Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes
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 connecte...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2020-06-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2020-06-08-280/pdf/ |
| _version_ | 1797448348828958720 |
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| author | Elie Wolfe David Schmid Ana Belén Sainz Ravi Kunjwal Robert W. Spekkens |
| author_facet | Elie Wolfe David Schmid Ana Belén Sainz Ravi Kunjwal Robert W. Spekkens |
| author_sort | Elie Wolfe |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T14:09:02Z |
| format | Article |
| id | doaj.art-ae4082b26f824ec39f6fd76a3a05e549 |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-09T14:09:02Z |
| publishDate | 2020-06-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-ae4082b26f824ec39f6fd76a3a05e5492023-11-29T15:30:43ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2020-06-01428010.22331/q-2020-06-08-28010.22331/q-2020-06-08-280Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause BoxesElie WolfeDavid SchmidAna Belén SainzRavi KunjwalRobert W. SpekkensWe 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.https://quantum-journal.org/papers/q-2020-06-08-280/pdf/ |
| spellingShingle | Elie Wolfe David Schmid Ana Belén Sainz Ravi Kunjwal Robert W. Spekkens Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes Quantum |
| title | Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes |
| title_full | Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes |
| title_fullStr | Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes |
| title_full_unstemmed | Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes |
| title_short | Quantifying Bell: the Resource Theory of Nonclassicality of Common-Cause Boxes |
| title_sort | quantifying bell the resource theory of nonclassicality of common cause boxes |
| url | https://quantum-journal.org/papers/q-2020-06-08-280/pdf/ |
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