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.

But this condition is not necessary as equality also holds when $B = cA$, $c \in \mathbb{C}$.