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 $\varrho(4,-\tfrac12)$ 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.

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

Quantum Physics (quant-ph); Rings and Algebras (math.RA)

arXiv:2607.24309 [quant-ph]

(or arXiv:2607.24309v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.24309 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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