(Almost-)Quantum Bell Inequalities and Device-Independent Applications

Investigations of the boundary of the quantum correlation set have gained increased attention in recent years. This is done through the derivation of quantum Bell inequalities, which are related to Tsirelson's problem and have significant applications in device-independent (DI) information processing. However, determining quantum Bell inequalities is a notoriously difficult task and only isolated examples are known. In this paper, we present families of (almost-)quantum Bell inequalities and highlight four foundational and DI applications. Firstly, it is known that quantum correlations on the non-signaling boundary are of crucial importance in the task of DI randomness extraction from weak sources. In the practical Bell scenario of two players with two <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math>-outcome measurements, we derive quantum Bell inequalities that demonstrate a separation between the quantum boundary and certain portions of the boundaries of the no-signaling polytope of dimension up to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>4</mml:mn><mml:mi>k</mml:mi><mml:mo>&amp;#x2212;</mml:mo><mml:mn>8</mml:mn></mml:math>, extending previous results from nonlocality distillation and the collapse of communication complexity. Secondly as an immediate by-product, we give a general proof of Aumann’s Agreement theorem for quantum systems as well as the almost-quantum correlations, which implies Aumann’s agreement theorem is a reasonable physical principle in the context of epistemics to pick out both quantum theory and almost-quantum correlations from general no-signaling theories. Thirdly, we present a family of quantum Bell inequalities in the two players with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> binary measurements scenarios, that we prove serve to self-test the two-qubit singlet and the corresponding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:math> measurements. Interestingly, this claim generalizes the result for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> discovered by Tsirelson-Landau-Masanes and shows an improvement over the state-of-the-art Device-Independent Randomness-Amplification (DIRA). Lastly, we use our quantum Bell inequalities to derive the general form of the principle of no advantage in nonlocal computation, which is an information-theoretic principle that serves to characterize the quantum correlation set.