Question: I am looking at a matrix reconstruction algorithm that, given singular values $\sigma\_i$ and quantum states $|u\_i\rangle$ and $|v\_i\rangle$ that are efficiently prepared on a quantum computer, produces a matrix $M$ such that

$M=\sum\_i \sigma\_i |u\_i \rangle\langle v\_i|$.

It seems like I can do two things:

I) Quantum State Tomography on both $|u\_i\rangle$ and $|v\_i\rangle$, and then classically reconstructing the outer product. This seems to take an exponential number of measurements because of the output problem. There exists <https://arxiv.org/abs/2111.11071> which says that you can measure $|u\_i\rangle$ and $|v\_i\rangle$ with a number of bases that is linear in the number of qubits.

II) Since $|u\_i\rangle\langle v\_i|$ is rank-1, there might exist an efficient way to reconstruct it altogether, though I haven't seen any literature that supports an efficient algorithm for doing this.

Are there any clues that support efficient reconstruction here? Please note that the vectors $|u\_i\rangle$ and $|v\_i\rangle$ have real entries only.

Answer: 
This outer product is, in general, not Hermitian and so does not correspond directly to a physical observable. Taking a lesson from $2\times 2$ matrices (ie from polarimetry), we can measure the two Hermitian observables
$$X=|a\rangle\langle b|+|b\rangle\langle a|$$ and $$Y=-i(|a\rangle\langle b|-|b\rangle\langle a|),$$ then use post-processing to say that the expectation values should satisfy
$$\big\langle |a\rangle \langle b|\big\rangle=\big\langle X\big\rangle+i\big\langle Y\big\rangle.$$
If you know $|u\_i\rangle\langle v\_i|$ then you can deduce $|u\_i\rangle$ and $|v\_i\rangle$ as well (up to a global phase). So, reconstructing the product can't be easier then reconstruction of the states.

But since $|u\_i\rangle$, $|v\_i\rangle$ are real it's enough to use measurements in the standard basis to reconstruct them.