Assume one is given two oracle circuits providing access to matrices $A_{ij}$ and $B_{ij}$ as follows (see eq. (6.2) here): \begin{equation} O_A |0\rangle|i\rangle|j\rangle=\left(A_{ij}|0\rangle+\sqrt{1-|A_{ij}|^2}|1\rangle\right)|i\rangle|j\rangle \, ,\\ O_B |0\rangle|i\rangle|j\rangle=\left(B_{ij}|0\rangle+\sqrt{1-|B_{ij}|^2}|1\rangle\right)|i\rangle|j\rangle \, , \end{equation} where indices $i$ and $j$ are encoded with qubits in a the binary form. The said representation will be referred to as block encoding.
I am wondering if one could construct a circuit $O_{A+B}$ implementing \begin{equation} O_{A+B} |0\rangle|i\rangle|j\rangle=\Bigl((A_{ij}+B_{ij})|0\rangle+\ldots|1\rangle\Bigr)|i\rangle|j\rangle \, . \end{equation}
I placed $\ldots$ instead of $\sqrt{1-|A_{ij}+B_{ij}|^2}$ just to indicate that using ancillas and measurements would be OK.
UPDATE: A BRUTE-FORCE SOLUTION
For completeness I add a straightforward implementation based on the conversion to query oracles.
Use the controlled rotation circuit (see Proposition 4.7 here)
to convert the block-encoded oracles to query-encoded oracles $\widetilde{O}_A$ and $\widetilde{O}_B$:
$$
\widetilde{O}_A |0\rangle|i\rangle|j\rangle = |\widetilde{A}_{ij}\rangle|i\rangle|j\rangle \, ,\\
\widetilde{O}_B |0\rangle|i\rangle|j\rangle = |\widetilde{B}_{ij}\rangle|i\rangle|j\rangle \, .
$$The consecutive application of oracles renders then $$ \widetilde{O}_A \widetilde{O}_B |0\rangle|i\rangle|j\rangle = \widetilde{O}_A |\widetilde{B}_{ij}\rangle|i\rangle|j\rangle = |\widetilde{A}_{ij} + \widetilde{B}_{ij}\rangle|i\rangle|j\rangle \, . $$
Would be nice to avoid converting to query oracles.
