a resource shared between some parties that allows for quantum communication between them without a predetermined causal order. These processes can be used to perform several tasks that are impossible in standard quantum mechanics: they allow for the violation of causal inequalities, and provide an advantage for computational and communication complexity. Nonetheless, no process that can be used to violate a causal inequality is known to be physically implementable. There is therefore considerable interest in determining which processes are physical and which are just mathematical artefacts of the framework. Here we make key progress in this direction by proposing a purification postulate: processes are physical only if they are purifiable. We derive necessary conditions for a process to be purifiable, and show that several known processes do not satisfy them.
Recently, a theoretical class of processes was found that do not respect causality, but nevertheless can not create logical paradoxes such as those where you travel back in time and kill your own grandfather. Whether such “non-causal” processes are physical and can be found in nature is an open question. In our paper we showed that there exists “non-causal” processes that do not generate paradoxes, but nevertheless violate the condition of reversibility. If reversibility is indeed respected in nature, then these processes must be unphysical.
 Ä. Baumelerand S. Wolf ``Perfect signaling among three parties violating predefined causal order'' Information Theory (ISIT), 2014 IEEE International Symposium on 526–530 (2014).
 Ämin Baumeler, Adrien Feix, and Stefan Wolf, ``Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios'' Phys. Rev. A 90, 042106 (2014).
 Cyril Branciard, Mateus AraÃºjo, Adrien Feix, Fabio Costa, and Časlav Brukner, ``The simplest causal inequalities and their violation'' New J. Phys. 18, 013008 (2015).
 Ä. Baumelerand S. Wolf ``The space of logically consistent classical processes without causal order'' New J. Phys. 18, 013036 (2016).
 O. Oreshkovand C. Giarmatzi ``Causal and causally separable processes'' New J. Phys. 18, 093020 (2015).
 A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, ``Multipartite Causal Correlations: Polytopes and Inequalities'' Phys. Rev. A 94, 032131 (2016).
 Mateus AraÃºjo, Cyril Branciard, Fabio Costa, Adrien Feix, Christina Giarmatzi, and Časlav Brukner, ``Witnessing causal nonseparability'' New J. Phys. 17, 102001 (2015).
 G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure'' Phys. Rev. A 88, 022318 (2013).
 G. Chiribella ``Perfect discrimination of no-signalling channels via quantum superposition of causal structures'' Phys. Rev. A 86, 040301 (2012).
 M. AraÃºjo, F. Costa, and Č. Brukner, ``Computational Advantage from Quantum-Controlled Ordering of Gates'' Phys. Rev. Lett. 113, 250402 (2014).
 A. Feix, M. Araújo, and Č. Brukner, ``Quantum superposition of the order of parties as a communication resource'' Phys. Rev. A 92, 052326 (2015).
 P. Allard Guérin, A. Feix, M. Araújo, and Č. Brukner, ``Exponential communication complexity advantage from quantum superposition of the direction of communication'' Phys. Rev. Lett. 117, 100502 (2016).
 Lorenzo M Procopio, Amir Moqanaki, Mateus AraÃºjo, Fabio Costa, Irati A Calafell, Emma G Dowd, Deny R Hamel, Lee A Rozema, Časlav Brukner, and Philip Walther, ``Experimental superposition of orders of quantum gates'' Nat. Commun. 6, 7913 (2015).
 G. Rubino, L. A. Rozema, A. Feix, M. Araújo, J. M. Zeuner, L. M. Procopio, Č. Brukner, and P. Walther, ``Experimental verification of an indefinite causal order'' Sci. Adv. 3 (2017).
 G. Brassard, H. Buhrman, N. Linden, A. A. Méthot, A. Tapp, and F. Unger, ``Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial'' Phys. Rev. Lett. 96, 250401 (2006).
 Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter, ``Quantum Nonlocality and Beyond: Limits from Nonlocal Computation'' Phys. Rev. Lett. 99, 180502 (2007).
 M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, ``Information causality as a physical principle'' Nature 461, 1101–1104 (2009).
 T. Fritz, A. B. Sainz, R. Augusiak, J. B. Brask, R. Chaves, A. Leverrier, and A. Acín, ``Local orthogonality as a multipartite principle for quantum correlations'' Nat. Commun. 4, 2263 (2013).
 Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti, ``Probabilistic theories with purification'' Phys. Rev. A 81, 062348 (2010).
