Custom Bell inequalities from formal sums of squares

Victor Barizien1, Pavel Sekatski2, and Jean-Daniel Bancal1

1Université Paris Saclay, CEA, CNRS, Institut de physique théorique, 91191 Gif-sur-Yvette, France
2Département de Physique Appliquée, Université de Genève, 1211 Genève, Suisse

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Bell inequalities play a key role in certifying quantum properties for device-independent quantum information protocols. It is still a major challenge, however, to devise Bell inequalities tailored for an arbitrary given quantum state. Existing approaches based on sums of squares provide results in this direction, but they are restricted by the necessity of first choosing measurement settings suited to the state. Here, we show how the sum of square property can be enforced for an arbitrary target state by making an appropriate choice of nullifiers, which is made possible by leaving freedom in the choice of measurement. Using our method, we construct simple Bell inequalities for several families of quantum states, including partially entangled multipartite GHZ states and qutrit states. In most cases we are able to prove that the constructed Bell inequalities achieve self-testing of the target state. We also use the freedom in the choice of measurement to self-test partially entangled two-qubit states with a family of settings with two parameters. Finally, we show that some statistics can be self-tested with distinct Bell inequalities, hence obtaining new insight on the shape of the set of quantum correlations.

In a Bell experiment, the probability distributions generated by local hidden variable models are limited by Bell inequalities. Measurements on a quantum system can however violate these inequalities. In fact, these violations can provide insights on the quantum device used in the experiment. Ultimately, the maximal violation of a well-chosen Bell inequality can provide a full characterization of the states and measurements used in the experiment. This property, called self-testing, establishes a fruitful relationship between Bell inequalities and quantum objects which can be utilized in many quantum information protocols. However, it is generally not known how to best devise a Bell inequality that would be suited to a specific quantum state. In this paper, we consider the problem of finding a Bell inequality suited to self-test a specific target state and/or measurement. We propose a constructive method, based on Sum of Squares of polynomials, which grants candidates verifying the necessary condition that the inequality be maximally violated by the target realization.

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Cited by

[1] Lewis Wooltorton, Peter Brown, and Roger Colbeck, "Device-independent quantum key distribution with arbitrarily small nonlocality", arXiv:2309.09650, (2023).

[2] Barizien Victor and Bancal Jean-Daniel, "Extremal Tsirelson inequalities", arXiv:2401.12791, (2024).

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