We study the notion of causal orders for the cases of (classical and quantum) circuits and spacetime events. We show that every circuit can be immersed into a classical spacetime, preserving the compatibility between the two causal structures. Using the process matrix formalism, we analyse the realisations of the quantum switch using 4 and 3 spacetime events in classical spacetimes with fixed causal orders, and the realisation of a gravitational switch with only 2 spacetime events that features superpositions of different gravitational field configurations and their respective causal orders. We show that the current quantum switch experimental implementations do not feature superpositions of causal orders between spacetime events, and that these superpositions can only occur in the context of superposed gravitational fields. We also discuss a recently introduced operational notion of an event, which does allow for superpositions of respective causal orders in flat spacetime quantum switch implementations. We construct two observables that can distinguish between the quantum switch realisations in classical spacetimes, and gravitational switch implementations in superposed spacetimes. Finally, we discuss our results in the light of the modern relational approach to physics.
 J. S. Bell, Physics Physique Fizika 1, 195 (1964).
 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, Nature Communications 6, 7913 (2015).
 L. Hardy, Journal of Physics A: Mathematical and Theoretical 40, 3081 (2007).
 M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi and Č. Brukner, New Journal of Physics 17, 102001 (2015).
 H. Lichtenegger and B. Mashhoon, in The Measurement of Gravitomagnetism: A Challenging Enterprise, edited by L. Iorio, 13, Nova Science Pub Inc, New York (2007).
 M. Blagojević, Gravitation and Gauge Symmetries, Institute of Physics Publishing, Bristol (2002).
 A. S. Eddington, Space time and gravitation, Cambridge University Press, Cambridge (1921).
 C. Misner, K. Thorne and J. Wheeler, Gravitation, W. H. Freeman, San Francisco (1973).
 C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge (2004).
 J. Janjić, N. Paunković and M. Vojinović, in preparation.
 C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge University Press, Cambridge (2014).
 R. A. Bertlmann and P. Krammer, Journal of Physics A: Mathematical and Theoretical 41, 235303 (2008).
 Laura J. Henderson, Alessio Belenchia, Esteban Castro-Ruiz, Costantino Budroni, Magdalena Zych, Časlav Brukner, and Robert B. Mann, "Quantum Temporal Superposition: The Case of Quantum Field Theory", Physical Review Letters 125 13, 131602 (2020).
 Jonathan Barrett, Robin Lorenz, and Ognyan Oreshkov, "Cyclic quantum causal models", Nature Communications 12 1, 885 (2021).
 Julian Wechs, Hippolyte Dourdent, Alastair A. Abbott, and Cyril Branciard, "Quantum circuits with classical versus quantum control of causal order", arXiv:2101.08796.
 Nicola Pinzani and Stefano Gogioso, "Giving Operational Meaning to the Superposition of Causal Orders", arXiv:2003.13306.
 Pablo Arrighi, Marios Christodoulou, and Amélia Durbec, "Quantum superpositions of graphs", arXiv:2010.13579.
 Ricardo Faleiro, Nikola Paunković, and Marko Vojinović, "Operational interpretation of the vacuum and process matrices for identical particles", arXiv:2010.16042.
The above citations are from Crossref's cited-by service (last updated successfully 2021-08-01 07:01:54) and SAO/NASA ADS (last updated successfully 2021-08-01 07:01:55). 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.