分离函数、概率与Dyson指数相关性的扩展研究
Extended Studies of Separability Functions and Probabilities and the Relevance of Dyson Indices
摘要：
我们报告了分离性函数的研究及其在计算实数、复数和四元数量子比特-量子比特和量子比特-量子比特系统的分离性概率中的应用的实质性进展。我们扩展了我们最近的工作（arXiv：0704.3723），其中戴森指数的随机矩阵理论发挥了重要作用，包括使用不仅体积元素的希尔伯特-施密特（HS）度量，但也Bures（最小单调）度量的措施，这些有限维量子系统。此外，我们现在采用欧拉角参数化的密度矩阵（ρ），除了布洛尔参数化。最小简并复双量子比特态的欧拉角可分性函数可以很好地拟合为参与比的六次方，R（ρ）=1/Tr（ρ）^2。此外，用Verstraete-Audenaert-De Moor函数（arXiv：quant-oh/0011111）的简单线性变换替换R（rho），我们发现真实和复杂（非简并）两量子比特场景与戴森指数行为密切相关。报告中的几个分析有助于加强我们的假设，即复杂双量子比特态的HS和Bures可分性概率分别为8/33 = 0.242424和1680（sqrt{2}-1）/pi^8 = 0.733389。在欧拉角可分性函数的作用下，采用某些正则化的β函数，我们紧密地再现了-与戴森指数分析一致-几个HS两量子比特可分性概率图。
英文摘要：
We report substantial progress in the study of separability functions and their application to the computation of separability probabilities for the real, complex and quaternionic qubit-qubit and qubit-qutrit systems. We expand our recent work (arXiv:0704.3723), in which the Dyson indices of random matrix theory played an essential role, to include the use of not only the volume element of the Hilbert-Schmidt (HS) metric, but also that of the Bures (minimal monotone) metric as measures over these finite-dimensional quantum systems. Further, we now employ the Euler-angle parameterization of density matrices (rho), in addition to the Bloore parameterization. The Euler-angle separability function for the minimally degenerate complex two-qubit states is well-fitted by the sixth-power of the participation ratio, R(rho)=1/Tr(rho)^2. Additionally, replacing R(rho) by a simple linear transformation of the Verstraete-Audenaert-De Moor function (arXiv:quant-oh/0011111), we find close adherence to Dyson-index behavior for the real and complex (nondegenerate) two-qubit scenarios. Several of the analyses reported help to fortify our conjectures that the HS and Bures separability probabilities of the complex two-qubit states are 8/33 = 0.242424 and 1680 (sqrt{2}-1)/pi^8 = 0.733389, respectively. Employing certain regularized beta functions in the role of Euler-angle separability functions, we closely reproduce--consistently with the Dyson-index ansatz--several HS two-qubit separability probability conjectures.
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