A purification postulate for quantum mechanics with indefinite causal order

Mateus Araújo1,2,3, Adrien Feix1,2, Miguel Navascués2, and Časlav Brukner1,2

1Faculty of Physics, University of Vienna, Boltzmanngasse 5 1090 Vienna, Austria
2Institute for Quantum Optics and Quantum Information (IQOQI), Boltzmanngasse 3 1090 Vienna, Austria
3Institute for Theoretical Physics, University of Cologne, Germany

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 the first step 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.

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Among the most fundamental concepts in physics are those of causality and reversibility. The first encapsulates the idea that events in the present are caused by events in the past and, in their turn, act as causes for events in the future. The second is the idea that physical processes are reversible, that is, that information is never created or destroyed.

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.

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[1] O. Oreshkov, F. Costa, and Č. Brukner, ``Quantum correlations with no causal order'' Nat. Commun. 3, 1092 (2012).
https://doi.org/10.1038/ncomms2076
arXiv:1105.4464

[2] A. Ashtekar ``Large Quantum Gravity Effects: Unforeseen Limitations of the Classical Theory'' Phys. Rev. Lett. 77, 4864-4867 (1996).
https://doi.org/10.1103/PhysRevLett.77.4864
arXiv:gr-qc/9610008

[3] Ä. Baumelerand S. Wolf ``Perfect signaling among three parties violating predefined causal order'' Information Theory (ISIT), 2014 IEEE International Symposium on 526-530 (2014).
https://doi.org/10.1109/ISIT.2014.6874888
arXiv:1312.5916

[4] Ä. Baumeler, A. Feix, and S. Wolf, ``Maximal incompatibility of locally classical behavior and global causal order in multi-party scenarios'' Phys. Rev. A 90, 042106 (2014).
https://doi.org/10.1103/PhysRevA.90.042106
arXiv:1403.7333

[5] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, ``The simplest causal inequalities and their violation'' New J. Phys. 18, 013008 (2015).
https://doi.org/10.1088/1367-2630/18/1/013008
arXiv:1508.01704

[6] Ä. Baumelerand S. Wolf ``The space of logically consistent classical processes without causal order'' New J. Phys. 18, 013036 (2016).
https://doi.org/10.1088/1367-2630/18/1/013036
arXiv:1507.01714

[7] O. Oreshkovand C. Giarmatzi ``Causal and causally separable processes'' New J. Phys. 18, 093020 (2015).
https://doi.org/10.1088/1367-2630/18/9/093020
arXiv:1506.05449

[8] A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, ``Multipartite Causal Correlations: Polytopes and Inequalities'' Phys. Rev. A 94, 032131 (2016).
https://doi.org/10.1103/PhysRevA.94.032131
arXiv:1608.01528

[9] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, ``Witnessing causal nonseparability'' New J. Phys. 17, 102001 (2015).
https://doi.org/10.1088/1367-2630/17/10/102001
arXiv:1506.03776

[10] G. Chiribella, G. M. D'Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure'' Phys. Rev. A 88, 022318 (2013).
https://doi.org/10.1103/PhysRevA.88.022318
arXiv:0912.0195

[11] G. Chiribella ``Perfect discrimination of no-signalling channels via quantum superposition of causal structures'' Phys. Rev. A 86, 040301 (2012).
https://doi.org/10.1103/PhysRevA.86.040301
arXiv:1109.5154

[12] M. Araújo, F. Costa, and Č. Brukner, ``Computational Advantage from Quantum-Controlled Ordering of Gates'' Phys. Rev. Lett. 113, 250402 (2014).
https://doi.org/10.1103/PhysRevLett.113.250402
arXiv:1401.8127

[13] A. Feix, M. Araújo, and Č. Brukner, ``Quantum superposition of the order of parties as a communication resource'' Phys. Rev. A 92, 052326 (2015).
https://doi.org/10.1103/PhysRevA.92.052326
arXiv:1508.07840

[14] 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).
https://doi.org/10.1103/PhysRevLett.117.100502
arXiv:1605.07372

[15] L. M. Procopio, A. Moqanaki, M. Araújo, F. Costa, I. A. Calafell, E. G. Dowd, D. R. Hamel, L. A. Rozema, Č. Brukner, and P. Walther, ``Experimental superposition of orders of quantum gates'' Nat. Commun. 6, 7913 (2015).
https://doi.org/10.1038/ncomms8913
arXiv:1412.4006

[16] 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).
https://doi.org/10.1126/sciadv.1602589
arXiv:1608.01683

[17] 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).
https://doi.org/10.1103/PhysRevLett.96.250401
arXiv:quant-ph/0508042

[18] N. Linden, S. Popescu, A. J. Short, and A. Winter, ``Quantum Nonlocality and Beyond: Limits from Nonlocal Computation'' Phys. Rev. Lett. 99, 180502 (2007).
https://doi.org/10.1103/PhysRevLett.99.180502
arXiv:quant-ph/0610097

[19] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski, ``Information causality as a physical principle'' Nature 461, 1101-1104 (2009).
https://doi.org/10.1038/nature08400
arXiv:0905.2292

