On the two-copy distillability of Werner states and a new partial trace inequality

Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state $\varrho(4,-\tfrac12)$ is two-copy distillable, where $\varrho(d,\alpha)=(I+\alpha F)/(d^2+\alpha d)$. We answer it in the negative. To this end, we show the following stronger statement: for all $C\in M_{d_1d_2}(\mathbb{C})$ of rank at most $r \le d_1 d_2$, $\mathrm{tr}_1(C)\|_F^2+\|\mathrm{tr}_2(C)\|_F^2 \le r\|C\|_F^2+\frac{1}{r}|\mathrm{tr}(C)|^2$. A result by Costa Rico on the equivalence of this inequality with two-copy undistillability at $r = 2$ then settles Problem 5: $\varrho(4,-\tfrac{1}{2})$ is not two-copy distillable. Furthermore, we show that $\varrho(d,\alpha)$ is two-copy undistillable for every $d\ge2$, if and only if $\alpha\ge-\tfrac{1}{2}$. Thus, the one and two-copy distillability regions of $\varrho(d,\alpha)$ coincide. These results have been found and written up with AI tools, pointing towards a structural change affecting the field of quantum information and computation.