Noise-robust preparation contextuality shared between any number of observers via unsharp measurements

Hammad Anwer; Natalie Wilson; Ralph Silva; Sadiq Muhammad; Armin Tavakoli; Mohamed Bourennane

doi:10.22331/q-2021-09-28-551

Noise-robust preparation contextuality shared between any number of observers via unsharp measurements

Hammad Anwer¹, Natalie Wilson¹, Ralph Silva², Sadiq Muhammad¹, Armin Tavakoli^3,4,5, and Mohamed Bourennane¹

¹Department of Physics, Stockholm University, S-10691 Stockholm, Sweden
²Institute for Theoretical Physics, ETH Zurich, Switzerland
³Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland
⁴Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
⁵Institute for Atomic and Subatomic Physics, Vienna University of Technology, 1020 Vienna, Austria

Published:	2021-09-28, volume 5, page 551
Eprint:	arXiv:1904.09766v2
Doi:	https://doi.org/10.22331/q-2021-09-28-551
Citation:	Quantum 5, 551 (2021).

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Abstract

Multiple observers who independently harvest nonclassical correlations from a single physical system share the system's ability to enable quantum correlations. We show that any number of independent observers can share the preparation contextual outcome statistics enabled by state ensembles in quantum theory. Furthermore, we show that even in the presence of any amount of white noise, there exists quantum ensembles that enable such shared preparation contextuality. The findings are experimentally realised by applying sequential unsharp measurements to an optical qubit ensemble which reveals three shared demonstrations of preparation contextuality.

Popular summary

When a quantum resource is measured, information is extracted from it at the price of distorting the state. Here, we aim to extract information from a quantum ensemble in such a way that classical noncontextual models cannot account for the lab observations, while simultaneously not distorting the ensemble too much to prevent us from doing such an extraction once again. We show that, in fact, one can extract contextual quantum correlations indefinitely many times from a single ensemble and we demonstrate three sequential violations of noncontextuality in an optics experiment.

► BibTeX data

@article{Anwer2021noiserobust,
  doi = {10.22331/q-2021-09-28-551},
  url = {https://doi.org/10.22331/q-2021-09-28-551},
  title = {Noise-robust preparation contextuality shared between any number of observers via unsharp measurements},
  author = {Anwer, Hammad and Wilson, Natalie and Silva, Ralph and Muhammad, Sadiq and Tavakoli, Armin and Bourennane, Mohamed},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {551},
  month = sep,
  year = {2021}
}

► References

[1] C. A. Fuchs, and A. Peres, Quantum-state disturbance versus information gain: Uncertainty relations for quantum information Phys. Rev. A 53, 2038 (1996).
https://doi.org/10.1103/PhysRevA.53.2038

[2] R. Gallego, L. E. Würflinger, R. Chaves, A. Acín, M. Navascués, Nonlocality in sequential correlation scenarios, New J. Phys. 16, 033037 (2014).
https://doi.org/10.1088/1367-2630/16/3/033037

[3] C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, Bounding Temporal Quantum Correlations, Phys. Rev. Lett. 111, 020403 (2013).
https://doi.org/10.1103/PhysRevLett.111.020403

[4] R. Silva, N. Gisin, Y. Guryanova, and S. Popescu, Multiple Observers Can Share the Nonlocality of Half of an Entangled Pair by Using Optimal Weak Measurements, Phys. Rev. Lett. 114, 250401 (2015).
https://doi.org/10.1103/PhysRevLett.114.250401

[5] A. Tavakoli, A. Cabello, Quantum predictions for an unmeasured system cannot be simulated with a finite-memory classical system, Phys. Rev. A 97, 032131 (2018).
https://doi.org/10.1103/PhysRevA.97.032131

[6] P. J. Brown and R. Colbeck Arbitrarily Many Independent Observers Can Share the Nonlocality of a Single Maximally Entangled Qubit Pair, Phys. Rev. Lett. 125, 090401 (2020).
https://doi.org/10.1103/PhysRevLett.125.090401

[7] F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, Unbounded randomness certification using sequences of measurements, Phys. Rev. A 95, 020102(R) (2017).
https://doi.org/10.1103/PhysRevA.95.020102

[8] B. Coyle, M. J. Hoban, and E. Kashefi, One-Sided Device-Independent Certification of Unbounded Random Numbers, EPTCS 273, 14-26 (2018).
https://doi.org/10.4204/EPTCS.273.2

