In many contexts one encounters Hermitian operators M on a Hilbert space whose dimension is so large that it is impossible to write down all matrix entries in an orthonormal basis. How does one determine whether such M is positive semidefinite? Here we approach this problem by deriving asymptotically optimal bounds to the distance to the positive semidefinite cone in Schatten p -norm for all integer $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) , assuming that we know the moments $$\mathbf {tr}(M^k)$$ tr ( M k ) up to a certain order $$k=1,\ldots , m$$ k = 1 , … , m . We then provide three methods to compute these bounds and relaxations thereof: the sos polynomial method (a semidefinite program), the Handelman method (a linear program relaxation), and the Chebyshev method (a relaxation not involving any optimization). We investigate the analytical and numerical performance of these methods and present a number of example computations, partly motivated by applications to tensor networks and to the theory of free spectrahedra.

中文翻译：

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几个时刻矩阵的正性的最优边界

在许多情况下，人们会在 Hilbert 空间上遇到 Hermitian 算子 M，该空间的维数太大以至于不可能在正交基中写下所有矩阵项。如何确定这样的 M 是否为半正定？在这里，我们通过为所有整数 $$p\in [1,\infty )$$ p ∈ [ 1 , ∞ ) 推导到 Schatten p 范数中到正半定锥的距离的渐近最优边界来解决这个问题，假设我们知道时刻 $$\mathbf {tr}(M^k)$$ tr ( M k ) 达到某个顺序 $$k=1,\ldots , m$$ k = 1 , ... , m 。然后，我们提供了三种方法来计算这些边界及其松弛：sos 多项式方法（半定程序）、Handelman 方法（线性程序松弛）和 Chebyshev 方法（不涉及任何优化的松弛）。