A SOLUTION TO 2-COPY DISTILLABILITY OF WERNER STATES

Entanglement distillation is the process of extracting high-quality entangled pairs from multiple copies of noisy, mixed quantum states using local operations and classical communication (LOCC). A central, long-standing problem in quantum information theory is whether every non-positive partial transpose (NPT) state is distillable. Because the distillability of general NPT states can be reduced to the study of the one-parameter family of Werner states, determining the distillability thresholds for these states is a canonical test for the general problem. This paper addresses the specific question of whether 2-copy distillability provides a larger range of distillable states than 1-copy distillability.

The authors analyze the 2-copy distillability of Werner states by examining the properties of Schmidt-rank-two test vectors. They establish a sharp geometric estimate for symmetric and antisymmetric tensor subspaces, specifically focusing on the projection onto the tensor product of two antisymmetric subspaces. By reformulating the 2-copy undistillability condition as a block-operator inequality, the authors relate the diagonal, off-diagonal, and transverse components of the problem to the variation of the Hilbert-Schmidt norm of the restricted antisymmetric projection. This allows them to prove that the 1-copy and 2-copy thresholds for Werner states coincide exactly.

The paper proves that for any dimension d ≥ 2, a Werner state is 2-copy undistillable if and only if it is 1-copy undistillable. This resolves a long-standing open question in the field. The result implies that for Werner states in the interval where they are NPT but 1-copy undistillable, having access to a second copy is insufficient to perform distillation. Consequently, if such states are distillable at all, they must require at least three copies, leaving the higher-copy and asymptotic questions as the next frontier for the NPT distillation problem.