The distillability conjecture of two-copy \(4\times 4\) Werner states is one of the main open problems in quantum information (https://arxiv.org/abs/2002.03233, P. Horodecki, L. Rudnicki, and K. Zyczkowski). We prove three special cases of the conjecture in terms of the \(4\times 4\) non-normal matrices A, B involved in the conjecture. The first case, namely the main result of this paper, occurs when A, B are monomial matrices. Then, we apply it to the remaining two cases. One case occurs when A, B both have at most two nonzero entries. The other case works for rank-one A and some rank-two B. Our results present the latest progress on the conjecture.

两拷贝\（4×4） Werner状态的可蒸馏性猜想是量子信息中的主要开放问题之一（https://arxiv.org/abs/2002.03233、P.Horodecki、L.Rudnicki和K Zyczkowski）。我们根据猜想所涉及的\（4 \ times 4 \）非正规矩阵A，  B证明了猜想的三种特殊情况。第一种情况，即本文的主要结果，发生在A，  B是单项式矩阵时。然后，我们将其应用于其余两种情况。当A，B都具有最多两个非零条目时， 会发生一种情况。另一种情况适用于排名第一的A和排名第二的B。我们的结果提出了猜想的最新进展。