Quantum states are represented by positive semidefinite Hermitian operators with unit trace, known as density matrices. An important subset of quantum states is that of separable states, the complement of which is the subset of entangled states. We show that the problem of deciding whether a quantum state is entangled can be seen as a moment problem in real analysis. Only a small number of such moments are accessible experimentally, and so in practice the question of quantum entanglement of a many-body system (e.g, a system consisting of several atoms) can be reduced to a truncated moment problem. By considering quantum entanglement of n identical atoms we arrive at the truncated moment problem defined for symmetric measures over a product of n copies of unit balls in $${\mathbb {R}}^d$$ . We work with moments up to degree 2 only, since these are most readily available experimentally. We derive necessary and sufficient conditions for belonging to the moment cone, which can be expressed via a linear matrix inequality of size at most $$2d+2$$ , which is independent of n. The linear matrix inequalities can be converted into a set of explicit semialgebraic inequalities giving necessary and sufficient conditions for membership in the moment cone, and show that the two conditions approach each other in the limit of large n. The inequalities are derived via considering the dual cone of nonnegative polynomials, and its sum-of-squares relaxation. We show that the sum-of-squares relaxation of the dual cone is asymptotically exact, and using symmetry reduction techniques (Blekherman and Riener: Symmetric nonnegative forms and sums of squares. arXiv:1205.3102 , 2012; Gatermann and Parrilo: J Pure Appl Algebra 192(1–3):95–128. https://doi.org/10.1016/j.jpaa.2003.12.011 , 2004), it can be written as a small linear matrix inequality of size at most $$2d+2$$ , which is independent of n. For the cone of symmetric nonnegative polynomials with the relevant support we also prove an analogue of the half-degree principle for globally nonnegative symmetric polynomials (Riener: J Pure Appl Algebra 216(4): 850–856. https://doi.org/10.1016/j.jpaa.2011.08.012 , 2012; Timofte: J Math Anal Appl 284(1):174–190. https://doi.org/10.1016/S0022-247X(03)00301-9 , 2003).

中文翻译：

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量子纠缠、对称非负二次多项式和矩问题

量子态由具有单位迹的正半正定 Hermitian 算子表示，称为密度矩阵。量子态的一个重要子集是可分离态，其补充是纠缠态的子集。我们表明，决定量子态是否纠缠的问题可以看作是实分析中的矩问题。只有少数这样的矩可以通过实验获得，因此在实践中，多体系统（例如，由多个原子组成的系统）的量子纠缠问题可以简化为截断矩问题。通过考虑 n 个相同原子的量子纠缠，我们得出了为 $${\mathbb {R}}^d$$ 中单位球的 n 个副本的乘积的对称度量定义的截断矩问题。我们只处理 2 级以下的时刻，因为这些最容易通过实验获得。我们推导出属于矩锥的充分必要条件，它可以通过大小至多 $$2d+2$$ 的线性矩阵不等式表示，它与 n 无关。线性矩阵不等式可以转化为一组显式的半代数不等式，给出矩锥成员资格的充分必要条件，并表明这两个条件在大 n 的极限内相互接近。不等式是通过考虑非负多项式的对偶锥及其平方和松弛得出的。我们证明了对偶锥的平方和松弛是渐近精确的，并且使用对称约简技术（Blekherman 和 Riener：对称非负形式和平方和。arXiv：1205.3102，2012；盖特曼和帕里洛：J Pure Appl Algebra 192(1–3):95–128。https://doi.org/10.1016/j.jpaa.2003.12.011 , 2004)，它可以写成一个大小最多为 $$2d+2$$ 的小线性矩阵不等式，它与 n 无关。对于具有相关支持的对称非负多项式锥，我们还证明了全局非负对称多项式的半度原理的类似物（Riener：J Pure Appl Algebra 216（4）：850-856。https://doi.org /10.1016/j.jpaa.2011.08.012 , 2012; Timofte: J Math Anal Appl 284(1):174–190. https://doi.org/10.1016/S0022-247X(03)00301-9) .