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.
 J. S. Bell, ``On the Einstein-Podolsky-Rosen paradox,'' Physics 1, 195 (1964).
 M. Giustina et al., ``Significant-Loophole-Free Test of Bell's Theorem with Entangled Photons,'' Phys. Rev. Lett. 115, 250401 (2015).
 L. Shalm et al, ``Strong Loophole-Free Test of Local Realism,'' Phys. Rev. Lett. 115, 250402 (2015).
 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).
 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).
 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).
 U. Vazirani and T. Vidick, ``Fully Device-Independent Quantum Key Distribution,'' Phys. Rev. Lett. 113, 140501 (2014).
 J. Kaniewski and S. Wehner, ``Device-independent two-party cryptography secure against sequential attacks,'' New J. Phys. 18, 055004 (2016).
 R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, ``Operational Framework for Nonlocality,'' Phys. Rev. Lett. 109, 070401 (2012).
 J. I. de Vicente, ``On nonlocality as a resource theory and nonlocality measures,'' J. Phys. A 47, 424017 (2014).
 J. Geller and M. Piani, ``Quantifying non-classical and beyond-quantum correlations in the unified operator formalism,'' J. Phys. A 47, 424030 (2014).
 K. Horodecki, A. Grudka, P. Joshi, W. Kłobus, and J. Łodyga, ``Axiomatic approach to contextuality and nonlocality,'' Phys. Rev. A 92, 032104 (2015).
 B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, ``Noncontextual wirings,'' Phys. Rev. Lett. 120, 130403 (2018).
 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).
 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).
 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).
 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).
 A. Shimony, ``Bell's Theorem,'' in The Stanford Encyclopedia of Philosophy (2017).
 B. d'Espagnat, ``The Quantum Theory and Reality,'' Scientific American 241, 158 (1979).
 H. M. Wiseman, ``The two Bell's theorems of John Bell,'' J. Phys. A 47, 424001 (2014).
 R. F. Werner, ``Comment on ‘What Bell did’,'' J. Phys. A 47, 424011 (2014).
 V. Scarani, ``The Device-Independent Outlook on Quantum Physics,'' Acta Physica Slovaca 62, 347 (2012).
 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).
 M. J. W. Hall, ``Local Deterministic Model of Singlet State Correlations Based on Relaxing Measurement Independence,'' Phys. Rev. Lett. 105, 250404 (2010).
 J. Barrett and N. Gisin, ``How Much Measurement Independence Is Needed to Demonstrate Nonlocality?'' Phys. Rev. Lett. 106, 100406 (2011).
 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).
 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).
 J. Henson, R. Lal, and M. F. Pusey, ``Theory-independent limits on correlations from generalized Bayesian networks,'' New J. Phys. 16, 113043 (2014).
 T. Fritz, ``Beyond Bell's theorem: correlation scenarios,'' New J. Phys. 14, 103001 (2012).
 P. Janotta and H. Hinrichsen, ``Generalized probability theories: what determines the structure of quantum theory?'' J. Phys. A 47, 323001 (2014).
 F. Costa and S. Shrapnel, ``Quantum causal modelling,'' New J. Phys 18, 063032 (2016).
 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).
 G. Chiribella, G. M. D'Ariano, and P. Perinotti, ``Quantum Circuit Architecture,'' Phys. Rev. Lett. 101, 060401 (2008).
 J. Selby et al., ``Contextuality Quantified: A Resource Theory Encompassing Prepare-and-Measure Processes,'' Forthcoming.
 C. Branciard, D. Rosset, N. Gisin, and S. Pironio, ``Bilocal versus nonbilocal correlations in entanglement-swapping experiments,'' Phys. Rev. A 85, 032119 (2012).
 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).
 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).
 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).
 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.
 O. Oreshkov and C. Giarmatzi, ``Causal and causally separable processes,'' New J. Phys. 18, 093020 (2016).
 D. Rosset, J.-D. Bancal, and N. Gisin, ``Classifying 50 years of Bell inequalities,'' J. Phys. A 47, 424022 (2014).
 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).
