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 $B = cA$, $c \in \mathbb{C}$.