Robust self-testing of steerable quantum assemblages and its applications on device-independent quantum certification

Shin-Liang Chen1,2,3, Huan-Yu Ku1, Wenbin Zhou4, Jordi Tura3,5, and Yueh-Nan Chen1

1Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan
2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
3Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany
4Graduate School of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan
5Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

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Given a Bell inequality, if its maximal quantum violation can be achieved only by a single set of measurements for each party or a single quantum state, up to local unitaries, one refers to such a phenomenon as self-testing. For instance, the maximal quantum violation of the Clauser-Horne-Shimony-Holt inequality certifies that the underlying state contains the two-qubit maximally entangled state and the measurements of one party contains a pair of anti-commuting qubit observables. As a consequence, the other party automatically verifies the set of states remotely steered, namely the "assemblage", is in the eigenstates of a pair of anti-commuting observables. It is natural to ask if the quantum violation of the Bell inequality is not maximally achieved, or if one does not care about self-testing the state or measurements, are we capable of estimating how close the underlying assemblage is to the reference one? In this work, we provide a systematic device-independent estimation by proposing a framework called "robust self-testing of steerable quantum assemblages". In particular, we consider assemblages violating several paradigmatic Bell inequalities and obtain the robust self-testing statement for each scenario. Our result is device-independent (DI), i.e., no assumption is made on the shared state and the measurement devices involved. Our work thus not only paves a way for exploring the connection between the boundary of quantum set of correlations and steerable assemblages, but also provides a useful tool in the areas of DI quantum certification. As two explicit applications, we show 1) that it can be used for an alternative proof of the protocol of DI certification of all entangled two-qubit states proposed by Bowles et al., and 2) that it can be used to verify all non-entanglement-breaking qubit channels with fewer assumptions compared with the work of Rosset et al.

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[2] Pavel Sekatski, Jean-Daniel Bancal, Marie Ioannou, Mikael Afzelius, and Nicolas Brunner, "Toward the Device-Independent Certification of a Quantum Memory", Physical Review Letters 131 17, 170802 (2023).

[3] Hong-Ming Wang, Huan-Yu Ku, Jie-Yien Lin, and Hong-Bin Chen, "Deep learning the hierarchy of steering measurement settings of qubit-pair states", Communications Physics 7 1, 72 (2024).

[4] Yuan-Yuan Zhao, Chao Zhang, Shuming Cheng, Xinhui Li, Yu Guo, Bi-Heng Liu, Huan-Yu Ku, Shin-Liang Chen, Qiaoyan Wen, Yun-Feng Huang, Guo-Yong Xiang, Chuan-Feng Li, and Guang-Can Guo, "Device-independent verification of Einstein–Podolsky–Rosen steering", Optica 10 1, 66 (2023).

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[7] Shin-Liang Chen and Jens Eisert, "Semi-Device-Independently Characterizing Quantum Temporal Correlations", Physical Review Letters 132 22, 220201 (2024).

[8] Shubhayan Sarkar, Debashis Saha, and Remigiusz Augusiak, "Certification of incompatible measurements using quantum steering", Physical Review A 106 4, L040402 (2022).

[9] Beata Zjawin, David Schmid, Matty J. Hoban, and Ana Belén Sainz, "Quantifying EPR: the resource theory of nonclassicality of common-cause assemblages", Quantum 7, 926 (2023).

[10] Huan-Yu Ku, Chung-Yun Hsieh, Shin-Liang Chen, Yueh-Nan Chen, and Costantino Budroni, "Complete classification of steerability under local filters and its relation with measurement incompatibility", Nature Communications 13 1, 4973 (2022).

[11] Beata Zjawin, David Schmid, Matty J. Hoban, and Ana Belén Sainz, "The resource theory of nonclassicality of channel assemblages", Quantum 7, 1134 (2023).

[12] Yuan-Yuan Zhao, Huan-Yu Ku, Shin-Liang Chen, Hong-Bin Chen, Franco Nori, Guo-Yong Xiang, Chuan-Feng Li, Guang-Can Guo, and Yueh-Nan Chen, "Experimental demonstration of measurement-device-independent measure of quantum steering", npj Quantum Information 6 1, 77 (2020).

[13] Kai-Siang Chen, Gelo Noel M. Tabia, Jebarathinam Chellasamy, Shiladitya Mal, Jun-Yi Wu, and Yeong-Cherng Liang, "Quantum correlations on the no-signaling boundary: self-testing and more", Quantum 7, 1054 (2023).

[14] Huan-Yu Ku, Josef Kadlec, Antonín Černoch, Marco Túlio Quintino, Wenbin Zhou, Karel Lemr, Neill Lambert, Adam Miranowicz, Shin-Liang Chen, Franco Nori, and Yueh-Nan Chen, "Quantifying Quantumness of Channels Without Entanglement", PRX Quantum 3 2, 020338 (2022).

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