If $A \in M_n(\mathbb{C})$, then the *field (of values)*, or *numerical range  of A*, is the compact, convex subset of the complex-plane defined by
$$
F(A)= \{z^* A z \mid z^*z = 1 \}.
$$

It is well-known that the field is *sub-additive* in the sense that 
\begin{equation}
F(A+B) \subseteq F(A) + F(B), \tag{1}\label{fov}
\end{equation}
where addition-symbol on the right denotes Minkowksi addition.

**Question:** Is there a characterization known for when equality holds in \eqref{fov}?

Notice that if $A = UDU^*$ and $B= U \hat{D} U^*$, with $U$ unitary and $D$, $\hat{D} $ diagonal, then it is easily shown that equality holds.

**EDIT:** <del>But this condition is not necessary as equality also holds when </del>$B = cA$, $c \in \mathbb{C}$.