The difference between two random mixed quantum states: exact and asymptotic spectral analysis
The difference between two random mixed quantum states: exact and asymptotic spectral analysis
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DOI：
10.1088/1751-8121/50/2/025301
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发表时间：
2015-11
期刊：
影响因子：
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通讯作者：
Jos'e Mej'ia;Camilo Zapata;A. Botero
中科院分区：
文献类型：
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作者：
Jos'e Mej'ia;Camilo Zapata;A. Botero
我们研究了两个密度矩阵的差异的谱统计性质，每个密度矩阵都是通过部分跟踪一个随机的二体纯量子态而独立获得的。我们首先展示了如何从差分矩阵的对角元素的联合概率密度函数中得到任意维的精确联合特征值概率密度函数的闭合形式，这是很容易计算的。随后，我们利用自由概率论的标准结果导出了差矩阵系综渐近特征值密度(AED)的相对简单的解析表达式，并利用Carlson定理得到了它的绝对矩的表达式。这些结果使我们可以用不同的距离度量来量化两个随机混合态之间的典型渐近距离；特别是，我们得到了算符范数距离和迹距离的几乎必然的渐近行为。
We investigate the spectral statistics of the difference of two density matrices, each of which is independently obtained by partially tracing a random bipartite pure quantum state. We first show how a closed-form expression for the exact joint eigenvalue probability density function for arbitrary dimensions can be obtained from the joint probability density function of the diagonal elements of the difference matrix, which is straightforward to compute. Subsequently, we use standard results from free probability theory to derive a relatively simple analytic expression for the asymptotic eigenvalue density (AED) of the difference matrix ensemble, and using Carlson’s theorem, we obtain an expression for its absolute moments. These results allow us to quantify the typical asymptotic distance between the two random mixed states using various distance measures; in particular, we obtain the almost sure asymptotic behavior of the operator norm distance and the trace distance.