  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.
