Abstract
Despite the monogamous nature of nonlocal correlations, in a Bell test involving three parties A, B, and C, the nonlocality in two bipartite subsystems (e.g., AB and BC) may force the remaining bipartite subsystem (e.g., AC) to exhibit nonlocality. Although this intriguing effect of nonlocality transitivity has been identified in the non-quantum non-signaling world since 2011, whether such transitivity could manifest within quantum theory has remained unresolved. Here, we answer this open problem affirmatively—both analytically and numerically—at the level of quantum states, thereby showing that a quantum-realizable notion of nonlocality transitivity exists. In our analytic construction, we prove and use the fact that copies of the W-state marginals *uniqu…
Abstract
Despite the monogamous nature of nonlocal correlations, in a Bell test involving three parties A, B, and C, the nonlocality in two bipartite subsystems (e.g., AB and BC) may force the remaining bipartite subsystem (e.g., AC) to exhibit nonlocality. Although this intriguing effect of nonlocality transitivity has been identified in the non-quantum non-signaling world since 2011, whether such transitivity could manifest within quantum theory has remained unresolved. Here, we answer this open problem affirmatively—both analytically and numerically—at the level of quantum states, thereby showing that a quantum-realizable notion of nonlocality transitivity exists. In our analytic construction, we prove and use the fact that copies of the W-state marginals uniquely determine the global compatible state, thus establishing another instance when the parts determine the whole. Moreover, we present a simple method to construct quantum states and correlations that are nonlocal in all their non-unipartite marginals. We also discuss the implications of our results in (semi-) device-independent cryptographic and certification tasks.
Data availability
All relevant data supporting the main conclusions are available upon request. Specifically, we provide the explicit form of some of the examples of Result 4 at https://github.com/ycliangTW/NonlocalityTransitivity.
References
Horodecki, R., Horodecki, P., Horodecki, M. & Horodecki, K. Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009).
Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S. Bell nonlocality. Rev. Mod. Phys. 86, 419–478 (2014).
Einstein, A. B., Born, M. & Born, H.The Born-Einstein letters: Correspondence between Albert Einstein and Max and Hedwig Born from 1916-1955, with commentaries by Max Born (Macmillan, 1971). 1.
Bell, J. S. On the Einstein Podolsky Rosen paradox. Physics 1, 195–200 (1964).
Bell, J. S.Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy 2 edn. (Cambridge University Press, 2004). 1.
Vidal, G. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett. 91, 147902 (2003).
Jozsa, R. & Linden, N. On the role of entanglement in quantum-computational speed-up. Proc. R. Soc. Lond. A 459, 2011–2032 (2003).
Ekert, A. K. Quantum cryptography based on Bell’s theorem. Phys. Rev. Lett. 67, 661–663 (1991).
Acín, A. et al. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. 98, 230501 (2007).
Vazirani, U. & Vidick, T. Fully device-independent quantum key distribution. Phys. Rev. Lett. 113, 140501 (2014).
Coffman, V., Kundu, J. & Wootters, W. K. Distributed entanglement. Phys. Rev. A 61, 052306 (2000).
Doherty, A. C. Entanglement and the shareability of quantum states. J. Phys. A Math. Theor. 47, 424004 (2014).
Popescu, S. & Rohrlich, D. Quantum nonlocality as an axiom. Found. Phys. 24, 379–385 (1994).
Barrett, J. et al. Nonlocal correlations as an information-theoretic resource. Phys. Rev. A 71, 022101 (2005).
Masanes, L., Acín, A. & Gisin, N. General properties of nonsignaling theories. Phys. Rev. A 73, 012112 (2006).
Collins, D. & Gisin, N. A relevant two qubit Bell inequality inequivalent to the CHSH inequality. J. Phys. A: Math. Gen. 37, 1775 (2004).
Scarani, V. & Gisin, N. Quantum communication between N partners and Bell’s inequalities. Phys. Rev. Lett. 87, 117901 (2001).