 L. Hardy ``Quantum Theory From Five Reasonable Axioms'' (2001).
 B. Dakićand Č. Brukner ``Deep Beauty: Understanding the Quantum World through Mathematical Innovation'' Cambridge University Press chapter Quantum Theory and Beyond: Is Entanglement Special? (2011).
 L. Masanesand M. P. Müller ``A derivation of quantum theory from physical requirements'' New J. Phys. 13, 063001 (2011).
 Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti, ``Informational derivation of quantum theory'' Phys. Rev. A 84, 012311 (2011).
 H. Barnum, M. P. Müller, and C. Ududec, ``Higher-order interference and single-system postulates characterizing quantum theory'' New J. Phys. 16, 123029 (2014).
 M. AraÃºjoand A. Feix (2014) Private communication.
 Ä. Baumelerand S. Wolf (2014) Private communication.
 A. Feix, M. Araújo, and Č. Brukner, ``Causally nonseparable processes admitting a causal model'' New J. Phys. 18, 083040 (2016).
 Gian Carlo Ghirardi, Philip Pearle, and Alberto Rimini, ``Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles'' Phys. Rev. A 42, 78–89 (1990).
 A. Bassi, K. Lochan, S. Satin, T. P. Singh, and H. Ulbricht, ``Models of wave-function collapse, underlying theories, and experimental tests'' Rev. Mod. Phys. 85, 471–527 (2013).
 M. Bojowald, D. Cartin, and G. Khanna, ``Lattice refining loop quantum cosmology, anisotropic models, and stability'' Phys. Rev. D 76, 064018 (2007).
 V.F. Mukhanov ``Physical Foundations of Cosmology'' Cambridge University Press (2005).
 Karol Å»yczkowskiand Ingmar Bengtsson ``Geometry of Quantum States'' Cambridge University Press (2006).
 Č. Brukner ``Bounding quantum correlations with indefinite causal order'' New J. Phys. 17, 083034 (2015).
 Samuel L. Braunstein, Giacomo M. D'Ariano, G. J. Milburn, and Massimiliano F. Sacchi, ``Universal Teleportation with a Twist'' Phys. Rev. Lett. 84, 3486–3489 (2000).
 Mateus Araújo, Philippe Allard Guérin, and Ämin Baumeler, "Quantum computation with indefinite causal structures", Physical Review A 96 5, 052315 (2017).
 Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, and Mio Murao, "Reversing Unknown Quantum Transformations: Universal Quantum Circuit for Inverting General Unitary Operations", Physical Review Letters 123 21, 210502 (2019).
 Germain Tobar and Fabio Costa, "Reversible dynamics with closed time-like curves and freedom of choice", Classical and Quantum Gravity 37 20, 205011 (2020).
 Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, "Communication through quantum-controlled noise", Physical Review A 99 6, 062317 (2019).
 Sally Shrapnel and Fabio Costa, "Causation does not explain contextuality", Quantum 2, 63 (2018).
 Ämin Baumeler, Amin Shiraz Gilani, and Jibran Rashid, "Unlimited non-causal correlations and their relation to non-locality", Quantum 6, 673 (2022).
 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, "Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity", Physical Review Letters 122 12, 120504 (2019).
 Ognyan Oreshkov, "Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics", arXiv:1801.07594, Quantum 3, 206 (2019).
 Márcio M. Taddei, Ranieri V. Nery, and Leandro Aolita, "Quantum superpositions of causal orders as an operational resource", Physical Review Research 1 3, 033174 (2019).
 Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12 1, 885 (2021).
 Simon Milz, Felix A Pollock, Thao P Le, Giulio Chiribella, and Kavan Modi, "Entanglement, non-Markovianity, and causal non-separability", New Journal of Physics 20 3, 033033 (2018).
 John H. Selby, Carlo Maria Scandolo, and Bob Coecke, "Reconstructing quantum theory from diagrammatic postulates", Quantum 5, 445 (2021).
 Fabio Costa, "A no-go theorem for superpositions of causal orders", Quantum 6, 663 (2022).
 Jessica Bavaresco, Mio Murao, and Marco Túlio Quintino, "Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies", Journal of Mathematical Physics 63 4, 042203 (2022).
 Manabendra N. Bera, "Quantifying superpositions of quantum evolutions", Physical Review A 100 4, 042307 (2019).
