Suppose two separated parties, Alice and Bob, share a bipartite quantum state or a classical correlation called a 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 canonical 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 canonical 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：假设两个分离方，Alice和Bob，共享一个称为种子的二分量子态或经典关联，他们试图通过对种子执行本地量子或经典操作来生成目标经典关联，即不允许任何通信。我们考虑关于此设置的基本问题：Alice和Bob能否利用给定种子生成目标经典关联？我们证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二分态，上述判定问题已经是NP-难的，且当种子是经典关联时也能得出类似结论，表明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，求解该问题等价于判定目标经典关联是否存在与种子纯态匹配的某种规范形式的半正定分解，揭示了当前问题与优化理论之间的有趣联系。基于这一观察及其他洞见，我们给出了种子纯态生成目标经典关联必须满足的几个必要条件，且这些条件可推广至种子为混合量子态的情形。最后，由于半正定分解的规范形式在求解该问题中起关键作用，我们开发了一种能为任意经典关联计算此类规范的算法，该算法在我们测试的案例中表现良好。