  Suppose two separated parties, Alice and Bob, share a bipartite quantum state
or a classical correlation called a \emph{seed}, and they try to generate a
target classical correlation by performing local quantum or classical
operations on the seed, i.e., any communications are not allowed. We consider
the following fundamental problem about this setting: whether Alice and Bob can
use a given seed to generate a target classical correlation. We show that this
problem has rich mathematical structures. Firstly, we prove that even if the
seed is a pure bipartite state, the above decision problem is already NP-hard
and a similar conclusion can also be drawn when the seed is also a classical
correlation, implying that this problem is hard to solve generally.
Furthermore, we prove that when the seed is a pure quantum state, solving the
problem is equivalent to finding out whether the target classical correlation
has some diagonal form of positive semi-definite factorizations that matches
the seed pure state, revealing an interesting connection between the current
problem and optimization theory. Based on this observation and other insights,
we give several necessary conditions where the seed pure state has to satisfy
to generate the target classical correlation, and it turns out that these
conditions can also be generalized to the case that the seed is a mixed quantum
state. Lastly, since diagonal forms of positive semi-definite factorizations
play a crucial role in solving the problem, we develop an algorithm that can
compute them for an arbitrary classical correlation, which has decent
performance on the cases we test.
