Characterising and bounding the set of quantum behaviours in contextuality scenarios

Anubhav Chaturvedi1,2, Máté Farkas3, and Victoria J Wright2,3

1Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gdansk, Poland
2International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308, Gdansk, Poland
3ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.


The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate $\textit{monogamy of preparation contextuality}$ in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.

► BibTeX data

► References

[1] Andris Ambainis, Manik Banik, Anubhav Chaturvedi, Dmitry Kravchenko, and Ashutosh Rai. Parity oblivious $d$-level random access codes and class of noncontextuality inequalities. Quantum Inf. Process., 18(4):111, 2019. doi:10.1007/​s11128-019-2228-3.

[2] Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani. Device-independent security of quantum cryptography against collective attacks. Phys. Rev. Lett. , 98:230501, 2007. doi:10.1103/​PhysRevLett.98.230501.

[3] Rotem Arnon-Friedman. Reductions to IID in Device-independent Quantum Information Processing. PhD thesis, ETH Zürich, 2018. doi:10.1007/​978-3-030-60231-4.

[4] Barbara Amaral and Marcelo Terra Cunha. Contextuality: The Compatibility-Hypergraph Approach, pages 13–48. Springer International Publishing, Cham, 2018. doi:10.1007/​978-3-319-93827-1_2.

[5] Manik Banik, Some Sankar Bhattacharya, Amit Mukherjee, Arup Roy, Andris Ambainis, and Ashutosh Rai. Limited preparation contextuality in quantum theory and its relation to the Cirel'son bound. Phys. Rev. A, 92(3):030103, 2015. doi:10.1103/​PhysRevA.92.030103.

[6] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86:419–478, 2014. doi:10.1103/​RevModPhys.86.419.

[7] Stephen Boyd and Lieven Vandenberghe. Convex Optimization. Cambridge University Press, 2004. doi:10.1017/​CBO9780511804441.

[8] André Chailloux, Iordanis Kerenidis, Srijita Kundu, and Jamie Sikora. Optimal bounds for parity-oblivious random access codes. New J. Phys., 18(4):045003, 2016. doi:10.1088/​1367-2630/​18/​4/​045003.

[9] Andrea Coladangelo and Jalex Stark. Unconditional separation of finite and infinite-dimensional quantum correlations. arXiv:1804.05116, 2018. URL: https:/​/​​abs/​1804.05116.

[10] Anubhav Chaturvedi and Debashis Saha. Quantum prescriptions are more ontologically distinct than they are operationally distinguishable. Quantum, 4:345, 2020. doi:10.22331/​q-2020-10-21-345.

[11] Imre Csiszár and János Körner. Broadcast channels with confidential messages. IEEE Trans. Inf. Theory, 24(3):339–348, 1978. doi:10.1109/​TIT.1978.1055892.

[12] Samuel Clarke, Gottfried Wilhelm von Leibniz, Isaac Newton, and H G Alexander. The Leibniz-Clarke Correspondence: Together With Extracts from Newton's Principia and Opticks. Manchester University Press, 1956. URL: https:/​/​​title/​leibniz-clarke-correspondence-together-with-extracts-from-newtons-principia-and-opticks/​oclc/​2181039.

[13] Pierre-Emmanuel Emeriau, Mark Howard, and Shane Mansfield. Quantum advantage in information retrieval. arXiv:2007.15643, 2020. URL: https:/​/​​abs/​2007.15643.

[14] Shouvik Ghorai and Alok Kumar Pan. Optimal quantum preparation contextuality in an n-bit parity-oblivious multiplexing task. Phys. Rev. A, 98(3):032110, 2018. doi:10.1103/​PhysRevA.98.032110.

[15] Ravi Kunjwal, Chris Heunen, and Tobias Fritz. Quantum realization of arbitrary joint measurability structures. Phys. Rev. A, 89:052126, 2014. doi:10.1103/​PhysRevA.89.052126.

