Henry Yuen

Robust Randomness Amplifiers: Upper and Lower Bounds

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.

Infinite randomness expansion with a constant number of devices

We present a device-independent randomness expansion protocol, involving only a constant number of non-signaling quantum devices, that achieves infinite expansion: starting with m bits of uniform private randomness, the protocol can produce an unbounded amount of certified randomness that is exp(--Ω(m1/3))-close to uniform and secure against a quantum adversary. The only parameters which depend on the size of the input are the soundness of the protocol and the security of the output (both are inverse exponential in m). This settles a long-standing open problem in the area of randomness expansion and device-independence. The analysis of our protocols involves overcoming fundamental challenges in the study of adaptive device-independent protocols. Our primary technical contribution is the design and analysis of device-independent protocols which are Input Secure; that is, their output is guaranteed to be secure against a quantum eavesdropper, even if the input randomness was generated by that same eavesdropper! The notion of Input Security may be of independent interest to other areas such as device-independent quantum key distribution.

A Parallel Repetition Theorem for All Entangled Games.

The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Razs classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game

Infinite Randomness Expansion and Amplification with a Constant Number of Devices

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Anchoring games for parallel repetition

repeated

Parallel repetition via fortification: analytic view and the quantum case

n

A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian

times in parallel is at most

Strong parallel repetition for free entangled games, with any number of players

c_G n^{-1/4} \log n

Quantum proof systems for iterated exponential time, and beyond.

for a constant

A simple proof of Renner's exponential de Finetti theorem

c_G