We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k-th condition involves comparing moments of the partially transposed density operator up to order k. Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.

我们提出了一组有序的实验可访问条件，用于检测混合状态下的纠缠。第k个条件涉及比较部分转置密度算子的矩直到k阶. 值得注意的是，所有矩不等式的并集再现了用于检测纠缠的 Peres-Horodecki 标准。我们的实证研究强调，前四种条件已经在各种量子架构中可靠地检测到混合态纠缠。利用对称性可以帮助进一步提高其检测能力。我们还展示了如何基于单个状态副本（经典阴影）的局部随机测量来估计矩不等式，并根据执行的测量次数得出统计上可靠的置信区间。我们的分析包括漂移源的实验相关情况，即不相同但独立的状态副本。