Two-copy distillability of Werner states

For every local dimension $d\geq 2$ , the Werner state $\rho_{\alpha}$ is two-copy undistillable if and only if $-\frac{1}{2}\leq\alpha\leq 1$ . Equivalently, $\rho_{\alpha}$ is two-copy distillable if and only if $\alpha<-\frac{1}{2}$ ; hence its one-copy and two-copy distillability thresholds coincide. In particular, for $d\geq 3$ and $-\frac{1}{2}\leq\alpha<-\frac{1}{d}$ , $\rho_{\alpha}$ is NPT but two-copy undistillable.

The problem asks whether any Werner state can be two-copy distillable without being one-copy distillable. It was identified as Problem 5 in the 2022 collection Five Open Problems in Quantum Information Theory.

Known results

Vianna and Doherty numerically established the qutrit interval $-\frac{1}{2}\leq\alpha< -\frac{1}{3}$ .
Costa Rico and Wolf proved two-copy undistillability for $\alpha\geq-\frac{1}{3}$ .
Earlier partial-trace inequalities established substantial additional ranges but not the full answer.

2026 claimed solution

Several arXiv preprints claim, for every $d\geq2$ , that Werner states are two-copy distillable exactly when $\alpha< -\frac{1}{2}$ . Thus the one-copy and two-copy thresholds coincide; in particular, the NPT interval $-\frac{1}{2}\leq\alpha< -\frac{1}{d}$ is claimed to be two-copy undistillable. Higher-copy and asymptotic distillability remain open.

Current status (as of July 2026): A complete negative answer is claimed in multiple arXiv preprints, but independent mathematical verification is still pending; higher-copy and asymptotic questions remain open.

Two-copy distillability of Werner states

Progress summary

Solutions 0