The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of $4\times4$ undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix $A\otimes I + I \otimes B$ does not exceed $(3d-4)/d^2$ with $A,B$ traceless $d\times d$ matrices and $||A||_F^2+||B||_F^2=1/d$ when $d=4$ . The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices $A$ and $B$ are normal. In this paper, we prove that the conjecture holds when one of matrices $A$ and $B$ is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of $4\times4$ undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix $A\otimes I + I \otimes B$ does not exceed $(3d-4)/d^2$ with $A,B$ traceless $d\times d$ matrices and $||A||_F^2+||B||_F^2=1/d$ when $d=4$. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices $A$ and $B$ are normal. In this paper, we prove that the conjecture holds when one of matrices $A$ and $B$ is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.

arXiv:1908.02428Aug 7, 20191 citation

A partial-trace matrix inequality and Werner-state distillability

Zhiwei Song, Lin Chen

Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.

quant-pharXiv:2607.23416Jul 28, 20260 citations

Entanglement distillation in terms of a conjectured matrix inequality

Yize Sun, Lin Chen

Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i \otimes S_i$ and Schmidt rank $K$ . We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$ .Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i \otimes S_i$ and Schmidt rank $K$. We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$.

arXiv:1909.02748Sep 6, 20190 citations

Towards feasible distillation of continuous-variable entanglement

Jens Eisert, Dan E. Browne, Martin Bodo Plenio +1 more

Jan 5, 20040 citations

4\times4

undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix

A\otimes I + I \otimes B

does not exceed

(3d-4)/d^2

with

A,B

traceless

d\times d

matrices and

||A||_F^2+||B||_F^2=1/d

when

d=4

. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices

A

and

B

are normal. In this paper, we prove that the conjecture holds when one of matrices

A

and

B

is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.

\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)

holds for any bipartite matrix

M=\sum_i R_i \otimes S_i

and Schmidt rank

K

. We prove that the conjecture holds for

M

with

K=3

and some special

M

with arbitrary

K