量子操縱已被證實是一種位於貝爾非局域性和量子糾纏之間的獨特量子關聯性。由於其基礎重要性，量子操縱已經被廣泛研究。為了證明可操縱性，人們依賴於被稱為可操縱集合的特定資源，其位於雙方系統的一側。然而，從二分的量子態中產生這樣的可操控資源通常不明確。為此，必須對所有可能的測量設置進行優化，以此建立了一個層次結構。在另一方面，隨著量子計算迅速演進，量子機器學習是一個嶄露頭角的領域，具有展示量子優勢的潛力。我們利用基於核的量子機器學習模型的強大能力來推斷操控測量設置的層次。為了完成這個任務，我們設計了一個計算協議來生成已標籤的訓練資料，並將訓練資料編碼為五種不同的特徵。然後，我們將已訓練的模型應用於分析隨機的量子態和三種不同類型的特定量子態。總括而言，這個工作任務利用經典和量子機器學習模型提供了關於操控測量設置層次結構和可操控性邊界的預測。

Quantum steering has been proven to be a unique quantum correlation sandwiched between Bell nonlocality and quantum entanglement. Due to its fundamental importance, quantum steering has been studied extensively. To demonstrate the steerability, one relies on a particular resource referred to as steerable assemble on one side of a two-party system. However, it is generically unclear how to reach such steerable resource from a bipartite quantum state. For this purpose, one must optimize over all possible measurement settings, which constitute a hierarchy structure. On the other hand, in light of the rapid development of quantum computing technology, quantum machine learning (QML) has emerged as a promising field with the potential to demonstrate quantum advantage. We harness the power of kernel-based QML models to infer the hierarchy of steering measurement settings. To achieve this, we design a computational protocol for generating a labeled training dataset, encoding the data into five distinct features. We then apply the well-trained models to analyze random quantum states and three different types of specific quantum states. In summary, this work offers predictions on the hierarchy of steering measurement settings and delineates the boundary between steerability and unsteerability using both classical and quantum machine learning models.

[1] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777 (1935).
[2] E. Schrödinger, “Discussion of probability relations between separated systems,” in Math. Proc. Camb. Phil. Soc., Vol. 31 (Cambridge University Press, 1935) pp. 555– 563.
[3] J. S. Bell, “On the einstein podolsky rosen paradox,” Physics 1, 195 (1964).
[4] S. J. Freedman and J. F. Clauser, “Experimental test of local hidden-variable theo- ries,” Phys. Rev. Lett. 28, 938 (1972).
[5] A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via bell’s theorem,” Phys. Rev. Lett. 47, 460 (1981).
[6] A. Aspect, P. Grangier, and G. Roger, “Experimental realization of einstein- podolsky-rosen-bohm gedankenexperiment: A new violation of bell’s inequalities,” Phys. Rev. Lett. 49, 91 (1982).
[7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entangle- ment,” Rev. Mod. Phys. 81, 865 (2009).
[8] R. Uola, A. C. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” Rev. Mod. Phys. 92, 015001 (2020).
[9] D. Cavalcanti and P. Skrzypczyk, “Quantum steering: a review with focus on semidefinite programming,” Rep. Prog. Phys. 80, 024001 (2016).
[10] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86.
[11] H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocal- ity, and the einstein-podolsky-rosen paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[12] K. Sun, X.-J. Ye, J.-S. Xu, X.-Y. Xu, J.-S. Tang, Y.-C. Wu, J.-L. Chen, C.-F. Li, and G.-C. Guo, “Experimental quantification of asymmetric einstein-podolsky-rosen steering,” Phys. Rev. Lett. 116, 160404 (2016).
[13] M. F. Pusey, “Negativity and steering: A stronger peres conjecture,” Phys. Rev. A 88, 032313 (2013).
[14] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880 (1969).
[15] H.-M. Wang, H.-Y. Ku, J.-Y. Lin, and H.-B. Chen, “Deep learning the hierarchy of steering measurement settings of qubit-pair states,” Commun. Phys. 7, 72 (2024).
[16] M. I. Jordan and T. M. Mitchell, “Machine learning: Trends, perspectives, and prospects,” Science 349, 255 (2015).
[17] D. C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to linear regression analysis (John Wiley & Sons, 2021).
[18] V. Vapnik, The nature of statistical learning theory (Springer, 1999).
[19] J. R. Quinlan, “Induction of decision trees,” Mach. Learn. 1, 81 (1986).
[20] C. M. Bishop, Neural networks for pattern recognition (Oxford university press, 1995).