[9] G. Foletto, L. Calderaro, A. Tavakoli, M. Schiavon, F. Picciariello, A. Cabello, P. Villoresi, and G. Vallone, Experimental Certification of Sustained Entanglement and Nonlocality after Sequential Measurements, Phys. Rev. Applied 13, 044008 (2020).
https://doi.org/10.1103/PhysRevApplied.13.044008

[10] M. Schiavon, L. Calderaro, M. Pittaluga, G. Vallone, and P. Villoresi, Three-observer Bell inequality violation on a two-qubit entangled state, Quantum Sci. Technol. 2 015010 (2017).
https://doi.org/10.1088/2058-9565/aa62be

[11] M-J. Hu, Z-Y. Zhou, X-M. Hu, C-F. Li, G-C. Guo, and Y-S. Zhang, Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement, npj Quantum Information 4, 63 (2018).
https://doi.org/10.1038/s41534-018-0115-x

[12] A. Bera, S. Mal, A. Sen De, and U. Sen, Witnessing bipartite entanglement sequentially by multiple observers, Phys. Rev. A 98, 062304 (2018).
https://doi.org/10.1103/PhysRevA.98.062304

[13] S. Sasmal, D. Das, S. Mal, and A.S. Majumdar, Steering a single system sequentially by multiple observers, Phys. Rev. A 98, 012305 (2018).
https://doi.org/10.1103/PhysRevA.98.012305

[14] A. Shenoy H, S. Designolle, F. Hirsch, R. Silva, N. Gisin, and N. Brunner, Unbounded sequence of observers exhibiting Einstein-Podolsky-Rosen steering, Phys. Rev. A 99, 022317 (2019).
https://doi.org/10.1103/PhysRevA.99.022317

[15] K. Mohan. A. Tavakoli, and N. Brunner, Sequential random access codes and self-testing of quantum measurement instruments, New J. Phys. 21 083034 (2019).
https://doi.org/10.1088/1367-2630/ab3773

[16] N. Miklin, J. Borkala, and M. Pawlowski, Semi-device-independent self-testing of unsharp measurements, Phys. Rev. Research 2, 033014 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033014

[17] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental Characterization of Unsharp Qubit Observables and Sequential Measurement Incompatibility via Quantum Random Access Codes, Phys. Rev. Lett. 125, 080403 (2020).
https://doi.org/10.1103/PhysRevLett.125.080403

[18] G. Foletto, L. Calderaro, G. Vallone, and P. Villoresi, Experimental demonstration of sequential quantum random access codes, Phys. Rev. Research 2, 033205 (2020).
https://doi.org/10.1103/PhysRevResearch.2.033205

[19] R. W. Spekkens, Contextuality for preparations, transformations, and unsharp measurements Phys. Rev. A 71, 052108 (2005).
https://doi.org/10.1103/PhysRevA.71.052108

[20] R. W. Spekkens, Negativity and Contextuality are Equivalent Notions of Nonclassicality, Phys. Rev. Lett. 101, 020401 (2008).
https://doi.org/10.1103/PhysRevLett.101.020401

[21] R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Preparation Contextuality Powers Parity-Oblivious Multiplexing Phys. Rev. Lett. 102, 010401 (2009).
https://doi.org/10.1103/PhysRevLett.102.010401

[22] M. S. Leifer, and O. J. E. Maroney, Maximally Epistemic Interpretations of the Quantum State and Contextuality, Phys. Rev. Lett. 110, 120401 (2013).
https://doi.org/10.1103/PhysRevLett.110.120401

[23] M. Banik, S. S. Bhattacharya, A. Mukherjee, A. Roy, A. Ambainis, and A. Rai, Limited preparation contextuality in quantum theory and its relation to the Cirel'son bound, Phys. Rev. A 92, 030103(R) (2015).
https://doi.org/10.1103/PhysRevA.92.030103

[24] S. Ghorai, A. K. Pan, Optimal quantum preparation contextuality in an n-bit parity-oblivious multiplexing task Phys. Rev. A 98, 032110 (2018).
https://doi.org/10.1103/PhysRevA.98.032110

[25] A. Tavakoli and R. Uola, Measurement incompatibility and steering are necessary and sufficient for operational contextuality, Phys. Rev. Research 2, 013011 (2020).
https://doi.org/10.1103/PhysRevResearch.2.013011