Toner, B. & Verstraete, F. Monogamy of Bell correlations and Tsirelson’s bound. arXiv: https://arxiv.org/abs/quant-ph/0611001 (2006). 1.
Toner, B. Monogamy of non-local quantum correlations. Proc. R. Soc. A 465, 59–69 (2009).
Pawłowski, M. & Brukner, Č Monogamy of Bell’s inequality violations in nonsignaling theories. Phys. Rev. Lett. 102, 030403 (2009).
Kurzyński, P., Paterek, T., Ramanathan, R., Laskowski, W. & Kaszlikowski, D. Correlation complementarity yields Bell monogamy relations. Phys. Rev. Lett. 106, 180402 (2011).
Ramanathan, R. & Horodecki, P. Strong monogamies of no-signaling violations for bipartite correlation Bell inequalities. Phys. Rev. Lett. 113, 210403 (2014).
Yang, Y.-H., Liu, X.-Z., Zheng, X.-Z., Fei, S.-M. & Luo, M.-X. Verification of Bell nonlocality by violating quantum monogamy relations. Cell Rep. Phys. Sci. 5, 101840 (2024).
Cui, D., Mehta, A. & Rochette, D. Monogamy of nonlocal games. Phys. Rev. Res. 7, L032003 (2025).
Tabia, G. N. M., Chen, K.-S., Hsieh, C.-Y., Yin, Y.-C. & Liang, Y.-C. Entanglement transitivity problems. npj Quantum Inf. 8, 98 (2022).
Coretti, S., Hänggi, E. & Wolf, S. Nonlocality is transitive. Phys. Rev. Lett. 107, 100402 (2011).
Yin, Y.-C. Transitivity of Quantum Nonlocality. Master’s thesis, National Cheng Kung University (2021). 1.
Scarani, V. & Gisin, N. Superluminal influences, hidden variables, and signaling. Phys. Lett. A 295, 167–174 (2002).
Scarani, V. & Gisin, N. Superluminal hidden communication as the underlying mechanism for quantum correlations: constraining models. Braz. J. Phys. 35, 328–332 (2005).
Bancal, J.-D. et al. Quantum non-locality based on finite-speed causal influences leads to superluminal signalling. Nat. Phys. 8, 867–870 (2012).
Barnea, T. J., Bancal, J.-D., Liang, Y.-C. & Gisin, N. Tripartite quantum state violating the hidden-influence constraints. Phys. Rev. A 88, 022123 (2013).
Chiribella, G. & Spekkens, R. W. (eds.) Quantum Theory: Informational Foundations and Foils (Springer, 2016). 1.
Brunner, N. & Vértesi, T. Persistency of entanglement and nonlocality in multipartite quantum systems. Phys. Rev. A 86, 042113 (2012).
Khot, S. & Vishnoi, N. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into l/sub 1/. in Proc. 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), 53–62 (IEEE, 2005). 1.
Palazuelos, C. Superactivation of quantum nonlocality. Phys. Rev. Lett. 109, 190401 (2012).
Cavalcanti, D., Acín, A., Brunner, N. & Vértesi, T. All quantum states useful for teleportation are nonlocal resources. Phys. Rev. A 87, 042104 (2013).
Nielsen, M. A. & Chuang, I. L.Quantum Computation and Quantum Information (Cambridge University Press, 2000). 1.
Cleve, R., Hoyer, P., Toner, B. & Watrous, J. Consequences and limits of nonlocal strategies. in Proc. 19th IEEE Annual Conference on Computational Complexity, 2004, 236–249 (IEEE, 2004). 1.
Clauser, J. F., Horne, M. A., Shimony, A. & Holt, R. A. Proposed experiment to test local hidden-variable theories. Phys. Rev. Lett. 23, 880–884 (1969).
Brassard, G. et al. Limit on nonlocality in any world in which communication complexity is not trivial. Phys. Rev. Lett. 96, 250401 (2006).
Pawłowski, M. et al. Information causality as a physical principle. Nature 461, 1101–1104 (2009).