[21] R. Xu and D. Wunsch, “Survey of clustering algorithms,” IEEE Trans. Neural Netw. 16, 645 (2005).
[22] K. Kourou, T. P. Exarchos, K. P. Exarchos, M. V. Karamouzis, and D. I. Fotiadis, “Machine learning applications in cancer prognosis and prediction,” Comput. Struct. Biotechnol. J. 13, 8 (2015).
[23] M. W. Libbrecht and W. S. Noble, “Machine learning applications in genetics and genomics,” Nat. Rev. Genet. 16, 321 (2015).
[24] A. Niculescu-Mizil and R. Caruana, “Predicting good probabilities with supervised learning,” in Proceedings of the 22nd international conference on Machine learning (2005) pp. 625–632.
[25] H. B. Barlow, “Unsupervised learning,” Neural Comput. 1, 295 (1989).
[26] L. P. Kaelbling, M. L. Littman, and A. W. Moore, “Reinforcement learning: A survey,” J. Artif. Intell. Res. 4, 237 (1996).
[27] X. Zhu and A. B. Goldberg, Introduction to semi-supervised learning (Springer Na- ture, 2022).
[28] C. Ciliberto, M. Herbster, A. D. Ialongo, M. Pontil, A. Rocchetto, S. Severini, and L. Wossnig, “Quantum machine learning: a classical perspective,” Proc. R. Soc. A Math. Phys. Eng. Sci 474, 20170551 (2018).
[29] V. Dunjko and H. J. Briegel, “Machine learning & artificial intelligence in the quan- tum domain: a review of recent progress,” Rep. Prog. Phys. 81, 074001 (2018).
[30] M. Schuld, “Supervised quantum machine learning models are kernel methods,” arXiv:2101.11020 (2021).
[31] F. Hirsch, M. T. Quintino, T. Vértesi, M. Navascués, and N. Brunner, “Better local hidden variable models for two-qubit werner states and an upper bound on the grothendieck constant k_g(3),” Quantum 1, 3 (2017).
[32] J. Bowles, F. Hirsch, M. T. Quintino, and N. Brunner, “Sufficient criterion for guaranteeing that a two-qubit state is unsteerable,” Phys. Rev. A 93, 022121 (2016).
[33] M. T. Quintino, T. Vértesi, D. Cavalcanti, R. Augusiak, M. Demianowicz, A. Acín, and N. Brunner, “Inequivalence of entanglement, steering, and bell nonlocality for general measurements,” Phys. Rev. A 92, 032107 (2015).
[34] S. Luo, “Quantum discord for two-qubit systems,” Phys. Rev. A 77, 042303 (2008).
[35] A. Costa and R. Angelo, “Quantification of einstein-podolsky-rosen steering for two- qubit states,” Phys. Rev. A 93, 020103 (2016).
[36] E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the einstein-podolsky-rosen paradox,” Phys. Rev. A 80, 032112 (2009).
[37] P. Skrzypczyk, M. Navascués, and D. Cavalcanti, “Quantifying einstein-podolsky- rosen steering,” Phys. Rev. Lett. 112, 180404 (2014).
[38] K. Jiráková, A. Černoch, K. Lemr, K. Bartkiewicz, and A. Miranowicz, “Experi- mental hierarchy and optimal robustness of quantum correlations of two-qubit states with controllable white noise,” Phys. Rev. A 104, 062436 (2021).
[39] S. Jevtic, M. Pusey, D. Jennings, and T. Rudolph, “Quantum steering ellipsoids,” Phys. Rev. Lett. 113, 020402 (2014).
[40] P. Rebentrost, M. Mohseni, and S. Lloyd, “Quantum support vector machine for big data classification,” Phys. Rev. Lett. 113, 130503 (2014).
[41] N. Chen, Support vector machine in chemistry (World Scientific, 2004).
[42] B. Schölkopf and A. J. Smola, Learning with kernels: support vector machines, regularization, optimiza (MIT press, 2002).
[43] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge university press, 2010).
[44] C. Cortes and V. Vapnik, “Support-vector networks,” Mach. Learn. 20, 273 (1995).
[45] R. Horodecki et al., “Information-theoretic aspects of inseparability of mixed states,” Phys. Rev. A 54, 1838 (1996).
[46] S. Jevtic, M. J. Hall, M. R. Anderson, M. Zwierz, and H. M. Wiseman, “Einstein– podolsky–rosen steering and the steering ellipsoid,” J. Opt. Soc. Am. B 32, A40 (2015).
[47] V. Havlíček, A. D. Córcoles, K. Temme, A. W. Harrow, A. Kandala, J. M. Chow, and J. M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nat. Phys 567, 209 (2019).