Duistermaat-Heckman measure and the mixture of quantum states
Duistermaat-Heckman measure and the mixture of quantum states
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Duistermaat-Heckman 测度和量子态混合
DOI：
10.1088/1751-8121/ab5297
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发表时间：
2019
影响因子：
2.1
通讯作者：
Wu Junde
中科院分区：
文献类型：
--
作者：
Zhang Lin;Jiang Yixin;Wu Junde
本文提出了一个通用的框架来解决随机矩阵理论中的一个基本问题，即描述两个独立随机Hermitian矩阵和的特征值的联合分布问题。关于量子态混合的一些考虑基本上包含在上述数学问题中。相反，我们专注于推导的频谱密度的混合伴随轨道的量子态的Duistermaat-Heckman措施，起源于辛几何理论。基于这种方法，我们可以得到独立随机态混合的谱密度。特别地，我们得到了随机量子比特混合的显式公式。我们还发现，在二能级量子系统中，从随机态系综中选择的n个随机密度矩阵的等概率混合物的平均熵（在文中指定）随着数量n而增加。因此，作为一个物理应用，我们的结果定量地解释了混合物的量子相干性随着混合物中组分数n的增加而统计地单调减小。此外，我们的方法还可以用来研究一类特殊的单位量子比特通道的统计性质。
In this paper, we present a general framework to solve a fundamental problem in random matrix theory (RMT), ie the problem of describing the joint distribution of eigenvalues of the sum of two independent random Hermitian matrices and. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of the Duistermaat–Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of n random density matrices chosen from a random state ensemble (specified in the text) increases with the number n. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components n in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.