The characterization of transformations among entangled pure states via local operations assisted by classical communication (LOCC) is a crucial problem in quantum information theory for both theoretical and practical reasons. As LOCC has a highly intricate structure, sometimes the larger set of separable (SEP) maps is considered, which has a mathematically much simpler description. In the literature, mainly SEP maps consisting of invertible Kraus operators have been taken into account. In this paper we show that the consideration of those maps is not sufficient when deciding whether a state can be mapped to another via general SEP transformations. This is done by providing explicit examples of transformations among pure three- and five-qubit states, which are feasible via SEP maps containing singular Kraus operators, however, not possible via SEP maps containing solely regular Kraus operators. The key point that allows to construct the SEP maps is to introduce projective measurements that occur with probability zero on the input state. The fact that it is not sufficient to consider SEP maps composed out of regular Kraus operators even in the case of pure state transformations, also affects the results on LOCC transformations among pure states. However, we show that non-invertible Kraus operators do not help in state transformations under LOCC with finitely many rounds of classical communication, i.e. the necessary and sufficient condition for SEP transformations with invertible Kraus operators is still a necessary condition for convertibility under finite-round LOCC. Moreover, we show that the results on transformations via SEP that are not possible with LOCC (including infinitely many rounds of classical communication) presented in Hebenstreit et al 2016 Phys. Rev. A 93, 012339 are not affected.

由于理论和实践的原因，通过经典通信（LOCC）辅助的局部操作来表征纠缠纯态之间的转换是量子信息理论中的一个关键问题。由于 LOCC 具有高度复杂的结构，有时会考虑更大的可分离 (SEP) 映射集，其在数学上的描述要简单得多。在文献中，主要考虑了由可逆克劳斯算子组成的 SEP 图。在本文中，我们表明在决定一个状态是否可以通过一般 SEP 转换映射到另一个状态时，对这些映射的考虑是不够的。这是通过提供纯三和五量子位状态之间转换的明确示例来完成的，这通过包含奇异克劳斯算子的 SEP 映射是可行的，但是，不可能通过仅包含常规 Kraus 算子的 SEP 映射。允许构建 SEP 映射的关键点是引入在输入状态上发生概率为零的投影测量。即使在纯状态转换的情况下，考虑由常规克劳斯算子组成的 SEP 映射也是不够的，这一事实也会影响纯状态之间 LOCC 转换的结果。然而，我们表明不可逆的克劳斯算子在有限多轮经典通信下的 LOCC 状态转换中没有帮助，即具有可逆克劳斯算子的 SEP 变换的充要条件仍然是有限轮下可转换的必要条件LOCC。而且，等人2016物理。Rev. A 93 , 012339 不受影响。