实值、小素数基Qubit-Qutrit和Rebit-Retrit秩-4/秩-6猜想的Hilbert-Schmidt可分性概率比
Rational-Valued, Small-Prime-Based Qubit-Qutrit and Rebit-Retrit Rank-4/Rank-6 Conjectured Hilbert-Schmidt Separability Probability Ratios
摘要：
我们实现了一个程序的基础上的Wishart-Laguerre分布-最近概述了{}yczkowski和Khvedelidze，Rogojin和Abgaryan，随机（复杂的或真实的）$N \times N$密度矩阵的秩$k \leq N$相对于希尔伯特-施密特（HS）措施的生成。在复杂的情况下，从维度为$k \times k+ 2 (N-k)$的Ginibre矩阵$A$开始，而对于真实场景，采用维度为$k \times k+1+ 2 (N-k)$的Ginibre矩阵$B$。然后，将$k \times k$乘积$A A^{\dagger}$或$B B^T$对角化-用零填充到大小为$N \times N$ -并旋转，从而获得随机密度矩阵。对于8亿个Ginibre矩阵实现，执行秩4重比特重试状态的过程，发现6，192，047个是可分离的，样本概率为.00774006-暗示精确值$\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^4}=.0774$。关于满秩重比特-重比特系统的HS可分性概率的一个猜想是$\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775$（两重比特对应系统已被证明是$\frac{29}{64}=\frac{29}{2^6}$）。根据这些假设，秩4概率与秩6概率的比率将是$\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049$，公因子43被取消。至于中间秩5概率，Szarek，Bengtsson和{}ycskowski在2006年的一个定理告诉我们，它必须是秩6概率的一半-本身被证明是$\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3}$，而对于秩3或更小，相关概率必须是0，这是Ruskai和Werner在2009年的结果。在这些方面，我们重新审视了2005年我们的量子位-四分之一分析，现在找到了$\frac{70}{2673}=\frac{2 \cdot 5 \cdot 7}{ 3^5 \cdot 11} \approx 0.0261878$等级-4与等级-6概率比的证据。
英文摘要：
We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) $N \times N$ density matrices of rank $k \leq N$ with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix $A$ of dimensions $k \times k+ 2 (N-k)$, while for a real scenario, one employs a Ginibre matrix $B$ of dimensions $k \times k+1+ 2 (N-k)$. Then, the $k \times k$ product $A A^{\dagger}$ or $B B^T$ is diagonalized-padded with zeros to size $N \times N$-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value $\frac{387}{5000} =\frac{3^2 \cdot 43}{2^3 \cdot 5^4}=.0774$. A conjecture for the HS separability probability of rebit-retrit systems of full rank is $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775$ (the two-rebit counterpart has been proven to be $\frac{29}{64}=\frac{29}{2^6}$). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be $\frac{59049}{1000000}=\frac{3^{10}}{2^6 \cdot 5^6} \approx 0.059049$, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {Ż}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be $\frac{27}{1000} =\frac{3^3}{2^3 \cdot 5^3}$, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a $\frac{70}{2673}=\frac{2 \cdot 5 \cdot 7}{ 3^5 \cdot 11} \approx 0.0261878$ rank-4 to rank-6 probability ratio.
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