Causal structure in the presence of sectorial constraints, with application to the quantum switch

Nick Ormrod1, Augustin Vanrietvelde1,2,3, and Jonathan Barrett1

1Quantum Group, Department of Computer Science, University of Oxford
2Department of Physics, Imperial College London
3HKU-Oxford Joint Laboratory for Quantum Information and Computation

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Existing work on quantum causal structure assumes that one can perform arbitrary operations on the systems of interest. But this condition is often not met. Here, we extend the framework for quantum causal modelling to situations where a system can suffer $\textit{sectorial constraints}$, that is, restrictions on the orthogonal subspaces of its Hilbert space that may be mapped to one another. Our framework (a) proves that a number of different intuitions about causal relations turn out to be equivalent; (b) shows that quantum causal structures in the presence of sectorial constraints can be represented with a directed graph; and (c) defines a fine-graining of the causal structure in which the individual sectors of a system bear causal relations. As an example, we apply our framework to purported photonic implementations of the quantum switch to show that while their coarse-grained causal structure is cyclic, their fine-grained causal structure is acyclic. We therefore conclude that these experiments realize indefinite causal order only in a weak sense. Notably, this is the first argument to this effect that is not rooted in the assumption that the causal relata must be localized in spacetime.

In science and in everyday life, we very commonly explain things using the concepts of cause and effect. When we see many puddles in the street, we assume they are all effects of the same cause — the rain. When we encourage people to quit smoking, it is because we believe it causes cancer.

And yet our most successful scientific theory — quantum theory — suggests our most basic ideas about causation and causal reasoning are somehow mistaken. The famous nonlocal correlations that violate Bell's inequalities resist causal explanation as traditionally understood, and the possibility of putting objects into superpositions seems to allow for situations in which there is no definite fact about the direction of causal influence.

As a result, there has been much effort in recent years to modify our causal notions for a quantum setting. Our paper extends the study of intrinsically quantum causal structures to a new range of scenarios. One of the consequences is that recent experiments that aim to create an indefinite direction of causal influence can be understood as "weakly" indefinite — even more strongly indefinite directions of influence are conceivable.

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► References

[1] L. Hardy, ``Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure,'' Journal of Physics A: Mathematical and Theoretical 40 no. 12, (2007) 3081, arXiv:gr-qc/​0608043.

[2] G. Chiribella, G. M. D’Ariano, P. Perinotti, and B. Valiron, ``Quantum computations without definite causal structure,'' Physical Review A 88 no. 2, (Aug, 2013) , arXiv:0912.0195 [quant-ph].

[3] O. Oreshkov, F. Costa, and Č. Brukner, ``Quantum correlations with no causal order,'' Nature communications 3 no. 1, (2012) 1–8, arXiv:1105.4464 [quant-ph].

[4] M. Araújo, C. Branciard, F. Costa, A. Feix, C. Giarmatzi, and Č. Brukner, ``Witnessing causal nonseparability,'' New Journal of Physics 17 no. 10, (2015) 102001, arXiv:1506.03776 [quant-ph].

[5] J. Barrett, R. Lorenz, and O. Oreshkov, ``Quantum causal models,'' (2020) , arXiv:1906.10726 [quant-ph].

[6] N. Paunković and M. Vojinović, ``Causal orders, quantum circuits and spacetime: distinguishing between definite and superposed causal orders,'' Quantum 4 (2020) 275, arXiv:1905.09682 [quant-ph].

[7] D. Felce and V. Vedral, ``Quantum refrigeration with indefinite causal order,'' Physical Review Letters 125 (Aug, 2020) 070603, arXiv:2003.00794 [quant-ph].

[8] J. Barrett, R. Lorenz, and O. Oreshkov, ``Cyclic quantum causal models,'' Nature Communications 12 no. 1, (2021) 1–15, arXiv:2002.12157 [quant-ph].

[9] A. Kissinger and S. Uijlen, ``A categorical semantics for causal structure,'' Logical Methods in Computer Science Volume 15, Issue 3 (2019) , arXiv:1701.04732 [quant-ph].

[10] R. Lorenz and J. Barrett, ``Causal and compositional structure of unitary transformations,'' Quantum 5 (2021) 511, arXiv:2001.07774 [quant-ph].

