近日，香港大学Saptarshi Ro团队研究了纯高斯态真多模纠缠的典型行为。这一研究成果发表在2025年6月10日出版的《物理评论A》杂志上。

研究组探讨了每模平均能量固定的Haar均匀生成多模纯高斯态中真实纠缠的趋势。基于距离的度量称为广义几何度量（GGM），用于量化真实纠缠。一个状态的GGM被定义为它与所有非完全纠缠态集的最小距离。首先，研究组推导出了在能量约束状态集上定义的任何函数的Haar平均值的表达式。随后，他们研究了具有大量模式的状态，并根据每种模式的平均能量为Haar平均GGM提供了一个闭式表达式。此外，还证明了典型状态非常接近其Haar平均GGM值，偏差概率受指数抑制极限的限制。 

然后，研究组分析了具有有限个模的典型态的GGM含量，并给出了GGM的分布。该研究结果表明，随着模数的增加，分布向更高的纠缠值转变，变得更加集中。他们通过计算Haar平均GGM和GGM分布的标准差来量化这些特征，揭示了前者随着模式数量的增加而增加，而后者随着模式数量减少而减少。最后，研究组比较了GGM的典型值及其与施加的能量界限一致的最大值。

附：英文原文

Title: Typical behavior of genuine multimode entanglement of pure Gaussian states

Author: Saptarshi Roy

Issue&Volume: 2025/06/10

Abstract: Trends of genuine entanglement in Haar uniformly generated multimode pure Gaussian states with fixed average energy per mode are explored. A distance-based metric known as the generalized geometric measure (GGM) is used to quantify genuine entanglement. The GGM of a state is defined as its minimum distance from the set of all nongenuinely entangled states. To begin with, we derive an expression for the Haar-averaged value of any function defined on the set of energy-constrained states. Subsequently, we investigate states with a large number of modes and provide a closed-form expression for the Haar-averaged GGM in terms of the average energy per mode. Furthermore, we demonstrate that typical states closely approximate their Haar-averaged GGM value, with deviation probabilities bounded by an exponentially suppressed limit. We then analyze the GGM content of typical states with a finite number of modes and present the distribution of GGM. Our findings indicate that, as the number of modes increases, the distribution shifts towards higher entanglement values and becomes more concentrated. We quantify these features by computing the Haar-averaged GGM and the standard deviation of the GGM distribution, revealing that the former increases while the latter decreases with the number of modes. Finally, we compare the typical value of the GGM with its maximal value consistent with the imposed energy bound.

DOI: 10.1103/8bbb-jhzh

Source: https://journals.aps.org/pra/abstract/10.1103/8bbb-jhzh