基于超几何/差分方程的随机诱导测度广义双量子比特态可分性概率公式及其渐近性
Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure
摘要：
我们找到了等价的基于超几何和差分方程的公式，$Q(k,α)= G_1^k(α) G_2^k(α)$，用于$k = -1, 0, 1,\ldots,9$，用于赋予随机诱导测度的广义双量子比特态的总可分性概率的（有理数值）部分，其中行列式不等式$|ρ^{PT}| >|ρ|$成立。这里$ρ$表示$4 \times 4$密度矩阵和$ρ^{PT}$，它的部分转置，而$α$是一个戴森指数样参数，$α= 1$用于标准（15维）凸集的两个量子比特状态。嵌入这些密度矩阵的空间的维数是$4 \times (4 +k)$。对于$k=0$的对称情况，我们得到了先前报道的希尔伯特-施密特公式，其中（两个re [al]比特的情况）$Q(0,\frac{1}{2}) = \frac{29}{128}$，（标准两个qubit的情况）$Q(0,1)=\frac{4}{33}$，和（两个quater [nionic]比特的情况）$Q(0,2)= \frac{13}{323}$。因子$G_2^k(α)$可以写为加权超几何函数$_{p}F_{p-1}$、$p \geq 7$的和，所有这些函数都具有自变量$\frac{27}{64} =(\frac{3}{4})^3$。我们发现这些功能的上，下参数集的公式，然后，等价地表示$G_2^k(α)$的一阶差分方程。因子$G_1^k(α)$等于$(\frac{27}{64})^{α-1}$乘以涉及所示参数的六个Pochhammer符号的乘积的比率。一些显著的$α-$和$k$特有的不变渐近性质（同样，涉及$\frac{27}{64}$和相关量）的可分性概率公式出现。
英文摘要：
We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,α)= G_1^k(α) G_2^k(α)$, for $k = -1, 0, 1,\ldots,9$, for that (rational-valued) portion of the total separability probability for generalized two-qubit states endowed with random induced measure, for which the determinantal inequality $|ρ^{PT}| >|ρ|$ holds. Here $ρ$ denotes a $4 \times 4$ density matrix and $ρ^{PT}$, its partial transpose, while $α$ is a Dyson-index-like parameter with $α= 1$ for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is $4 \times (4 +k)$. For the symmetric case of $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard two-qubit case) $Q(0,1)=\frac{4}{33}$, and (the two-quater[nionic]bit case) $Q(0,2)= \frac{13}{323}$. The factors $G_2^k(α)$ can be written as the sum of weighted hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $\frac{27}{64} =(\frac{3}{4})^3$. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express $G_2^k(α)$ in terms of first-order difference equations. The factors $G_1^k(α)$ are equal to $(\frac{27}{64})^{α-1}$ times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable $α-$ and $k$-specific invariant asymptotic properties (again, involving $\frac{27}{64}$ and related quantities) of separability probability formulas emerge.
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