Correlations constrained by composite measurements

John H. Selby1, Ana Belén Sainz1, Victor Magron2, Łukasz Czekaj1, and Michał Horodecki1

1International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
2LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 Avenue du Colonel Roche, 31077 Toulouse Cédex 4, France

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


How to understand the set of correlations admissible in nature is one outstanding open problem in the core of the foundations of quantum theory. Here we take a complementary viewpoint to the device-independent approach, and explore the correlations that physical theories may feature when restricted by some particular constraints on their measurements. We show that demanding that a theory exhibits a composite measurement imposes a hierarchy of constraints on the structure of its sets of states and effects, which translate to a hierarchy of constraints on the allowed correlations themselves. We moreover focus on the particular case where one demands the existence of a correlated measurement that reads out the parity of local fiducial measurements. By formulating a non-linear Optimisation Problem, and semidefinite relaxations of it, we explore the consequences of the existence of such a parity reading measurement for violations of Bell inequalities. In particular, we show that in certain situations this assumption has surprisingly strong consequences, namely, that Tsirelson's bound can be recovered.

► BibTeX data

► References

[1] John S Bell. ``On the Einstein-Podolsky-Rosen paradox''. Physics 1, 195–200 (1964).

[2] Elie Wolfe, David Schmid, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens. ``Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes''. Quantum 4, 280 (2020).

[3] Jonathan Barrett, Lucien Hardy, and Adrian Kent. ``No Signaling and Quantum Key Distribution''. Phys. Rev. Lett. 95, 010503 (2005).

[4] Antonio Acín, Nicolas Gisin, and Lluis Masanes. ``From Bell's Theorem to Secure Quantum Key Distribution''. Phys. Rev. Lett. 97, 120405 (2006).

[5] Valerio Scarani, Nicolas Gisin, Nicolas Brunner, Lluis Masanes, Sergi Pino, and Antonio Acín. ``Secrecy extraction from no-signaling correlations''. Phys. Rev. A 74, 042339 (2006).

[6] A. Acín $et$ $al.$ ``Device-Independent Security of Quantum Cryptography against Collective Attacks''. Phys. Rev. Lett. 98, 230501 (2007).

[7] Umesh Vazirani and Thomas Vidick. ``Fully Device-Independent Quantum Key Distribution''. Phys. Rev. Lett. 113, 140501 (2014).

[8] Jȩdrzej Kaniewski and Stephanie Wehner. ``Device-independent two-party cryptography secure against sequential attacks''. New J. Phys. 18, 055004 (2016).

[9] Roger Colbeck and Renato Renner. ``Free randomness can be amplified''. Nat. Phys. 8, 450 EP – (2012).

[10] S. Pironio $et$ $al.$ ``Random numbers certified by Bell's theorem''. Nature 464, 1021 EP – (2010).

[11] Matej Pivoluska and Martin Plesch. ``Device Independent Random Number Generation''. Acta Physica Slovaca 64, 600–663 (2015).

[12] Chirag Dhara, Giuseppe Prettico, and Antonio Acín. ``Maximal quantum randomness in Bell tests''. Phys. Rev. A 88, 052116 (2013).

[13] Anne Broadbent and André Allan Méthot. ``On the power of non-local boxes''. Theo. Comp. Sci. 358, 3 – 14 (2006).

[14] Carlos Palazuelos and Thomas Vidick. ``Survey on nonlocal games and operator space theory''. J. Math. Phys. 57, 015220 (2016).

[15] Nathaniel Johnston, Rajat Mittal, Vincent Russo, and John Watrous. ``Extended non-local games and monogamy-of-entanglement games''. Proc. Roy. Soc. A 472, 20160003 (2016).

[16] Sandu Popescu and Daniel Rohrlich. ``Quantum nonlocality as an axiom''. Foundations of Physics 24, 379–385 (1994).

[17] Gilles Brassard, Harry Buhrman, Noah Linden, André Allan Méthot, Alain Tapp, and Falk Unger. ``Limit on nonlocality in any world in which communication complexity is not trivial''. Physical Review Letters 96, 250401 (2006).

[18] Marcin Pawłowski, Tomasz Paterek, Dagomir Kaszlikowski, Valerio Scarani, Andreas Winter, and Marek Żukowski. ``Information causality as a physical principle''. Nature 461, 1101–1104 (2009).

[19] Miguel Navascués and Harald Wunderlich. ``A glance beyond the quantum model''. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, 881–890 (2010).

[20] Joe Henson and Ana Belén Sainz. ``Macroscopic noncontextuality as a principle for almost-quantum correlations''. Physical Review A 91, 042114 (2015).

[21] Tobias Fritz, Ana Belén Sainz, Remigiusz Augusiak, Jonatan Bohr Brask, Rafael Chaves, Anthony Leverrier, and Antonio Acín. ``Local orthogonality as a multipartite principle for quantum correlations''. Nature communications 4, 1–7 (2013).

