The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|_1$ is the 1-norm and given by $|X|_1 = \text{Tr}(\sqrt{X^\dagger X})$. The point here is that I do not need to know $\rho$ or $\sigma$ to compute the trace distance between them. All I need to know is $\rho - \sigma$.
Can one also compute $F(\rho,\sigma)$ where $F$ is the fidelity if one is only given $\rho - \sigma$? I am aware of bounds that can be placed using the trace distance on $F(\rho,\sigma)$ but was wondering if it could be exactly computed.