AI 中文总结
该研究区分密度算子贝叶斯网络的两种构造，证明内在构造的马尔可夫性质等价性等结论，揭示外在构造的阻碍，明确序不变性对传递DAG的充分性，相关量在DAG马尔可夫等价下不变。
AI 中文摘要
我们研究有限维张量积希尔伯特空间上正定密度算子的有向无环图（DAG）贝叶斯网络的量子类似物，区分两种构造方式：内在构造从联合态及其条件独立性性质出发；外在构造遵循与DAG箭头兼容的序，从规定的局域量子核逐步组装态。对于内在构造，我们证明了有序、局域、全局有向马尔可夫性质的等价性，以及熵、递归因式分解和对数表征。外在构造总能给出归一化态，并将每个核恢复为所有前置系统上的条件，但同一核未必能从顶点与其父节点的边缘态中恢复，一个三量子比特示例展示了这种阻碍。我们证明，所选拓扑序的独立性恰好是传递DAG的充分条件：此时每个序不变核族都会产生内在有向马尔可夫态。最后，我们为每个正定态和DAG关联一个对数候选量，证明其是次归一化的，并证明迹为1的候选量是有向马尔可夫态，该候选量与过剩全局信息在DAG马尔可夫等价下不变。
英文摘要
We study quantum analogues of Bayesian networks on a directed acyclic graph (DAG), distinguishing two constructions for positive definite density operators on finite-dimensional tensor-product Hilbert spaces. The intrinsic construction starts from a joint state and its conditional-independence properties. The extrinsic construction assembles a state sequentially from prescribed local quantum kernels, following an ordering compatible with the arrows of the DAG. For the intrinsic construction, we prove the equivalence of the ordered, local, and global directed Markov properties, together with entropy, recursive-factorization, and logarithmic characterizations. The extrinsic construction always gives a normalized state and recovers each kernel as a conditional on all preceding systems. The same kernel, however, need not be recovered from the marginal on the vertex and its parents; a three-qubit example exhibits this obstruction. We prove that independence of the chosen topological ordering is sufficient exactly for transitive DAGs: every order-invariant kernel family then yields an intrinsically directed Markov state. Finally, we associate a logarithmic candidate with every positive definite state and DAG, prove that it is subnormalized, and show that the trace-one candidate is a directed Markov state. Both the candidate and the excess global information are invariant under DAG Markov equivalence.
