All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences

Within the framework of generalized noncontextuality, we introduce a general technique for systematically deriving noncontextuality inequalities for any experiment involving finitely many preparations and finitely many measurements, each of which has a finite number of outcomes. Given any fixed sets of operational equivalences among the preparations and among the measurements as input, the algorithm returns a set of noncontextuality inequalities whose satisfaction is necessary and sufficient for a set of operational data to admit of a noncontextual model. Additionally, we show that the space of noncontextual data tables always defines a polytope. Finally, we provide a computationally efficient means for testing whether any set of numerical data admits of a noncontextual model, with respect to any fixed operational equivalences. Together, these techniques provide complete methods for characterizing arbitrary noncontextuality scenarios, both in theory and in practice.

[1]  R. Spekkens,et al.  Preparation contextuality powers parity-oblivious multiplexing. , 2008, Physical review letters.

[2]  ERLING D. ANDERSEN,et al.  Certificates of Primal or Dual Infeasibility in Linear Programming , 2001, Comput. Optim. Appl..

[3]  Reformulating noncontextuality inequalities in an operational approach , 2015, 1509.06027.

[4]  Adan Cabello,et al.  Proposed Experimental Tests of the Bell-Kochen-Specker Theorem , 1998 .

[5]  Ravi Kunjwal,et al.  Beyond the Cabello-Severini-Winter framework: Making sense of contextuality without sharpness of measurements , 2017, Quantum.

[6]  Phillipp Kaestner,et al.  Linear And Nonlinear Programming , 2016 .

[7]  Egon Balas.  Projection with a Minimal System of Inequalities , 1998, Comput. Optim. Appl..

[8]  Debashis Saha,et al.  State independent contextuality advances one-way communication , 2017, New Journal of Physics.

[9]  N. Yu. Zolotykh,et al.  New modification of the double description method for constructing the skeleton of a polyhedral cone , 2012 .

[10]  Earl T. Campbell,et al.  Non-adaptive measurement-based quantum computation and multi-party Bell inequalities , 2010, 1009.5213.

[11]  D. V. Shapot,et al.  Solution Building for Arbitrary System of Linear Inequalities in an Explicit Form , 2012 .

[12]  Costantino Budroni,et al.  Bell inequalities from variable-elimination methods , 2011, 1112.5876.

[13]  Matthew F. Pusey.  The robust noncontextuality inequalities in the simplest scenario , 2015 .

[14]  M. Elliott.  A linear program for testing local realism , 2009, 0905.2950.

[15]  D. Browne,et al.  Computational power of correlations. , 2008, Physical review letters.

[16]  Antonio Acín,et al.  Certified randomness in quantum physics , 2016, Nature.

[17]  M. A. Can,et al.  Simple test for hidden variables in spin-1 systems. , 2007, Physical review letters.

[18]  Ravi Kunjwal,et al.  From the Kochen-Specker theorem to noncontextuality inequalities without assuming determinism. , 2015, Physical review letters.

[19]  Anirudh Krishna.  Experimentally Testable Noncontextuality Inequalities Via Fourier-Motzkin Elimination , 2015 .

[20]  Ian C. Percival,et al.  A general computer program for the Bell detection loophole , 2000 .

[21]  R. Raussendorf.  Quantum computation, discreteness, and contextuality , 2009 .

[22]  S. Wehner,et al.  Bell Nonlocality , 2013, 1303.2849.

[23]  Y. Ye,et al.  Duality and Complementarity , 2021, Linear and Nonlinear Programming.

[24]  N. Mermin.  Hidden variables and the two theorems of John Bell , 1993, 1802.10119.

[25]  Gerhard Reinelt,et al.  PANDA: a software for polyhedral transformations , 2015, EURO J. Comput. Optim..

[26]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[27]  J. Maciejowski,et al.  Equality Set Projection: A new algorithm for the projection of polytopes in halfspace representation , 2004 .

[28]  R. Spekkens,et al.  Specker’s parable of the overprotective seer: A road to contextuality, nonlocality and complementarity , 2010 .

[29]  R. Spekkens,et al.  Contextual Advantage for State Discrimination , 2017, 1706.04588.

[30]  Robert W Spekkens,et al.  Negativity and contextuality are equivalent notions of nonclassicality. , 2006, Physical review letters.

