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arXiv:1707.03250

The Exclusivity Principle Determines the Correlation Monogamy

Zhian Jia, Gao-Di Cai, Yu-Chun Wu, Guang‐Can Guo, Adán Cabello

Published Jul 11, 2017• 6 citations
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Abstract

Adopting the graph-theoretic approach to the correlation experiments, we analyze the origin of monogamy and prove that it can be recognized as a consequence of the exclusivity principle(EP). We provide an operational criterion for monogamy: if the fractional packing number of the graph corresponding to the union of event sets of several physical experiments does not exceed the sum of independence numbers of each individual experiment graph, then these experiments are monogamous. As applications of this observation, several examples are provided, including the monogamy for experiments of Clauser-Horne-Shimony-Holt (CHSH) type, Klyachko-Can-Binicioğlu-Shumovsky (KCBS) type, and for the first time, we give some monogamy relations of Svetlichny's genuine nonlocality. We also give the necessary and sufficient conditions for several experiments to be monogamous: several experiments are monogamous if and only if the Lovász number of the union exclusive graph is less than or equal to the sum of independence numbers of each exclusive graph.

DOI
10.48550/arxiv.1707.03250

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Adopting the graph-theoretic approach to the correlation experiments, we analyze the origin of monogamy and prove that it can be recognized as a consequence of the exclusivity principle(EP). We provide an operational criterion for monogamy: if the fractional packing number of the graph corresponding to the union of event sets of several physical experiments does not exceed the sum of independence numbers of each individual experiment graph, then these experiments are monogamous. As applications of this observation, several examples are provided, including the monogamy for experiments of Clauser-Horne-Shimony-Holt (CHSH) type, Klyachko-Can-Binicioğlu-Shumovsky (KCBS) type, and for the first time, we give some monogamy relations of Svetlichny's genuine nonlocality. We also give the necessary and sufficient conditions for several experiments to be monogamous: several experiments are monogamous if and only if the Lovász number of the union exclusive graph is less than or equal to the sum of independence numbers of each exclusive graph.
Correlations in Bell and noncontextuality inequalities can be expressed as a positive linear combination of probabilities of events. Exclusive events can be represented as adjacent vertices of a graph, so correlations can be associated to a subgraph. We show that the maximum value of the correlations for classical, quantum, and more general theories is the independence number, the Lovász number, and the fractional packing number of this subgraph, respectively. We also show that, for any graph, there is always a correlation experiment such that the set of quantum probabilities is exactly the Grötschel-Lovász-Schrijver theta body. This identifies these combinatorial notions as fundamental physical objects and provides a method for singling out experiments with quantum correlations on demand.
We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.
This work explores the implications of the exclusivity principle (EP) in the context of quantum and postquantum correlations. We first establish a key technical result demonstrating that given the set of correlations for a complementary experiment, the EP restricts the maximum set of correlations for the original experiment to the antiblocking set. Based on it, we can prove our central result: if all quantum behaviors are accessible in Nature, the EP guarantees that no postquantum behaviors can be realized. This can be seen as a generalization of the result of B. Amaral et al. [Phys. Rev. A 89, 030101(R) (2014)], to a wider range of scenarios. It also provides novel insights into the structure of quantum correlations and their limitations.
We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number.The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated to G, and exclusive events are adjacent.Mathematically, the graph H(G) corresponds to the orbit of G under local complementation.Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics.In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G).The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number.Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum non-locality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.
A unified view on the phenomenon of monogamy exhibited by Bell inequalities and non-contextuality inequalities arising from the no-signaling and no-disturbance principles is developed using the graph-theoretic method introduced in \textit{Phys. Rev. Lett. 109, 050404 (2012)}. The sufficient condition for the existence of a monogamy between any two Bell or non-contextuality inequalities in terms of a decomposition of the commutation graph representing all the observables in the setup into induced chordal graphs, is studied. The condition is used to show that any two cycle inequalities, whether studied as non-contextuality or Bell inequalities and for any length of the cycle, are monogamous under certain conditions with each other. As a main application, a novel type of trade-off is then derived, namely Bell inequalities that do not exhibit monogamy features of their own can be activated to be monogamous by the addition of a local contextuality term. This is illustrated by means of the well-known I3322I_{3322}I3322​ inequality. The non-necessity of the condition for the existence of a monogamy relation is shown by means of an explicit counter-example. In the derivation of novel no-signaling monogamies, we uncover a new feature, namely that two-party Bell expressions that are trivially classically saturated can become non-trivial upon the addition of an expression involving a third party with a single measurement input. This implies that the study of Bell inequalities (and the corresponding classical polytopes) with single inputs for some of the parties becomes interesting.
We show that the no-disturbance principle imposes a tradeoff between locally contextual correlations violating the Klyachko-Can-Biniciogˇlu-Shumovski inequality and spatially separated correlations violating the Clauser-Horne-Shimony-Holt inequality. The violation of one inequality forbids the violation of the other. We also obtain the corresponding monogamy relation imposed by quantum theory for a qutrit-qubit system. Our results show the existence of fundamental monogamy relations between contextuality and nonlocality that suggest that entanglement might be a particular form of a more fundamental resource.