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arXiv:2203.01837 · Physical Review A

Quantum value for a family of I3322 -like Bell functionals

N. Gigena, Jędrzej Kaniewski

Published Jul 1, 2022• 3 citations
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Abstract

We introduce a three-parameter family of Bell functionals that extends those studied previously [Kaniewski, Phys. Rev. Res. 2, 033420 (2020)] by including a marginal contribution. An analysis of their largest value achievable by quantum realizations 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 characterize the realizations 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 optimal value can be attained with two-dimensional quantum systems and characterize the state and measurements attaining this value. Finally, we show that the set of realizations introduced by P\'al and V\'ertesi [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 analyze and discuss the main features of the optimal realizations.

DOI
10.1103/physreva.106.012401

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We introduce a three-parameter family of Bell functionals that extends those studied previously [Kaniewski, Phys. Rev. Res. 2, 033420 (2020)] by including a marginal contribution. An analysis of their largest value achievable by quantum realizations 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 characterize the realizations 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 I3322{I}_{3322}I3322​ functional, is studied numerically. We identify the region in the parameter space where the optimal value can be attained with two-dimensional quantum systems and characterize the state and measurements attaining this value. Finally, we show that the set of realizations introduced by P'al and V'ertesi [Phys. Rev. A 82, 022116 (2010)] to obtain the maximal violation of the I3322{I}_{3322}I3322​ inequality succeeds in approaching the optimal value for a large subset of the functionals in this branch. In these cases we analyze and discuss the main features of the optimal realizations.
The I3322I_{3322}I3322​ Bell inequality requires infinite dimensions
The I3322I_{3322}I3322​ inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for I3322I_{3322}I3322​, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture. An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.
We present two novel families of bipartite Bell inequalities designed to achieve optimal global randomness certification for an arbitrary number of outputs ddd. We first use symmetry arguments to argue that their maximal quantum violations certify 2log⁡d2\log d2logd random bits. For the first family, we construct a quantum realization using d×dd\times dd×d maximally entangled states which provides a quantum violation that we conjecture to be optimal for any ddd. It is then numerically shown that the obtained quantum violation certifies optimal global randomness, up to numerical precision, for d=3,4d=3,4d=3,4. For the second family, we provide the optimal quantum violation and its quantum realization for any ddd, again using d×dd\times dd×d maximally entangled states and projective measurements over at least two unbiased bases on one of the parties. We self-test this realization for d=3d=3d=3, which implies the optimal certification of two fully random trits.
We introduce two families of sum-of-squares (SOS) decompositions for the Bell operators associated with the tilted Clauser-Horne-Shimony-Holt (CHSH) expressions introduced in Ac\'{\i}n et al. [Phys. Rev. Lett. 108, 100402 (2012)]. These SOS decompositions provide tight upper bounds on the maximal quantum value of these Bell expressions. Moreover, they establish algebraic relations that are necessarily satisfied by any quantum state and observables yielding the optimal quantum value. These algebraic relations are then used to show that the tilted CHSH expressions provide robust self-tests for any partially entangled two-qubit state. This application to self-testing follows closely the approach of Yang and Navascu\'es [Phys. Rev. A 87, 050102(R) (2013)], where we identify and correct two nontrivial flaws.
We define a family of binary-outcome nnn-party $m\ensuremath{\le}n$ settings per party Bell inequalities whose members require the least detection efficiency for their violation among all known inequalities of the same type. This gives upper bounds for the minimum value of the critical efficiency---below which no violation is possible---achievable for such inequalities. For m=2m=2m=2, our family reduces to the one given by Larsson and Semitecolos in 2001 [Phys. Rev. A 63, 022117 (2001)]. For m>2m>2m>2, a gap remains between these bounds and the best lower bounds. The violating state near the threshold efficiency always approaches a product state of nnn qubits.