Quantum value for a family of $I_{3322}$-like Bell functionals

Abstract

We introduce a three-parameter family of Bell functionals that extends those studied in reference [Phys. Rev. Research 2, 033420 (2020)] by including a marginal contribution. An analysis of their quantum value naturally splits the family into two branches, and for the first of them we show that this value is given by a simple function of the parameters defining the functionals. In this case we completely characterise the realisations attaining the optimal value and show that these functionals can be used to self-test any partially entangled state of two qubits. The optimal measurements, however, are not unique and form a one-parameter family of qubit measurements. The second branch, which includes the well-known $I_{3322}$ functional, is studied numerically. We identify the region in the parameter space where the quantum value can be attained, with two-dimensional systems and characterise the state and measurements attaining this value. Finally, we show that the set of realisations introduced in reference [Phys. Rev. A 82, 022116 (2010)] to obtain the maximal violation of the $I_{3322}$ inequality succeeds in approaching the optimal value for a large subset of the functionals in this branch. In these cases we analyse and discuss the main features of the optimal realisations.

References in corpus (28)

• Bell nonlocality
• Advances in Quantum Cryptography
• Device-independent security of quantum cryptography against collective attacks
• Random Numbers Certified by Bell's Theorem
• No Signalling and Quantum Key Distribution
• A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations
• From Bell's Theorem to Secure Quantum Key Distribution
• Bounding the set of quantum correlations
• Self-testing of quantum systems: a review
• A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality
• Randomness vs Non Locality and Entanglement
• Certified randomness in quantum physics
• Sum-of-squares decompositions for a family of CHSH-like inequalities and their application to self-testing
• Symmetries of the Bell correlation inequalities
• Device-independent entanglement quantification and related applications
• Popescu-Rohrlich correlations as a unit of nonlocality
• Maximal violation of the I3322 inequality using infinite dimensional quantum systems
• Geometry of the set of quantum correlations
• Analytic and nearly optimal self-testing bounds for the Clauser-Horne-Shimony-Holt and Mermin inequalities
• Interconversion of Nonlocal Correlations
• Randomness in Quantum Mechanics: Philosophy, Physics and Technology
• Bell inequalities tailored to maximally entangled states
• Self-testing protocols based on the chained Bell inequalities
• Scalable Bell inequalities for qubit graph states and robust self-testing
• Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments
• Device-independent characterization of quantum instruments
• Self-testing properties of Gisin's elegant Bell inequality
• A weak form of self-testing

Cited by in corpus (3)

• Quantum correlations on the no-signaling boundary: self-testing and more
• On the Fidelity Robustness of CHSH--Bell Inequality via Filtered Random States
• Trading symmetry for Hilbert-space dimension in Bell-inequality violation