Background Independence and Quantum Causal Structure

Lachlan Parker and Fabio Costa

Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, QLD 4072 Australia

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Abstract

One of the key ways in which quantum mechanics differs from relativity is that it requires a fixed background reference frame for spacetime. In fact, this appears to be one of the main conceptual obstacles to uniting the two theories. Additionally, a combination of the two theories is expected to yield non-classical, or "indefinite", causal structures. In this paper, we present a background-independent formulation of the process matrix formalism – a form of quantum mechanics that allows for indefinite causal structure – while retaining operationally well-defined measurement statistics. We do this by postulating an arbitrary probability distribution of measurement outcomes across discrete "chunks" of spacetime, which we think of as physical laboratories, and then requiring that this distribution be invariant under any permutation of laboratories. We find (a) that one still obtains nontrivial, indefinite causal structures with background independence, (b) that we lose the idea of local operations in distinct laboratories, but can recover it by encoding a reference frame into the physical states of our system, and (c) that permutation invariance imposes surprising symmetry constraints that, although formally similar to a superselection rule, cannot be interpreted as such.

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► References

[1] Jeremy Butterfield and Christopher J. Isham. ``Space-time and the philosophical challenge of quantum gravity''. Pages 33–89. Cambridge University Press. (1999). arXiv:gr-qc/​9903072.
https:/​/​doi.org/​10.1017/​CBO9780511612909.003
arXiv:gr-qc/9903072

[2] Lucien Hardy. ``Probability theories with dynamic causal structure: A New framework for quantum gravity'' (2005). arXiv:gr-qc/​0509120.
arXiv:gr-qc/0509120

[3] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. ``Quantum correlations with no causal order''. Nat. Commun. 3 (2012).
https:/​/​doi.org/​10.1038/​ncomms2076

[4] C. Rovelli. ``What is observable in classical and quantum gravity?''. Class. Quantum Grav. 8 (1991).
https:/​/​doi.org/​10.1088/​0264-9381/​8/​2/​011

[5] John D. Norton. ``The Hole Argument''. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University (2019). Summer 2019 edition.
https:/​/​plato.stanford.edu/​archives/​sum2019/​entries/​spacetime-holearg/​

[6] Abhay Ashtekar and Jerzy Lewandowski. ``Background independent quantum gravity: a status report''. Classical and Quantum Gravity 21, R53–R152 (2004).
https:/​/​doi.org/​10.1088/​0264-9381/​21/​15/​r01

[7] Lee Smolin. ``The Case for Background Independence''. In The Structural Foundations of Quantum Gravity. Oxford University Press (2006). arXiv:hep-th/​0507235.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199269693.003.0007
arXiv:hep-th/0507235

[8] L. Procopio, A. Moqnaki, M. Araujo, F. Costa, I. Calafell, E. Dowd, D. Hamel, L. Rozema, C. Brukner, and P. Walther. ``Experimental superposition of orders of quantum gates''. Nat. Comms. 6 (2015).
https:/​/​doi.org/​10.1038/​ncomms8913

[9] Giulia Rubino, Lee A Rozema, Adrien Feix, Mateus Araújo, Jonas M Zeuner, Lorenzo M Procopio, Časlav Brukner, and Philip Walther. ``Experimental verification of an indefinite causal order''. Science Advances 3, e1602589 (2017).
https:/​/​doi.org/​10.1126/​sciadv.1602589

[10] Giulia Rubino, Lee A. Rozema, Francesco Massa, Mateus Araújo, Magdalena Zych, Časlav Brukner, and Philip Walther. ``Experimental entanglement of temporal order''. Quantum 6, 621 (2022).
https:/​/​doi.org/​10.22331/​q-2022-01-11-621

[11] 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 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.090503

[12] K. Goswami, Y. Cao, G. A. Paz-Silva, J. Romero, and A. G. White. ``Increasing communication capacity via superposition of order''. Phys. Rev. Research 2, 033292 (2020).
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.033292