[22] Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter. ``Quantum nonlocality and beyond: Limits from nonlocal computation''. Phys. Rev. Lett. 99, 180502 (2007).

[23] L. Czekaj, M. Horodecki, and T. Tylec. ``Bell measurement ruling out supraquantum correlations''. Phys. Rev. A 98, 032117 (2018).

[24] Jonathan Barrett. ``Information processing in generalized probabilistic theories''. Physical Review A 75, 032304 (2007).

[25] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt. ``Proposed Experiment to Test Local Hidden-Variable Theories''. Phys. Rev. Lett. 23, 880–884 (1969).

[26] Lucien Hardy. ``Quantum theory from five reasonable axioms'' (2001). arXiv:quant-ph/​0101012.

[27] G. Ludwig. ``An axiomatic basis of quantum mechanics. 1. derivation of hilbert space''. Springer-Verlag. (1985).

[28] E Brian Davies and John T Lewis. ``An operational approach to quantum probability''. Communications in Mathematical Physics 17, 239–260 (1970).

[29] CH Randall and DJ Foulis. ``An approach to empirical logic''. The American Mathematical Monthly 77, 363–374 (1970).

[30] C. Piron. ``Axiomatique quantique''. Helvetia Physica Acta 37, 439–468 (1964).

[31] G. W. Mackey. ``The mathematical foundations of quantum mechanics''. W. A. Benjamin. New York (1963).

[32] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Probabilistic theories with purification''. Physical Review A 81, 062348 (2010).

[33] Lucien Hardy. ``Reformulating and reconstructing quantum theory'' (2011). arXiv:1104.2066.

[34] David Schmid, John H Selby, Matthew F Pusey, and Robert W Spekkens. ``A structure theorem for generalized-noncontextual ontological models'' (2020). arXiv:2005.07161.

[35] Antonio Acín, Serge Massar, and Stefano Pironio. ``Randomness versus Nonlocality and Entanglement''. Phys. Rev. Lett. 108, 100402 (2012). arXiv:1107.2754.

[36] Miguel Navascués, Yelena Guryanova, Matty J. Hoban, and Antonio Acín. ``Almost quantum correlations''. Nature Communications 6, 6288 (2015). arXiv:1403.4621.

[37] Daniel Collins and Nicolas Gisin. ``A relevant two qubit bell inequality inequivalent to the chsh inequality''. Journal of Physics A: Mathematical and General 37, 1775 (2004).

[38] Marius Krumm and Markus P Müller. ``Quantum computation is the unique reversible circuit model for which bits are balls''. npj Quantum Information 5, 7 (2019).

[39] Howard Barnum, Ciarán M Lee, and John H Selby. ``Oracles and query lower bounds in generalised probabilistic theories''. Foundations of physics 48, 954–981 (2018).

[40] Andrew JP Garner. ``Interferometric computation beyond quantum theory''. Foundations of Physics 48, 886–909 (2018).

[41] Jonathan Barrett, Niel de Beaudrap, Matty J Hoban, and Ciarán M Lee. ``The computational landscape of general physical theories''. npj Quantum Information 5, 41 (2019).

[42] Ciarán M Lee and John H Selby. ``Deriving Grover's lower bound from simple physical principles''. New Journal of Physics 18, 093047 (2016).

[43] Ciarán M Lee and Matty J Hoban. ``Bounds on the power of proofs and advice in general physical theories''. Proc. R. Soc. A 472, 20160076 (2016).

[44] Ciarán M Lee and John H Selby. ``Generalised phase kick-back: the structure of computational algorithms from physical principles''. New Journal of Physics 18, 033023 (2016).

[45] Ciarán M Lee and Jonathan Barrett. ``Computation in generalised probabilisitic theories''. New Journal of Physics 17, 083001 (2015).

[46] Ciarán M Lee and John H Selby. ``Higher-order interference in extensions of quantum theory''. Foundations of Physics 47, 89–112 (2017).

[47] Jamie Sikora and John Selby. ``Simple proof of the impossibility of bit commitment in generalized probabilistic theories using cone programming''. Physical review A 97, 042302 (2018).

[48] John H Selby and Jamie Sikora. ``How to make unforgeable money in generalised probabilistic theories''. Quantum 2, 103 (2018).

[49] Ludovico Lami, Carlos Palazuelos, and Andreas Winter. ``Ultimate data hiding in quantum mechanics and beyond''. Communications in Mathematical Physics 361, 661–708 (2018).

[50] Howard Barnum and Alexander Wilce. ``Information processing in convex operational theories''. Electronic Notes in Theoretical Computer Science 270, 3–15 (2011).

[51] Howard Barnum, Oscar CO Dahlsten, Matthew Leifer, and Ben Toner. ``Nonclassicality without entanglement enables bit commitment''. In Information Theory Workshop, 2008. ITW'08. IEEE. Pages 386–390. IEEE (2008).

