QuIC, Ecole polytechnique de Bruxelles, C.P. 165, Université libre de Bruxelles, 1050 Brussels, Belgium
Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.
Abstract
It has been shown that it is theoretically possible for there to exist higher-order quantum processes in which the operations performed by separate parties cannot be ascribed a definite causal order. Some of these processes are believed to have a physical realization in standard quantum mechanics via coherent control of the times of the operations. A prominent example is the quantum SWITCH, which was recently demonstrated experimentally. However, the interpretation of such experiments as realizations of a process with indefinite causal structure as opposed to some form of simulation of such a process has remained controversial. Where exactly are the local operations of the parties in such an experiment? On what spaces do they act given that their times are indefinite? Can we probe them directly rather than assume what they ought to be based on heuristic considerations? How can we reconcile the claim that these operations really take place, each once as required, with the fact that the structure of the presumed process implies that they cannot be part of any acyclic circuit? Here, I offer a precise answer to these questions: the input and output systems of the operations in such a process are generally nontrivial subsystems of Hilbert spaces that are tensor products of Hilbert spaces associated with systems at different times—a fact that is directly experimentally verifiable. With respect to these time-delocalized subsystems, the structure of the process is one of a circuit with a causal cycle. This provides a rigorous sense in which processes with indefinite causal structure can be said to exist within the known quantum mechanics. I also identify a whole class of isometric processes, of which the quantum SWITCH is a special case, that admit a physical realization on time-delocalized subsystems. These results unveil a novel structure within quantum mechanics, which may have important implications for physics and information processing.
► BibTeX data
► References
[1] L. Hardy, Probability Theories with Dynamic Causal Structure: A New Framework for Quantum Gravity, (2005).
arXiv:gr-qc/0509120
[2] L. Hardy, Quantum Gravity Computers: On the Theory of Computation with Indefinite Causal Structure, in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle, The Western Ontario Series in Philosophy of Science, vol. 73 (Springer, Dordrecht, 2009); DOI: https://doi.org/10.1007/978-1-4020-9107-0_21; (2007).
https://doi.org/10.1007/978-1-4020-9107-0_21
arXiv:quant-ph/0701019
[3] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013); DOI: https://doi.org/10.1103/PhysRevA.88.022318; (2009).
https://doi.org/10.1103/PhysRevA.88.022318
[4] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Transforming quantum operations: quantum supermaps, Europhys. Lett. 83, 30004 (2008); DOI: https://doi.org/10.1209/0295-5075/83/30004; (2008).
https://doi.org/10.1209/0295-5075/83/30004
arXiv:0804.0180
[5] G. Chiribella, G. M. D’Ariano, and P. Perinotti, Theoretical framework for quantum networks, Phys. Rev. A 80, 022339 (2009); DOI: https://doi.org/10.1103/PhysRevA.80.022339; (2009).
https://doi.org/10.1103/PhysRevA.80.022339
arXiv:0904.4483
[6] The idea of quantum computation beyond causal circuits was notably first considered by Deutsch Deutsch who studied modifications of quantum theory in the vicinity of closed timelike curves, which led to the study of the computational power of such models (see Ref. Aaronson). Although linked to time travel in a different sense Chiribella12, the quantum SWITCH is motivated by the idea of `quantum superpositions of different causal structures’ as opposed to classically definite backgrounds with timelike cycles.
[7] D. Deutsch, Quantum mechanics near closed timelike lines, Phys. Rev. D 44, 3197 (1991); DOI: https://doi.org/10.1103/PhysRevD.44.3197.
https://doi.org/10.1103/PhysRevD.44.3197
[8] S. Aaaronson and J. Watrous, Closed timelike curves make quantum and classical computing equivalent, Proc. R. Soc. A 465, 631-647 (2009); DOI: https://doi.org/10.1098/rspa.2008.0350; (2008).
https://doi.org/10.1098/rspa.2008.0350
arXiv:0808.2669
[9] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Commun. 3, 1092 (2012); DOI: https://doi.org/10.1038/ncomms2076; (2011).
