Complete Extension: The Non-signaling Analog Of Quantum Purification

Marek Winczewski^1,2, Tamoghna Das^1,3, John H. Selby¹, Karol Horodecki^4,1, Paweł Horodecki^1,5,6, Łukasz Pankowski⁷, Marco Piani^8,9, and Ravishankar Ramanathan¹⁰

¹International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
²Institute of Theoretical Physics and Astrophysics and National Quantum Information Centre in Gdańsk, University of Gdańsk, 80–952 Gdańsk, Poland
³Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
⁴Institute of Informatics and National Quantum Information Centre in Gdańsk, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80–952 Gdańsk, Poland
⁵Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80–233 Gdańsk, Poland
⁶National Quantum Information Centre, University of Gdańsk, ul. Jana Bażyńskiego 8, 80-309 Gdańsk, Poland
⁷VOICELAB.AI, Al. Grunwaldzka 135A; 80-264 Gdańsk, Poland,
⁸evolutionQ Inc., Waterloo, Ontario, N2L 3L3, Canada
⁹SUPA and Department of Physics, University of Strathclyde, Glasgow, G4 0NG, UK
¹⁰Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong

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

Abstract

Deriving quantum mechanics from information-theoretic postulates is a recent research direction taken, in part, with the view of finding a beyond-quantum theory; once the postulates are clear, we can consider modifications to them. A key postulate is the purification postulate, which we propose to replace by a more generally applicable postulate that we call the complete extension postulate (CEP), i.e., the existence of an extension of a physical system from which one can generate any other extension. This new concept leads to a plethora of open questions and research directions in the study of general theories satisfying the CEP (which may include a theory that hyper-decoheres to quantum theory). For example, we show that the CEP implies the impossibility of bit-commitment. This is exemplified by a case study of the theory of non-signalling behaviors which we show satisfies the CEP. We moreover show that in certain cases the complete extension will not be pure, highlighting the key divergence from the purification postulate.

Featured image: Schematic depiction of the relationship between the set of all theories, those are satisfying the complete extension postulate and those are satisfying the purification postulate. Also shown is the hyperdecoherence mechanism (purple arrows) relating hypothetical beyond quantum theories and quantum theory in analogy to the relationship between quantum and classical theory by a decoherence mechanism (blue arrow).

Popular summary

Quantum mechanics can be derived from a few information-theoretic postulates. However, if there exists a theory of Nature with more explanatory power than quantum mechanics, it must not satisfy one of these. We study the consequences of replacing a particular postulate with a less restrictive one. Namely, we replace the purification postulate (PP) with a less restrictive complete extension postulate (CEP). This new postulate requires, for all states, the existence of extensions from which any other extension can be generated, i.e., complete extensions. We do so by studying the properties of theories satisfying CEP in the framework of the so-called generalized probabilistic theories (GPTs). First, we show that PP can not hold in any discrete convex theory. Second, we show that the mentioned replacement does not trivialize important cryptographic tasks (i.e., bit commitment). Second, we show that CEP, contrary to PP, may not exclude (post-quantum) theories that hyper-decohere to quantum mechanics. As a case study, we construct non-signaling complete extensions in the theory of non-signaling behaviors. We also show how the structure of the famous Popescu-Rohrlich behavior (PR box) arises as a non-signaling complete extension of the maximally mixed behavior. Finally, we proved that classical probability theory, quantum theory, and theory of non-signaling behaviors satisfy CEP. Therefore, we say that complete extensions are non-signaling analogs of quantum-mechanical purifications. In this way, our findings might establish a step towards finding a more fundamental theory of Nature.