[20] M. Navascuésand H. Wunderlich ``A glance beyond the quantum model'' Proc. Royal Soc. A 466, 881-890 (2009).
https://doi.org/10.1098/rspa.2009.0453
arXiv:0907.0372

[21] 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).
https://doi.org/10.1038/ncomms3263
arXiv:1210.3018

[22] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification'' Phys. Rev. A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348
arXiv:0908.1583

[23] L. Hardy ``Quantum Theory From Five Reasonable Axioms'' (2001).
arXiv:quant-ph/0101012

[24] B. Dakićand Č. Brukner ``Deep Beauty: Understanding the Quantum World through Mathematical Innovation'' Cambridge University Press (2011).
arXiv:0911.0695

[25] L. Masanesand M. P. Müller ``A derivation of quantum theory from physical requirements'' New J. Phys. 13, 063001 (2011).
https://doi.org/10.1088/1367-2630/13/6/063001
arXiv:1004.1483

[26] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Informational derivation of quantum theory'' Phys. Rev. A 84, 012311 (2011).
https://doi.org/10.1103/PhysRevA.84.012311
arXiv:1011.6451

[27] H. Barnum, M. P. Müller, and C. Ududec, ``Higher-order interference and single-system postulates characterizing quantum theory'' New J. Phys. 16, 123029 (2014).
https://doi.org/10.1088/1367-2630/16/12/123029
arXiv:1403.4147

[28] P. A Höhn ``Toolbox for reconstructing quantum theory from rules on information acquisition'' (2014).
arXiv:1412.8323

[29] P. A Höhnand C. Wever ``Quantum theory from questions'' Phys. Rev. A 95, 012102 (2017).
https://doi.org/10.1103/PhysRevA.95.012102
arXiv:1511.01130

[30] Ä. Baumelerand S. Wolf Private communication (2015).

[31] A. Feix, M. Araújo, and Č. Brukner, ``Causally nonseparable processes admitting a causal model'' New J. Phys. 18, 083040 (2016).
https://doi.org/10.1088/1367-2630/18/8/083040
arXiv:1604.03391

[32] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks'' Phys. Rev. A 80, 022339 (2009).
https://doi.org/10.1103/PhysRevA.80.022339
arXiv:0904.4483

[33] G. C. Ghirardi, A. Rimini, and T. Weber, ``Unified dynamics for microscopic and macroscopic systems'' Phys. Rev. D 34, 470-491 (1986).
https://doi.org/10.1103/PhysRevD.34.470

[34] G. C. Ghirardi, P. Pearle, and A. Rimini, ``Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles'' Phys. Rev. A 42, 78-89 (1990).
https://doi.org/10.1103/PhysRevA.42.78

[35] 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).
https://doi.org/10.1103/RevModPhys.85.471
arXiv:1204.4325

[36] S. W. Hawking ``Breakdown of predictability in gravitational collapse'' Phys. Rev. D 14, 2460-2473 (1976).
https://doi.org/10.1103/PhysRevD.14.2460

[37] D. Harlow ``Jerusalem Lectures on Black Holes and Quantum Information'' Rev. Mod. Phys. 88, 015002 (2016).
https://doi.org/10.1103/RevModPhys.88.015002
arXiv:1409.1231

[38] T. Jacobson ``Trans-Planckian Redshifts andthe Substance of the Space-Time River'' Prog. Theor. Phys. 136, 1-17 (1999).
https://doi.org/10.1143/PTPS.136.1
arXiv:hep-th/0001085

[39] M. Bojowald, D. Cartin, and G. Khanna, ``Lattice refining loop quantum cosmology, anisotropic models, and stability'' Phys. Rev. D 76, 064018 (2007).
https://doi.org/10.1103/PhysRevD.76.064018
arXiv:0704.1137

[40] S. Gielenand L. Sindoni ``Quantum Cosmology from Group Field Theory Condensates: a Review'' SIGMA 12, 082 (2016).
https://doi.org/10.3842/SIGMA.2016.082
arXiv:1602.08104

[41] P. A. Höhn ``Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces'' J. Math. Phys. 55, 083508 (2014).
https://doi.org/10.1063/1.4890558
arXiv:1401.6062

[42] V. Mukhanov ``Physical Foundations of Cosmology'' Cambridge University Press (2005).

[43] K. Życzkowskiand I. Bengtsson ``Geometry of Quantum States'' Cambridge University Press (2006).

[44] Č. Brukner ``Bounding quantum correlations with indefinite causal order'' New J. Phys. 17, 083034 (2015).
https://doi.org/10.1088/1367-2630/17/8/083034
arXiv:1404.0721

[45] A. Royer ``Wigner function in Liouville space: A canonical formalism'' Phys. Rev. A 43, 44-56 (1991).
https://doi.org/10.1103/PhysRevA.43.44

[46] S. L. Braunstein, G. M. D'Ariano, G. J. Milburn, and M. F. Sacchi, ``Universal Teleportation with a Twist'' Phys. Rev. Lett. 84, 3486-3489 (2000).
https://doi.org/10.1103/PhysRevLett.84.3486
arXiv:quant-ph/9908036