[26] D. Saha, and A. Chaturvedi, Preparation contextuality: the ground of quantum communication advantage, Phys. Rev. A 100, 022108 (2019).
https://doi.org/10.1103/PhysRevA.100.022108

[27] A. Tavakoli, E. Zambrini Cruzeiro, R. Uola, and A. A. Abbott, Bounding and Simulating Contextual Correlations in Quantum Theory, PRX Quantum 2, 020334 (2021).
https://doi.org/10.1103/PRXQuantum.2.020334

[28] A. Chaturvedi, M. Farkas, and V. Wright, Characterising and bounding the set of quantum behaviours in contextuality scenarios, Quantum 5, 484 (2021).
https://doi.org/10.22331/q-2021-06-29-484

[29] A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane, Communication games reveal preparation contextuality, Phys. Rev. Lett. 119, 220402 (2017).
https://doi.org/10.1103/PhysRevLett.119.220402

[30] M. D. Mazurek, M. F. Pusey, R. Kunjwal, K. J. Resch, and R. W. Spekkens, An experimental test of noncontextuality without unphysical idealizations, Nature Communications 7, 11780 (2016).
https://doi.org/10.1038/ncomms11780

[31] S. Kochen, and E. P. Specker, The Problem of Hidden Variables in Quantum Mechanics, Indiana University Mathematics Journal, 17, 59 (1967).

[32] N. Harrigan, and R. W. Spekkens, Einstein, Incompleteness, and the Epistemic View of Quantum States, Found Phys (2010) 40, 125 (2010).
https://doi.org/10.1007/s10701-009-9347-0

[33] A. Ambainis, A. Nayak, A. Ta-Shma, U. Vazirani, Dense quantum coding and a lower bound for 1-way quantum automata, Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC'99), 376-383 (1999).
https://doi.org/10.1145/301250.301347

[34] A. Tavakoli, A. Hameedi, B. Marques, and M. Bourennane, Quantum random access codes using single d-Level systems, Phys. Rev. Lett. 114, 170502 (2015).
https://doi.org/10.1103/PhysRevLett.114.170502

[35] A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, Optimal bounds for parity-oblivious random access codes, New J. Phys. 18, 045003 (2016).
https://doi.org/10.1088/1367-2630/18/4/045003

[36] One could alternatively consider the Bobs' measurement devices inducing the noise. However, this is less detrimental than noisy preparations. The reason is that if Alice's preparations are noisy the correlations due to all Bobs' measurements are weaker, whereas if instead one (or many) of the Bobs sometimes fail to perform the intended measurement, the state relayed to the next Bob retains a higher degree of coherence and leads to him observering stronger correlations.

[37] A. Kumari and A. K. Pan, Sharing nonlocality and nontrivial preparation contextuality using the same family of Bell expressions, Phys. Rev. A 100, 062130 (2019).
https://doi.org/10.1103/PhysRevA.100.062130

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[1] Abhyoudai S. S., Sumit Mukherjee, and A. K. Pan, "Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game", Physical Review A 107 1, 012411 (2023).

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[3] Xianzhi Huang, Liyao Zhan, Liang Li, Suhui Bao, Zipeng Tao, and Jiayu Ying, "Quantum Correlation Resource Recycling via Sequential Measurements: Theoretical Models and Optical Experiments", Photonics 10 12, 1314 (2023).

[4] Pritam Halder, Ratul Banerjee, Shiladitya Mal, and Aditi Sen(De), "Limits of network nonlocality probed by timelike-separated observers", Physical Review A 106 5, 052413 (2022).

[5] Xin-Miao Yu, Shu-Yuan Yang, and Kan He, "Sharing entanglement of the Werner state by arbitrarily many independent observers", Acta Physica Sinica 72 7, 070301 (2023).

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[8] Meng Yan, Hao-Zhan Zou, and Xiang Zhan, "Qubit measurements for testing contextuality through the violation of Liang-Spekkens-Wiseman-Yu-Oh inequality", New Journal of Physics 25 8, 083032 (2023).

[9] Zinuo Cai and Changliang Ren, "Full network nonlocality sharing in extended bilocal scenario via weak measurements with the optimal pointer", Journal of Physics A: Mathematical and Theoretical 57 19, 195305 (2024).

