Tilted Hardy paradoxes for device-independent randomness extraction

Shuai Zhao1, Ravishankar Ramanathan1, Yuan Liu1, and Paweł Horodecki2,3

1Department of Computer Science, The University of Hong Kong, Pokfulam Road, Hong Kong
2International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
3Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland

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The device-independent paradigm has had spectacular successes in randomness generation, key distribution and self-testing, however most of these results have been obtained under the assumption that parties hold trusted and private random seeds. In efforts to relax the assumption of measurement independence, Hardy's non-locality tests have been proposed as ideal candidates. In this paper, we introduce a family of tilted Hardy paradoxes that allow to self-test general pure two-qubit entangled states, as well as certify up to $1$ bit of local randomness. We then use these tilted Hardy tests to obtain an improvement in the generation rate in the state-of-the-art randomness amplification protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement independence. Our result shows that device-independent randomness amplification is possible for arbitrarily biased SV sources and from almost separable states. Finally, we introduce a family of Hardy tests for maximally entangled states of local dimension $4, 8$ as the potential candidates for DI randomness extraction to certify up to the maximum possible $2 \log d$ bits of global randomness.

We introduce a family of tilted Hardy paradoxes that enable the self-testing of general pure two-qubit entangled states and the certification of up to $1$ bit of local randomness. Utilizing these tilted Hardy tests, we achieve enhanced generation rates in the state-of-the-art randomness amplification protocols for Santha-Vazirani (SV) sources with arbitrarily limited measurement independence. Our findings show that device-independent randomness amplification is possible for arbitrarily biased SV sources and from almost separable states.

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Cited by

[1] Kai-Siang Chen, Shiladitya Mal, Gelo Noel M. Tabia, and Yeong-Cherng Liang, "Hardy-type paradoxes for an arbitrary symmetric bipartite Bell scenario", Physical Review A 109 4, 042206 (2024).

[2] Abhishek Sadhu and Siddhartha Das, "Testing of quantum nonlocal correlations under constrained free will and imperfect detectors", Physical Review A 107 1, 012212 (2023).

[3] Ravishankar Ramanathan, "Finite Device-Independent Extraction of a Block Min-Entropy Source against Quantum Adversaries", arXiv:2304.09643, (2023).

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