Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes

Elie Wolfe1, David Schmid1,2, Ana Belén Sainz1,3, Ravi Kunjwal1,4, and Robert W. Spekkens1

1Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada
2Institute for Quantum Computing and Dept. of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
3International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
4Centre for Quantum Information and Communication, Ecole polytechnique de Bruxelles, CP 165, Université libre de Bruxelles, 1050 Brussels, Belgium

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

Abstract

We take a resource-theoretic approach to the problem of quantifying nonclassicality in Bell scenarios. The resources are conceptualized as probabilistic processes from the setting variables to the outcome variables having a particular causal structure, namely, one wherein the wings are only connected by a common cause. We term them "common-cause boxes". We define the distinction between classical and nonclassical resources in terms of whether or not a classical causal model can explain the correlations. One can then quantify the relative nonclassicality of resources by considering their interconvertibility relative to the set of operations that can be implemented using a classical common cause (which correspond to local operations and shared randomness). We prove that the set of free operations forms a polytope, which in turn allows us to derive an efficient algorithm for deciding whether one resource can be converted to another. We moreover define two distinct monotones with simple closed-form expressions in the two-party binary-setting binary-outcome scenario, and use these to reveal various properties of the pre-order of resources, including a lower bound on the cardinality of any complete set of monotones. In particular, we show that the information contained in the degrees of violation of facet-defining Bell inequalities is not sufficient for quantifying nonclassicality, even though it is sufficient for witnessing nonclassicality. Finally, we show that the continuous set of convexly extremal quantumly realizable correlations are all at the top of the pre-order of quantumly realizable correlations. In addition to providing new insights on Bell nonclassicality, our work also sets the stage for quantifying nonclassicality in more general causal networks.

► BibTeX data

► References

[1] J. S. Bell, ``On the Einstein-Podolsky-Rosen paradox,'' Physics 1, 195 (1964).
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[2] J. S. Bell, ``On the Problem of Hidden Variables in Quantum Mechanics,'' Rev. Mod. Phys. 38, 447 (1966).
https:/​/​doi.org/​10.1103/​RevModPhys.38.447

[3] B. Hensen et al., ``Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,'' Nature 526, 682 EP (2015).
https:/​/​doi.org/​10.1038/​nature15759

[4] M. Giustina et al., ``Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons,'' Phys. Rev. Lett. 115, 250401 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250401

[5] L. Shalm et al, ``Strong Loophole-Free Test of Local Realism,'' Phys. Rev. Lett. 115, 250402 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.115.250402

[6] J. Barrett, L. Hardy, and A. Kent, ``No Signaling and Quantum Key Distribution,'' Phys. Rev. Lett. 95, 010503 (2005a).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.010503

[7] A. Acín, 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

[8] V. Scarani, N. Gisin, N. Brunner, L. Masanes, S. Pino, and A. Acín, ``Secrecy extraction from no-signaling correlations,'' Phys. Rev. A 74, 042339 (2006).
https:/​/​doi.org/​10.1103/​PhysRevA.74.042339

[9] A. Acín, 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).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.230501

[10] R. Colbeck and R. Renner, ``Free randomness can be amplified,'' Nat. Phys. 8, 450 EP (2012).
https:/​/​doi.org/​10.1038/​nphys2300

[11] S. Pironio, A. Acín, S. Massar, A. B. de la Giroday, D. N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T. A. Manning, and C. Monroe, ``Random numbers certified by Bell's theorem,'' Nature 464, 1021 EP (2010).
https:/​/​doi.org/​10.1038/​nature09008

[12] C. Dhara, G. Prettico, and A. Acín, ``Maximal quantum randomness in Bell tests,'' Phys. Rev. A 88, 052116 (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.88.052116

[13] U. Vazirani and T. Vidick, ``Fully Device-Independent Quantum Key Distribution,'' Phys. Rev. Lett. 113, 140501 (2014).
https:/​/​doi.org/​10.1103/​PhysRevLett.113.140501

