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.
Highlights
In his seminal work Bell showed that performing measurements on spatially separated quantum systems may give rise to correlations inconsistent with any local-realistic description of the world [1]
In this work we derive a new type of self-testing statement which allows us to certify the state but not the measurements
Until now self-testing of the state or randomness certification have only been shown for rigid Bell inequalities, and so one might have conjectured rigidity to be necessary for these purposes
Summary
In his seminal work Bell showed that performing measurements on spatially separated quantum systems may give rise to correlations inconsistent with any local-realistic description of the world [1] (see Ref. [2] for a comprehensive review). In addition to its foundational importance self-testing has immediate applications to cryptography: if the Bell violation alone essentially determines the quantum realization, one can certify that the randomness generated in the experiment is intrinsically quantum and cannot be known to an external eavesdropper This is precisely the idea behind deviceindependent cryptography [36,37,38,39,40] We study a 1-parameter family of Bell inequalities and show that observing the maximal violation certifies the presence of a maximally entangled state of two qubits even though the measurements cannot be uniquely determined To understand how this phenomenon is affected by noise, we focus on a particular member of the family and derive an analytic robust self-testing result for the state. We study the amount of randomness that can be certified from the observed violation
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