[13] Kejin Wei, Nora Tischler, Si-Ran Zhao, Yu-Huai Li, Juan Miguel Arrazola, Yang Liu, Weijun Zhang, Hao Li, Lixing You, Zhen Wang, et al. ``Experimental quantum switching for exponentially superior quantum communication complexity''. Phys. Rev. Lett. 122, 120504 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.120504

[14] 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''. PRX Quantum 2, 010320 (2021).
https:/​/​doi.org/​10.1103/​PRXQuantum.2.010320

[15] 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''. Phys. Rev. Lett. 124, 030502 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.030502

[16] Giulia Rubino, Lee A. Rozema, Daniel Ebler, Hlér Kristjánsson, Sina Salek, Philippe Allard Guérin, Alastair A. Abbott, Cyril Branciard, Časlav Brukner, Giulio Chiribella, and Philip Walther. ``Experimental quantum communication enhancement by superposing trajectories''. Phys. Rev. Research 3, 013093 (2021).
https:/​/​doi.org/​10.1103/​PhysRevResearch.3.013093

[17] Philippe Allard Guérin and Časlav Brukner. ``Observer-dependent locality of quantum events''. New Journal of Physics 20, 103031 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aae742

[18] Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019).
https:/​/​doi.org/​10.22331/​q-2019-12-02-206

[19] 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

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

[21] T. Colnaghi, G. D'Ariano, S. Facchini, and P. Perinotti. ``Quantum computation with programmable connections between gates''. Phys. Lett. A 376 (2012).
https:/​/​doi.org/​10.1016/​j.physleta.2012.08.028

[22] 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

[23] Adrien Feix, Mateus Araújo, and Časlav 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

[24] Philippe Allard Guérin, Adrien Feix, Mateus Araújo, and Časlav 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

[25] Daniel Ebler, Sina Salek, and Giulio Chiribella. ``Enhanced communication with the assistance of indefinite causal order''. Phys. Rev. Lett. 120, 120502 (2018).
https:/​/​doi.org/​10.1103/​physrevlett.120.120502

[26] Sina Salek, Daniel Ebler, and Giulio Chiribella. ``Quantum communication in a superposition of causal orders'' (2018). arXiv:1809.06655v2.
arXiv:1809.06655v2

[27] Manish K. Gupta and Ujjwal Sen. ``Transmitting quantum information by superposing causal order of mutually unbiased measurements'' (2019). arXiv:1909.13125v1.
arXiv:1909.13125v1

[28] S. Shrapnel, F. Costa, and G. Milburn. ``Updating the Born Rule''. New J. Phys. 20 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aabe12

[29] M. Araujo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and C. Brukner. ``Witnessing causal noseparability''. New. J. Phys. 17 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​10/​102001

[30] A. Jamiołkowski. ``Linear transformations which preserve trace and positive semidefiniteness of operators''. Rep. Math. Phys 3, 275–278 (1972).
https:/​/​doi.org/​10.1016/​0034-4877(72)90011-0

[31] Man-Duen Choi. ``Completely positive linear maps on complex matrices''. Linear Algebra Appl. 10, 285–290 (1975).
https:/​/​doi.org/​10.1016/​0024-3795(75)90075-0

[32] T. Heinosaari and M. Ziman. ``The mathematical language of quanum theory''. Cambridge University Press. (2012).
https:/​/​doi.org/​10.1017/​CBO9781139031103

[33] A. Abbott, G. Giarmatzi, F. Costa, and C. Branciard. ``Multipartite causal correlations: polytopes and inequalities''. Phys. Rev. A 94 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.032131

[34] Alastair Abbott, Julian Wechs, Fabio Costa, and Cyril Branciard. ``Genuinely multipartite noncausality''. Quantum 1, 39 (2017).
https:/​/​doi.org/​10.22331/​q-2017-12-14-39

[35] J. Wechs, A. Abbott, and C. Branciard. ``On the definition and characterisation of multipartite causal (non)separability''. New J. Phys. 21 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​aaf352

