A weak form of self-testing

TL;DR

This paper introduces a family of Bell inequalities with multiple quantum realisations achieving maximal violation, revealing a weak form of self-testing where the state can be identified but measurements are not fully determined.

Contribution

The work characterizes a new class of Bell inequalities exhibiting a weak form of self-testing, expanding understanding of quantum realisations and robustness in device-independent scenarios.

Findings

Maximal violation achieved by multiple inequivalent quantum realisations.

The set of probability points saturating the quantum bound forms a line segment.

Robustness analysis shows non-maximal violations still provide quantitative state information.

Abstract

The concept of self-testing (or rigidity) refers to the fact that for certain Bell inequalities the maximal violation can be achieved in an essentially unique manner. In this work we present a family of Bell inequalities which are maximally violated by multiple inequivalent quantum realisations. We completely characterise the quantum realisations achieving the maximal violation and we show that each of them requires a maximally entangled state of two qubits. This implies the existence of a new, weak form of self-testing in which the maximal violation allows us to identify the state, but does not fully determine the measurements. From the geometric point of view the set of probability points that saturate the quantum bound is a line segment. We then focus on a particular member of the family and show that the self-testing statement is robust, i.e. that observing a non-maximal violation allows us to make a quantitative statement about the unknown state. To achieve this we present a new construction of extraction channels and analyse their performance. For completeness we provide two independent approaches: analytical and numerical. The noise robustness, i.e. the amount of white noise at which the bound becomes trivial, of the analytical bound is rather small (~0.06%), but the numerical method takes us into an experimentally-relevant regime (~5%). We conclude by investigating the amount of randomness that can be certified using these Bell violations. Perhaps surprisingly, we find that the qualitative behaviour resembles the behaviour of rigid inequalities like the Clauser-Horne-Shimony-Holt inequality. This shows that at least for some device-independent applications rigidity is not a necessary ingredient.