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arXiv:1201.5836 · Physical Review Letters

Generalized Monogamy of Contextual Inequalities from the No-Disturbance Principle

Ravishankar Ramanathan, Akihito Soeda, Paweł Kurzyński, Dagomir Kaszlikowski

Published Aug 1, 2012• 83 citations
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Abstract

In this Letter, we demonstrate that the property of monogamy of Bell violations seen for no-signaling correlations in composite systems can be generalized to the monogamy of contextuality in single systems obeying the Gleason property of no disturbance. We show how one can construct monogamies for contextual inequalities by using the graph-theoretic technique of vertex decomposition of a graph representing a set of measurements into subgraphs of suitable independence numbers that themselves admit a joint probability distribution. After establishing that all the subgraphs that are chordal graphs admit a joint probability distribution, we formulate a precise graph-theoretic condition that gives rise to the monogamy of contextuality. We also show how such monogamies arise within quantum theory for a single four-dimensional system and interpret violation of these relations in terms of a violation of causality. These monogamies can be tested with current experimental techniques.

DOI
10.1103/physrevlett.109.050404

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In this Letter, we demonstrate that the property of monogamy of Bell violations seen for no-signaling correlations in composite systems can be generalized to the monogamy of contextuality in single systems obeying the Gleason property of no disturbance. We show how one can construct monogamies for contextual inequalities by using the graph-theoretic technique of vertex decomposition of a graph representing a set of measurements into subgraphs of suitable independence numbers that themselves admit a joint probability distribution. After establishing that all the subgraphs that are chordal graphs admit a joint probability distribution, we formulate a precise graph-theoretic condition that gives rise to the monogamy of contextuality. We also show how such monogamies arise within quantum theory for a single four-dimensional system and interpret violation of these relations in terms of a violation of causality. These monogamies can be tested with current experimental techniques.
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.
A unified view of the phenomenon of monogamy exhibited by Bell inequalities and noncontextuality inequalities arising from the no-signaling and no-disturbance principles is presented using the graph-theoretic method introduced by Ramanathan et al. [Phys. Rev. Lett. 109, 050404 (2012)]. We propose a hitherto unexplored tradeoff relation, 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 I3322{\mathcal{I}}_{3322}I3322​ inequality and reveals a resource trade-off between bipartite correlations and the local purity of a single system. In the derivation of no-signaling monogamies, we uncover a unique feature, namely, that two-party Bell expressions that are trivially classically saturated can become nontrivial upon the addition of an expression involving a third party with a single measurement input.
The celebrated Bell-Kochen-Specker no-go theorem asserts that quantum mechanics does not present the property of realism; the essence of the theorem is the lack of a joint probability distribution for some experiment settings. We exploit the information theoretic form of the theorem using information measure instead of probabilistic measure and indicate that quantum mechanics does not present such kind of entropic realism either. The entropic form of Gleason's no-disturbance principle is developed and characterized by the intersection of several entropic cones. Entropic contextuality and entropic nonlocality are investigated in depth in this framework as well. We show how one can construct monogamy relations using entropic cone and basic Shannon-type inequalities. The general criterion for several entropic tests to be monogamous is also developed; using the criterion, we demonstrate that entropic nonlocal correlations, entropic contextuality tests, and entropic nonlocality and entropic contextuality are monogamous. Finally, we analyze the entropic monogamy relations for the multiparty and many-test case, which may play a crucial role in quantum network communication.
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.
We show that the perfect commutation graph is the sufficient tight condition for admitting the noncontextual description of each observable set satisfying it in the yes-no question scenario. With this condition, we propose a method for proving the monogamy relation between two informationtheoretic contextuality inequalities by decomposing the total commutation graph into perfect subgraphs. The results offer a powerful tool to investigate the contextuality and to understand quantum information theory. This theoretical work can be experimentally verified in current laboratorial technology.
Monogamy is a fundamental property of Bell nonlocality and contextuality. In this article, we study the nnn-cycle noncontextual inequalities and generalized Clauser-Horne-Shimony-Holt (CHSH) inequalities in detail and find sufficient conditions for those inequalities to hold. According to those conditions, we provide several kinds of tradeoff relations: monogamy of generalized Bell inequalities in a nonsignaling framework, monogamy of cycle-type noncontextual inequalities, and monogamy between Bell inequalities and noncontextual inequalities in a general no-disturbance framework. Finally, some generic tradeoff relations of generalized CHSH inequalities for nnn-party physical systems, which are beyond the one-to-many scenario, are discussed.