[36] Jan Myrheim. ``Statistical geometry''. Technical Report CERN-TH-2538. CERN (1978). url: http:/​/​cds.cern.ch/​record/​293594.
http:/​/​cds.cern.ch/​record/​293594

[37] Luca Bombelli, Joohan Lee, David Meyer, and Rafael D. Sorkin. ``Space-time as a causal set''. Phys. Rev. Lett. 59, 521–524 (1987).
https:/​/​doi.org/​10.1103/​PhysRevLett.59.521

[38] G 't Hooft. ``Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness''. Class. and Quantum Grav. 13, 1023–1039 (1996).
https:/​/​doi.org/​10.1088/​0264-9381/​13/​5/​018

[39] Renate Loll. ``Discrete approaches to quantum gravity in four dimensions''. Living Reviews in Relativity 1, 13 (1998).
https:/​/​doi.org/​10.12942/​lrr-1998-13

[40] Fay Dowker. ``Causal sets as discrete spacetime''. Contemp. Phys. 47, 1–9 (2006).
https:/​/​doi.org/​10.1080/​17445760500356833

[41] Pablo Arrighi, Marios Christodoulou, and Amélia Durbec. ``Quantum superpositions of graphs'' (2020). arXiv:2010.13579.
arXiv:2010.13579

[42] S. Bartlett, T. Rudolph, and R. Spekkens. ``Reference frames, superselection rules, and quantum information''. Rev. Mod. Phys. 79 (2007).
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[43] Lachlan Parker. ``Permutation invariance in quantum processes.''. Honours thesis. The University of Queensland. (2020).
https:/​/​doi.org/​10.14264/​4ab01e5

[44] C. Branciard, M. Araujo, A. Feix, F. Costa, and C. Brukner. ``The simplest causal inequalities and their violation''. New J. Phys. 27 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​1/​013008

[45] Don N. Page and William K. Wootters. ``Evolution without evolution: Dynamics described by stationary observables''. Phys. Rev. D 27, 2885–2892 (1983).
https:/​/​doi.org/​10.1103/​PhysRevD.27.2885

[46] Carlo Rovelli. ``Relational quantum mechanics''. International Journal of Theoretical Physics 35, 1637–1678 (1996).
https:/​/​doi.org/​10.1007/​BF02302261

[47] David Poulin. ``Toy model for a relational formulation of quantum theory''. International Journal of Theoretical Physics 45, 1189–1215 (2006).
https:/​/​doi.org/​10.1007/​s10773-006-9052-0

[48] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum time''. Phys. Rev. D 92, 045033 (2015).
https:/​/​doi.org/​10.1103/​PhysRevD.92.045033

[49] Takayuki Miyadera, Leon Loveridge, and Paul Busch. ``Approximating relational observables by absolute quantities: a quantum accuracy-size trade-off''. Journal of Physics A: Mathematical and Theoretical 49, 185301 (2016).
https:/​/​doi.org/​10.1088/​1751-8113/​49/​18/​185301

[50] Flaminia Giacomini, Esteban Castro-Ruiz, and Časlav Brukner. ``Quantum mechanics and the covariance of physical laws in quantum reference frames''. Nature Communications 10, 494 (2019).
https:/​/​doi.org/​10.1038/​s41467-018-08155-0

[51] Alexander R. H. Smith and Mehdi Ahmadi. ``Quantizing time: Interacting clocks and systems''. Quantum 3, 160 (2019).
https:/​/​doi.org/​10.22331/​q-2019-07-08-160

[52] Lucien Hardy. ``Operational general relativity: Possibilistic, probabilistic, and quantum'' (2016). arXiv:1608.06940.
arXiv:1608.06940

[53] Magdalena Zych, Fabio Costa, and Timothy C. Ralph. ``Relativity of quantum superpositions'' (2018). arXiv:1809.04999.
arXiv:1809.04999

Cited by

[1] Matheus Capela, Harshit Verma, Fabio Costa, and Lucas Chibebe Céleri, "Indefinite causal order is not always a resource for thermodynamic processes", arXiv:2208.03205, (2022).

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