https://doi.org/10.1038/ncomms2076
arXiv:1105.4464
[10] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes, New J. Phys. 18, 093020 (2016); DOI: https://doi.org/10.1088/1367-2630/18/9/093020; (2015).
https://doi.org/10.1088/1367-2630/18/9/093020
arXiv:1506.05449
[11] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, Witnessing causal nonseparability, New J. Phys. 17, 102001 (2015); DOI: https://doi.org/10.1088/1367-2630/17/10/102001; (2015).
https://doi.org/10.1088/1367-2630/17/10/102001
arXiv:1506.03776
[12] J. Wechs, A. A. Abbott, and C. Branciard, On the definition and characterisation of multipartite causal (non)separability, New J. Phys. 21, 013027 (2019); DOI: https://doi.org/10.1088/1367-2630/aaf352; (2018).
https://doi.org/10.1088/1367-2630/aaf352
arXiv:1807.10557
[13] Ä. Baumeler and S. Wolf, Perfect signaling among three parties violating predefined causal order, Proceedings of International Symposium on Information Theory (ISIT) 2014, 526-530, 2014; DOI: https://doi.org/10.1109/ISIT.2014.6874888; (2013).
https://doi.org/10.1109/ISIT.2014.6874888
arXiv:1312.5916
[14] 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); DOI: https://doi.org/10.1103/PhysRevA.90.042106; (2014).
https://doi.org/10.1103/PhysRevA.90.042106
arXiv:1403.7333
[15] Ä. Baumeler and S. Wolf, The space of logically consistent classical processes without causal order, New J. Phys. 18, 013036 (2016); DOI: https://doi.org/10.1088/1367-2630/18/1/013036; (2015).
https://doi.org/10.1088/1367-2630/18/1/013036
arXiv:1507.01714
[16] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, The simplest causal inequalities and their violation, New J. Phys. 18, 013008 (2016); DOI: https://doi.org/10.1088/1367-2630/18/1/013008; (2015).
https://doi.org/10.1088/1367-2630/18/1/013008
arXiv:1508.01704
[17] S. S. Bhattacharya and M. Banik, Biased Non-Causal Game, (2015).
arXiv:1509.02721
[18] A. Feix, M. Araújo, and Č. Brukner, Causally nonseparable processes admitting a causal model, New J. Phys. 18, 083040 (2016); DOI: https://doi.org/10.1088/1367-2630/18/8/083040; (2016).
https://doi.org/10.1088/1367-2630/18/8/083040
arXiv:1604.03391
[19] A. A. Abbott, C. Giarmatzi, F. Costa, and C. Branciard, Multipartite Causal Correlations: Polytopes and Inequalities, Phys. Rev. A 94, 032131 (2016); DOI: https://doi.org/10.1103/PhysRevA.94.032131; (2016).
https://doi.org/10.1103/PhysRevA
.94.032131
arXiv:1608.01528
[20] N. Miklin, A. A. Abbott, C. Branciard, R. Chaves, C. Budroni, The entropic approach to causal correlations, New J. Phys. 19, 113041 (2017); DOI: https://doi.org/10.1088/1367-2630/aa8f9f; (2017).
https://doi.org/10.1088/1367-2630/aa8f9f
arXiv:1706.10270
[21] J. S. Bell, On the Einstein Podolsky Rosen Paradox, Physics 1, 3, 195-200 (1964); DOI: https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195.
https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
[22] O. Oreshkov and N. J. Cerf, Operational quantum theory without predefined time, New J. Phys. 18, 073037 (2016); DOI: https://doi.org/10.1088/1367-2630/18/7/073037; (2014).
https://doi.org/10.1088/1367-2630/18/7/073037
arXiv:1406.3829
[23] R. Silva, Y. Guryanova, A. J. Short, P. Skrzypczyk, N. Brunner, and S. Popescu, Connecting processes with indefinite causal order and multi-time quantum states, New J. Phys. 19 , 103022 (2017); DOI: https://doi.org/10.1088/1367-2630/aa84fe; (2017).