 F. Buscemi, ``All Entangled Quantum States Are Nonlocal,'' Phys. Rev. Lett. 108, 200401 (2012).
 P. Bierhorst, ``Geometric decompositions of Bell polytopes with practical applications,'' J. Phys. A 49, 215301 (2016).
 D. Cavalcanti and P. Skrzypczyk, ``Quantitative relations between measurement incompatibility, quantum steering, and nonlocality,'' Phys. Rev. A 93, 052112 (2016).
 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).
 M. W. Girard and G. Gour, ``Computable entanglement conversion witness that is better than the negativity,'' New J. Phys. 17, 093013 (2015).
 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).
 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).
 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).
 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).
 A. Acín, S. Massar, and S. Pironio, ``Randomness versus Nonlocality and Entanglement,'' Phys. Rev. Lett. 108, 100402 (2012).
 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).
 N. Brunner and P. Skrzypczyk, ``Nonlocality Distillation and Postquantum Theories with Trivial Communication Complexity,'' Phys. Rev. Lett. 102, 160403 (2009).
 B. Lang, T. Vértesi, and M. Navascués, ``Closed sets of correlations: answers from the zoo,'' J. Phys. A 47, 424029 (2014).
 T. C. Fraser and E. Wolfe, ``Causal compatibility inequalities admitting quantum violations in the triangle structure,'' Phys. Rev. A 98, 022113 (2018).
 C. Branciard, N. Gisin, and S. Pironio, ``Characterizing the Nonlocal Correlations Created via Entanglement Swapping,'' Phys. Rev. Lett. 104, 170401 (2010).
 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).
 A. Tavakoli, P. Skrzypczyk, D. Cavalcanti, and A. Acín, ``Nonlocal correlations in the star-network configuration,'' Phys. Rev. A 90, 062109 (2014).
 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).
 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.
 B. Bonet, ``Instrumentality Tests Revisited,'' in Proc. 17th Conf. Uncertainty in Artificial Intelligence (2001) pp. 48–55.
 R. J. Evans, ``Graphical methods for inequality constraints in marginalized DAGs,'' in IEEE International Workshop on Machine Learning for Signal Processing (2012).
 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).
 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).
 Andrés F. Ducuara and Paul Skrzypczyk, "Operational Interpretation of Weight-Based Resource Quantifiers in Convex Quantum Resource Theories", Physical Review Letters 125 11, 110401 (2020).
 John H. Selby and Ciarán M. Lee, "Compositional resource theories of coherence", Quantum 4, 319 (2020).
 V. Vilasini and Roger Colbeck, "Limitations of entropic inequalities for detecting nonclassicality in the postselected Bell causal structure", Physical Review Research 2 3, 033096 (2020).
 Yunchao Liu and Xiao Yuan, "Operational resource theory of quantum channels", Physical Review Research 2 1, 012035 (2020).
 David Schmid, Denis Rosset, and Francesco Buscemi, "The type-independent resource theory of local operations and shared randomness", arXiv:1909.04065.
 Denis Rosset, David Schmid, and Francesco Buscemi, "Characterizing nonclassicality of arbitrary distributed devices", arXiv:1911.12462.
 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.
 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.
 Tomáš Gonda and Robert W. Spekkens, "Monotones in General Resource Theories", arXiv:1912.07085.
 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.
 Gilad Gour and Marco Tomamichel, "Optimal Extensions of Resource Measures and their Applications", arXiv:2006.12408.
 Patricia Contreras-Tejada, Carlos Palazuelos, and Julio I. de Vicente, "Genuine multipartite nonlocality is intrinsic to quantum networks", arXiv:2004.01722.
 Andrés F. Ducuara, Patryk Lipka-Bartosik, and Paul Skrzypczyk, "Multi-object operational tasks for convex quantum resource theories", arXiv:2004.12898.
 Debasis Mondal and Dagomir Kaszlikowski, "Self-testing of quantum states using symmetric local hidden state model", arXiv:1911.07517.
 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).
 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.
 Chung-Yun Hsieh, "Resource Preservability", arXiv:1910.02464.
 C. Jebarathinam and Debarshi Das, "Equivalence of the quantumness of sequential correlations and spatial correlations", arXiv:1912.01270.
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