[14] J. Kaniewski and S. Wehner, ``Device-independent two-party cryptography secure against sequential attacks,'' New J. Phys. 18, 055004 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​5/​055004

[15] R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, ``Operational Framework for Nonlocality,'' Phys. Rev. Lett. 109, 070401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.109.070401

[16] J. I. de Vicente, ``On nonlocality as a resource theory and nonlocality measures,'' J. Phys. A 47, 424017 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424017

[17] J. Geller and M. Piani, ``Quantifying non-classical and beyond-quantum correlations in the unified operator formalism,'' J. Phys. A 47, 424030 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424030

[18] R. Gallego and L. Aolita, ``Nonlocality free wirings and the distinguishability between Bell boxes,'' Phys. Rev. A 95 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.032118

[19] K. Horodecki, A. Grudka, P. Joshi, W. Kłobus, and J. Łodyga, ``Axiomatic approach to contextuality and nonlocality,'' Phys. Rev. A 92, 032104 (2015).
https:/​/​doi.org/​10.1103/​physreva.92.032104

[20] B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, ``Noncontextual wirings,'' Phys. Rev. Lett. 120, 130403 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.130403

[21] E. Kaur, M. M. Wilde, and A. Winter, ``Fundamental limits on key rates in device-independent quantum key distribution,'' https:/​/​arxiv.org/​abs/​1810.05627 arXiv:1810.05627 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​ab6eaa
arXiv:1810.05627

[22] S. G. A. Brito, B. Amaral, and R. Chaves, ``Quantifying Bell nonlocality with the trace distance,'' Phys. Rev. A 97, 022111 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.022111

[23] D. Schmid, D. Rosset, and F. Buscemi, ``Type-independent resource theory of local operations and shared randomness,'' https:/​/​arxiv.org/​abs/​1909.04065 arXiv:1909.04065 (2019a).
https:/​/​doi.org/​10.22331/​q-2020-04-30-262
arXiv:1909.04065

[24] D. Rosset, D. Schmid, and F. Buscemi, ``Characterizing nonclassicality of arbitrary distributed devices,'' arXiv:2004.09194 (2020a).
arXiv:1911.12462

[25] D. Schmid, T. C. Fraser, R. Kunjwal, A. B. Sainz, E. Wolfe, and R. W. Spekkens, ``Why standard entanglement theory is inappropriate for the study of Bell scenarios,'' arXiv:1911.12462 (2019b).
arXiv:2004.09194

[26] B. Coecke, T. Fritz, and R. W. Spekkens, ``A mathematical theory of resources,'' Info. & Comp. 250, 59 (2016).
https:/​/​doi.org/​10.1016/​j.ic.2016.02.008

[27] 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 (1969).
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[28] A. Shimony, ``Bell's Theorem,'' in The Stanford Encyclopedia of Philosophy (2017).
https:/​/​plato.stanford.edu/​archives/​fall2017/​entries/​bell-theorem/​

[29] B. d'Espagnat, ``The Quantum Theory and Reality,'' Scientific American 241, 158 (1979).
https:/​/​doi.org/​10.1038/​scientificamerican1179-158

[30] H. M. Wiseman, ``The two Bell's theorems of John Bell,'' J. Phys. A 47, 424001 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424001

[31] R. F. Werner, ``Comment on ‘What Bell did’,'' J. Phys. A 47, 424011 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424011

[32] V. Scarani, ``The Device-Independent Outlook on Quantum Physics,'' Acta Physica Slovaca 62, 347 (2012).
http:/​/​www.physics.sk/​aps/​pub.php?y=2012&pub=aps-12-04

[33] T. Maudlin, Quantum Non-Locality and Relativity : Metaphysical Intimations of Modern Physics (Blackwell Publishers, 2002).
https:/​/​doi.org/​10.1002/​9780470752166

[34] T. Norsen, ``Bell Locality and the Nonlocal Character of Nature,'' Found. Phys. Lett. 19, 633 (2006).
https:/​/​doi.org/​10.1007/​s10702-006-1055-9

