Quantifying the randomness of copies of noisy Popescu-Rohrlich correlations

TL;DR

This paper investigates how the unpredictability of outputs from noisy Popescu-Rohrlich (PR) boxes, used in quantum information, decreases when multiple copies are generated from a single device, revealing a reduction in min-entropy.

Contribution

It demonstrates that the min-entropy per PR-box diminishes with more copies when generated from a single device, contrasting with independent implementations.

Findings

Min-entropy per PR-box decreases with more copies from a single device.

Independent PR-boxes maintain constant min-entropy regardless of number of copies.

Sequential or single-run implementations reduce unpredictability compared to independent setups.

Abstract

In a no-signaling world, the outputs of a nonlocal box cannot be completely predetermined, a feature that is exploited in many quantum information protocols exploiting non-locality, such as device-independent randomness generation and quantum key distribution. This relation between non-locality and randomness can be formally quantified through the min-entropy, a measure of the unpredictability of the outputs that holds conditioned on the knowledge of any adversary that is limited only by the no-signaling principle. This quantity can easily be computed for the noisy Popescu-Rohrlich (PR) box, the paradigmatic example of non-locality. In this paper, we consider the min-entropy associated to several copies of noisy PR boxes. In the case where n noisy PR-boxes are implemented using n non-communicating pairs of devices, it is known that each PR-box behaves as an independent biased coin: the min-entropy per PR-box is constant with the number of copies. We show that this doesn't hold in more general scenarios where several noisy PR-boxes are implemented from a single pair of devices, either used sequentially n times or producing n outcome bits in a single run. In this case, the min-entropy per PR-box is smaller than the min-entropy of a single PR-box, and it decreases as the number of copies increases.