Abstract
The nonlocal behavior of quantum mechanics can be used to generate guaranteed fresh randomness from an untrusted device that consists of two nonsignalling components; since the generation process requires some initial fresh randomness to act as a catalyst, one also speaks of randomness expansion. Colbeck and Kent proposed the first method for generating randomness from untrusted devices, however, without providing a rigorous analysis. This was addressed subsequently by Pironio et al. [Nature 464 (2010)], who aimed at deriving a lower bound on the min-entropy of the data extracted from an untrusted device, based only on the observed non-local behavior of the device. Although that article succeeded in developing important tools towards the acquired goal, it failed in putting the tools together in a rigorous and correct way, and the given formal claim on the guaranteed amount of min-entropy needs to be revisited. In this paper we show how to combine the tools provided by Pironio et al., as to obtain a meaningful and correct lower bound on the min-entropy of the data produced by an untrusted device, based on the observed non-local behavior of the device. Our main result confirms the essence of the improperly formulated claims of Pironio et al., and puts them on solid ground. We also address the question of composability and show that different untrusted devices can be composed in an alternating manner under the assumption that they are not entangled. This enables for superpolynomial randomness expansion based on two untrusted yet unentangled devices.
Highlights
Anticipated by Einstein, Rosen, and Podolsky [1], it was Bell [2] who put this property on firm ground by proposing an inequality that is satisfied by any classical correlation but is violated when the correlation is obtained from measuring entangled quantum states
An important example of such a Bell inequality was proposed by Clauser, Horne, Shimony, and Holt (CHSH) [3] and states that if X and Y are independent uniformly distributed bits and if bit A is obtained by “processing” X without knowing Y and bit B is obtained by “processing” Y without knowing X, the probability that A ⊕ B = X ∧ Y is at most 75%. This bound on the probability holds if the processing is done classically with shared randomness but can be violated when the processing involves measuring an entangled quantum state; in this latter case, a probability of roughly 85% can be achieved
We address the question of the composability of untrusted devices
Summary
One of the counterintuitive features of quantum mechanics is its nonlocality: measuring possibly far apart quantum systems in randomly selected bases (chosen out of some given class) may lead to correlations that are impossible to obtain classically. An important example of such a Bell inequality was proposed by Clauser, Horne, Shimony, and Holt (CHSH) [3] and states that if X and Y are independent uniformly distributed bits and if bit A is obtained by “processing” X without knowing Y and bit B is obtained by “processing” Y without knowing X, the probability that A ⊕ B = X ∧ Y is at most 75% This bound on the probability holds if the processing is done classically with shared randomness but can be violated when the processing involves measuring an entangled quantum state; in this latter case, a probability of roughly 85% can be achieved. The claimed bound in [6] does not hold in general; there is a flaw in its derivation, which is without an obvious fix. even though the necessary tools are provided in [6], they are not put together in the right way to be able to control the min-entropy of (A1,B1), . . . ,(An,Bn) produced by an untrusted device D
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