Volume of the set of separable states
Volume of the set of separable states
复制标题
DOI：
10.1103/physreva.58.883
复制
发表时间：
1998-08-01
影响因子：
2.9
通讯作者：
Lewenstein, M
中科院分区：
文献类型：
--
作者：
Zyczkowski, K;Horodecki, P;Lewenstein, M
考虑到所有量子状态集中有多少纠缠或可分离状态的问题。我们在密度矩阵RHO中提出了一种自然度量，描述了N维量子系统。我们证明，在此度量下，可分离状态的集合具有非零的体积。该体积的分析下限和上限也用于n = 2x2和n = 2x3的情况。最后，数值蒙特卡洛计算使我们能够估算可分离状态的体积，从而提供了数值证据，表明它随复合系统的尺寸呈指数下降。我们还分析了在固定纯度条件下的有条件可分离性度量。我们的结果表现出纯度和可分离性之间的清晰二元论：纠缠是纯状态的典型特征，而可分离性与量子混合物相连。特别是，纯度足够低的状态必然是可分离的。 [S1050-2947（98）02808-X]。
The question of how many entangled or, respectively, separable states there are in the set of all quantum states is considered. We propose a natural measure in the space of density matrices rho describing N-dimensional quantum systems. We prove that, under this measure, the set of separable states possesses a nonzero volume. Analytical lower and upper bounds of this volume are also derived for N=2X2 and N=2X3 cases. Finally, numerical Monte Carlo calculations allow us to estimate the volume of separable states, providing numerical evidence that it decreases exponentially with the dimension of the composite system. We have also analyzed a conditional measure of separability under the condition of fixed purity. Our results display a clear dualism between purity and separability: entanglement is typical of pure states, while separability is connected with quantum mixtures. In particular, states of sufficiently low purity are necessarily separable. [S1050-2947(98)02808-X].