[35] R. Chaves, R. Kueng, J. B. Brask, and D. Gross, ``Unifying Framework for Relaxations of the Causal Assumptions in Bell's Theorem,'' Phys. Rev. Lett. 114, 140403 (2015).
https:/​/​doi.org/​10.1103/​PhysRevLett.114.140403

[36] R. Chaves, D. Cavalcanti, and L. Aolita, ``Causal hierarchy of multipartite Bell nonlocality,'' Quantum 1, 23 (2017a).
https:/​/​doi.org/​10.22331/​q-2017-08-04-23

[37] T. Maudlin, ``Bell's Inequality, Information Transmission, and Prism Models,'' in Philosophy of Science Association, 1 (1992) pp. 404–417.
https:/​/​www.jstor.org/​stable/​192771

[38] B. F. Toner and D. Bacon, ``Communication Cost of Simulating Bell Correlations,'' Phys. Rev. Lett. 91, 187904 (2003).
https:/​/​doi.org/​10.1103/​PhysRevLett.91.187904

[39] G. Hooft, ``The Fate of the Quantum,'' arXiv:1308.1007 (2013), report numbers: ITP-UU-13/​22, SPIN-13/​15.
arXiv:1308.1007

[40] M. J. W. Hall, ``Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence,'' Phys. Rev. Lett. 105, 250404 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.105.250404

[41] J. Barrett and N. Gisin, ``How Much Measurement Independence Is Needed to Demonstrate Nonlocality?'' Phys. Rev. Lett. 106, 100406 (2011).
https:/​/​doi.org/​10.1103/​PhysRevLett.106.100406

[42] J. Pearl, Causality: Models, Reasoning, and Inference (Cambridge University Press, 2009).
https:/​/​doi.org/​10.1017/​CBO9780511803161

[43] C. J. Wood and R. W. Spekkens, ``The lesson of causal discovery algorithms for quantum correlations: causal explanations of Bell-inequality violations require fine-tuning,'' New J. Phys. 17, 033002 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​3/​033002

[44] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee, and R. W. Spekkens, ``Quantum Common Causes and Quantum Causal Models,'' Phys. Rev. X 7, 031021 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.031021

[45] J. Henson, R. Lal, and M. F. Pusey, ``Theory-independent limits on correlations from generalized Bayesian networks,'' New J. Phys. 16, 113043 (2014).
https:/​/​doi.org/​10.1088/​1367-2630/​16/​11/​113043

[46] T. Fritz, ``Beyond Bell's theorem: correlation scenarios,'' New J. Phys. 14, 103001 (2012).
https:/​/​doi.org/​10.1088/​1367-2630/​14/​10/​103001

[47] L. Hardy, ``Quantum Theory From Five Reasonable Axioms,'' quant-ph/​0101012 (2001).
arXiv:quant-ph/0101012

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

[49] P. Janotta and H. Hinrichsen, ``Generalized probability theories: what determines the structure of quantum theory?'' J. Phys. A 47, 323001 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​32/​323001

[50] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Probabilistic theories with purification,'' Phys. Rev. A 81, 062348 (2010).
https:/​/​doi.org/​10.1103/​PhysRevA.81.062348

[51] G. M. Ariano, Quantum Theory from First Principles: An Informational Approach (Cambridge University Press, 2019).
https:/​/​books.google.com/​books?id=ywizwwEACAAJ

[52] F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys 18, 063032 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​6/​063032

[53] J. Barrett, R. Lorenz, and O. Oreshkov, ``Quantum Causal Models,'' arXiv:1906.10726 (2019).
arXiv:1906.10726

[54] D. Schmid, H. Du, M. Mudassar, G. C. de Wit, D. Rosset, and M. J. Hoban, ``Postquantum common-cause channels: the resource theory of local operations and shared entanglement,'' arXiv:2004.06133 (2020).
arXiv:2004.06133

[55] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum Circuit Architecture,'' Phys. Rev. Lett. 101, 060401 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.060401

[56] G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Theoretical framework for quantum networks,'' Phys. Rev. A 80, 022339 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.022339

[57] S. Popescu and D. Rohrlich, ``Quantum nonlocality as an axiom,'' Found. Phys. 24, 379 (1994).
https:/​/​doi.org/​10.1007/​BF02058098

[58] J. Selby et al., ``Contextuality Quantified: A Resource Theory Encompassing Prepare-and-Measure Processes,'' Forthcoming.