[31]  Matthew F. Pusey,et al.  Robust preparation noncontextuality inequalities in the simplest scenario , 2015, Physical Review A.

[32]  V. Scarani,et al.  Extremal correlations of the tripartite no-signaling polytope , 2011, 1101.2477.

[33]  Mathieu Dutour Sikiric,et al.  Polyhedral representation conversion up to symmetries , 2007, ArXiv.

[34]  Rafael Chaves,et al.  Entropic Inequalities and Marginal Problems , 2011, IEEE Transactions on Information Theory.

[35]  Robert W. Spekkens,et al.  The Inflation Technique for Causal Inference with Latent Variables , 2016, Journal of Causal Inference.

[36]  Victor Veitch,et al.  Contextuality supplies the ‘magic’ for quantum computation , 2014, Nature.

[37]  A. Cabello.  Experimentally testable state-independent quantum contextuality. , 2008, Physical review letters.

[38]  N. Gisin,et al.  From Bell's theorem to secure quantum key distribution. , 2005, Physical review letters.

[39]  D. Gross,et al.  Causal structures from entropic information: geometry and novel scenarios , 2013, 1310.0284.

[40]  N. Gisin,et al.  A relevant two qubit Bell inequality inequivalent to the CHSH inequality , 2003, quant-ph/0306129.

[41]  J. Bell.  On the Problem of Hidden Variables in Quantum Mechanics , 1966 .

[42]  E. Nash.  From the . . . , 2013 .

[43]  Rui Soares Barbosa,et al.  Contextual Fraction as a Measure of Contextuality. , 2017, Physical review letters.

[44]  Sergey Bastrakov,et al.  Fast method for verifying Chernikov rules in Fourier-Motzkin elimination , 2015 .

[45]  D. Avis.  A Revised Implementation of the Reverse Search Vertex Enumeration Algorithm , 2000 .

[46]  Christopher Ferrie,et al.  Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations , 2007, 0711.2658.

[47]  Robert W. Spekkens,et al.  What is the appropriate notion of noncontextuality for unsharp measurements in quantum theory , 2013 .

[48]  Marco T'ulio Quintino,et al.  All noncontextuality inequalities for the n-cycle scenario , 2012, 1206.3212.

[49]  Christian Majenz,et al.  Information–theoretic implications of quantum causal structures , 2014, Nature Communications.

[50]  Matthew F Pusey.  Anomalous weak values are proofs of contextuality. , 2014, Physical review letters.

[51]  T. Fritz,et al.  A Combinatorial Approach to Nonlocality and Contextuality , 2012, Communications in Mathematical Physics.

[52]  Raimund Seidel,et al.  How Good Are Convex Hull Algorithms? , 1997, Comput. Geom..

[53]  Robert W. Spekkens,et al.  Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the Peres–Mermin square , 2017, 1704.01153.

[54]  George B. Dantzig,et al.  Fourier-Motzkin Elimination and Its Dual , 1973, J. Comb. Theory A.

[55]  Andris Ambainis,et al.  Parity oblivious d-level random access codes and class of noncontextuality inequalities , 2016, Quantum Inf. Process..

[56]  N. Gisin,et al.  Looking for symmetric Bell inequalities , 2010, 1004.4146.

[57]  Adrian Kent,et al.  No signaling and quantum key distribution. , 2004, Physical review letters.

[58]  Christopher Ferrie,et al.  Quasi-probability representations of quantum theory with applications to quantum information science , 2010, 1010.2701.

[59]  R. Spekkens.  Contextuality for preparations, transformations, and unsharp measurements , 2004, quant-ph/0406166.

[60]  Ravi Kunjwal,et al.  An experimental test of noncontextuality without unphysical idealizations , 2015, Nature Communications.

[61]  Iordanis Kerenidis,et al.  Optimal bounds for parity-oblivious random access codes , 2014, TQC.

[62]  Ravi Kunjwal.  Fine's theorem, noncontextuality, and correlations in Specker's scenario , 2014, 1410.7760.

[63]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[64]  Robert W. Spekkens,et al.  From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities , 2017, 1708.04793.

[65]  N. Gisin,et al.  Partial list of bipartite Bell inequalities with four binary settings , 2007, 0711.3362.