► BibTeX data

@article{Winczewski2023completeextension, doi = {10.22331/q-2023-11-03-1159}, url = {https://doi.org/10.22331/q-2023-11-03-1159}, title = {Complete extension: the non-signaling analog of quantum purification}, author = {Winczewski, Marek and Das, Tamoghna and Selby, John H. and Horodecki, Karol and Horodecki, Pawe{l{}} and Pankowski, {L{}}ukasz and Piani, Marco and Ramanathan, Ravishankar}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1159}, month = nov, year = {2023}<br /> }

► References

[1] Asher Peres. “Karl Popper and the Copenhagen interpretation”. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 33, 23–34 (2002).
https://doi.org/10.1016/s1355-2198(01)00034-x

[2] Paul Adrian Dirac. “The principles of quantum mechanics (third ed.)”. Clarendon Press Oxford. (1948).
https://doi.org/10.1038/136411a0

[3] J. von Neumann. “Mathematische grundlagen der quantenmachanik”. Springer, Berlin. (1932).
https://doi.org/10.1007/978-3-642-61409-5

[4] Johann von Neumann. “Mathematische grundlagen der quantenmechanik”. Springer. (1932).
https://doi.org/10.1007/978-3-642-61409-5

[5] S. Popescu and D. Rohrlich. “Quantum nonlocality as an axiom”. Found. Phys. 24, 379–385 (1994). url: doi.org/10.1007/BF02058098.
https://doi.org/10.1007/BF02058098

[6] Wolfgang Bertram. “An Essay on the Completion of Quantum Theory. I: General Setting” (2017). arXiv:1711.08643.
arXiv:1711.08643

[7] Wolfgang Bertram. “An Essay on the Completion of Quantum Theory. II: Unitary Time Evolution” (2018). arXiv:1807.04650.
arXiv:1807.04650

[8] Lucien Hardy. “Quantum theory from five reasonable axioms” (2001). url: arxiv.org/abs/quant-ph/0101012.
arXiv:quant-ph/0101012

[9] Karol Życzkowski. “Quartic quantum theory: an extension of the standard quantum mechanics”. J. Phys. A: Math. Theor. 41, 355302 (2008). url: doi.org/10.1088/1751-8113/41/35/355302.
https://doi.org/10.1088/1751-8113/41/35/355302

[10] Lee Smolin. “Could quantum mechanics be an approximation to another theory?” (2006). arXiv:quant-ph/060910.
arXiv:quant-ph/0609109v1

[11] Marshall H Stone. “On one-parameter unitary groups in hilbert space”. Ann. Math. 33, 643–648 (1932). url: doi.org/10.2307/1968538.
https://doi.org/10.2307/1968538

[12] Andrew M Gleason. “Measures on the closed subspaces of a hilbert space”. In The Logico-Algebraic Approach to Quantum Mechanics. Pages 123–133. Springer (1975).
https://doi.org/10.1007/978-94-010-1795-4_7

[13] Lluís Masanes, Thomas D Galley, and Markus P Müller. “The measurement postulates of quantum mechanics are operationally redundant”. Nat. Comm. 10, 1–6 (2019).
https://doi.org/10.1038/s41467-019-09348-x

[14] Borivoje Dakic and Caslav Brukner. “Quantum theory and beyond: Is entanglement special?”. Deep Beauty: Understanding the Quantum World Through Mathematical Innovation (ed. Halvorson, H.) (Cambridge Univ. Press, 2011) (2011).
https://doi.org/10.48550/arXiv.0911.0695

[15] G. Chiribella, G. M. D’Ariano, and P. Perinotti. “Probabilistic theories with purification”. Phys. Rev. A 81, 062348 (2010). arXiv:0908.1583.
https://doi.org/10.1103/PhysRevA.81.062348
arXiv:0908.1583

[16] Lucien Hardy. “Reformulating and Reconstructing Quantum Theory” (2011). arXiv:1104.2066.
arXiv:1104.2066

[17] Rob Clifton, Jeffrey Bub, and Hans Halvorson. “Characterizing quantum theory in terms of information-theoretic constraints”. Foundations of Physics 33, 1561–1591 (2003).
https://doi.org/10.1023/a:1026056716397

[18] Philip Goyal. “Information-geometric reconstruction of quantum theory”. Phys. Rev. A 78, 052120 (2008).
https://doi.org/10.1103/physreva.78.052120