[52] Jonathan Barrett, Lucien Hardy, and Adrian Kent. ``No signaling and quantum key distribution''. Physical Review Letters 95, 010503 (2005).

[53] Samuel Fiorini, Serge Massar, Manas K Patra, and Hans Raj Tiwary. ``Generalized probabilistic theories and conic extensions of polytopes''. Journal of Physics A: Mathematical and Theoretical 48, 025302 (2014).

[54] Anna Jenčová and Martin Plávala. ``Conditions on the existence of maximally incompatible two-outcome measurements in general probabilistic theory''. Physical Review A 96, 022113 (2017).

[55] Joonwoo Bae, Dai-Gyoung Kim, and Leong-Chuan Kwek. ``Structure of optimal state discrimination in generalized probabilistic theories''. Entropy 18, 39 (2016).

[56] Bob Coecke and Aleks Kissinger. ``Picturing quantum processes: A first course in quantum theory and diagrammatic reasoning''. Cambridge University Press. (2017).

[57] Stefano Gogioso and Carlo Maria Scandolo. ``Categorical probabilistic theories''. EPTCS 266, 367 (2018).

[58] John H. Selby, Carlo Maria Scandolo, and Bob Coecke. ``Reconstructing quantum theory from diagrammatic postulates''. Quantum 5, 445 (2021).

[59] Peter Janotta and Raymond Lal. ``Generalized probabilistic theories without the no-restriction hypothesis''. Physical Review A 87, 052131 (2013).

[60] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. ``Bell nonlocality''. Reviews of Modern Physics 86, 419 (2014).

[61] Bob Coecke. ``Terminality implies non-signalling''. EPTCS 172, 27 (2014).

[62] Aleks Kissinger, Matty Hoban, and Bob Coecke. ``Equivalence of relativistic causal structure and process terminality'' (2017). arXiv:1708.04118.

[63] Borivoje Dakic and Caslav Brukner. ``Quantum Theory and Beyond: Is Entanglement Special?'' (2009). arXiv:0911.0695.

[64] Jamie Sikora and John H. Selby. ``Impossibility of coin flipping in generalized probabilistic theories via discretizations of semi-infinite programs''. Phys. Rev. Res. 2, 043128 (2020).

[65] Jean B Lasserre. ``Global optimization with polynomials and the problem of moments''. SIAM Journal on optimization 11, 796–817 (2001).

[66] Peter Wittek. ``Algorithm 950: Ncpol2sdpa—sparse semidefinite programming relaxations for polynomial optimization problems of noncommuting variables''. ACM Transactions on Mathematical Software (2015). arXiv:1308.6029.

[67] ``http:/​/​​''.

[68] Jie Wang, Victor Magron, and Jean-Bernard Lasserre. ``Tssos: A moment-sos hierarchy that exploits term sparsity''. SIAM Journal on Optimization 31, 30–58 (2021).

[69] Victor Magron and Jie Wang. ``Sparse polynomial optimization: theory and practice''. Series on Optimization and Its Applications, World Scientific Press. (2023).

[70] Erling D Andersen and Knud D Andersen. ``The mosek interior point optimizer for linear programming: an implementation of the homogeneous algorithm''. High performance optimizationPages 197–232 (2000).

[71] Wolfram Research, Inc. ``Mathematica, Version 12.1''. Champaign, IL, 2020.

[72] Bob Coecke and Eric Oliver Paquette. ``Categories for the practising physicist''. In New structures for physics. Pages 173–286. Springer (2010).

[73] Saunders Mac Lane. ``Categories for the working mathematician''. Volume 5. Springer Science & Business Media. (2013).

Cited by

[1] Giorgos Eftaxias, Mirjam Weilenmann, and Roger Colbeck, "Multisystem measurements in generalized probabilistic theories and their role in information processing", Physical Review A 108 6, 062212 (2023).

[2] Martin Plávala, "General probabilistic theories: An introduction", Physics Reports 1033, 1 (2023).

[3] Thomas D. Galley, Flaminia Giacomini, and John H. Selby, "A no-go theorem on the nature of the gravitational field beyond quantum theory", Quantum 6, 779 (2022).

[4] John H. Selby, David Schmid, Elie Wolfe, Ana Belén Sainz, Ravi Kunjwal, and Robert W. Spekkens, "Accessible fragments of generalized probabilistic theories, cone equivalence, and applications to witnessing nonclassicality", Physical Review A 107 6, 062203 (2023).

[5] Paulo J. Cavalcanti, John H. Selby, Jamie Sikora, and Ana Belén Sainz, "Decomposing all multipartite non-signalling channels via quasiprobabilistic mixtures of local channels in generalised probabilistic theories", Journal of Physics A Mathematical General 55 40, 404001 (2022).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 15:10:35) and SAO/NASA ADS (last updated successfully 2024-05-21 15:10:36). The list may be incomplete as not all publishers provide suitable and complete citation data.