Realistic noise-tolerant randomness amplification using finite number of devices

Secret and perfect randomness is an essential resource in cryptography. Yet, it is not even clear that such exists. It is well known that the tools of classical computer science do not allow us to create secret and perfect randomness from a single weak public source. Quantum physics, on the other hand, allows for such a process, even in the most paranoid cryptographic sense termed "device-independent quantum cryptography". We propose and prove the security of a new device-independent protocol that takes any single public Santha-Vazirani source as input and creates a secret close to uniform string in the presence of a quantum adversary. Our work is the first to achieve randomness amplification with all the following properties: (1) amplification and "privatization" of a public Santha-Vazirani source with arbitrary bias (2) the use of a device with only two components (3) non-vanishing extraction rate and (4) maximal noise tolerance. In particular, this implies that our protocol is the first protocol that can possibly be implemented with reachable parameters. We achieve these by combining three new tools: a particular family of Bell inequalities, a proof technique to lower bound entropy in the device-independent setting, and a framework for quantum-proof multi-source extractors.

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“…Thus, Equation 1holds. Several previous works, e.g., [15], [16], [17], considered similar settings as well.…”

Section: A Results and Contributionsmentioning

confidence: 99%

Device-Independent Randomness Amplification and Privatization

Kessler

Arnon-Friedman

2020

IEEE J. Sel. Areas Inf. Theory

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“…The first results [14][15][16] were obtained for a slightly stronger model of random sources, the so-called Santha-Vazirani sources, but subsequent results have claimed the possibility of amplifying even a min-entropy source [17,18]. However, all these protocols require multi-partite entanglement and are not robust to deviations from an ideal quantum state; it is currently an open problem to devise a randomness amplification protocol that can be implemented with existing devices.…”

Section: Resultsmentioning

confidence: 99%

“…This task is provably impossible with classical information processing, but it becomes possible if the weak source is used to choose the inputs (including the state) in a Bell test, whose outcomes are taken as the new random numbers [14][15][16][17][18]. One may wonder why the optimised approach of Pütz and coworkers has not yet been applied to improve the bounds on randomness amplification.…”

Section: Introductionmentioning

confidence: 99%

Measurement-dependent locality beyond independent and identically distributed runs

2016

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When conducting a Bell test, it is normal to assume that the preparation of the quantum state is independent of the measurements performed on it. Remarkably, the violation of local realism by entangled quantum systems can be certified even if this assumption is partially relaxed. Here, we allow such measurement dependence to correlate multiple runs of the experiment, going beyond previous studies that considered independent and identically distributed (i.i.d.) runs. To do so, we study the polytope that defines block-i.i.d. measurement-dependent local models. We prove that non-i.i.d. models are strictly more powerful than i.i.d. ones, and comment on the relevance of this work for the study of randomness amplification in simple Bell scenarios with suitably optimised inequalities. INTRODUCTIONSince their introduction, by John Bell in 1964 [1], Bell inequalities have been a subject of extensive study, as they highlight the fact that quantum theory is incompatible with local realism. Numerous experimental tests of Bell inequalities have been carried out, with the results being overwhelmingly in favour of the quantum predictions. Of particular note are the recent loophole-free Bell tests [2][3][4], which simultaneously addressed several loopholes that had been raised regarding previous experiments. All these tests were conducted under the assumption that the choice of measurements and the state of the source are independent in each run. This observation should not be taken as a reservation: such measurement independence is an essential piece of the scientific method, and its negation would be rightly considered conspiratorial. This makes it all the more remarkable that quantum theory can be proved incompatible with local realism even if this assumption is relaxed to some extent.Indeed, while unrestricted measurement dependence would lead to an unfalsifiable superdeterminism [5], it was noted by Hall [6] that the violation of Bell inequalities keeps its meaning if some restrictions are made. This led to the study of measurement-dependent local (MDL) scenarios, where some correlation is allowed between the measurement choices and the source. A few subsequent works refined our understanding of measurement dependence [7][8][9][10], all sticking to known inequalities. A significant breakthrough was achieved when Pütz and coworkers noticed that the traditional Bell inequalities are no longer optimal: other linear constraints, suitably named MDL inequalities, more tightly define the conditions under which local realism holds in the MDL scenario. Their works [11,12] developed the mathematical framework to study these inequalities. Their most celebrated discovery is the following: there exist quantum correlations that violate local realism with "arbitrarily low measurement independence", that is, as long as the MDL model does not trivially allow us to reproduce all no-signalling correlations. The corresponding inequality has been tested in an experiment [13].Measurement dependence in Bell-type tests is also central in the tas...

