Bounds on semi-device-independent quantum random number expansion capabilities

TL;DR

This paper establishes fundamental limits on the amount of certifiable randomness in semi-device-independent quantum protocols, providing explicit bounds, optimal settings, and practical robustness insights.

Contribution

It proves the maximum certifiable entropy achievable in SDI protocols is independent of dimension witnesses and identifies minimal settings for optimal entropy generation.

Findings

Maximum certifiable entropy is $-rac{1}{2}ig(1+rac{1}{ oot3 elax 3}ig)$ bits.

An SDI protocol with minimal input settings achieves this maximum entropy.

Certifiable entropy is achievable once the dimension witness exceeds the classical bound.

Abstract

The randomness expansion capabilities of semi-device-independent (SDI) prepare and measure protocols are analyzed under the sole assumption that the Hilbert state dimension is known. It's explicitly proved that the maximum certifiable entropy that can be obtained through this set of protocols is $-\log_2\left[\frac{1}{2}\left(1+\frac{1}{\sqrt{3}}\right)\right]$ and the same is independent of the dimension witnesses used to certify the protocol. The minimum number of preparation and measurement settings required to achieve this entropy is also proven. An SDI protocol that generates the maximum output entropy with the least amount of input setting is provided. An analytical relationship between the entropy generated and the witness value is obtained. It's also established that certifiable entropy can be generated as soon as dimension witness crosses the classical bound, making the protocol noise-robust and useful in practical applications.