Semi-device-independent certification of indefinite causal order

Jessica Bavaresco1, Mateus Araújo2, Časlav Brukner1,3, and Marco Túlio Quintino4

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
2Institute for Theoretical Physics, University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany
3Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
4Department of Physics, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan

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Updated version: The authors have uploaded version v6 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
17 + 22 pages. Code available at this https URL. v5: Correction to the proof of Lemma 3. v6: Equation numbering amended in order to update the published version


When transforming pairs of independent quantum operations according to the fundamental rules of quantum theory, an intriguing phenomenon emerges: some such higher-order operations may act on the input operations in an indefinite causal order. Recently, the formalism of process matrices has been developed to investigate these noncausal properties of higher-order operations. This formalism predicts, in principle, statistics that ensure indefinite causal order even in a device-independent scenario, where the involved operations are not characterised. Nevertheless, all physical implementations of process matrices proposed so far require full characterisation of the involved operations in order to certify such phenomena. Here we consider a semi-device-independent scenario, which does not require all operations to be characterised. We introduce a framework for certifying noncausal properties of process matrices in this intermediate regime and use it to analyse the quantum switch, a well-known higher-order operation, to show that, although it can only lead to causal statistics in a device-independent scenario, it can exhibit noncausal properties in semi-device-independent scenarios. This proves that the quantum switch generates stronger noncausal correlations than it was previously known.

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[2] Huan Cao, Ning-Ning Wang, Zhian Jia, Chao Zhang, Yu Guo, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo, "Quantum simulation of indefinite causal order induced quantum refrigeration", Physical Review Research 4 3, L032029 (2022).

[3] Huan Cao, Jessica Bavaresco, Ning-Ning Wang, Lee A. Rozema, Chao Zhang, Yun-Feng Huang, Bi-Heng Liu, Chuan-Feng Li, Guang-Can Guo, and Philip Walther, "Semi-device-independent certification of indefinite causal order in a photonic quantum switch", Optica 10 5, 561 (2023).

[4] Ricardo Faleiro, Nikola Paunkovic, and Marko Vojinovic, "Operational interpretation of the vacuum and process matrices for identical particles", Quantum 7, 986 (2023).

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