Navascués, M. & Wunderlich, H. A glance beyond the quantum model. Proc. R. Soc. A Math. Phys. Eng. Sci. 466, 881–890 (2009).
Sainz, A. B., Guryanova, Y., Acín, A. & Navascués, M. Almost-quantum correlations violate the no-restriction hypothesis. Phys. Rev. Lett. 120, 200402 (2018).
Werner, R. F. Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277–4281 (1989).
Terhal, B. M., Doherty, A. C. & Schwab, D. Symmetric extensions of quantum states and local hidden variable theories. Phys. Rev. Lett. 90, 157903 (2003).
Liang, Y.-C. & Doherty, A. C. Bounds on quantum correlations in Bell-inequality experiments. Phys. Rev. A 75, 042103 (2007).
Hirsch, F., Quintino, M. T., Vértesi, T., Pusey, M. F. & Brunner, N. Algorithmic construction of local hidden variable models for entangled quantum states. Phys. Rev. Lett. 117, 190402 (2016).
Cavalcanti, D., Guerini, L., Rabelo, R. & Skrzypczyk, P. General method for constructing local hidden variable models for entangled quantum states. Phys. Rev. Lett. 117, 190401 (2016).
Hsieh, C.-Y., Liang, Y.-C. & Lee, R.-K. Quantum steerability: Characterization, quantification, superactivation, and unbounded amplification. Phys. Rev. A 94, 062120 (2016).
Horodecki, R., Horodecki, P. & Horodecki, M. Violating Bell inequality by mixed spin-12 states: necessary and sufficient condition. Phys. Lett. A 200, 340–344 (1995).
Dür, W., Vidal, G. & Cirac, J. I. Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000).
Tóth, G., Knapp, C., Gühne, O. & Briegel, H. J. Optimal spin squeezing inequalities detect bound entanglement in spin models. Phys. Rev. Lett. 99, 250405 (2007).
Würflinger, L. E., Bancal, J.-D., Acín, A., Gisin, N. & Vértesi, T. Nonlocal multipartite correlations from local marginal probabilities. Phys. Rev. A 86, 032117 (2012).
Liang, Y.-C., Curchod, F. J., Bowles, J. & Gisin, N. Anonymous quantum nonlocality. Phys. Rev. Lett. 113, 130401 (2014).
Navascués, M., Baccari, F. & Acín, A. Entanglement marginal problems. Quantum 5, 589 (2021).
Bender, E. A. & Williamson, S. G.Lists, Decisions and Graphs, 171 (UC San Diego, 2010). 1.
Parashar, P. & Rana, S. N-qubit W states are determined by their bipartite marginals. Phys. Rev. A 80, 012319 (2009).
Wu, X. et al. Determination of W states equivalent under stochastic local operations and classical communication by their bipartite reduced density matrices with tree form. Phys. Rev. A 90, 012317 (2014).
Shen, Y. & Chen, L. Additivity of states uniquely determined by marginals. Phys. Rev. A 108, 062418 (2023).
Liu, M.-E., Chen, K.-S., Hsieh, C.-Y., Tabia, G. N. M. & Liang, Y.-C. Large parts are generically entangled across all cuts. Qantum Sci. Technol. (in print). Also available at arXiv: https://arxiv.org/abs/2505.20420v1 (2025). 1.
Chen, J. et al. Uniqueness of quantum states compatible with given measurement results. Phys. Rev. A 88, 012109 (2013).
Liang, Y.-C., Spekkens, R. W. & Wiseman, H. M. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Phys. Rep. 506, 1 – 39 (2011).
Coretti, S. Is (quantum) Non-locality Transitive? Semester Thesis, Swiss Federal Institute of Technology Zürich (2009). 1.
Cieśliński, P. et al. Unmasking the polygamous nature of quantum nonlocality. Proc. Natl. Acad. Sci. USA 121, e2404455121 (2024).
Christandl, M., Ferrara, R. & Horodecki, K. Upper bounds on device-independent quantum key distribution. Phys. Rev. Lett. 126, 160501 (2021).
