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.
 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.
 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.
 Abhay Ashtekar and Jerzy Lewandowski. ``Background independent quantum gravity: a status report''. Classical and Quantum Gravity 21, R53–R152 (2004).
 Lee Smolin. ``The Case for Background Independence''. In The Structural Foundations of Quantum Gravity. Oxford University Press (2006). arXiv:hep-th/0507235.
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 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).
 Ognyan Oreshkov. ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics''. Quantum 3, 206 (2019).
 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(R) (2012).
 T. Colnaghi, G. D'Ariano, S. Facchini, and P. Perinotti. ``Quantum computation with programmable connections between gates''. Phys. Lett. A 376 (2012).
 M. Araújo, F. Costa, and Č. Brukner. ``Computational Advantage from Quantum-Controlled Ordering of Gates''. Phys. Rev. Lett. 113, 250402 (2014).
 Adrien Feix, Mateus Araújo, and Časlav Brukner. ``Quantum superposition of the order of parties as a communication resource''. Phys. Rev. A 92, 052326 (2015).
 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).
 Daniel Ebler, Sina Salek, and Giulio Chiribella. ``Enhanced communication with the assistance of indefinite causal order''. Phys. Rev. Lett. 120, 120502 (2018).
 M. Araujo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and C. Brukner. ``Witnessing causal noseparability''. New. J. Phys. 17 (2015).
 G 't Hooft. ``Quantization of point particles in (2 + 1)-dimensional gravity and spacetime discreteness''. Class. and Quantum Grav. 13, 1023–1039 (1996).
 C. Branciard, M. Araujo, A. Feix, F. Costa, and C. Brukner. ``The simplest causal inequalities and their violation''. New J. Phys. 27 (2015).
 Don N. Page and William K. Wootters. ``Evolution without evolution: Dynamics described by stationary observables''. Phys. Rev. D 27, 2885–2892 (1983).
 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).
 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).
 Matheus Capela, Harshit Verma, Fabio Costa, and Lucas C. Céleri, "Reassessing thermodynamic advantage from indefinite causal order", Physical Review A 107 6, 062208 (2023).
The above citations are from Crossref's cited-by service (last updated successfully 2023-12-07 06:14:31) and SAO/NASA ADS (last updated successfully 2023-12-07 06:14:32). 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.