[59] C. Branciard, D. Rosset, N. Gisin, and S. Pironio, ``Bilocal versus nonbilocal correlations in entanglement-swapping experiments,'' Phys. Rev. A 85, 032119 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.85.032119

[60] A. Acín, R. Augusiak, D. Cavalcanti, C. Hadley, J. K. Korbicz, M. Lewenstein, L. Masanes, and M. Piani, ``Unified Framework for Correlations in Terms of Local Quantum Observables,'' Phys. Rev. Lett. 104, 140404 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.140404

[61] S. W. Al-Safi and A. J. Short, ``Simulating all Nonsignaling Correlations via Classical or Quantum Theory with Negative Probabilities,'' Phys. Rev. Lett. 111, 170403 (2013).
https:/​/​doi.org/​10.1103/​PhysRevLett.111.170403

[62] J.-D. Bancal, S. Pironio, A. Acín, Y.-C. Liang, V. Scarani, and N. Gisin, ``Quantum non-locality based on finite-speed causal influences leads to superluminal signalling,'' Nat. Phys. 8, 867 (2012).
https:/​/​doi.org/​10.1038/​nphys2460

[63] J. S. Bell, ``La nouvelle cuisine,'' in Quantum Mechanics, High Energy Physics And Accelerators: Selected Papers Of John S Bell (With Commentary) (World Scientific, 1995) pp. 910–928.
https:/​/​doi.org/​10.1142/​9789812386540_0022

[64] O. Oreshkov, F. Costa, and Č. Brukner, ``Quantum correlations with no causal order,'' Nat. Comm. 3, 1092 EP (2012).
https:/​/​doi.org/​10.1038/​ncomms2076

[65] O. Oreshkov and C. Giarmatzi, ``Causal and causally separable processes,'' New J. Phys. 18, 093020 (2016).
https:/​/​doi.org/​10.1088/​1367-2630/​18/​9/​093020

[66] D. Rosset, J.-D. Bancal, and N. Gisin, ``Classifying 50 years of Bell inequalities,'' J. Phys. A 47, 424022 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424022

[67] A. Seress, Permutation Group Algorithms (Cambridge University Press, 2003).
https:/​/​doi.org/​10.1017/​CBO9780511546549

[68] S. Pironio, ``Lifting Bell inequalities,'' J. Math. Phys. 46, 062112 (2005).
https:/​/​doi.org/​10.1063/​1.1928727

[69] D. Rosset, Ämin Baumeler, J.-D. Bancal, N. Gisin, A. Martin, M.-O. Renou, and E. Wolfe, ``Algebraic and geometric properties of local transformations,'' arXiv:2004.09405 (2020b).
arXiv:2004.09405

[70] A. Fine, ``Hidden Variables, Joint Probability, and the Bell Inequalities,'' Phys. Rev. Lett. 48, 291 (1982).
https:/​/​doi.org/​10.1103/​PhysRevLett.48.291

[71] T. Gonda and R. W. Spekkens, ``Monotones in General Resource Theories,'' arXiv:1912.07085 (2019).
arXiv:1912.07085

[72] F. Buscemi, ``All Entangled Quantum States Are Nonlocal,'' Phys. Rev. Lett. 108, 200401 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.200401

[73] S. Beigi and A. Gohari, ``Monotone Measures for Non-Local Correlations,'' IEEE T. Inform. Theory 61, 5185 (2015).
https:/​/​doi.org/​10.1109/​tit.2015.2452253

[74] P. Bierhorst, ``Geometric decompositions of Bell polytopes with practical applications,'' J. Phys. A 49, 215301 (2016).
https:/​/​doi.org/​10.1088/​1751-8113/​49/​21/​215301

