title: Randomness Amplification under Minimal Fundamental Assumptions on the Devices.
abstract: Recently, the physically realistic protocol amplifying the randomness of Santha-Vazirani sources producing cryptographically secure random bits was proposed; however, for reasons of practical relevance, the crucial question remained open regarding whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling components and in a situation where the violation of a Bell inequality only guarantees that some outcomes of the device for specific inputs exhibit randomness? Here, we solve this question and present a device-independent protocol for randomness amplification of Santha-Vazirani sources using a device consisting of two nonsignaling components. We show that the protocol can amplify any such source that is not fully deterministic into a fully random source while tolerating a constant noise rate and prove the composable security of the protocol against general no-signaling adversaries. Our main innovation is the proof that even the partial randomness certified by the two-party Bell test [a single input-output pair (u^{*}, x^{*}) for which the conditional probability P(x^{*}|u^{*}) is bounded away from 1 for all no-signaling strategies that optimally violate the Bell inequality] can be used for amplification. We introduce the methodology of a partial tomographic procedure on the empirical statistics obtained in the Bell test that ensures that the outputs constitute a linear min-entropy source of randomness. As a technical novelty that may be of independent interest, we prove that the Santha-Vazirani source satisfies an exponential concentration property given by a recently discovered generalized Chernoff bound.
authors.affiliation: Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland.
authors.forename: Ravishankar
authors.initials: R
authors.lastname: Ramanathan
authors.affiliation: Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, USA.|Department of Computer Science, University College London, WC1E 6BT London, United Kingdom.
authors.forename: Fernando G S L
authors.initials: FG
authors.lastname: Brandão
authors.affiliation: Institute of Informatics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland.
authors.forename: Karol
authors.initials: K
authors.lastname: Horodecki
authors.affiliation: Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland.
authors.forename: Michał
authors.initials: M
authors.lastname: Horodecki
authors.affiliation: Faculty of Applied Physics and Mathematics, National Quantum Information Center, Gdańsk University of Technology, 80-233 Gdańsk, Poland.
authors.forename: Paweł
authors.initials: P
authors.lastname: Horodecki
authors.affiliation: Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, 80-308 Gdańsk, Poland.|Institute of Mathematics, Faculty of Mathematics, Physics and Chemistry, University of Silesia, Bankowa 14, 40-007 Katowice, Poland.
authors.forename: Hanna
authors.initials: H
authors.lastname: Wojewódka