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

• URL: http://arxiv.org/abs/2607.24309v1
• Date: Mon, 27 Jul 2026 11:51:18 GMT
• Title: On the two-copy distillability of Werner states and a new partial trace inequality
• Abstract summary: Problem 5 in it Five Open Problems in Quantum Information Theory [PRX Quantum 3, 01 r (2022), asks whether the two-ququart Werner state $varrho(4,-tfrac12)$ is two-copy distillable, where $varrho(d,)=(I+F)/(d2+d)$.<n>We show that $varrho(d,) is two-copy undistillable for every $dge2$, if and only if $ge-frac12$.
• Score: 1.3999481573773072
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state $\varrho(4,-\tfrac12)$ is two-copy distillable, where $\varrho(d,α)=(I+αF)/(d^2+αd)$. 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 $\varrho(d,α)$ is two-copy undistillable for every $d\ge2$, if and only if $α\ge-\tfrac{1}{2}$. Thus, the one and two-copy distillability regions of $\varrho(d,α)$ 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.

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