Farkas, M., Balanzó-Juandó, M., Łukanowski, K., Kołodyński, J. & Acín, A. Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution protocols. Phys. Rev. Lett. 127, 050503 (2021).
Wiseman, H. M., Jones, S. J. & Doherty, A. C. Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. Phys. Rev. Lett. 98, 140402 (2007).
Uola, R., Costa, A. C. S., Nguyen, H. C. & Gühne, O. Quantum steering. Rev. Mod. Phys. 92, 015001 (2020).
Horodecki, M. & Horodecki, P. Reduction criterion of separability and limits for a class of distillation protocols. Phys. Rev. A 59, 4206–4216 (1999).
Acknowledgements
We thank Antonio Acín, Nicolas Brunner, Marwan Haddara, Mu-En Liu, Hakop Pashayan, Valerio Scarani, Pavel Sekatski, Rob Spekkens, and Yujie Zhang for helpful discussions. Discussion with Pavel Sekatski, in particular, has inspired us to prove Result 2, thereby leading to a stronger result of nonlocality transitivity for quantum states. K.S.C. is grateful for the hospitality of the Institut Néel. This work was supported by the National Science and Technology Council, Taiwan (Grants No. 109-2112-M-006-010-MY3, 112-2628-M-006-007-MY4, 113-2917-I-006-023, 113-2918-I-006-001), the Foxconn Research Institute, Taipei, Taiwan, and in part by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Colleges and Universities. CYH acknowledges support from ICFOstepstone (the Marie Sk\l odowska-Curie Co-fund GA665884), the Spanish MINECO (Severo Ochoa SEV-2015-0522), the Government of Spain (FIS2020-TRANQI and Severo Ochoa CEX2019-000910-S), Fundació Cellex, Fundaci Mir-Puig, Generalitat de Catalunya (SGR1381 and CERCA Programme), the ERC Advanced Grant (CERQUTE and FLQuant), the AXA Chair in Quantum Information Science, the Royal Society through Enhanced Research Expenses (NFQI), and the Leverhulme Trust Early Career Fellowship (``Quantum complementarity: a novel resource for quantum science and technologies’’ with Grant No.~ECF-2024-310).
Author information
Authors and Affiliations
Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan, Taiwan
Kai-Siang Chen, Gelo Noel M. Tabia, Yu-Chun Yin & Yeong-Cherng Liang 1.
Hon Hai (Foxconn) Research Institute, Taipei, Taiwan
Gelo Noel M. Tabia 1.
Physics Division, National Center for Theoretical Sciences, Taipei, Taiwan
Gelo Noel M. Tabia & Yeong-Cherng Liang 1.
H. H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom
Chung-Yun Hsieh 1.
ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Castelldefels, Spain
Chung-Yun Hsieh 1.
Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu, Taiwan
Yu-Chun Yin 1.
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
Yeong-Cherng Liang
Authors
- Kai-Siang Chen
- Gelo Noel M. Tabia
- Chung-Yun Hsieh
- Yu-Chun Yin
- Yeong-Cherng Liang
Contributions
Y.C.L. conceived the research, formalized the various definitions, and supervised, together with G.N.M.T., the initial investigation by Y.C.Y. The key results of Result 2, Result 3, and Result 5 in the Supplementary Information were developed mostly by C.Y.H. and K.S.C. with help from Y.C.L. and G.M.N.T. All numerical results were obtained by K.S.C. and independently verified by Y.C.L. and G.N.M.T. All authors contributed to the discussions and to the preparation of the manuscript.
Corresponding authors
Correspondence to Chung-Yun Hsieh or Yeong-Cherng Liang.
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Chen, KS., Tabia, G.N.M., Hsieh, CY. et al. Nonlocality of quantum states can be transitive. npj Quantum Inf (2026). https://doi.org/10.1038/s41534-025-01173-z
Received: 30 July 2025
Accepted: 18 December 2025
Published: 10 January 2026
DOI: https://doi.org/10.1038/s41534-025-01173-z