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arXiv:2201.02330 · Europhysics Letters (EPL)

Monogamy relations of entropic non-contextual inequalities and their experimental demonstration

Dileep Singh, Jaskaran Singh, Kavita Dorai, Arvind

Published May 26, 2023• 2 citations
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Abstract

Abstract We analyze monogamous relationships of abritrary entropic non-contextuality (ENC) inequalities via a graph theoretic approach. While ENC inequalities are important in quantum information theory, a theoretical and experimental understanding of their monogamous nature is still elusive. We analyze conditions for ENC inequalities to exhibit a monogamous relationship and derive the same explicitly for an entropic version of the Bell-CHSH scenario. Next, we consider two different sets of three qubit states and experimentally show that they exhibit monogamy of entropic Bell-CHSH inequality. The monogamous relationship of these inequalities is theoretically exhibited via an insightful proof and is experimentally demonstrated on an NMR quantum information processor. We also develop a novel and easy-to-implement experimental method to evaluate entropies on an NMR quantum information processor using only the information about expectation values of the observables.

DOI
10.1209/0295-5075/acd954

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•Jun 13, 2016•2 citations
Abstract We analyze monogamous relationships of abritrary entropic non-contextuality (ENC) inequalities via a graph theoretic approach. While ENC inequalities are important in quantum information theory, a theoretical and experimental understanding of their monogamous nature is still elusive. We analyze conditions for ENC inequalities to exhibit a monogamous relationship and derive the same explicitly for an entropic version of the Bell-CHSH scenario. Next, we consider two different sets of three qubit states and experimentally show that they exhibit monogamy of entropic Bell-CHSH inequality. The monogamous relationship of these inequalities is theoretically exhibited via an insightful proof and is experimentally demonstrated on an NMR quantum information processor. We also develop a novel and easy-to-implement experimental method to evaluate entropies on an NMR quantum information processor using only the information about expectation values of the observables.
The quantum no-disturbance principle imposes a fundamental monogamy relation on quantum contextuality and quantum nonlocality. We experimentally demonstrated the simulation of this monogamy relation on a ququart-qubit system using three NMR qubits. We constructed experimental tests of the violation of the Klyachko-Can-Binicioglu-Shumovski (KCBS) inequality and the Clauser-Horne-Shimony-Holt (CHSH) inequality. We measured the average CHSH and KCBS values at the single point where the quantum boundary meets the no-disturbance boundary, at which point the tradeoff inequality becomes an equality. The tradeoff relation is always satisfied, implying that the violation of either the KCBS or the CHSH inequality automatically forbids the violation of the other.
Tests for quantum contextuality in terms of qqq-entropies
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.
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.
The information-theoretic approach to Bell's theorem is developed with use of the conditional qqq-entropies. The qqq-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional qqq-entropies for all q≥1q\geq1q≥1. Quantum violations of the new inequalities are exemplified within the Clauser--Horne--Shimony--Holt (CHSH) and Klyachko--Can--Binicio\v{g}lu--Shumovsky (KCBS) scenarios. An extension to the case of nnn-cycle scenario is briefly mentioned. The new inequalities with conditional qqq-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The qqq-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the qqq-entropic formulation.
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.