Qualification of quantum states judges which of two sets of quantum states an unknown state lies in, where the two sets are labeled by two distinct parameter regions. We formulate this problem as a composite quantum hypothesis test and uncover universal scaling laws for the minimum error probability for $N$ copies. Taking polarization-direction qualification and purity qualification as examples, we show that the $N$-copy permutation symmetry and the geometric symmetries of the parameter regions identify the optimal measurements and the"worst pairwise states". The minimum error probability scales as $N^{-3/2}\exp(-N\xi)$ for disjoint regions and as $(NF)^{-1/2}$ for adjacent regions, where $\xi$ and $F$ are the quantum Chernoff divergence and quantum Fisher information associated with the"worst pairwise states", respectively. With the minimum error probability serving as an order parameter, the transition between the scaling behaviors becomes a second-order phase transition as $N\to\infty$. Our approach determines whether a quantum state belongs to a given set without full state tomography, thereby enabling qualification of large ensembles using finite samples.
Zhaoyu Fei, Yaotian Li, Weicheng Huang et al.· 0 citations
Predicting the percolation threshold of highly clustered networks from local statistics remains difficult, because short loops break the independence assumption underlying tree-like message passing. Existing remedies address loopy connectivity either through prescribed local motifs in random-graph ensembles or through a single network's realized topology, leaving an ensemble-level treatment of arbitrary connectivity patterns absent. Here, we develop a loopy message-passing framework for random clustered graph ensembles based on generalized-edge statistics, which characterize overlap patterns among the neighborhoods of different nodes. This yields a progressively refined approximation scheme based on neighborhoods of increasing size around each node. The low-order approximations recover previous equations for random network ensembles, and the new result that yields refined threshold prediction is developed by the second-order approximation. We show that the effectiveness of this framework depends not only on short-cycle density but also on the internal consistency of generalized edges. To diagnose this effectiveness, we introduce the generalized-edge closure coefficient (GECC) to quantify this consistency. Because GECC is computed entirely from local statistics and does not rely on any percolation calculation, it serves as an a priori diagnostic for the reliability of the approximation. Using synthetic and real networks, the threshold is evaluated via the second-order and lower-order approximations. Comparisons with Monte Carlo simulations show that GECC captures key structural features that strongly affect the percolation threshold. These results establish ensemble-based loopy message passing as an efficient route for predicting the percolation threshold in large clustered networks.
L.-H. Wang, Y.-M. Du· 0 citations
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