[11] C. Branciard, M. Araújo, A. Feix, F. Costa, and Č. Brukner, ``The simplest causal inequalities and their violation,'' New Journal of Physics 18 no. 1, (2015) 013008, arXiv:1508.01704 [quant-ph].

[12] M. Araújo, F. Costa, and i. c. v. Brukner, ``Computational advantage from quantum-controlled ordering of gates,'' Physical Review Letters 113 (Dec, 2014) 250402, arXiv:1401.8127 [quant-ph].

[13] D. Felce, N. T. Vidal, V. Vedral, and E. O. Dias, ``Indefinite causal orders from superpositions in time,'' Physical Review A 105 no. 6, (2022) 062216, arXiv:2107.08076 [quant-ph].

[14] 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,'' Nature communications 6 no. 1, (2015) 1–6, arXiv:1412.4006 [quant-ph].

[15] 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,'' Science advances 3 no. 3, (2017) e1602589, arXiv:1608.01683 [quant-ph].

[16] K. Goswami, C. Giarmatzi, M. Kewming, F. Costa, C. Branciard, J. Romero, and A. G. White, ``Indefinite causal order in a quantum switch,'' Physical review letters 121 no. 9, (2018) 090503, arXiv:1803.04302 [quant-ph].

[17] G. Rubino, L. A. Rozema, F. Massa, M. Araújo, M. Zych, v. Brukner, and P. Walther, ``Experimental entanglement of temporal order,'' Quantum 6 (2022) 621, arXiv:1712.06884 [quant-ph].

[18] X. Nie, X. Zhu, C. Xi, X. Long, Z. Lin, Y. Tian, C. Qiu, X. Yang, Y. Dong, J. Li, T. Xin, and D. Lu, ``Experimental realization of a quantum refrigerator driven by indefinite causal orders,'' Physical Review Letters 129 no. 10, (2022) 100603, arXiv:2011.12580 [quant-ph].

[19] H. Cao, N.-n. Wang, Z.-A. Jia, C. Zhang, Y. Guo, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, ``Experimental demonstration of indefinite causal order induced quantum heat extraction,'' (2021) , arXiv:2101.07979 [quant-ph].

[20] K. Goswami and J. Romero, ``Experiments on quantum causality,'' AVS Quantum Science 2 no. 3, (Oct, 2020) 037101, arXiv:2009.00515 [quant-ph].

[21] L. Hardy, ``Quantum gravity computers: On the theory of computation with indefinite causal structure,'' Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle (2009) 379–401, arXiv:quant-ph/​0701019.

[22] G. Chiribella, G. M. D’Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Physical Review A 80 no. 2, (Aug, 2009) , arXiv:0904.4483 [quant-ph].

[23] G. Chiribella, G. D’Ariano, P. Perinotti, and B. Valiron, ``Beyond quantum computers,'' (2009) , arXiv:0912.0195v1 [quant-ph].

[24] G. Chiribella, ``Perfect discrimination of no-signalling channels via quantum superposition of causal structures,'' Physical Review A 86 no. 4, (Oct, 2012) , arXiv:1109.5154 [quant-ph].

[25] T. Colnaghi, G. M. D'Ariano, S. Facchini, and P. Perinotti, ``Quantum computation with programmable connections between gates,'' Physics Letters A 376 no. 45, (Oct, 2012) 2940–2943, arXiv:1109.5987 [quant-ph].

[26] Ä. Baumeler and S. Wolf, ``The space of logically consistent classical processes without causal order,'' New Journal of Physics 18 no. 1, (2016) 013036, arXiv:1507.01714 [quant-ph].

[27] Ä. Baumeler, A. Feix, and S. Wolf, ``Maximal incompatibility of locally classical behavior and global causal order in multiparty scenarios,'' Physical Review A 90 no. 4, (2014) 042106, arXiv:1403.7333 [quant-ph].

[28] M. Araújo, A. Feix, M. Navascués, and Č. Brukner, ``A purification postulate for quantum mechanics with indefinite causal order,'' Quantum 1 (Apr, 2017) 10, arXiv:1611.08535 [quant-ph].

[29] A. Vanrietvelde, N. Ormrod, H. Kristjánsson, and J. Barrett, ``Consistent circuits for indefinite causal order,'' (2022) , arXiv:2206.10042 [quant-ph].

[30] H. Reichenbach, The direction of time, vol. 65. Univ of California Press, 1956.

