# A no-go theorem for superpositions of causal orders

Fabio Costa

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

### Abstract

The causal order of events need not be fixed: whether a bus arrives before or after another at a certain stop can depend on other variables – like traffic. Coherent quantum control of causal order is possible too and is a useful resource for several tasks. However, quantum control implies that a controlling system carries the which-order information – if the control is traced out, the order of events remains in a probabilistic mixture. Can the order of two events be in a pure superposition, uncorrelated with any other system? Here we show that this is not possible for a broad class of processes: a pure superposition of any pair of Markovian, unitary processes with equal local dimensions and different causal orders is not a valid process, namely it results in non-normalised probabilities when probed with certain operations. The result imposes constraints on novel resources for quantum information processing and on possible processes in a theory of quantum gravity.

Quantum theory famously departs from our classical intuition, allowing for objects to be in “superposition”: A particle that is in a superposition of two locations can be found in either location, but, as long as it’s not measured, it cannot be said to be in either place: its position is indefinite. Recent studies have discovered that even the causal order between events can be indefinite: there are situations that cannot be interpreted as A being before B, nor as B before A. It is natural to think that indefinite causal order extends the idea of quantum superposition and, indeed, several experiments have claimed the realisation of “superpositions of causal orders”. However, these experiments involve, apart from the events A, B whose causal order is under investigation, an additional system that determines the order of the two events. We can think of the extra system as a coin flip: if it lands heads, we perform A before B, if it’s tail, B is before A. By preparing the coin in a superposition of heads and tails, we obtain an indefinite order between A and B. However, a careful analysis shows that, although the coin and the events are all in a superposition, the events alone cannot be considered, by themselves, in a superposition of orders. This makes us wonder: can we actually realise a superposition of causal orders, without relying on the quantum coin flip?

This work shows that, in general, indefinite causal order and quantum superpositions are quite distinct concepts. In particular, when we consider the experiments performed so far, we cannot simply remove the coin flip: it is essential for the whole scheme to be even possible. More generally, this work shows that quantum superpositions of causal orders are not possible, at least in a very broad range of situations. The result provides new tools to investigate indefinite causal order, which will help in the search of novel quantum causal structures.

### ► References

[1] R. P. Feynman, Space-Time Approach to Non-Relativistic Quantum Mechanics,'' Rev. Mod. Phys. 20, 367–387 (1948).
https:/​/​doi.org/​10.1103/​RevModPhys.20.367

[2] J. Butterfield and C. Isham, Spacetime and the philosophical challenge of quantum gravity, p. 33–89. Cambridge University Press. arXiv:gr-qc/​9903072.
https:/​/​doi.org/​10.1017/​CBO9780511612909.003
arXiv:arXiv:gr-qc/9903072

[3] L. Hardy, Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure,'' J. Phys. A: Math. Gen. 40, 3081–3099 (2007).
https:/​/​doi.org/​10.1088/​1751-8113/​40/​12/​S12

[4] M. Zych, F. Costa, I. Pikovski, and Č. Brukner, Bell's theorem for temporal order,'' Nat. Commun. 10, 3772 (2019).
https:/​/​doi.org/​10.1038/​s41467-019-11579-x

[5] 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

[6] 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).

[7] G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, Č. Brukner, and P. Walther, Experimental entanglement of temporal order,'' Quantum 6, 621 (2022).
https:/​/​doi.org/​10.22331/​q-2022-01-11-621

[8] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.121.090503

[9] 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

[10] Y. Guo, X.-M. Hu, Z.-B. Hou, H. Cao, J.-M. Cui, B.-H. Liu, Y.-F. Huang, C.-F. Li, G.-C. Guo, and G. 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

[11] K. Wei, N. Tischler, S.-R. Zhao, Y.-H. Li, J. M. Arrazola, Y. Liu, W. Zhang, H. Li, L. You, Z. Wang, Y.-A. Chen, B. C. Sanders, Q. Zhang, G. J. Pryde, F. Xu, and J.-W. Pan, Experimental Quantum Switching for Exponentially Superior Quantum Communication Complexity,'' Phys. Rev. Lett. 122, 120504 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.120504

[12] M. M. Taddei, J. Cariñe, D. Martínez, T. García, N. Guerrero, A. A. Abbott, M. Araújo, C. Branciard, E. S. Gómez, S. P. Walborn, L. Aolita, and G. 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

[13] 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

[14] 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

[15] T. Colnaghi, G. M. D'Ariano, S. Facchini, and P. Perinotti, Quantum computation with programmable connections between gates,'' Phys. Lett. A 376, 2940–2943 (2012).
https:/​/​doi.org/​10.1016/​j.physleta.2012.08.028

[16] 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

[17] 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

[18] 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.117.100502

[19] D. Ebler, S. Salek, and G. Chiribella, Enhanced Communication with the Assistance of Indefinite Causal Order,'' Phys. Rev. Lett. 120, 120502 (2018).
https:/​/​doi.org/​10.1103/​physrevlett.120.120502

[20] S. Salek, D. Ebler, and G. Chiribella, Quantum communication in a superposition of causal orders,'' arXiv:1809.06655v2 [quant-ph].
arXiv:1809.06655v2

[21] M. K. Gupta and U. Sen, Transmitting quantum information by superposing causal order of mutually unbiased measurements,'' arXiv:1909.13125v1 [quant-ph].
arXiv:arXiv:1909.13125v1

[22] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman, Superpositions of time evolutions of a quantum system and a quantum time-translation machine,'' Phys. Rev. Lett. 64, 2965 (1990).
https:/​/​doi.org/​10.1103/​PhysRevLett.64.2965

