随机诱导态的绝对正部分转置性质
The absolute positive partial transpose property for random induced states
摘要：
本文首先得到了中心Wishart矩阵矩的一个代数公式，并应用它得到了分布的两个参数以不同的速度趋于无穷大时在大维数极限下的新的收敛结果。我们利用这个结果研究了绝对正部分转置（APPT）量子态.我们表明，$\C^d=\C^{d_1} \otimes \C^{d_2}$上的二分随机诱导态的阈值，通过部分跟踪$\C^d \otimes \C^s$上的随机纯态，是APPT发生，如果环境维度$s$是顺序$s_0=\min(d_1, d_2)^3 \max(d_1, d_2)$。也就是说，当$s \geq Cs_0$时，这种随机诱导状态是具有大概率的APPT，而当$s \leq cs_0 $时，这种随机状态不是具有大概率的APPT。此外，我们还有效地计算了$C$和$c$，并证明了当$\min(d_1, d_2)^{2}\ll d$时，用相同的锐跃迁常数代替它们是可能的。
英文摘要：
In this paper, we first obtain an algebraic formula for the moments of a centered Wishart matrix, and apply it to obtain new convergence results in the large dimension limit when both parameters of the distribution tend to infinity at different speeds. We use this result to investigate APPT (absolute positive partial transpose) quantum states. We show that the threshold for a bipartite random induced state on $\C^d=\C^{d_1} \otimes \C^{d_2}$, obtained by partial tracing a random pure state on $\C^d \otimes \C^s$, being APPT occurs if the environmental dimension $s$ is of order $s_0=\min(d_1, d_2)^3 \max(d_1, d_2)$. That is, when $s \geq Cs_0$, such a random induced state is APPT with large probability, while such a random states is not APPT with large probability when $s \leq cs_0 $. Besides, we compute effectively $C$ and $c$ and show that it is possible to replace them by the same sharp transition constant when $\min(d_1, d_2)^{2}\ll d$.
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