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arXiv:2105.08561 · Physical Review A

Quantum sets of the multicolored-graph approach to contextuality

Lina Vandré, Marcelo Terra Cunha

Published Dec 16, 2022• 5 citations
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Abstract

The Clauser-Horne-Shimony-Holt (CHSH) inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the Cabello-Severini-Winter (CSW) approach, one may think that the theta body of the CHSH graph $\mathrm{TH}({G}_{\text{CHSH}})$ is equal to the quantum set of the CHSH Bell inequality ${\mathcal{Q}}_{\text{CHSH}}$, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while the CSW approach only demands the exclusivity structure of a noncontextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like the CHSH one, the colored-graph approach was introduced. Does it make any difference to think about the CHSH scenario as a Bell one or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph $\mathcal{G}$ and the NC CHSH inequality by a simple graph $G$, which is the shadow of the colored graph $\mathcal{G}$. In general, we have that the theta body of the colored graph ${\ensuremath{\theta}}_{c}(\mathcal{G})$ is a subset of the theta body of its shadow graph $\mathrm{TH}(G)$ in the same way that the Lov\'asz number, which corresponds to the quantum bound, of the simple graph $\ensuremath{\vartheta}(G)$ is greater than or equal to the Lov\'asz number of the colored graph ${\ensuremath{\vartheta}}_{c}(\mathcal{G})$. In the case of the CHSH inequality, we have that $\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})$. Does this accident also hold for the corresponding quantum sets? Is it true that $\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})$, which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negative. We show that ${\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)$ and therefore that there are quantum correlations which cannot be obtained under Bell restrictions.