[16] Ravi Kunjwal, Matteo Lostaglio, and Matthew F. Pusey. Anomalous weak values and contextuality: Robustness, tightness, and imaginary parts. Phys. Rev. A, 100:042116, 2019. doi:10.1103/​PhysRevA.100.042116.

[17] Ravi Kunjwal and Robert W. Spekkens. From statistical proofs of the kochen-specker theorem to noise-robust noncontextuality inequalities. Phys. Rev. A, 97:052110, 2018. doi:10.1103/​PhysRevA.97.052110.

[18] Ravi Kunjwal. Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements. Quantum, 3:184, 2019. doi:10.22331/​q-2019-09-09-184.

[19] Johan Löfberg. YALMIP : A toolbox for modeling and optimization in MATLAB. In In Proceedings of the CACSD Conference, Taipei, Taiwan, 2004. doi:10.1109/​CACSD.2004.1393890.

[20] Yeong-Cherng Liang, Robert W. Spekkens, and Howard M. Wiseman. Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity. Phys. Rep., 506(1):1 – 39, 2011. doi:10.1016/​j.physrep.2011.05.001.

[21] Iman Marvian. Inaccessible information in probabilistic models of quantum systems, non-contextuality inequalities and noise thresholds for contextuality. arXiv:2003.05984, 2020. URL: https:/​/​​abs/​2003.05984.

[22] Piotr Mironowicz. Applications of semi-definite optimization in quantum information protocols. PhD thesis, Gdansk University of Technology, 2015. URL: https:/​/​​abs/​1810.05145.

[23] Michael D. Mazurek, Matthew F. Pusey, Ravi Kunjwal, Kevin J. Resch, and Robert W. Spekkens. An experimental test of noncontextuality without unphysical idealizations. Nat. Commun., 7(1):11780, 2016. doi:10.1038/​ncomms11780.

[24] Miguel Navascués, Yelena Guryanova, Matty J. Hoban, and Antonio Acín. Almost quantum correlations. Nat. Commun., 6(1):6288, 2015. doi:10.1038/​ncomms7288.

[25] Miguel Navascués, Stefano Pironio, and Antonio Acín. A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys., 10(7):073013, 2008. doi:10.1088/​1367-2630/​10/​7/​073013.

[26] Matthew F. Pusey. Robust preparation noncontextuality inequalities in the simplest scenario. Phys. Rev. A, 98:022112, 2018. doi:10.1103/​PhysRevA.98.022112.

[27] Jaskaran Singh, Kishor Bharti, and Arvind. Quantum key distribution protocol based on contextuality monogamy. Phys. Rev. A, 95:062333, 2017. doi:10.1103/​PhysRevA.95.062333.

[28] Robert W. Spekkens, D. H. Buzacott, A. J. Keehn, Ben Toner, and Geoff J. Pryde. Preparation contextuality powers parity-oblivious multiplexing. Phys. Rev. Lett., 102:010401, 2009. doi:10.1103/​PhysRevLett.102.010401.

[29] Debashis Saha and Anubhav Chaturvedi. Preparation contextuality as an essential feature underlying quantum communication advantage. Phys. Rev. A, 100:022108, 2019. doi:10.1103/​PhysRevA.100.022108.

[30] Debashis Saha, Paweł Horodecki, and Marcin Pawłowski. State independent contextuality advances one-way communication. New J. Phys., 21(9):093057, 2019. doi:10.1088/​1367-2630/​ab4149.

[31] Robert W. Spekkens. Contextuality for preparations, transformations, and unsharp measurements. Phys. Rev. A, 71(5):052108, 2005. doi:10.1103/​PhysRevA.71.052108.

[32] Robert W. Spekkens. The ontological identity of empirical indiscernibles: Leibniz's methodological principle and its significance in the work of Einstein. arXiv:1909.04628, 2019. URL: https:/​/​​abs/​1909.04628.

[33] David Schmid and Robert W. Spekkens. Contextual advantage for state discrimination. Phys. Rev. X, 8(1):011015, 2018. doi:10.1103/​PhysRevX.8.011015.