[31] C. J. Wood and R. W. Spekkens, ``The lesson of causal discovery algorithms for quantum correlations: causal explanations of bell-inequality violations require fine-tuning,'' New Journal of Physics 17 no. 3, (Mar, 2015) 033002, arXiv:1208.4119 [quant-ph].

[32] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, ``Quantum common causes and quantum causal models,'' Physical Review X 7 no. 3, (Jul, 2017) , arXiv:1609.09487 [quant-ph].

[33] J. Pearl, Causality. Cambridge university press, 2009.

[34] J. Pienaar and Č. Brukner, ``A graph-separation theorem for quantum causal models,'' New Journal of Physics 17 no. 7, (2015) 073020, arXiv:1406.0430v3 [quant-ph].

[35] F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New Journal of Physics 18 no. 6, (June, 2016) 063032, arXiv:1512.07106 [quant-ph].

[36] J. Pienaar, ``A time-reversible quantum causal model,'' (2019) , arXiv:1902.00129 [quant-ph].

[37] J. Pienaar, ``Quantum causal models via quantum bayesianism,'' Physical Review A 101 no. 1, (2020) 012104, arXiv:1806.00895 [quant-ph].

[38] S. Gogioso and N. Pinzani, ``The topology and geometry of causality,'' (2022) . https:/​/​​abs/​2206.08911.

[39] G. Chiribella and H. Kristjánsson, ``Quantum shannon theory with superpositions of trajectories,'' Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475 no. 2225, (May, 2019) 20180903, arXiv:1812.05292 [quant-ph].

[40] Y. Aharonov and D. Bohm, ``Significance of electromagnetic potentials in the quantum theory,'' Physical Review 115 (Aug, 1959) 485–491.

[41] N. Erez, ``AB effect and aharonov–susskind charge non-superselection,'' Journal of Physics A: Mathematical and Theoretical 43 no. 35, (Aug, 2010) 354030, arXiv:1003.1044 [quant-ph].

[42] F. D. Santo and B. Dakić, ``Two-way communication with a single quantum particle,'' Physical Review Letters 120 no. 6, (Feb, 2018) , arXiv:1706.08144 [quant-ph].

[43] L.-Y. Hsu, C.-Y. Lai, Y.-C. Chang, C.-M. Wu, and R.-K. Lee, ``Carrying an arbitrarily large amount of information using a single quantum particle,'' Physical Review A 102 (Aug, 2020) 022620, arXiv:2002.10374 [quant-ph].

[44] F. Massa, A. Moqanaki, Ämin Baumeler, F. D. Santo, J. A. Kettlewell, B. Dakić , and P. Walther, ``Experimental two-way communication with one photon,'' Advanced Quantum Technologies 2 no. 11, (Sep, 2019) 1900050, arXiv:1802.05102 [quant-ph].

[45] R. Faleiro, N. Paunkovic, and M. Vojinovic, ``Operational interpretation of the vacuum and process matrices for identical particles,'' Quantum 7 (2023) 986, arXiv:2010.16042 [quant-ph].

[46] I. Marvian and R. W. Spekkens, ``A generalization of Schur–Weyl duality with applications in quantum estimation,'' Communications in Mathematical Physics 331 no. 2, (2014) 431–475, arXiv:1112.0638 [quant-ph].

[47] A. W. Harrow, Applications of coherent classical communication and the Schur transform to quantum information theory. PhD thesis, Massachusetts Institute of Technology, 2005. arXiv:quant-ph/​0512255.

[48] G. M. Palma, K.-A. Suominen, and A. K. Ekert, ``Quantum computers and dissipation,'' Proceedings of the Royal Society A 452 (1996) 567–584, arXiv:quant-ph/​9702001.

[49] L.-M. Duan and G.-C. Guo, ``Preserving coherence in quantum computation by pairing the quantum bits,'' Physical Review Letters 79 (1997) 1953–1956, arXiv:quant-ph/​9703040.

[50] P. Zanardi and M. Rasetti, ``Noiseless quantum codes,'' Physical Review Letters 79 no. 17, (1997) 3306, arXiv:quant-ph/​9705044.

[51] D. A. Lidar, I. L. Chuang, and K. B. Whaley, ``Decoherence-free subspaces for quantum computation,'' Physical Review Letters 81 no. 12, (1998) 2594, arXiv:quant-ph/​9807004.