[75] D. Cavalcanti and P. Skrzypczyk, ``Quantitative relations between measurement incompatibility, quantum steering, and nonlocality,'' Phys. Rev. A 93, 052112 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.052112

[76] K. T. Goh, J. Kaniewski, E. Wolfe, T. Vértesi, X. Wu, Y. Cai, Y.-C. Liang, and V. Scarani, ``Geometry of the set of quantum correlations,'' Phys. Rev. A 97, 022104 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.022104

[77] M. W. Girard and G. Gour, ``Computable entanglement conversion witness that is better than the negativity,'' New J. Phys. 17, 093013 (2015).
https:/​/​doi.org/​10.1088/​1367-2630/​17/​9/​093013

[78] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality,'' Rev. Mod. Phys. 86, 419 (2014a).
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[79] J. Barrett, N. Linden, S. Massar, S. Pironio, S. Popescu, and D. Roberts, ``Nonlocal correlations as an information-theoretic resource,'' Phys. Rev. A 71, 022101 (2005b).
https:/​/​doi.org/​10.1103/​PhysRevA.71.022101

[80] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, ``Bell nonlocality,'' Rev. Mod. Phys. 86, 419 (2014b).
https:/​/​doi.org/​10.1103/​RevModPhys.86.419

[81] J. Barrett and S. Pironio, ``Popescu-Rohrlich Correlations as a Unit of Nonlocality,'' Phys. Rev. Lett. 95, 140401 (2005).
https:/​/​doi.org/​10.1103/​PhysRevLett.95.140401

[82] V. L. Popov, Algebraic Geometry IV (Springer-Verlag, 1994) Chap. 4: Quotients.
https:/​/​doi.org/​10.1007/​978-3-662-03073-8

[83] D. Collins and N. Gisin, ``A relevant two qubit Bell inequality inequivalent to the CHSH inequality,'' J. Phys. A 37, 1775 (2004).
https:/​/​doi.org/​10.1088/​0305-4470/​37/​5/​021

[84] T. H. Yang and M. Navascués, ``Robust self-testing of unknown quantum systems into any entangled two-qubit states,'' Phys. Rev. A 87, 050102(R) (2013).
https:/​/​doi.org/​10.1103/​PhysRevA.87.050102

[85] C. Bamps and S. Pironio, ``Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing,'' Phys. Rev. A 91, 052111 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.052111

[86] L. Masanes, ``Necessary and sufficient condition for quantum-generated correlations,'' quant-ph/​0309137 (2003).
arXiv:quant-ph/0309137

[87] J. Allcock, N. Brunner, M. Pawlowski, and V. Scarani, ``Recovering part of the boundary between quantum and nonquantum correlations from information causality,'' Phys. Rev. A 80, 040103(R) (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.80.040103

[88] A. Acín, S. Massar, and S. Pironio, ``Randomness versus Nonlocality and Entanglement,'' Phys. Rev. Lett. 108, 100402 (2012).
https:/​/​doi.org/​10.1103/​PhysRevLett.108.100402

[89] E. Wolfe and S. F. Yelin, ``Quantum bounds for inequalities involving marginal expectation values,'' Phys. Rev. A 86, 012123 (2012).
https:/​/​doi.org/​10.1103/​PhysRevA.86.012123

[90] M. A. Nielsen, ``Conditions for a class of entanglement transformations,'' Phys. Rev. Lett. 83, 436 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.436

[91] C. Bamps, S. Massar, and S. Pironio, ``Device-independent randomness generation with sublinear shared quantum resources,'' Quantum 2, 86 (2018).
https:/​/​doi.org/​10.22331/​q-2018-08-22-86

[92] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Y. Halpern, ``The resource theory of informational nonequilibrium in thermodynamics,'' Phys. Rep. 583, 1 (2015).
https:/​/​doi.org/​10.1016/​j.physrep.2015.04.003

