Quantum entanglement, symmetric nonnegative quadratic polynomials and moment problems
Quantum entanglement, symmetric nonnegative quadratic polynomials and moment problems
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DOI：
10.1007/s10107-020-01596-w
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发表时间：
2020-11
影响因子：
2.7
通讯作者：
Grigoriy Blekherman;Bharath Hebbe Madhusudhana
中科院分区：
文献类型：
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作者：
Grigoriy Blekherman;Bharath Hebbe Madhusudhana
量子态由具有单位迹（称为密度矩阵）的正半定埃尔米特算子表示。量子态的一个重要子集是可分离态，其补集是纠缠态的子集。我们证明，决定量子态是否纠缠的问题可以看作是实际分析中的矩问题。只有少量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。通过考虑异质原子的量子纠缠，我们得到了截断矩问题，该问题是为单位球的n个副本的乘积上的对称测量定义的。我们仅使用高达2阶的矩，因为这些矩最容易通过实验获得。我们推导出属于力矩锥的充分必要条件，它最多可以通过大小不等式的线性矩阵来表示，该矩阵不等式与n无关。线性矩阵不等式可以转换为一组显式半代数不等式，给出了矩锥隶属度的充要条件，并表明这两个条件在放大极限上相互接近。通过考虑非负多项式的双锥及其平方和松弛来导出不等式。我们证明，双锥的平方和松弛是渐近精确的，并且使用对称性约简技术（Blekherman 和 Riener：对称非负形式和平方和。arXiv：1205.3102，2012；Gatermann 和 Parrilo：J Pure Appl Algebra 192（1-3）：95-128。 https://doi.org/10.1016/j.jpaa.2003.12.011, 2004)，它最多可以写成一个大小最大的小的线性矩阵不等式，它与n无关。对于具有相关支持的对称非负多项式圆锥，我们还证明了全局非负对称多项式的半次原理的类似物（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)00301-9，2003)。
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 ofentangledstates. 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 ofnidentical atoms we arrive at the truncated moment problem defined for symmetric measures over a product ofncopies of unit balls in. 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, which is independent ofn. 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 largen. 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, which is independent ofn. 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).