Quantum Randomness Expansion: Upper and Lower Bounds

Masahito Hayashi

To discuss secure key generation from imperfect random numbers, we address the secure key generation length. There are several studies for its asymptotic expansion up to the order √n or log n. However, these expansions have errors of the order o(√n) or o(log n), which does not go to zero asymptotically. To resolve this problem, we derive the asymptotic expansion up to the constant order for upper and lower bounds of these optimal values. While the expansions of upper and lower bonds do not match, they clarify the ranges of these optimal values, whose errors go to zero asymptotically.

Rutvij Bhavsar, Sammy Ragy, Roger Colbeck

Abstract A device-independent randomness expansion (DIRE) 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 lower bounds the extractable randomness of a protocol with many rounds in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on the von Neumann entropy are known for the one-sided randomness (i.e. where the randomness from only one party is used) when using the Clauser–Horne–Shimony–Holt game. Here we investigate the possible improvement that could be gained using the two-sided randomness. We generate upper bounds on this randomness by attempting to find the optimal eavesdropping strategy, providing analytic formulae in two cases. We additionally compute lower bounds that outperform previous ones and can be made arbitrarily tight (at the expense of more computation time). These bounds get close to our upper bounds, and hence we conjecture that our upper bounds are tight. We also consider a modified protocol in which the input randomness is recycled. This modified protocol shows the possibility of rate gains of several orders of magnitude based on recent experimental parameters, making DIRE significantly more practical. It also enables the locality loophole to be closed while expanding randomness in a way that typical spot-checking protocols do not.

Matthew Coudron, Thomas Vidick, Henry Yuen

A recent sequence of works, initially motivated by the study of the nonlocal properties of entanglement, demonstrate that a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two black-box devices do not communicate (i.e. are non-signaling). We call protocols achieving such certified amplification of a short random seed randomness amplifiers. We introduce a simple framework in which we initiate the systematic study of the possibilities and limitations of randomness amplifiers. Our main results include a new, improved analysis of a robust randomness amplifier with exponential expansion, as well as the first upper bounds on the maximum expansion achievable by a broad class of randomness amplifiers. In particular, we show that non-adaptive randomness amplifiers that are robust to noise cannot achieve more than doubly exponential expansion. Finally, we show that a wide class of protocols based on the use of the CHSH game can only lead to (singly) exponential expansion if adversarial devices are allowed the full power of non-signaling strategies. Our upper bound results apply to all known non-adaptive randomness amplifier constructions to date.

Igor M. Arbekov, S. N. Molotkov