[19] Lluís Masanes and Markus P Müller. “A derivation of quantum theory from physical requirements”. New J. Phys. 13, 063001 (2011).
https://doi.org/10.1088/1367-2630/13/6/063001

[20] Howard Barnum, Markus P Müller, and Cozmin Ududec. “Higher-order interference and single-system postulates characterizing quantum theory”. New Journal of Physics 16, 123029 (2014).
https://doi.org/10.1088/1367-2630/16/12/123029

[21] Alexander Wilce. “A Royal Road to Quantum Theory (or Thereabouts)” (2016). arXiv:1606.09306.
arXiv:1606.09306

[22] Philipp Höhn. “Quantum theory from rules on information acquisition”. Entropy 19, 98 (2017).
https://doi.org/10.3390/e19030098

[23] Agung Budiyono and Daniel Rohrlich. “Quantum mechanics as classical statistical mechanics with an ontic extension and an epistemic restriction”. Nature Communications 8 (2017).
https://doi.org/10.1038/s41467-017-01375-w

[24] John H. Selby, Carlo Maria Scandolo, and Bob Coecke. “Reconstructing quantum theory from diagrammatic postulates”. Quantum 5, 445 (2021).
https://doi.org/10.22331/q-2021-04-28-445

[25] Sean Tull. “A categorical reconstruction of quantum theory”. Logical Methods in Computer Science ; Volume 16Pages Issue 1 ; 1860–5974 (2020).
https://doi.org/10.23638/LMCS-16(1:4)2020

[26] John van de Wetering. “An effect-theoretic reconstruction of quantum theory”. Compositionality 1, 1 (2019).
https://doi.org/10.32408/compositionality-1-1

[27] Kenji Nakahira. “Derivation of quantum theory with superselection rules”. Physical Review A 101 (2020).
https://doi.org/10.1103/physreva.101.022104

[28] G. Chiribella, G. M. D’Ariano, and P. Perinotti. “Informational derivation of quantum theory”. Phys. Rev. A 84, 012311 (2011). arXiv:1011.6451.
https://doi.org/10.1103/PhysRevA.84.012311
arXiv:1011.6451

[29] Ciarán M Lee and John H Selby. “Generalised phase kick-back: the structure of computational algorithms from physical principles”. New J. Phys. 18, 033023 (2016). url: doi.org/10.1088/1367-2630/18/3/033023.
https://doi.org/10.1088/1367-2630/18/3/033023

[30] Ciarán M Lee and John H Selby. “Deriving grover’s lower bound from simple physical principles”. New J. Phys. 18, 093047 (2016).
https://doi.org/10.1088/1367-2630/18/9/093047

[31] Howard Barnum, Ciarán M Lee, and John H Selby. “Oracles and query lower bounds in generalised probabilistic theories”. Found. Phys. 48, 954–981 (2018).
https://doi.org/10.1007/s10701-018-0198-4

[32] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti. “Probabilistic theories with purification”. Physical Review A 81, 062348 (2010).
https://doi.org/10.1103/PhysRevA.81.062348

[33] Jamie Sikora and John Selby. “Simple proof of the impossibility of bit commitment in generalized probabilistic theories using cone programming”. Phys. Rev. A 97, 042302 (2018).
https://doi.org/10.1103/PhysRevA.97.042302

[34] Giulio Chiribella and Carlo Maria Scandolo. “Entanglement and thermodynamics in general probabilistic theories”. New J. Phys. 17, 103027 (2015).
https://doi.org/10.1088/1367-2630/17/10/103027

[35] Giulio Chiribella and Carlo Maria Scandolo. “Microcanonical thermodynamics in general physical theories”. New J. Phys. 19, 123043 (2017).
https://doi.org/10.1088/1367-2630/aa91c7

[36] Howard Barnum, Ciarán M Lee, Carlo Maria Scandolo, and John H Selby. “Ruling out higher-order interference from purity principles”. Entropy 19, 253 (2017).
https://doi.org/10.3390/e19060253