https://doi.org/10.1088/1367-2630/aa84fe
arXiv:1701.08638
[24] M. Araújo, P. A. Guérin, and Ä. Baumeler, Quantum computation with indefinite causal structures, Phys. Rev. A 96, 052315 (2017); DOI: https://doi.org/10.1103/PhysRevA.96.052315; (2017).
https://doi.org/10.1103/PhysRevA.96.052315
arXiv:1706.09854
[25] S. Milz, F. A. Pollock, T. P. Le, G. Chiribella, and K. Modi, Entanglement, non-Markovianity, and causal non-separability, New J. Phys. 20, 033033 (2018); DOI: https://doi.org/10.1088/1367-2630/aaafee; (2017).
https://doi.org/10.1088/1367-2630/aaafee
arXiv:1711.04065
[26] G. Chiribella, Perfect discrimination of no-signalling channels via quantum superposition of causal structures, Phys. Rev. A 86, 040301 (2012); DOI: https://doi.org/10.1103/PhysRevA.86.040301; (2011).
https://doi.org/10.1103/PhysRevA.86.040301
arXiv:1109.5154
[27] C. Branciard, Witnesses of causal nonseparability: an introduction and a few case studies, Sci. Rep. 6, 26018 (2016); DOI: https://doi.org/10.1038/srep26018; (2016).
https://doi.org/10.1038/srep26018
arXiv:1603.00043
[28] T. Colnaghi, G. M. D’Ariano, P. Perinotti, and S. Facchini, Quantum computation with programmable connections between gates, Phys. Lett. A 376, 2940 – 2943 (2012); DOI: https://doi.org/10.1016/j.physleta.2012.08.028; (2011).
https://doi.org/10.1016/j.physleta.2012.08.028
arXiv:1109.5987
[29] M. Araújo, F. Costa, and Č. Brukner, Computational advantage from quantum-controlled ordering of gates, Phys. Rev. Lett. 113, 250402 (2014); DOI: https://doi.org/10.1103/PhysRevLett.113.250402; (2014).
https://doi.org/10.1103/PhysRevLett.113.250402
arXiv:1401.8127
[30] A. Feix, M. Araújo, and Č. Brukner, Quantum superposition of the order of parties as a communication resource, Phys. Rev. A 92, 052326 (2015); DOI: https://doi.org/10.1103/PhysRevA.92.052326; (2016).
https://doi.org/10.1103/PhysRevA.92.052326
arXiv:1508.07840
[31] P. A. 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); DOI: https://doi.org/10.1103/PhysRevLett.117.100502; (2016).
https://doi.org/10.1103/PhysRevLett.117.100502
arXiv:1605.07372
[32] N. Friis, V. Dunjko, W. Dür, and H. J. Briegel, Implementing quantum control for unknown subroutines, Phys. Rev. A 89, 030303(R) (2014); DOI: https://doi.org/10.1103/PhysRevA.89.030303; (2014).
https://doi.org/10.1103/PhysRevA.89.030303
arXiv:1401.8128
[33] L. M. Procopio et al., Experimental Superposition of Orders of Quantum Gates, Nat. Commun. 6, 7913 (2015); DOI: https://doi.org/10.1038/ncomms8913; (2014).
https://doi.org/10.1038/ncomms8913
arXiv:1412.4006
[34] N. Friis, A. A. Melnikov, G. Kirchmair, and H. J. Briegel, Coherent controlization using superconducting qubits, Sci. Rep. 5, 18036 (2015); DOI: https://doi.org/10.1038/srep18036; (2015).
https://doi.org/10.1038/srep18036
arXiv:1508.00447
[35] 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, e1602589 (2017); DOI: https://doi.org/10.1126/sciadv.1602589; (2016).
https://doi.org/10.1126/sciadv.1602589
arXiv:1608.01683
[36] G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, Experimental Entanglement of Temporal Orders, (2017).
arXiv:1712.06884
[37] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, Indefinite Causal Order in a Quantum Switch, Phys. Rev. Lett. 121, 090503 (2018); DOI: https://doi.org/10.1103/PhysRevLett.121.090503; (2018).