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

[24] R. Oeckl, A “general boundary” formulation for quantum mechanics and quantum gravity,'' Phys. Lett. B 575, 318–324 (2003).
https:/​/​doi.org/​10.1016/​j.physletb.2003.08.043

[25] Y. Aharonov, S. Popescu, J. Tollaksen, and L. Vaidman, Multiple-time states and multiple-time measurements in quantum mechanics,'' Phys. Rev. A 79, 052110 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.052110

[26] J. Cotler, C.-M. Jian, X.-L. Qi, and F. Wilczek, Superdensity operators for spacetime quantum mechanics,'' J. High Energ. Phys. 2018, 93 (2018).
https:/​/​doi.org/​10.1007/​jhep09(2018)093

[27] 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).
https:/​/​doi.org/​10.1088/​1367-2630/​aa84fe

[28] J. Barrett, R. Lorenz, and O. Oreshkov, Quantum Causal Models,'' arXiv:1906.10726v2 [quant-ph].
arXiv:1906.10726v2

[29] P. A. Guérin and Č. Brukner, Observer-dependent locality of quantum events,'' New J. Phys. 20, 103031 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aae742

[30] O. 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

[31] T. Heinosaari and M. Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement. Cambridge University Press, 2011.
https:/​/​doi.org/​10.1017/​CBO9781139031103

[32] M.-D. 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

[33] 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

[34] S. Shrapnel, F. Costa, and G. Milburn, Updating the Born rule,'' New J. Phys. 20, 053010 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aabe12

[35] O. Oreshkov and N. J. Cerf, Operational quantum theory without predefined time,'' New J. Phys. 18, 073037 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​7/​073037

[36] 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).
https:/​/​doi.org/​10.1088/​1367-2630/​aaafee

[37] O. Oreshkov and C. Giarmatzi, Causal and causally separable processes,'' New J. of Phys. 18, 093020 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​9/​093020

[38] J. Wechs, A. A. Abbott, and C. Branciard, On the definition and characterisation of multipartite causal (non)separability,'' New J. of Phys. 21, 013027 (2019).
https:/​/​doi.org/​10.1088/​1367-2630/​aaf352

[39] 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

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

[41] Ä. Baumeler, F. Costa, T. C. Ralph, S. Wolf, and M. Zych, Reversible time travel with freedom of choice,'' Class. Quantum Grav. 36, 224002 (2019).
https:/​/​doi.org/​10.1088/​1361-6382/​ab4973

[42] G. Tobar and F. Costa, Reversible dynamics with closed time-like curves and freedom of choice,'' Classical and Quantum Gravity 37, 205011 (2020).
https:/​/​doi.org/​10.1088/​1361-6382/​aba4bc

[43] 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

[44] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, A purification postulate for quantum mechanics with indefinite causal order,'' Quantum 1, 10 (2017).
https:/​/​doi.org/​10.22331/​q-2017-04-26-10

[45] F. Costa and S. Shrapnel, Quantum causal modelling,'' New J. of Phys. 18, 063032 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​6/​063032

[46] F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, Operational Markov Condition for Quantum Processes,'' Phys. Rev. Lett. 120, 040405 (2018).
https:/​/​doi.org/​10.1103/​physrevlett.120.040405

[47] C. Giarmatzi and F. Costa, Witnessing quantum memory in non-Markovian processes,'' Quantum 5, 440 (2021).
https:/​/​doi.org/​10.22331/​q-2021-04-26-440

[48] W. Yokojima, M. T. Quintino, A. Soeda, and M. Murao, Consequences of preserving reversibility in quantum superchannels,'' Quantum 5, 441 (2021).
https:/​/​doi.org/​10.22331/​q-2021-04-26-441

[49] F. Costa, A no-go theorem for superpositions of causal order.'' Space-time and Information, Manitoulin Island, Ontario, Canada, 2017.
https:/​/​uwaterloo.ca/​spacetime-information-workshop/​abstracts

[50] T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio, Resource Theory of Superposition,'' Phys. Rev. Lett. 119, 230401 (2017).
https:/​/​doi.org/​10.1103/​PhysRevLett.119.230401

[51] F. Bischof, H. Kampermann, and D. Bruß, Resource Theory of Coherence Based on Positive-Operator-Valued Measures,'' Phys. Rev. Lett. 123, 110402 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.110402

### Cited by

[1] Carlos Sabín, "Causality in a Qubit-Based Implementation of a Quantum Switch", Universe 8 5, 269 (2022).

[2] Llorenç Escolà-Farràs and Daniel Braun, "Quantifying causal influence in quantum mechanics", Physical Review A 106 6, 062415 (2022).

[3] Tom Purves and Anthony J. Short, "Quantum Theory Cannot Violate a Causal Inequality", Physical Review Letters 127 11, 110402 (2021).

[4] Anne-Catherine de la Hamette, Viktoria Kabel, Marios Christodoulou, and Časlav Brukner, "Quantum diffeomorphisms cannot make indefinite causal order definite", arXiv:2211.15685, (2022).

[5] Matt Wilson, Giulio Chiribella, and Aleks Kissinger, "Quantum Supermaps are Characterized by Locality", arXiv:2205.09844, (2022).

[6] Simon Milz, Dominic Jurkschat, Felix A. Pollock, and Kavan Modi, "Delayed-choice causal order and nonclassical correlations", Physical Review Research 3 2, 023028 (2021).

[7] Carlos Sabín, "Causality in a qubit-based quantum switch", arXiv:2109.14951, (2021).

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