[37] Ciarán M Lee and John H Selby. “A no-go theorem for theories that decohere to quantum mechanics”. Proc. R. Soc. A: Math. Phys. Eng. Sci. 474, 20170732 (2018).
https://doi.org/10.1098/rspa.2017.0732

[38] Roman V. Buniy, Stephen D.H. Hsu, and A. Zee. “Is Hilbert space discrete?”. Physics Letters B 630, 68–72 (2005).
https://doi.org/10.1016/j.physletb.2005.09.084

[39] Markus Mueller. “Does probability become fuzzy in small regions of spacetime?”. Physics Letters B 673, 166–167 (2009).
https://doi.org/10.1016/j.physletb.2009.02.017

[40] T. N. Palmer. “Discretisation of the Bloch Sphere, Fractal Invariant Sets and Bell’s Theorem” (2020). arXiv:1804.01734.
arXiv:1804.01734

[41] Bas Westerbaan and John van de Wetering. “A computer scientist’s reconstruction of quantum theory”. J. Phys. A: Math. Theor. 55, 384002 (2022).
https://doi.org/10.1088/1751-8121/ac8459

[42] L. Hardy. “Probability theories with dynamic causal structure: a new framework for quantum gravity” (2005). arXiv:gr-qc/0509120.
arXiv:gr-qc/0509120

[43] L. Hardy. “Towards quantum gravity: a framework for probabilistic theories with non-fixed causal structure”. J. Phys. A 40, 3081–3099 (2007).
https://doi.org/10.1088/1751-8113/40/12/S12

[44] Ognyan Oreshkov, Fabio Costa, and Časlav Brukner. “Quantum correlations with no causal order”. Nat. Comm. 3, 1–8 (2012).
https://doi.org/10.1038/ncomms2076

[45] Giulio Chiribella, Giacomo Mauro D’Ariano, Paolo Perinotti, and Benoit Valiron. “Quantum computations without definite causal structure”. Phys. Rev. A 88, 022318 (2013).
https://doi.org/10.1103/PhysRevA.88.022318

[46] Mateus Araújo, Adrien Feix, Miguel Navascués, and Časlav Brukner. “A purification postulate for quantum mechanics with indefinite causal order”. Quantum 1, 10 (2017).
https://doi.org/10.22331/q-2017-04-26-10

[47] M. A. Nielsen and I. L. Chuang. “Quantum computation and quantum information”. Cambridge University Press,Cambridge. (2000).
https://doi.org/10.1017/CBO9780511976667

[48] J. Barrett. “Information processing in generalized probabilistic theories”. Phys. Rev. A 75, 032304 (2007).
https://doi.org/10.1103/PhysRevA.75.032304

[49] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner. “Bell nonlocality”. Rev. Mod. Phys. 86, 839 (2014).
https://doi.org/10.1103/RevModPhys.86.419

[50] Howard Barnum, Oscar CO Dahlsten, Matthew Leifer, and Ben Toner. “Nonclassicality without entanglement enables bit commitment”. In 2008 IEEE Information Theory Workshop. Pages 386–390. IEEE (2008).
https://doi.org/10.1109/ITW.2008.4578692

[51] Marek Winczewski, Tamoghna Das, and Karol Horodecki. “Limitations on device independent secure key via squashed non-locality” (2019). arXiv:1903.12154.
arXiv:1903.12154

[52] Martin Plávala. “General probabilistic theories: An introduction” (2021).
https://doi.org/10.21468/SciPostPhys.11.4.082

[53] Markus Müller. “Probabilistic theories and reconstructions of quantum theory”. SciPost Phys. Lect. NotesPage 28 (2021).
https://doi.org/10.21468/SciPostPhysLectNotes.28

[54] Ludovico Lami. “Non-classical correlations in quantum mechanics and beyond” (2018).
https://doi.org/10.1039/C7NR07218J

[55] Bob Coecke. “Terminality implies non-signalling” (2014). url: arxiv.org/abs/1405.3681v3.
arXiv:1405.3681v3