https://doi.org/10.1103/PhysRevLett.121.090503
arXiv:1803.04302
[38] L. Viola, E. Knill, and R. Laflamme, Constructing Qubits in Physical Systems, J. Phys. A 34, 7067 (2001); DOI https://doi.org/10.1088/0305-4470/34/35/331; (2001).
https://doi.org/10.1088/0305-4470/34/35/331
arXiv:quant-ph/0101090
[39] E. Knill, Protected realizations of quantum information, Phys. Rev. A 74, 042301 (2006); DOI: https://doi.org/10.1103/PhysRevA.74.042301; (2006).
https://doi.org/10.1103/PhysRevA.74.042301
arXiv:quant-ph/0603252
[40] D. W. Kribs and R. W. Spekkens, Quantum Error Correcting Subsystems are Unitarily Recoverable Subsystems, Phys. Rev. A 74, 042329 (2006); DOI: https://doi.org/10.1103/PhysRevA.74.042329; (2006).
https://doi.org/10.1103/PhysRevA.74.042329
arXiv:quant-ph/0608045
[41] P. Zanardi, Virtual Quantum Subsystems, Phys. Rev. Lett. 87, 077901 (2001); DOI: https://doi.org/10.1103/PhysRevLett.87.077901; (2001).
https://doi.org/10.1103/PhysRevLett.87.077901
arXiv:quant-ph/0103030
[42] P. Zanardi, D. Lidar, and S. Lloyd, Quantum Tensor Product Structures are Observable Induced, Phys. Rev. Lett. 92, 060402 (2004); DOI: https://doi.org/10.1103/PhysRevLett.92.060402; (2003).
https://doi.org/10.1103/PhysRevLett.92.060402
arXiv:quant-ph/0308043
[43] M. Araújo, Adrien Feix, Miguel Navascués, and Časlav Brukner, A purification postulate for quantum mechanics with indefinite causal order, Quantum 1, 10 (2017); DOI: https://doi.org/10.22331/q-2017-04-26-10; (2016).
https://doi.org/10.22331/q-2017-04-26-10
arXiv:1611.08535
[44] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Rep. Math. Phys. 3, 4, 275-278 (1972); DOI https://doi.org/10.1016/0034-4877(72)90011-0.
https://doi.org/10.1016/0034-4877(72)90011-0
[45] M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. Appl. 10, 285-290 (1975); DOI: https://doi.org/10.1016/0024-3795(75)90075-0.
https://doi.org/10.1016/0024-3795(75)90075-0
[46] O. Oreshkov and N. J. Cerf, Operational formulation of time reversal in quantum theory, Nature Phys. 11, 853-858 (2015); DOI: https://doi.org/10.1038/nphys3414; (2015).
https://doi.org/10.1038/nphys3414
arXiv:1507.07745
[47] P. Perinotti, Causal Structures and the Classification of Higher Order Quantum Computations, in Time in physics, R. Renner and S. Stupar (eds), Tutorials, Schools, and Workshops in the Mathematical Sciences, (Birkhäuser, Cham, 2017); DOI: https://doi.org/10.1007/978-3-319-68655-4_7;.
https://doi.org/10.1007/978-3-319-68655-4_7
arXiv:1612.05099
[48] A. Kissinger and S. Uijlen, A categorical semantics for causal structure, Logical Methods in Computer Science, Volume 15, Issue 3 (August 9, 2019), lmcs:5681; DOI: https://doi.org/10.23638/LMCS-15(3:15)2019; (2017).
https://doi.org/10.23638/LMCS-15(3:15)2019
arXiv:1701.04732
[49] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, (Cambridge University Press, Cambridge, 2000); DOI https://doi.org/10.1017/CBO9780511976667.
https://doi.org/10.1017/CBO9780511976667
[50] E. Castro-Ruiz, F. Giacomini, and Časlav Brukner, Dynamics of Quantum Causal Structures, Phys. Rev. X 8, 011047 (2018); DOI: https://doi.org/10.1103/PhysRevX.8.011047; (2017).
