Finding the closest separable state to a given target state is a notoriously difficult task, even more difficult than deciding whether a state is entangled or separable. To tackle this task, we parametrize separable states with a neural network and train it to minimize the distance to a given target state, with respect to a differentiable distance, such as the trace distance or Hilbert-Schmidt distance. By examining the output of the algorithm, we can deduce whether the target state is entangled or not, and construct an approximation for its closest separable state. We benchmark the method on a variety of well-known classes of bipartite states and find excellent agreement, even up to local dimension of $d=10$. Moreover, we show our method to be efficient in the multipartite case, considering different notions of separability. Examining three and four-party GHZ and W states we recover known bounds and obtain novel ones, for instance for triseparability. Finally, we show how to use the neural network's results to gain analytic insight.

翻译：找到与特定目标国最接近的可分离状态是一项众所周知的困难任务, 甚至比决定一个国家是纠缠的还是分解的更困难。 为了完成这项任务, 我们将具有神经网络的可分离状态与神经网络对齐, 并训练它以尽量减少与特定目标国的距离, 如追踪距离或Hilbert- Schmidt距离。 通过检查算法的输出, 我们可以推断目标国是否是纠结的, 并为它最接近的可分离状态构建一个近似点。 我们把该方法以已知的两边国家类别为基准, 并找到极好的一致点, 甚至达到当地1 000美元。 此外, 我们还在多边案例中展示了我们的方法效率, 考虑不同的分离概念 。 检查三方和四方 GHZ 和 W 状态, 我们找到已知的界限, 并获得新的界限, 例如, 以三角性为例 。 最后, 我们展示如何使用神经网络的结果获得解析的洞察力 。