 Christina Giarmatzi, Springer Theses 7 (2019) ISBN:978-3-030-31929-8.
 Lorenzo Maccone, "A Fundamental Problem in Quantizing General Relativity", Foundations of Physics 49 12, 1394 (2019).
 Laurie Letertre, "Causal nonseparability and its implications for spatiotemporal relations", Studies in History and Philosophy of Science 95, 64 (2022).
 Ding Jia and Fabio Costa, "Causal order as a resource for quantum communication", Physical Review A 100 5, 052319 (2019).
 Marco Túlio Quintino and Daniel Ebler, "Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality", Quantum 6, 679 (2022).
 Simon Milz, Dominic Jurkschat, Felix A. Pollock, and Kavan Modi, "Delayed-choice causal order and nonclassical correlations", Physical Review Research 3 2, 023028 (2021).
 Ämin 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).
 Márcio M. Taddei, Jaime Cariñe, Daniel Martínez, Tania García, Nayda Guerrero, Alastair A. Abbott, Mateus Araújo, Cyril Branciard, Esteban S. Gómez, Stephen P. Walborn, Leandro Aolita, and Gustavo Lima, "Computational Advantage from the Quantum Superposition of Multiple Temporal Orders of Photonic Gates", arXiv:2002.07817, PRX Quantum 2 1, 010320 (2021).
 Ciarán M. Lee and John H. Selby, "A no-go theorem for theories that decohere to quantum mechanics", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474 2214, 20170732 (2018).
 Kaumudibikash Goswami and Fabio Costa, "Classical communication through quantum causal structures", Physical Review A 103 4, 042606 (2021).
 Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum Circuits with Classical Versus Quantum Control of Causal Order", PRX Quantum 2 3, 030335 (2021).
 Marco Túlio Quintino, Qingxiuxiong Dong, Atsushi Shimbo, Akihito Soeda, and Mio Murao, "Probabilistic exact universal quantum circuits for transforming unitary operations", Physical Review A 100 6, 062339 (2019).
 Qingxiuxiong Dong, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, "Success-or-Draw: A Strategy Allowing Repeat-Until-Success in Quantum Computation", Physical Review Letters 126 15, 150504 (2021).
 Alastair A. Abbott, Julian Wechs, Fabio Costa, and Cyril Branciard, "Genuinely multipartite noncausality", Quantum 1, 39 (2017).
 Philippe Allard Guérin and Časlav Brukner, "Observer-dependent locality of quantum events", New Journal of Physics 20 10, 103031 (2018).
 Simon Milz and Kavan Modi, "Quantum Stochastic Processes and Quantum non-Markovian Phenomena", PRX Quantum 2 3, 030201 (2021).
 Juan Gu, Longsuo Li, and Zhi Yin, "Two Multi-Setting Causal Inequalities and Their Violations", International Journal of Theoretical Physics 59 1, 97 (2020).
 Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner, "Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems", Nature Communications 11 1, 2672 (2020).
 Roope Uola, Tristan Kraft, and Alastair A. Abbott, "Quantification of quantum dynamics with input-output games", Physical Review A 101 5, 052306 (2020).
 Veronika Baumann, Marius Krumm, Philippe Allard Guérin, and Časlav Brukner, "Noncausal Page-Wootters circuits", Physical Review Research 4 1, 013180 (2022).
 Marco Túlio Quintino and Daniel Ebler, "Deterministic transformations between unitary operations: Exponential advantage with adaptive quantum circuits and the power of indefinite causality", arXiv:2109.08202.
 Augustin Vanrietvelde, Nick Ormrod, Hlér Kristjánsson, and Jonathan Barrett, "Consistent circuits for indefinite causal order", arXiv:2206.10042.
 Julian Wechs, Cyril Branciard, and Ognyan Oreshkov, "Existence of processes violating causal inequalities on time-delocalised subsystems", arXiv:2201.11832.
 Qingxiuxiong Dong, Marco Túlio Quintino, Akihito Soeda, and Mio Murao, "The quantum switch is uniquely defined by its action on unitary operations", arXiv:2106.00034.
The above citations are from Crossref's cited-by service (last updated successfully 2022-10-04 21:17:00) and SAO/NASA ADS (last updated successfully 2022-10-04 21:17:01). The list may be incomplete as not all publishers provide suitable and complete citation data.
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.