[56] Aleks Kissinger, Matty Hoban, and Bob Coecke. “Equivalence of relativistic causal structure and process terminality” (2017). url: doi.org/10.48550/arXiv.1708.04118.
https://doi.org/10.48550/arXiv.1708.04118

[57] Stefano Gogioso and Carlo Maria Scandolo. “Categorical probabilistic theories” (2017). url: doi.org/10.4204/EPTCS.266.23.
https://doi.org/10.4204/EPTCS.266.23

[58] C. Pfister and S. Wehner. “If no information gain implies no disturbance, then any discrete physical theory is classical”. Nat. Comm. 4, 1851 (2013). url: doi.org/10.1038/ncomms2821.
https://doi.org/10.1038/ncomms2821

[59] Ł. Czekaj, M. Horodecki, P. Horodecki, and R. Horodecki. “Information content of systems as a physical principle”. Phys. Rev. A 95, 022119 (2017).
https://doi.org/10.1103/PhysRevA.95.022119

[60] P. Janotta, C. Gogolin, J. Barrett, and N. Brunner. “Limits on non-local correlations from the structure of the local state space”. New J. Phys. 13, 063024 (2011).
https://doi.org/10.1088/1367-2630/13/6/063024

[61] Howard Barnum and Alexander Wilce. “Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory” (2009). arXiv:0908.2354.
arXiv:0908.2354

[62] Peter Janotta and Raymond Lal. “Generalized probabilistic theories without the no-restriction hypothesis”. Phys. Rev. A 87, 052131 (2013). url: doi.org/10.1103/PhysRevA.87.052131.
https://doi.org/10.1103/PhysRevA.87.052131

[63] K. Kuratowski. “Introduction to set theory & topology”. Volume 101 of International series of monographs in pure and applied mathematics. PWN. Warsaw (1961).
https://doi.org/10.1002/zamm.19620421218

[64] Kenta Cho and Bart Jacobs. “Disintegration and bayesian inversion, both abstractly and concretely”. Math. Struct. Comput. Sci. 29, 938–971 (2017). url: doi.org/10.1017/S0960129518000488.
https://doi.org/10.1017/S0960129518000488

[65] Manuel Blum. “Coin flipping by telephone”. In Advances in Cryptology: A Report on CRYPTO 81, IEEE Workshop on Communications Security. Pages 11–15. (1981).
https://doi.org/10.1145/1008908.1008911

[66] Shafi Goldwasser, Silvio Micali, and Charles Rackoff. “The knowledge complexity of interactive proof systems”. SIAM J. Comput. 18, 186–208 (1989).

[67] Dominic Mayers. “Unconditionally secure quantum bit commitment is impossible”. Phys. Rev. Lett. 78, 3414–3417 (1997).
https://doi.org/10.1103/PhysRevLett.78.3414

[68] Hoi-Kwong Lo and Hoi Fung Chau. “Why quantum bit commitment and ideal quantum coin tossing are impossible”. Physica D: Nonlinear Phenomena 120, 177–187 (1998).
https://doi.org/10.1016/S0167-2789(98)00053-0

[69] Stephen Boyd and Lieven Vandenberghe. “Convex optimization”. Cambridge University Press. (2004).
https://doi.org/10.1017/CBO9780511804441

[70] Sevag Gharibian, Jamie Sikora, and Sarvagya Upadhyay. “QMA variants with polynomially many provers”. Quantum Information & Computation 13, 0135–0157 (2013). arXiv:1108.0617.
arXiv:1108.0617

[71] Somshubhro Bandyopadhyay, Alessandro Cosentino, Nathaniel Johnston, Vincent Russo, John Watrous, and Nengkun Yu. “Limitations on separable measurements by convex optimization”. IEEE Transactions on Information Theory 61, 3593–3604 (2015). url: doi.org/10.1109/TIT.2015.2417755.
https://doi.org/10.1109/TIT.2015.2417755

[72] Monique Laurent and Teresa Piovesan. “Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone”. Siam J. Optim. 25, 2461–2493 (2015). url: doi.org/10.1137/14097865X.
https://doi.org/10.1137/14097865X

