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.

翻译：假设两个分离的参与方Alice和Bob共享一个称为“种子”的双方向量子态或经典关联，他们尝试通过对种子进行局域量子或经典操作（即不允许任何通信）来生成目标经典关联。我们考虑该设定下的基本问题：Alice和Bob能否利用给定种子生成目标经典关联？研究表明该问题具有丰富的数学结构。首先，我们证明即使种子是纯双方向态，上述判定问题已经是NP困难的，而当种子也是经典关联时可得类似结论，这意味着该问题通常难以求解。进一步，我们证明当种子是纯量子态时，求解该问题等价于寻找目标经典关联是否具有与种子纯态匹配的半正定分解的对角形式，揭示了当前问题与优化理论之间的有趣联系。基于此观察及其他洞见，我们给出了种子纯态生成目标经典关联所需满足的若干必要条件，且这些条件可推广至种子为混合量子态的情形。最后，由于半正定分解的对角形式在求解该问题中起关键作用，我们开发了一种算法，可为任意经典关联计算此类形式，并在测试案例中表现良好。