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“…Starting with the work by Nisan and Ta-Shma [32] and followed by Trevisan's breakthrough result [47] there has been a lot of progress in this direction, and there are now many constructions that almost achieve the converse bounds in (4) (see the review articles [41,48]). For applications in classical and quantum cryptography (see, e.g., [30,39]), for constructing device independent randomness amplification and expansion schemes (see, e.g., [11,14,15,31]), and for applications in quantum Shannon theory (see, e.g., [9,19]) it is important to find out if extractor constructions also work when the input source is correlated to another (possibly quantum) system Q. That is, we would like that for all classical-quantum input density matrices ρ QN with conditional min-entropy H min (N |Q) ρ := − log p guess (N |Q) ρ ≥ k ,…”

Section: A Randomness Extractorsmentioning

confidence: 99%

Quantum-Proof Randomness Extractors via Operator Space Theory

Berta

Fawzi

Scholz

2017

IEEE Trans. Inform. Theory

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Quantum-proof randomness extractors are an important building block for classical and quantum cryptography as well as device independent randomness amplification and expansion. Furthermore they are also a useful tool in quantum Shannon theory. It is known that some extractor constructions are quantum-proof whereas others are provably not [Gavinsky et al., STOC'07]. We argue that the theory of operator spaces offers a natural framework for studying to what extent extractors are secure against quantum adversaries: we first phrase the definition of extractors as a bounded norm condition between normed spaces, and then show that the presence of quantum adversaries corresponds to a completely bounded norm condition between operator spaces. From this we show that very high min-entropy extractors as well as extractors with small output are always (approximately) quantumproof.We also study a generalization of extractors called randomness condensers. We phrase the definition of condensers as a bounded norm condition and the definition of quantum-proof condensers as a completely bounded norm condition. Seeing condensers as bipartite graphs, we then find that the bounded norm condition corresponds to an instance of a well studied combinatorial problem, called bipartite densest subgraph. Furthermore, using the characterization in terms of operator spaces, we can associate to any condenser a Bell inequality (two-player game) such that classical and quantum strategies are in one-to-one correspondence with classical and quantum attacks on the condenser. Hence, we get for every quantum-proof condenser (which includes in particular quantum-proof extractors) a Bell inequality that can not be violated by quantum mechanics. * berta@caltech.edu † omar.fawzi@ens-lyon.fr ‡ scholz@phys.ethz.ch 1 Other notions for weaker quantum adversaires have also been discussed in the literature, e.g., in the bounded storage model (see [16, Section 1] for a detailed overview). 2 Note that the dimension of Q is unbounded and that it is a priori unclear if there exist any extractor constructions that are quantumproof (even with arbitrarily worse parameters). Furthermore, and in contrast to some claims in the literature, we believe that the question to what extent extractors are quantum-proof is already interesting for weak extractors. In particular, if weak extractors were perfectly quantum-proof then strong extractors would be perfectly quantum-proof as well (and we know that the latter is wrong [21]). 3 Since the quality of the output randomness of Gavinsky et al.'s construction is bad to start with, the decrease ε → Ω(m ′ ε) for quantum Q already makes the extractor fail completely in this case.

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Realistic noise-tolerant randomness amplification using finite number of devices

Cited by 61 publications

References 20 publications

Device-Independent Randomness Amplification and Privatization

Device-Independent Randomness Amplification and Privatization

Measurement-dependent locality beyond independent and identically distributed runs

Quantum-Proof Randomness Extractors via Operator Space Theory

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