AI 中文总结
本文针对量子树网络，提出源转移模型并证明其与混合量子模型等价，进而建立两个收敛的半定规划层级（源转移层级与膨胀-NPA层级），解决了关联约束的无维度方法问题，并证明博弈值承诺问题为coRE-完备。
AI 中文摘要
理解量子系统的能力与极限是量子信息处理的核心问题。这包括确定由独立源分布的量子系统网络中能够产生哪些关联。然而，对于一般的量子网络，目前尚无系统性的、无维度限制的方法来任意好地约束这些关联。我们通过一种新的源转移模型解决了所有量子树网络的这一问题，并证明了该模型等价于一种混合量子模型。混合模型对维度不加限制，并在有限维情况下退化为标准的张量积模型；而源转移模型则通过$C^*$-代数上的态以及沿树递归复合的完全正映射来描述这些关联。我们通过将局部测量重构为网络所规定的空间张量积中的Radon-Nikodym导数来证明这一等价性。基于这一刻画，我们建立了两个收敛的外部半定规划（SDP）层级：一个新颖的源转移层级和膨胀-NPA层级。对于源转移层级，我们将非交换实代数几何扩展到具有完全正映射的多态多项式，证明了一个重构定理以确立其收敛性，并给出了提取有限维实现的一个充分停止准则。对于膨胀-NPA层级，我们从源副本平均的弱算子极限重构源转移实现。因此，混合模型中量子树网络的博弈值承诺问题是$\mathsf{coRE}$-完备的。
英文摘要
Understanding the capabilities and limits of quantum systems is central to quantum information processing. This includes determining which correlations can arise in networks of quantum systems distributed by independent sources. Yet for general quantum networks, no systematic, dimension-free method is known to bound these correlations arbitrarily well. We resolve this problem for all quantum tree networks through a new source-transfer model, which we prove equivalent to a mixed quantum model. The mixed model places no restriction on dimension and reduces to the standard tensor-product model in finite dimensions; the source-transfer model instead describes these correlations through states on $C^*$-algebras and completely positive maps composed recursively along the tree. We prove the equivalence by reconstructing local measurements as Radon-Nikodym derivatives in the spatial tensor products prescribed by the network. Building on this characterization, we establish two convergent outer hierarchies of semidefinite programs (SDPs): a new source-transfer hierarchy and the inflation-NPA hierarchy. For the source-transfer hierarchy, we extend noncommutative real algebraic geometry to multi-state polynomials with completely positive maps, prove a reconstruction theorem establishing its convergence, and give a sufficient stopping criterion for extracting finite-dimensional realizations. For the inflation-NPA hierarchy, we reconstruct source-transfer realizations from weak-operator limits of averages over source copies produced by quantum inflation. Consequently, the game-value promise problem for quantum tree networks in the mixed model is $\mathsf{coRE}$-complete.
Comments84 pages, 7 figures. v2: more informative title, typos corrected. Comments welcome!
