自伴迹类算子基于矩的有效特征值近似
Effective eigenvalue approximation from moments for self-adjoint trace-class operators
AI总结：
针对自伴迹类算子，提出仅依赖前n个矩的谱近似集合$\Lambda_n$，可超指数收敛到极值特征值并构造严格界，无需对角化，适用于量子信息等多领域。
摘要：
有界线性算子的谱性质在数学和物理学的多个领域中发挥着关键作用。针对每个自伴迹类算子$O$，我们定义了一个集合$\Lambda_n\subset \mathbb{R}$，并证明在温和条件下，该集合在豪斯多夫度量下收敛于$O$的谱。我们提出的集合$\Lambda_n$仅依赖于$O$的前$n$个矩。我们证明，对于物理相关的算子，该集合可以被有效计算，且无需对角化就能很好地近似谱。\n 我们证明，使用上述方法，我们可以以超指数速度收敛到最小和最大特征值。\n 我们还构造了最小特征值的单调递增下界$q_n$（或最大特征值的单调递减上界）。该序列仅依赖于$O$的矩及其1-范数的具体上估计；我们还证明，对于一大类物理相关算子，$q_n$可以被有效计算。若$O$不是半正定的，这个严格的下界$q_n$会以超指数速度趋近于最小特征值。作为副产品，我们还得到了$O$的1-范数的可计算上界。\n 数值例子证明了我们的近似方法在估计熵和负性（negativity）方面的实用性，这一方法尤其在量子光学和开放量子系统模型等领域具有应用价值。这些结果可以直接应用于量子信息、统计力学和量子热力学中的问题，在这些领域中，使用基于对角化的传统技术是不切实际的。
英文摘要：
Spectral properties of bounded linear operators play a crucial role in several areas of mathematics and physics. For each self-adjoint, trace-class operator $O$ we define a set $Λ_n\subset \mathbb{R}$, and we show that it converges to the spectrum of $O$ in the Hausdorff metric under mild conditions. Our set $Λ_n$ only depends on the first $n$ moments of $O$. We show that it can be effectively calculated for physically relevant operators, and it approximates the spectrum well without diagonalization. We prove that using the above method we can converge to the minimal and maximal eigenvalues with super-exponential speed. We also construct monotone increasing lower bounds $q_n$ for the minimal eigenvalue (or decreasing upper bounds for the maximal eigenvalue). This sequence only depends on the moments of $O$ and a concrete upper estimate of its $1$-norm; we also demonstrate that $q_n$ can be effectively calculated for a large class of physically relevant operators. This rigorous lower bound $q_n$ tends to the minimal eigenvalue with super-exponential speed provided that $O$ is not positive semidefinite. As a by-product, we obtain computable upper bounds for the $1$-norm of $O$, too. Numerical examples demonstrate the relevance of our approximation in estimating entropy and negativity, which is useful, among others, in quantum optical and in open quantum system models. The results can be directly applicable to problems in quantum information, statistical mechanics, and quantum thermodynamics, where using traditional techniques based on diagonalization is impractical.
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