基于贝尔测试的高维纠缠直接自适应认证
Direct Adaptive Certification of High-Dimensional Entanglement with Bell Tests
AI总结：
本文将CSPSA扩展至高维贝尔测试，基于CGLMP不等式通过数值研究实现高维纠缠直接自适应认证，在d=2-8范围内验证了其有效性，且测量成本与维度无关。
AI中文摘要：
纠缠光子在量子应用中发挥关键作用，确定并表征其纠缠是有效利用它们的核心。高维纠缠态提供了更丰富的可能性，但额外的测量自由度使其表征难度显著提升。此前基于复杂同时扰动随机近似（CSPSA）的自适应贝尔测试方法主要聚焦于量子比特。本文通过数值研究一种基于贝尔不等式违背的未知量子态纠缠检测方法，将CSPSA扩展至高维贝尔测试，采用适用于二分qudit的Collins-Gisin-Linden-Massar-Popescu（CGLMP）不等式。所得协议无需先重构未知纠缠态的密度矩阵，即可检测纯态或混合态中的贝尔非局域关联。针对每个维度d，在100次独立有限采样运行中，每次运行进行100次优化迭代，我们展示了d=2至8范围内的已认证CGLMP违背情况；对于在可见度仅高于标准傅里叶违背阈值0.05的各向同性混合态，同样在研究的d范围内观察到置信度认证的CGLMP违背。我们将该直接随机方法与作为未知态表征标准方法的量子态层析成像进行比较：在匹配基准中，从d=6开始CSPSA每次尝试使用的测量配置更少，而层析成像在d=8以内每个认证结果所需的探测对更少。此外，我们推导了CGLMP参数的相位依赖关系，阐明了其决定自适应搜索的特征。由于每次CSPSA迭代的测量设置成本与维度无关，该方法为高维纠缠认证提供了极具吸引力的途径。
英文摘要：
Entangled photons play a crucial role in quantum applications, and determining and characterising their entanglement is vital to using them effectively. High-dimensional entangled states offer richer possibilities, but their additional measurement degrees of freedom make them increasingly demanding to characterise. However, adaptive Bell-test methods based on complex simultaneous perturbation stochastic approximation (CSPSA) have so far focused mainly on qubits. Here we numerically investigate a Bell-inequality-violation-based method for detecting entanglement in unknown quantum states. We extend CSPSA to high-dimensional Bell testing by using the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality for bipartite qudits. The resulting protocol can detect Bell-nonlocal correlations in unknown entangled states, whether pure or mixed, without first reconstructing their density matrix. Using 100 optimisation iterations in each of 100 independent finite-shot runs per number of dimensions d, we demonstrate certified CGLMP violations throughout d=2-8. For isotropic mixed states tested at a visibility of just 0.05 above the standard-Fourier violation threshold, we likewise observe confidence-certified CGLMP violations throughout the range of d studied. We compare this direct stochastic approach with quantum state tomography, the standard method for characterising an unknown state. In the matched benchmark, CSPSA uses fewer measurement configurations per attempt from d=6, whereas tomography requires fewer detected pairs per certified result through d=8. We also derive the phase dependence of the CGLMP parameter and clarify the features of its landscape that govern the adaptive search. Because the measurement-setting cost of each CSPSA iteration is independent of dimension, the method offers a particularly attractive route to the certification of high-dimensional entanglement.
