Proving the distillability problem of two-copy $$4\times 4$$ 4 × 4 Werner states for monomial matrices

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.