Self-testing quantum systems of arbitrary local dimension with minimal number of measurements

Shubhayan Sarkar,Debashis Saha,Remigiusz Augusiak,Jędrzej Kaniewski

doi:10.1038/s41534-021-00490-3

Shubhayan Sarkar, Debashis Saha + Show 2 more

Open Access

https://doi.org/10.1038/s41534-021-00490-3

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Abstract

Bell nonlocality as a resource for device-independent certification schemes has been studied extensively in recent years. The strongest form of device-independent certification is referred to as self-testing, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single d-outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, {{{mathrm{log}}},}_{2}d perfect random bits, while it requires only one random bit to encode the measurement choice.

Highlights

The advent of quantum theory has not just changed the understanding of physics but has given rise to new phenomena that would have never been possible in the classical world
One of the most interesting features of quantum theory is the existence of quantum correlations, which cannot be explained by any local hidden-variable model, a phenomenon commonly referred to as Bell nonlocality[1,2]
An example of such a DI-certification scheme would be verifying whether a quantum device produces entanglement[8] or certification of the dimension of a quantum system[9], both based on a violation of some Bell inequality by the corresponding quantum system

Summary

INTRODUCTION

The advent of quantum theory has not just changed the understanding of physics but has given rise to new phenomena that would have never been possible in the classical world. First introduced in[6], it allows one to provide a full description, up to certain well-understood equivalences, of the considered quantum system and the measurements performed on it based on observing the maximal violation of some Bell inequality Such a form of certification is interesting from the application point of view as it provides a way of verifying that a given quantum device functions properly without the need of knowing its internal working. In Ref. 17, the two-qubit results[11,12] were combined to design a self-testing protocol for any entangled state of arbitrary local dimension Still, these results are based on a violation of many two–outcome Bell inequalities and the question whether one can design a self–testing statement for qudit quantum systems relying on violation of a single and truly doutcome Bell inequality remains open. A straightforward implication of our self–testing statement is a novel and simpler, as compared with the previous copies of the atpwpor–oqauchbeits2m1,a22x,imscahlleymeentfaonrgplaerdalsletal tseelfφ–þ2tes.tAinngoothfeNr implication is that the outcomes of local measurements maximally violating the respective Bell inequality are perfectly random, which allows us to propose a quantum protocol for unbounded expansion of quantum randomness

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