Two-copy non-distillability of the two-ququart Werner state ρ(4, −1/2)

Show that the Werner state ρ(4, −1/2) on C^4 ⊗ C^4 is not 2-copy distillable; specifically, prove that there do not exist rank-2 projectors P and Q acting on (C^4)⊗2 such that (P ⊗ Q)[ρ(4, −1/2)Γ ⊗ ρ(4, −1/2)Γ](P ⊗ Q) has a negative eigenvalue.

Background

Werner states are U ⊗ U-invariant mixed states parameterized by α and dimension d. For α ∈ [−2/d, 1], they are 1-copy nondistillable, and for α ∈ [−1, −1/d), they are NPT. The specific state ρ(4, −1/2) is distinguished because its partial transpose is proportional to a dichotomic unitary.

A conjecture posits that Werner states which are not 1-copy distillable are not distillable at all, implying 2-copy non-distillability. Proving non-distillability for ρ(4, −1/2) would advance the broader NPT bound entanglement question and provide new insights into matrix-analytic inequalities related to Kronecker sums.