[93] T. Fritz, ``Resource convertibility and ordered commutative monoids,'' Math. Struct. Comp. Sci. 27, 850–938 (2017).
https:/​/​doi.org/​10.1017/​S0960129515000444

[94] N. Brunner and P. Skrzypczyk, ``Nonlocality Distillation and Postquantum Theories with Trivial Communication Complexity,'' Phys. Rev. Lett. 102, 160403 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.160403

[95] B. Lang, T. Vértesi, and M. Navascués, ``Closed sets of correlations: answers from the zoo,'' J. Phys. A 47, 424029 (2014).
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424029

[96] Y. R. Sanders and G. Gour, ``Necessary conditions for entanglement catalysts,'' Phys. Rev. A 79, 054302 (2009).
https:/​/​doi.org/​10.1103/​PhysRevA.79.054302

[97] D. Jonathan and M. B. Plenio, ``Entanglement-Assisted Local Manipulation of Pure Quantum States,'' Phys. Rev. Lett. 83, 3566 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.3566

[98] W. van Dam and P. Hayden, ``Universal entanglement transformations without communication,'' Phys. Rev. A 67, 060302 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.060302

[99] B. Steudel and N. Ay, ``Information-Theoretic Inference of Common Ancestors,'' Entropy 17, 2304 (2015).
https:/​/​doi.org/​10.3390/​e17042304

[100] E. Wolfe, R. W. Spekkens, and T. Fritz, ``The Inflation Technique for Causal Inference with Latent Variables,'' J. Causal Inference 7 (2019).
https:/​/​doi.org/​10.1515/​jci-2017-0020

[101] N. Gisin, ``The Elegant Joint Quantum Measurement and some conjectures about N-locality in the Triangle and other Configurations,'' arXiv:1708.05556 (2017).
arXiv:1708.05556

[102] T. C. Fraser and E. Wolfe, ``Causal compatibility inequalities admitting quantum violations in the triangle structure,'' Phys. Rev. A 98, 022113 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.98.022113

[103] C. Branciard, N. Gisin, and S. Pironio, ``Characterizing the Nonlocal Correlations Created via Entanglement Swapping,'' Phys. Rev. Lett. 104, 170401 (2010).
https:/​/​doi.org/​10.1103/​PhysRevLett.104.170401

[104] F. Andreoli, G. Carvacho, L. Santodonato, R. Chaves, and F. Sciarrino, ``Maximal violation of $n$-locality inequalities in a star-shaped quantum network,'' New J. Phys. 19, 113020 (2017).
https:/​/​doi.org/​10.1088/​1367-2630/​aa8b9b

[105] A. Tavakoli, P. Skrzypczyk, D. Cavalcanti, and A. Acín, ``Nonlocal correlations in the star-network configuration,'' Phys. Rev. A 90, 062109 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.062109

[106] D. Rosset, C. Branciard, T. J. Barnea, G. Pütz, N. Brunner, and N. Gisin, ``Nonlinear Bell inequalities tailored for quantum networks,'' Phys. Rev. Lett. 116, 010403 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.010403

[107] A. Tavakoli, ``Bell-type inequalities for arbitrary noncyclic networks,'' Phys. Rev. A 93, 030101(R) (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.93.030101

[108] J. Pearl, ``On the Testability of Causal Models with Latent and Instrumental Variables,'' in Proc. 11th Conf. Uncertainty in Artificial Intelligence (1995) pp. 435–443.
http:/​/​singapore.cs.ucla.edu/​LECTURE/​lecture_sec1.htm

[109] B. Bonet, ``Instrumentality Tests Revisited,'' in Proc. 17th Conf. Uncertainty in Artificial Intelligence (2001) pp. 48–55.
https:/​/​pdfs.semanticscholar.org/​e397/​89b514f1a059e90fabada35aaaf7e6ef3bc9.pdf

[110] R. J. Evans, ``Graphical methods for inequality constraints in marginalized DAGs,'' in IEEE International Workshop on Machine Learning for Signal Processing (2012).
https:/​/​doi.org/​10.1109/​mlsp.2012.6349796