https://doi.org/10.1103/PhysRevX.8.011047
arXiv:1710.03139
[51] Strictly speaking, in Ref. Chiribella12 it was shown that if there exists a realization of the quantum SWITCH such that Alice’s operation is in the past of Bob’s operation so that the output ancilla of Alice could be connected to the input ancilla of Bob, this would allow deterministic transmission of information back in time. In the realization discussed here, this condition is not satisfied-the ancillary systems of Alice and Bob cannot be connected to each other as they occupy space-like separated regions. Nevertheless, the full experiment still has the structure of a circuit with a `timelike’ cycle, albeit not permitting deterministic time travel, as any quantum process matrix is equivalent to a channel from the output systems of all parties to their input systems OCB.
[52] More precisely, C-SWAP$^{XYZ} = |0ranglelangle 0|^{X}otimes mathbb{I}^{YZ} + |1ranglelangle 1|^{X}otimes$SWAP$^{YZ}$, where SWAP$^{YZ}$ is the SWAP operator on $Y$ and $Z$, which can be defined as follows. Consider two systems $Y$ and $Z$ with Hilbert spaces of the same dimension and a linear isomorphism between the states in these Hilbert spaces. An arbitrary vector in the joint system $YZ$ can be written in the form $|psirangle^{YZ} = sum_{i,j} psi_{ij}|irangle^Y |jrangle^Z$, where ${|irangle^Y}$ are orthonormal bases for $Y$ and $Z$, respectively. The action of the operator SWAP$^{YZ}$ on the vector $|psirangle^{YZ}$ is then given by SWAP$^{YZ} |psirangle^{YZ} = sum_{i,j} psi_{ij}|jrangle^Y |irangle^Z$.
[53] Of course, if during the working of the device, an adversary turns on unwanted interactions, such as a Hamiltonian on the control qubit that is not diagonal in the logical basis, this could prevent the device from implementing the correct operation on the systems of interest. But this is the case for any physical device implementing an operation, irrespectively of whether the operation is localized or delocalized in time.
[54] Very recently, after the submission of this paper, the author and colleagues J. Barrett and R. Lorenz showed via different methods that all bipartite processes that are unitarily extendible are causally separable, and hence their unitary extensions are variations of the quantum SWITCH (in preparation). Nevertheless, we believe that the proof of realizability presented here has a particular value since it is based on a different idea that could have wider applications. In particular, it provides the basis for the generalization in Sec. 7, and might be useful in the search for realizations of more complicated unitary processes.
[55] W. F. Stinespring, Positive functions on C*-algebras, Proc. Amer. Math. Soc. 6, 211 (1955); DOI: https://doi.org/10.2307/2032342.
https://doi.org/10.2307/2032342
[56] Throughout this paper, when we speak about iso
morphic mapping between two Hilbert spaces, we understand isometric isomorphism.
[57] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014); DOI: https://doi.org/10.1103/RevModPhys.86.419; (2013).
https://doi.org/10.1103/RevModPhys.86.419
arXiv:1303.2849
[58] After this paper appeared, a subsequent paper AllardGuerin claimed to show that all unitary processes admit a representation on time-delocalized subsystems. However, this claim is based on a misunderstanding of the concept of time-delocalized subsystems. The proof claimed in AllardGuerin amounts to the observation (discussed in this paper) that if we have a unitary process, the unitary maps isomorphically the output system of any one party, say Alice, onto a subsystem of the input systems of the rest of the parties, and similarly maps a subsystem of the output systems of the rest of the parties onto the input system of Alice. This by itself does not imply that we can associate the input and output systems of Alice with time-delocalized subsystems (which are subsystems of tensor products of Hilbert spaces associated with concrete physical systems at concrete times).