[73] Ashwin Nayak, Jamie Sikora, and Levent Tunçel. “A search for quantum coin-flipping protocols using optimization techniques”. Math. Program. 156, 581–613 (2016). url: doi.org/10.1007/s10107-015-0909-y.
https://doi.org/10.1007/s10107-015-0909-y

[74] Jamie Sikora and Antonios Varvitsiotis. “Linear conic formulations for two-party correlations and values of nonlocal games”. Math. Program. 162, 431–463 (2017). url: doi.org/10.1007/s10107-016-1049-8.
https://doi.org/10.1007/s10107-016-1049-8

[75] Samuel Fiorini, Serge Massar, Manas K Patra, and Hans Raj Tiwary. “Generalized probabilistic theories and conic extensions of polytopes”. J. Phys. A: Math. Theor. 48, 025302 (2014). url: doi.org/10.1088/1751-8113/48/2/025302.
https://doi.org/10.1088/1751-8113/48/2/025302

[76] Anna Jenčová and Martin Plávala. “Conditions on the existence of maximally incompatible two-outcome measurements in general probabilistic theory”. Phys. Rev. A 96, 022113 (2017). url: doi.org/10.1103/PhysRevA.96.022113.
https://doi.org/10.1103/PhysRevA.96.022113

[77] Joonwoo Bae, Dai-Gyoung Kim, and Leong-Chuan Kwek. “Structure of optimal state discrimination in generalized probabilistic theories”. Entropy 18, 39 (2016). url: doi.org/10.3390/e18020039.
https://doi.org/10.3390/e18020039

[78] L. Lami, C. Palazuelos, and A. Winter. “Ultimate data hiding in quantum mechanics and beyond”. Commun. Math. Phys. 361, 661–708 (2018).
https://doi.org/10.1007/s00220-018-3154-4

[79] Jamie Sikora and John H. Selby. “Impossibility of coin flipping in generalized probabilistic theories via discretizations of semi-infinite programs”. Phys. Rev. Research 2, 043128 (2020).
https://doi.org/10.1103/PhysRevResearch.2.043128

[80] John H Selby and Jamie Sikora. “How to make unforgeable money in generalised probabilistic theories”. Quantum 2, 103 (2018). url: doi.org/10.22331/q-2018-11-02-103.
https://doi.org/10.22331/q-2018-11-02-103

[81] Bob Coecke, John Selby, and Sean Tull. “Two roads to classicality” (2017). url: doi.org/10.4204/EPTCS.266.7.
https://doi.org/10.4204/EPTCS.266.7

[82] John Selby and Bob Coecke. “Leaks: quantum, classical, intermediate and more”. Entropy 19, 174 (2017). url: doi.org/10.3390/e19040174.
https://doi.org/10.3390/e19040174

[83] M. Pawłowski, T. Paterek, D. Kaszlikowski, V. Scarani, A. Winter, and M. Żukowski. “Information causality as a physical principle”. Nature 461, 1101–1104 (2009). url: doi.org/10.1038/nature08400.
https://doi.org/10.1038/nature08400

[84] J. Barrett. “Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality”. Phys. Rev. A 65, 042302 (2002). url: doi.org/10.1103/PhysRevA.65.042302.
https://doi.org/10.1103/PhysRevA.65.042302

[85] A. J. Short, S. Popescu, and N. Gisin. “Entanglement swapping for generalized nonlocal correlations”. Phys. Rev. A 73, 012101 (2006). url: doi.org/10.1103/PhysRevA.73.012101.
https://doi.org/10.1103/PhysRevA.73.012101

[86] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters. “Teleporting an unknown quantum state via dual classical and einstein–podolsky–rosen channels”. Phys. Rev. Lett. 70, 1895 (1993).
https://doi.org/10.1103/PhysRevLett.70.1895

[87] M. Żukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert. “Event-ready deterctors bell experiment via entanglement swapping”. Phys. Rev. Lett. 71, 4287 (1993).
https://doi.org/10.1103/PhysRevLett.71.4287

