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

Abstract

Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state $\varrho (4,-\frac{1}{2})$ 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 $C\in M_{d_1{d}_2}(\mathbb{C})$ of rank at most $r\le d_1{d}_2$, ${\mathrm{tr}}_1(C){\Vert }_F^2+\Vert {\mathrm{tr}}_2(C){\Vert }_F^2\le r\Vert C{\Vert }_F^2+\frac{1}{r}|\mathrm{tr}(C){|}^2$. A result by Costa Rico on the equivalence of this inequality with two-copy undistillability at $r=2$ then settles Problem 5: $\varrho (4,-\frac{1}{2})$ is not two-copy distillable. Furthermore, we show that $\varrho (d,\alpha )$ is two-copy undistillable for every $d\ge 2$, if and only if $\alpha \ge -\frac{1}{2}$. 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.

13 pages. Found and written up with GPT Sol 5.6, Claude Fable, and Claude Opus. Comments welcome!