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A device-independent randomness expansion protocol aims to take an initial random string and generate a longer one, where the security of the protocol does not rely on knowing the inner workings of the devices used to run it. In order to do so, the protocol tests that the devices violate a Bell inequality and one then needs to bound the amount of extractable randomness in terms of the observed violation. The entropy accumulation theorem gives a bound in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on this are known for the one-sided randomness when using the Clauser-Horne-Shimony-Holt (CHSH) game. Here we find the minimum von Neumann entropies for a given CHSH score relevant for one and two sided randomness that can be applied to various protocols. In particular, we show the gain that can be made by using the two-sided randomness and by using a protocol without spot-checking where the input randomness is recycled. We also discuss protocols that fully close the locality loophole while expanding randomness. Although our bounds are mostly numerical, we conjecture analytic formulae for the curves in two cases.
With the growing availability of experimental loophole-free Bell tests, it has become possible to implement a new class of device-independent random number generators whose output can be certified to be uniformly random without requiring a detailed m
Randomness expansion where one generates a longer sequence of random numbers from a short one is viable in quantum mechanics but not allowed classically. Device-independent quantum randomness expansion provides a randomness resource of the highest se
The ability to produce random numbers that are unknown to any outside party is crucial for many applications. Device-independent randomness generation (DIRNG) allows new randomness to be provably generated, without needing to trust the devices used f
Quantum Bell nonlocality allows for the design of protocols that amplify the randomness of public and arbitrarily biased Santha-Vazirani sources, a classically impossible task. Information-theoretical security in these protocols is certified in a dev
Applications of randomness such as private key generation and public randomness beacons require small blocks of certified random bits on demand. Device-independent quantum random number generators can produce such random bits, but existing quantum-pr