[111] R. Chaves, G. Carvacho, I. Agresti, V. D. Giulio, L. Aolita, S. Giacomini, and F. Sciarrino, ``Quantum violation of an instrumental test,'' Nat. Phy. 14, 291 (2017b).
https:/​/​doi.org/​10.1038/​s41567-017-0008-5

[112] T. Van Himbeeck, J. Bohr Brask, S. Pironio, R. Ramanathan, A. Belén Sainz, and E. Wolfe, ``Quantum violations in the Instrumental scenario and their relations to the Bell scenario,'' Quantum 3, 186 (2019).
https:/​/​doi.org/​10.22331/​q-2019-09-16-186

Cited by

[1] Yunchao Liu and Xiao Yuan, "Operational resource theory of quantum channels", Physical Review Research 2 1, 012035 (2020).

[2] David Schmid, Denis Rosset, and Francesco Buscemi, "The type-independent resource theory of local operations and shared randomness", arXiv:1909.04065.

[3] Elie Wolfe, Alejandro Pozas-Kerstjens, Matan Grinberg, Denis Rosset, Antonio Acín, and Miguel Navascues, "Quantum Inflation: A General Approach to Quantum Causal Compatibility", arXiv:1909.10519.

[4] Denis Rosset, David Schmid, and Francesco Buscemi, "Characterizing nonclassicality of arbitrary distributed devices", arXiv:1911.12462.

[5] Tomáš Gonda and Robert W. Spekkens, "Monotones in General Resource Theories", arXiv:1912.07085.

[6] David Schmid, Thomas C. Fraser, Ravi Kunjwal, Ana Belen Sainz, Elie Wolfe, and Robert W. Spekkens, "Why standard entanglement theory is inappropriate for the study of Bell scenarios", arXiv:2004.09194.

[7] David Schmid, Haoxing Du, Maryam Mudassar, Ghi Coulter-de Wit, Denis Rosset, and Matty J. Hoban, "Postquantum common-cause channels: the resource theory of local operations and shared entanglement", arXiv:2004.06133.

[8] Patricia Contreras-Tejada, Carlos Palazuelos, and Julio I. de Vicente, "Genuine multipartite nonlocality is intrinsic to quantum networks", arXiv:2004.01722.

[9] V. Vilasini and Roger Colbeck, "On the insufficiency of entropic inequalities for detecting non-classicality in the Bell causal structure", arXiv:1912.01031.

[10] John H. Selby and Ciarán M. Lee, "Compositional resource theories of coherence", arXiv:1911.04513.

[11] Shiv Akshar Yadavalli and Ravi Kunjwal, "Contextuality in entanglement-assisted one-shot classical communication", arXiv:2006.00469.

[12] Gilad Gour and Marco Tomamichel, "Optimal Extensions of Resource Measures and their Applications", arXiv:2006.12408.

[13] C. Jebarathinam and Debarshi Das, "Equivalence of the quantumness of sequential correlations and spatial correlations", arXiv:1912.01270.

[14] Debasis Mondal and Dagomir Kaszlikowski, "Self-testing of quantum states using symmetric local hidden state model", arXiv:1911.07517.

[15] M. M. Taddei, T. L. Silva, R. V. Nery, G. H. Aguilar, S. P. Walborn, and L. Aolita, "Exposure of subtle multipartite quantum nonlocality", arXiv:1910.12884.

[16] Andrés F. Ducuara, Patryk Lipka-Bartosik, and Paul Skrzypczyk, "Multi-object operational tasks for convex quantum resource theories", arXiv:2004.12898.

[17] Chung-Yun Hsieh, "Resource Preservability", arXiv:1910.02464.

[18] I. S. Eliëns, S. G. A. Brito, and R. Chaves, "Bell nonlocality using tensor networks and sparse recovery", Physical Review Research 2 2, 023198 (2020).

The above citations are from SAO/NASA ADS (last updated successfully 2020-07-14 04:22:15). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref's cited-by service no data on citing works was found (last attempt 2020-07-14 04:22:14).