量子态的Ulam重建问题
Some Ulam's reconstruction problems for quantum states
摘要：
假设一个多体量子态有一套完整的假定的$k$ -体约化，我们能否确定一个联合态是否存在？我们推导出这为真的必要条件。与所谓的量子边际问题相反，我们考虑一个子系统的标签是未知的设置。这个问题可以类比于图论中的乌拉姆重建猜想。猜想-仍然没有解决-声称每个图至少有三个顶点可以唯一地重建从它的顶点删除子图的集合。当考虑量子态时，我们证明了在某些情况下，联合态的不存在已经可以从一组边缘的大小刚刚超过一半的缔约方推断出来。我们将这些方法应用于图状态，其中许多约束可以通过知道出现在减少中的某些权重的稳定器元素的数量来评估。这种观点与量子纠错码和多项式不变量中的约束相联系。这些约束中的一些可以被解释为单偶关系，其限制了由量子态产生的相关性。最后，我们给出了一般量子态的Ulam重建问题的答案。
英文摘要：
Provided a complete set of putative $k$-body reductions of a multipartite quantum state, can one determine if a joint state exists? We derive necessary conditions for this to be true. In contrast to what is known as the quantum marginal problem, we consider a setting where the labeling of the subsystems is unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture in graph theory. The conjecture - still unsolved - claims that every graph on at least three vertices can uniquely be reconstructed from the set of its vertex-deleted subgraphs. When considering quantum states, we demonstrate that the non-existence of joint states can, in some cases, already be inferred from a set of marginals having the size of just more than half of the parties. We apply these methods to graph states, where many constraints can be evaluated by knowing the number of stabilizer elements of certain weights that appear in the reductions. This perspective links with constraints that were derived in the context of quantum error-correcting codes and polynomial invariants. Some of these constraints can be interpreted as monogamy-like relations that limit the correlations arising from quantum states. Lastly, we provide an answer to Ulam's reconstruction problem for generic quantum states.
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