[88] A. Acin, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani. “Device-independent security of quantum cryptography against collective attacks”. Phys. Rev. Lett. 98, 230501 (2007). url: doi.org/10.1103/PhysRevLett.98.230501.
https://doi.org/10.1103/PhysRevLett.98.230501

[89] E. Hänggi, R. Renner, and S. Wolf. “Efficient quantum key distribution based solely on bell’s theorem”. EUROCRYPTPages 216–234 (2010). arXiv:org:0911.4171.
arXiv:0911.4171v1

[90] J. Barrett, L. Hardy, and A. Kent. “No signaling and quantum key distribution”. Phys. Rev. Lett 95, 010503 (2005).
https://doi.org/10.1103/PhysRevLett.95.010503

[91] A. Acin, N. Gisin, and L. Masanes. “From bell’s theorem to secure quantum key distribution”. Phys. Rev. Lett 97, 120405 (2006).
https://doi.org/10.1103/PhysRevLett.97.120405

[92] E. Hänggi. “Device-independent quantum key distribution”. PhD thesis. PhD Thesis, 2010. (2010). url: doi.org/10.48550/arXiv.1012.3878.
https://doi.org/10.48550/arXiv.1012.3878

[93] R. Colbeck and R. Renner. “Free randomness can be amplified”. Nat. Phys. 8, 450–454 (2012). url: doi.org/10.1038/nphys2300.
https://doi.org/10.1038/nphys2300

[94] R. Gallego, L. Masanes, G. DeLaTorre, C. Dhara, L. Aolita, and A. Acin. “Full randomness from arbitrarily deterministic events”. Nat. Comm. 4, 2654 (2013). url: doi.org/10.1038/ncomms3654.
https://doi.org/10.1038/ncomms3654

[95] P. Mironowicz, R. Gallego, and M. Pawłowski. “Amplification of arbitrarily weak randomness”. Phys. Rev. A 91, 032317 (2015). url: doi.org/10.1103/PhysRevA.91.032317.
https://doi.org/10.1103/PhysRevA.91.032317

[96] F. G. S. L. Brandão, R. Ramanathan, A. Grudka, K. Horodecki, P. Horodecki M. Horodecki, T. Szarek, and H. Wojewódka. “Robust device-independent randomness amplification with few devices”. Nat. Comm. 7, 11345 (2016). url: doi.org/10.1038/ncomms11345.
https://doi.org/10.1038/ncomms11345

[97] R. Ramanathan, F. G. S. L. Brandão, K. Horodecki, M. Horodecki, P. Horodecki, and H. Wojewódka. “Randomness amplification against no-signaling adversaries using two devices”. Phys. Rev. Lett. 117, 230501 (2016). url: doi.org/10.1103/PhysRevLett.117.230501.
https://doi.org/10.1103/PhysRevLett.117.230501

[98] H. Wojewódka, F. G. S. L. Brandão, A. Grudka, M. Horodecki, K. Horodecki, P. Horodecki, M. Pawlowski, and R. Ramanathan. “Randomness amplification against no-signaling adversaries using two devices”. IEEE Trans. Inf. Theory 63, 7592 (2017). url: doi.org/10.1109/TIT.2017.2738010.
https://doi.org/10.1109/TIT.2017.2738010

[99] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt. “Proposed experiment to test local hidden-variable theories”. Phys. Rev. Lett. 23, 880–884 (1969).
https://doi.org/10.1103/PhysRevLett.23.880

[100] Marek Winczewski, Tamoghna Das, and Karol Horodecki. “Limitations on device independent secure key via squashed non-locality” (2020). arXiv:1903.12154.
arXiv:1903.12154

[101] P. Horodecki and R. Ramanathan. “The relativistic causality versus no-signaling paradigm for multi-party correlations”. Nat Commun 10, 1701 (2019).
https://doi.org/10.1038/s41467-019-09505-2

[102] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts. “Non-local correlations as an information theoretic resource”. Phys. Rev. A 71, 022101 (2005).
https://doi.org/10.1103/PhysRevA.71.022101

