This paper studies the diameter of the numerical range of bounded operators on Hilbert space and the induced seminorm, called the numerical diameter, on bounded linear maps between operator systems which is sensible in the case of unital maps and their scalar multiples. It is shown that the completely bounded numerical diameter is a norm that is comparable but not equal to the completely bounded norm. This norm is particularly interesting in the case of unital completely positive maps and their sections.

This work began whilst N.dB. was affiliated with the University of Oxford, where he was supported by the EPSRC National Hub in Networked Quantum Information Technologies (NQIT.org). C.R. was supported by the NSERC Discovery grant 2019-05430.