DOI
10.1103/physreva.106.062210

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The Clauser-Horne-Shimony-Holt (CHSH) inequalities are the most famous examples of Bell inequalities. Cabello, Severini, and Winter came up with a graph approach to noncontextuality inequalities, which connects some graph-theoretic concepts to quantum and classical correlations. For example, the theta body of the exclusivity graph can be associated with the set of correlations achieved by quantum theory. Following the Cabello-Severini-Winter (CSW) approach, one may think that the theta body of the CHSH graph TH(GCHSH)\mathrm{TH}({G}_{\text{CHSH}})TH(GCHSH​) is equal to the quantum set of the CHSH Bell inequality QCHSH{\mathcal{Q}}_{\text{CHSH}}QCHSH​, but is this really true? All assumptions about the CHSH inequalities come from Bell scenarios, while the CSW approach only demands the exclusivity structure of a noncontextuality (NC) scenario. To deal with the extra structure related to the presence of different players in a Bell scenario like the CHSH one, the colored-graph approach was introduced. Does it make any difference to think about the CHSH scenario as a Bell one or a more general NC scenario? The Bell CHSH inequality is represented by a bicolored graph G\mathcal{G}G and the NC CHSH inequality by a simple graph GGG, which is the shadow of the colored graph G\mathcal{G}G. In general, we have that the theta body of the colored graph \ensuremathθc(G){\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremathθc​(G) is a subset of the theta body of its shadow graph TH(G)\mathrm{TH}(G)TH(G) in the same way that the Lov'asz number, which corresponds to the quantum bound, of the simple graph \ensuremathϑ(G)\ensuremath{\vartheta}(G)\ensuremathϑ(G) is greater than or equal to the Lov'asz number of the colored graph \ensuremathϑc(G){\ensuremath{\vartheta}}_{c}(\mathcal{G})\ensuremathϑc​(G). In the case of the CHSH inequality, we have that \ensuremathϑ(G)=\ensuremathϑc(G)\ensuremath{\vartheta}(G)={\ensuremath{\vartheta}}_{c}(\mathcal{G})\ensuremathϑ(G)=\ensuremathϑc​(G). Does this accident also hold for the corresponding quantum sets? Is it true that TH(G)=\ensuremathθc(G)\mathrm{TH}(G)={\ensuremath{\theta}}_{c}(\mathcal{G})TH(G)=\ensuremathθc​(G), which would mean that every correlation reached by quantum theory applied to the CHSH NC scenario could also be obtained at the in principle more restrictive CHSH Bell scenario? In this paper our answer to such a question is negative. We show that \ensuremathθc(G)\ensuremath⊈TH(G){\ensuremath{\theta}}_{c}(\mathcal{G})\ensuremath{\nsubseteq}\mathrm{TH}(G)\ensuremathθc​(G)\ensuremath⊈TH(G) and therefore that there are quantum correlations which cannot be obtained under Bell restrictions.
It is well known that certain measurement scenarios behave in a way which can not be explained by classical theories but by quantum theories. This behaviours are usually studied by Bell or non-contextuality (NC) inequalities. Knowing the maximal classical and quantum bounds of this inequalities is interesting, but tells us little about the quantum set Q of all quantum behaviours P. Despite having a constructive description of the quantum set associated to a given inequality, the freedom to choose quantum dimension, quantum states, and quantum measurements makes the shape of such convex bodies quite elusive. It is well known that a NC-inequality can be associated to a graph and the quantum set is a combinatorial object. Extra conditions, like Bell concept of parts, may restrict the behaviours achievable within quantum theory for a given scenario. For the simplest case, CHSH inequality, the NC and Bell versions share the same quantum bound. By exploring a family of intermediate coloured graphs, we show that CHSH NC quantum set is strictly larger than CHSH Bell quantum set. In graph language, the coloured-THETA body of the coloured CHSH graph is a proper subset of the THETA body of its shadow graph.
Non-contextuality (NC) and Bell inequalities can be expressed as bounds ΩΩΩ for positive linear combinations SSS of probabilities of events, S≤ΩS \leq ΩS≤Ω. Exclusive events in SSS can be represented as adjacent vertices of a graph called the exclusivity graph of SSS. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Grötschel-Lovász-Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing SSS over the theta body and calculating the Lovász number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given NNN-partite Bell inequality, an edge-coloured multigraph composed of NNN single-colour graphs can be used to encode the relationships of exclusivity between each party's parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lovász number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for I3I_3I3​, I3322I_{3322}I3322​ and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles.
Non-contextuality (NC) and Bell inequalities can be expressed as bounds Ω for positive linear combinations S of probabilities of events, S≤Ω. Exclusive events in S can be represented as adjacent vertices of a graph called the exclusivity graph of S. In the case that events correspond to the outcomes of quantum projective measurements, quantum probabilities are intimately related to the Grötschel-Lovász-Schrijver theta body of the exclusivity graph. Then, one can easily compute an upper bound to the maximum quantum violation of any NC or Bell inequality by optimizing S over the theta body and calculating the Lovász number of the corresponding exclusivity graph. In some cases, this upper bound is tight and gives the exact maximum quantum violation. However, in general, this is not the case. The reason is that the exclusivity graph does not distinguish among the different ways exclusivity can occur in Bell-inequality (and similar) scenarios. An interesting question is whether there is a graph-theoretical concept which accounts for this problem. Here we show that, for any given N-partite Bell inequality, an edge-coloured multigraph composed of N single-colour graphs can be used to encode the relationships of exclusivity between each party's parts of the events. Then, the maximum quantum violation of the Bell inequality is exactly given by a refinement of the Lovász number that applies to these edge-coloured multigraphs. We show how to calculate upper bounds for this number using a hierarchy of semi-definite programs and calculate upper bounds for I3, I3322 and the three bipartite Bell inequalities whose exclusivity graph is a pentagon. The multigraph-theoretical approach introduced here may remove some obstacles in the program of explaining quantum correlations from first principles.
Bell nonlocality and quantum contextuality are two important resources which, however, behave very differently in many tasks where the quantum correlation has an edge over its classical counterpart. In this work, we directly compare these two behaviors, which via the Cabello-Severini-Winter graph-theoretical approach, can be associated with a same exclusivity graph. In particular, we consider the exclusivity graph that leads to I3322{I}_{3322}I3322​-type inequalities in the Bell and the contextual scenarios, respectively. We find that maximal violation of Bell inequality is around 6.251, and maximal violation of the noncontextuality inequality is around 6.588 for a five-dimensional system and 6.571 for a four-dimensional system, respectively. The results predict a gap of $\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}0.32$ between quantum contextuality and Bell nonlocality. We then present an experimental observation of the gap by employing both the maximally entangled photon pairs from the spontaneous parametric down-conversion process and the single photons encoded into qudits from an intrinsic defect in gallium nitride. Our results will further deepen the understanding of different types of quantum correlations.
Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of Abramsky-Brandenburger) both provide classes of Bell and contextuality inequalities for very general experimental scenarios using vastly different mathematical techniques. We review both approaches, carefully detail the links between them, and give simple, graph-theoretic methods for finding inequality-free proofs of nonlocality and contextuality and for finding states exhibiting strong nonlocality and/or contextuality. Finally, we apply these methods to concrete examples in stabilizer quantum mechanics relevant to understanding contextuality as a resource in quantum computation.
So far, most of the literature on (quantum) contextuality and the Kochen–\nSpecker theorem seems either to concern particular examples of contextuality, or be\nconsidered as quantum logic. Here, we develop a general formalism for contextuality\nscenarios based on the combinatorics of hypergraphs, which significantly refines a similar\nrecent approach by Cabello, Severini and Winter (CSW). In contrast to CSW, we\nexplicitly include the normalization of probabilities, which gives us a much finer control\nover the various sets of probabilistic models like classical, quantum and generalized\nprobabilistic. In particular, our framework specializes to (quantum) nonlocality in the\ncase of Bell scenarios, which arise very naturally from a certain product of contextuality\nscenarios due to Foulis and Randall. In the spirit of CSW, we find close relationships to\nseveral graph invariants. The recently proposed Local Orthogonality principle turns out\nto be a special case of a general principle for contextuality scenarios related to the Shannon\ncapacity of graphs. Our results imply that it is strictly dominated by a low level of\nthe Navascués–Pironio–Acín hierarchy of semidefinite programs, which we also apply\nto contextuality scenarios.\nWe derive a wealth of results in our framework, many of these relating to quantum\nand supraquantum contextuality and nonlocality, and state numerous open problems.\nFor example, we show that the set of quantum models on a con