[103] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. “Quantum entanglement”. Rev. Mod. Phys. 81, 865 (2009). url: doi.org/10.1103/RevModPhys.81.865.
https://doi.org/10.1103/RevModPhys.81.865

[104] S. Pironio. “Lifting bell inequalities”. Journal of Mathematical Physics 46, 062112 (2005). arXiv:1210.0194.
https://doi.org/10.1063/1.1928727
arXiv:1210.0194

[105] A. Schrijver. “Combinatorial optimization polyhedra and efficiency”. Springer. Berlin (2003). url: link.springer.com/book/9783540443896.
https://link.springer.com/book/9783540443896

[106] C. Carathéodory. “Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen”. Aus: Rendiconti del Circolo Matematico di Palermo. Direzione e Redazione. (1911). url: books.google.co.in/books?id=n4SkjwEACAAJ.
https://books.google.co.in/books?id=n4SkjwEACAAJ

[107] Günter M. Ziegler. “Lectures on polytopes”. Springer New York. (1995).
https://doi.org/10.1007/978-1-4613-8431-1

[108] D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu. “Bell inequalities for arbitrarily high-dimensional systems”. Phys. Rev. Lett. 88, 040404 (2002). url: doi.org/10.1103/PhysRevLett.88.040404.
https://doi.org/10.1103/PhysRevLett.88.040404

[109] P. McMullen. “The maximum numbers of faces of a convex polytope”. Mathematika 17, 179–184 (1970). arXiv:https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/S0025579300002850.
https://doi.org/10.1112/S0025579300002850
arXiv:https://londmathsoc.onlinelibrary.wiley.com/doi/pdf/10.1112/S0025579300002850

[110] Khaled Elbassioni, Zvi Lotker, and Raimund Seidel. “Upper bound on the number of vertices of polyhedra with 0,1-constraint matrices”. Information Processing Letters 100, 69 – 71 (2006).
https://doi.org/10.1016/j.ipl.2006.05.011

[111] Samson Abramsky and Adam Brandenburger. “The sheaf-theoretic structure of non-locality and contextuality”. New J. Phys. 13, 113036 (2011). url: doi.org/10.1088/1367-2630/13/11/113036.
https://doi.org/10.1088/1367-2630/13/11/113036

[112] M. Araújo, M. Túlio Quintino, C. Budroni, M. Terra Cunha, and A. Cabello. “All noncontextuality inequalities for the n-cycle scenario”. Phys. Rev. A 88, 022118 (2013). url: doi.org/10.1103/PhysRevA.88.022118.
https://doi.org/10.1103/PhysRevA.88.022118

[113] Ernst Specker. “Die logik nicht gleichzeitig entscheidbarer aussagen”. In Ernst Specker Selecta. Pages 175–182. Springer (1990).
https://doi.org/10.1007/978-3-0348-9259-9_14

[114] 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, 1–39 (2011). url: doi.org/10.1016/j.physrep.2011.05.001.
https://doi.org/10.1016/j.physrep.2011.05.001

[115] Ravi Kunjwal, Chris Heunen, and Tobias Fritz. “Quantum realization of arbitrary joint measurability structures”. Phys. Rev. A 89, 052126 (2014). url: doi.org/10.1103/PhysRevA.88.022118.
https://doi.org/10.1103/PhysRevA.88.022118

[116] B. Tsirelson. “Quantum generalizations of Bell’s inequality”. Lett. Math. Phys. 4, 93–100 (1980). url: doi.org/10.1007/BF00417500.
https://doi.org/10.1007/BF00417500

[117] A. Grudka, K. Horodecki, M. Horodecki, P. Horodecki, R. Horodecki, P. Joshi, W. Kłobus, and A. Wójcik. “Quantifying Contextuality”. Phys. Rev. Lett. 112, 120401 (2014). url: doi.org/10.1103/PhysRevLett.112.120401.
https://doi.org/10.1103/PhysRevLett.112.120401