[59] P. Allard Guérin and Č. Brukner, Observer-dependent locality of quantum events, New J. Phys. 20, 103031 (2018); DOI: https://doi.org/10.1088/1367-2630/aae742; (2018).
https://doi.org/10.1088/1367-2630/aae742
arXiv:1805.12429
[60] D. Ebler, S. Salek, and G. Chiribella, Enhanced Communication with the Assistance of Indefinite Causal Order, Phys. Rev. Lett. 120, 120502 (2018); DOI: https://doi.org/10.1103/PhysRevLett.120.120502; (2017).
https://doi.org/10.1103/PhysRevLett.120.120502
arXiv:1711.10165
[61] M. Zych, F. Costa, I. Pikovski, and Časlav Brukner, Bell’s Theorem for Temporal Order, Nat. Commun. 10, 3772 (2019); DOI: https://doi.org/10.1038/s41467-019-11579-x; (2017).
https://doi.org/10.1038/s41467-019-11579-x
arXiv:1708.00248
[62] A. Dimić, M. Milivojević, D. Gočanin, and Časlav Brukner, Simulating spacetime with indefinite causal order via Rindler observers, (2017).
arXiv:1712.02689
Cited by
[1] Christina Giarmatzi, Springer Theses 1 (2019) ISBN:978-3-030-31929-8.
[2] Daniel Ebler, Sina Salek, and Giulio Chiribella, “Enhanced Communication with the Assistance of Indefinite Causal Order”, Physical Review Letters 120 12, 120502 (2018).
[3] Giulio Chiribella and Hlér Kristjánsson, “Quantum Shannon theory with superpositions of trajectories”, Proceedings of the Royal Society of London Series A 475 2225, 20180903 (2019).
[4] 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).
[5] Philippe Allard Guérin and Časlav Brukner, “Observer-dependent locality of quantum events”, New Journal of Physics 20 10, 103031 (2018).
[6] Sina Salek, Daniel Ebler, and Giulio Chiribella, “Quantum communication in a superposition of causal orders”, arXiv:1809.06655.
[7] Giulio Chiribella, Manik Banik, Some Sankar Bhattacharya, Tamal Guha, Mir Alimuddin, Arup Roy, Sutapa Saha, Sristy Agrawal, and Guruprasad Kar, “Indefinite causal order enables perfect quantum communication with zero capacity channel”, arXiv:1810.10457.
[8] Lucien Hardy, “Implementation of the Quantum Equivalence Principle”, arXiv:1903.01289.
[9] 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] Lucien Hardy, “The Construction Interpretation: Conceptual Roads to Quantum Gravity”, arXiv:1807.10980.
[11] Alastair A. Abbott, Julian Wechs, Dominic Horsman, Mehdi Mhalla, and Cyril Branciard, “Communication through coherent control of quantum channels”, arXiv:1810.09826.
[12] Yu Guo, Xiao-Min Hu, Zhi-Bo Hou, Huan Cao, Jin-Ming Cui, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, and Giulio Chiribella, “Experimental transmission of quantum information using a superposition of causal orders”, arXiv:1811.07526.
[13] Philippe Allard Guérin, Marius Krumm, Costantino Budroni, and Časlav Brukner, “Composition rules for quantum processes: a no-go theorem”, New Journal of Physics 21 1, 012001 (2019).
[14] Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, “Communication through quantum-controlled noise”, arXiv:1812.06848.
[15] C. T. Marco Ho, Fabio Costa, Christina Giarmatzi, and Timothy C. Ralph, “Violation of a causal inequality in a spacetime with definite causal order”, arXiv:1804.05498.
[16] Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, “Quantum Causal Models”, arXiv:1906.10726.
[17] Philippe Allard Guérin, Giulia Rubino, and Časlav Brukner, “Communication through quantum-controlled noise”, Physical Review A 99 6, 062317 (2019).
[18] Esteban Castro-Ruiz, Flaminia Giacomini, Alessio Belenchia, and Časlav Brukner, “Time reference frames and gravitating quantum clocks”, arXiv:1908.10165.
[19] Nicolas Loizeau and Alexei Grinbaum, “Channel capacity enhancement with indefinite causal order”, arXiv:1906.08505.
The above citations are from Crossref’s cited-by service (last updated successfully 2020-01-22 19:34:08) and SAO/NASA ADS (last updated successfully 2020-01-22 19:34:09). 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.
Source: https://quantum-journal.org/papers/q-2019-12-02-206/