[34] David Schmid, John H. Selby, Matthew F. Pusey, and Robert W. Spekkens. A structure theorem for generalized-noncontextual ontological models. arXiv:2005.07161, 2020. URL: https:/​/​​abs/​2005.07161.

[35] David Schmid, John H. Selby, and Robert W. Spekkens. Unscrambling the omelette of causation and inference: The framework of causal-inferential theories. arXiv:2009.03297, 2020. URL: https:/​/​​abs/​2009.03297.

[36] David Schmid, Robert W. Spekkens, and Elie Wolfe. All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences. Phys. Rev. A, 97:062103, 2018. doi:10.1103/​PhysRevA.97.062103.

[37] David Schmid, John H. Selby, Elie Wolfe, Ravi Kunjwal, and Robert W. Spekkens. Characterization of noncontextuality in the framework of generalized probabilistic theories. PRX Quantum, 2:010331, 2021. doi:10.1103/​PRXQuantum.2.010331.

[38] Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Roope Uola, and Alastair A. Abbott. Bounding and simulating contextual correlations in quantum theory. PRX Quantum, 2:020334, 2021. doi:10.1103/​PRXQuantum.2.020334.

[39] James E. Troupe and Jacob M. Farinholt. A contextuality based quantum key distribution protocol. arXiv:1512.02256, 2015. URL: https:/​/​​abs/​1512.02256.

[40] Kim-Chuan Toh, Michael J. Todd, and Reha H. Tütüncü. Sdpt3—a matlab software package for semidefinite programming, version 1.3. Optim. Methods Softw. , 11(1-4):545–581, 1999. doi:10.1080/​10556789908805762.

[41] Peter Wittek. Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables. ACM Trans. Math. Softw., 41(3):1–12, 2015. doi:10.1145/​2699464.

[42] Zhen-Peng Xu, Debashis Saha, Hong-Yi Su, Marcin Pawłowski, and Jing-Ling Chen. Reformulating noncontextuality inequalities in an operational approach. Phys. Rev. A, 94:062103, 2016. doi:10.1103/​PhysRevA.94.062103.

[43] Shiv Akshar Yadavalli and Ravi Kunjwal. Contextuality in entanglement-assisted one-shot classical communication. arXiv:2006.00469, 2020. URL: https:/​/​​abs/​2006.00469.

Cited by

[1] Vaisakh M, Ram krishna Patra, Mukta Janpandit, Samrat Sen, Manik Banik, and Anubhav Chaturvedi, "Mutually unbiased balanced functions and generalized random access codes", Physical Review A 104 1, 012420 (2021).

[2] Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou, "Bell nonlocality in networks", Reports on Progress in Physics 85 5, 056001 (2022).

[3] Hammad Anwer, Natalie Wilson, Ralph Silva, Sadiq Muhammad, Armin Tavakoli, and Mohamed Bourennane, "Noise-robust preparation contextuality shared between any number of observers via unsharp measurements", Quantum 5, 551 (2021).

[4] Costantino Budroni, Adán Cabello, Otfried Gühne, Matthias Kleinmann, and Jan-Åke Larsson, "Quantum Contextuality", arXiv:2102.13036.

[5] Armin Tavakoli, Emmanuel Zambrini Cruzeiro, Roope Uola, and Alastair A. Abbott, "Bounding and Simulating Contextual Correlations in Quantum Theory", PRX Quantum 2 2, 020334 (2021).

[6] Rafael Wagner, Roberto D. Baldijão, Alisson Tezzin, and Bárbara Amaral, "Using a resource theoretic perspective to witness and engineer quantum generalized contextuality for prepare-and-measure scenarios", arXiv:2102.10469.

The above citations are from Crossref's cited-by service (last updated successfully 2022-05-17 06:08:16) and SAO/NASA ADS (last updated successfully 2022-05-17 06:08:17). The list may be incomplete as not all publishers provide suitable and complete citation data.