[10] S. Sasmal, S. Kanjilal, and A. K. Pan, "Unbounded Sharing of Nonlocality Using Qubit Projective Measurements", Physical Review Letters 133 17, 170201 (2024).

[11] Jaskaran Singh, Rajendra Singh Bhati, and Arvind, "Revealing quantum contextuality using a single measurement device", Physical Review A 107 1, 012201 (2023).

[12] Zinuo Cai, Changliang Ren, Tianfeng Feng, Xiaoqi Zhou, and Jingling Chen, "A review of quantum correlation sharing: The recycling of quantum correlations triggered by quantum measurements", Physics Reports 1098, 1 (2025).

[13] Yiyang Xu, Hao Sun, Fenzhuo Guo, Haifeng Dong, and Qiaoyan Wen, "Sharing tripartite nonlocality sequentially using only projective measurements", Quantum Information Processing 24 1, 22 (2025).

[14] Bingzi Huo, Kunkun Wang, Dengke Qu, Junjing Xing, Xiang Zhan, Xiaojian Huang, Huixia Gao, Lei Xiao, and Peng Xue, "Sequential Sharing of Quantum Nonlocality via Projective Measurements", Physical Review Letters 136 10, 100201 (2026).

[15] Shufen Dong, Zinuo Cai, Chunfeng Wu, and Changliang Ren, "Sharing quantum steering via standard projective measurements", Physical Review A 110 1, 012203 (2024).

[16] Shuming Cheng, Lijun Liu, Travis J. Baker, and Michael J. W. Hall, "Recycling qubits for the generation of Bell nonlocality between independent sequential observers", Physical Review A 105 2, 022411 (2022).

[17] Debarshi Das, Arkaprabha Ghosal, Ananda G. Maity, Som Kanjilal, and Arup Roy, "Ability of unbounded pairs of observers to achieve quantum advantage in random access codes with a single pair of qubits", Physical Review A 104 6, L060602 (2021).

[18] Yao Xiao, Fen-Zhuo Guo, Hai-Feng Dong, and Su-Juan Qin, "Sharing high-dimensional preparation contextuality via weak measurements with different pointers", Physica Scripta 100 4, 045107 (2025).

[19] Qian Zhang, Kai-Yu Yuan, Yan-Xin Rong, Zhen Shang, Yong-Jian Gu, and Ya Xiao, "Robust device-independent characterization of sharpness and incompatibility of unsharp instruments", Physical Review A 114 2, 022411 (2026).

[20] Wenlin Hou, Xiaowei Liu, and Changliang Ren, "Network nonlocality sharing via weak measurements in the extended bilocal scenario", Physical Review A 105 4, 042436 (2022).

[21] Tinggui Zhang, Hong Yang, and Shao-Ming Fei, "Sharing Bell nonlocality of bipartite high-dimensional pure states using only projective measurements", Physical Review A 109 2, 022419 (2024).

[22] Wan-Guan Chang, Chia-Yi Ju, Guang-Yin Chen, Yueh-Nan Chen, and Huan-Yu Ku, "Visually quantifying single-qubit quantum memory", Physical Review Research 6 2, 023035 (2024).

[23] Ya Xiao, Yan Xin Rong, Shuo Wang, Xin Hong Han, Jin Shi Xu, and Yong Jian Gu, "Experimental sharing of Bell nonlocality with projective measurements", New Journal of Physics 26 5, 053019 (2024).

[24] Sudipta Mondal, Pritam Halder, Amit Kumar Pal, and Aditi Sen(De), "Recovery of resources through sequential noisy measurements", Physical Review A 111 4, 042401 (2025).

[25] Shishir Khandelwal and Armin Tavakoli, "Simulating Quantum Instruments with Projective Measurements and Quantum Postprocessing", Physical Review Letters 135 4, 040202 (2025).

[26] Yan Xin Rong, Shuo Wang, Zhen Fei Zhang, Yong Jian Gu, and Ya Xiao, "Sharing asymmetric Einstein–Podolsky–Rosen steering with projective measurements", New Journal of Physics 26 8, 083014 (2024).

[27] Xin-Hong Han, Hui-Chao Qu, Xuan Fan, Ya Xiao, and Yong-Jian Gu, "Manipulating the quantum steering direction with sequential unsharp measurements", Physical Review A 106 4, 042416 (2022).

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