从秩 $k$ 投影子的认证到矩阵幂迹的非正性、纠缠与非厄米见证者
From Certifying Rank $k$ Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers
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中文总结 AI 辅助
本文系统研究了通过矩阵幂迹判定厄米矩阵特征值属于指定集合的充要条件，并利用这些刻画结果构造出能自动认证量子纠缠、非正性和非厄米性的见证者，通过数值与解析方法评估了其性能与资源需求。
中文摘要 AI 辅助
众所周知，密度矩阵 $\rho$ 是纯态当且仅当 $\mathrm{Tr} \\,\rho^2 = 1$。但这是否是利用其幂的迹来证明 $\rho$ 纯性的唯一方法？在此，我系统地研究了更一般问题的充要条件，即通过厄米矩阵幂的迹来保证其所有特征值属于指定集合。更重要的是，这些刻画结果自动成为见证者，用于认证量子态为纠缠态、厄米算子至少有一个负特征值以及线性算子为非厄米算子。我通过数值模拟和解析工作展示了这些见证者的有效性，分析了它们的性能，并研究了它们的优势、劣势以及资源需求。我还简要讨论了数值稳定性、测量不确定性和舍入误差对这些问题的效应。
英文摘要
We all know that a density matrix $ρ$ is pure if and only if $\mathrm{Tr} \,ρ^2 = 1$. But is this the only way to prove the purity of $ρ$ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.
发表机构
- University of Hong Kong（香港大学）
机构由 AI 辅助整理，请以论文原文为准。