[52] A. Beige, D. Braun, B. Tregenna, and P. L. Knight, ``Quantum computing using dissipation to remain in a decoherence-free subspace,'' Physical Review Letters 85 no. 8, (2000) 1762.

[53] P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. White, ``Experimental verification of decoherence-free subspaces,'' Science 290 no. 5491, (2000) 498–501.

[54] O. Oreshkov, ``Time-delocalized quantum subsystems and operations: on the existence of processes with indefinite causal structure in quantum mechanics,'' Quantum 3 (2019) 206, arXiv:1801.07594 [quant-ph].

[55] A. Vanrietvelde, H. Kristjánsson, and J. Barrett, ``Routed quantum circuits,'' Quantum 5 (Jul, 2021) 503, arXiv:2011.08120 [quant-ph].

[56] A. Vanrietvelde and G. Chiribella, ``Universal control of quantum processes using sector-preserving channels,'' Quantum Information and Computation 21 no. 15-16, (Dec, 2021) 1320–1352, arXiv:2106.12463 [quant-ph].

[57] M. Wilson and A. Vanrietvelde, ``Composable constraints,'' (2021) , arXiv:2112.06818 [math.CT].

[58] A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, and C. Branciard, ``Communication through coherent control of quantum channels,'' Quantum 4 (Sep, 2020) 333, arXiv:1810.09826 [quant-ph].

[59] H. Kristjánsson, G. Chiribella, S. Salek, D. Ebler, and M. Wilson, ``Resource theories of communication,'' New Journal of Physics 22 no. 7, (Jul, 2020) 073014, arXiv:1910.08197 [quant-ph].

[60] I. Friend, ``Private communication,'' (2022).

[61] G. Chiribella, G. M. D’Ariano, and P. Perinotti, ``Transforming quantum operations: Quantum supermaps,'' EPL (Europhysics Letters) 83 no. 3, (Jul, 2008) 30004, arXiv:0804.0180 [quant-ph].

[62] M. Zych, F. Costa, I. Pikovski, and Č. Brukner, ``Bell’s theorem for temporal order,'' Nature communications 10 no. 1, (2019) 1–10, arXiv:1708.00248 [quant-ph].

[63] N. S. Móller, B. Sahdo, and N. Yokomizo, ``Quantum switch in the gravity of Earth,'' Physical Review A 104 no. 4, (2021) 042414, arXiv:2012.03989 [quant-ph].

[64] J. Wechs, C. Branciard, and O. Oreshkov, ``Existence of processes violating causal inequalities on time-delocalised subsystems,'' Nature Communications 14 no. 1, (2023) 1471, arXiv:2201.11832 [quant-ph].

[65] V. Vilasini, ``An introduction to causality in quantum theory (and beyond) (master's thesis),'' (2017) . https:/​/​​wp-content/​uploads/​2019/​07/​vilasini_master_thesis-v2.pdf.

[66] V. Vilasini, ``Causality in definite and indefinite space-times (extended abstract for qpl 2020),'' (2020) . https:/​/​​users/​valiron/​qplmfps/​papers/​qs01t3.pdf.

[67] C. Portmann, C. Matt, U. Maurer, R. Renner, and B. Tackmann, ``Causal boxes: quantum information-processing systems closed under composition,'' IEEE Transactions on Information Theory 63 no. 5, (2017) 3277–3305. https:/​/​​10.1109/​TIT.2017.2676805.

[68] B. d'Espagnat, ``An elementary note about `mixtures','' Preludes in Theoretical Physics in honor of VF Weisskopf (1966) 185.

[69] B. d’Espagnat, Conceptual foundations of quantum mechanics. CRC Press, 2018.

[70] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, ``Reference frames, superselection rules, and quantum information,'' Review of Modern Physics 79 (Apr, 2007) 555–609, arXiv:quant-ph/​0610030.

[71] V. Vilasini and R. Renner, ``Embedding cyclic causal structures in acyclic spacetimes: no-go results for process matrices,'' (2022) , arXiv:2203.11245 [quant-ph].

[72] B. Schumacher and M. D. Westmoreland, ``Locality and information transfer in quantum operations,'' Quantum Information Processing 4 no. 1, (2005) 13–34, arXiv:quant-ph/​0406223.

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