Two-copy distillability of the two-ququart Werner state at $\alpha = -1/2$
In plain words
A sharper test case: one specific entangled state of two four-level systems. The task is to show that even two copies of it cannot be processed into an entangled pair.
Precise statement
Show that the $d = 4$ Werner state $\rho(4, -1/2)$ is not 2-copy distillable: for all rank-2 projectors P, Q acting on two copies of each party's space, $\left(P\,\otimes\,Q\right)\left[(\rho^\Gamma)^{\otimes 2}\right]\left(P\,\otimes\,Q\right)$ has no negative eigenvalue; or exhibit P, Q giving a negative eigenvalue. Its partial transpose is proportional to a unitary (Problem 5 of Horodecki, Rudnicki and Zyczkowski 2022).
What would settle it
A proof of positivity over all rank-2 projector pairs, or explicit projectors giving a negative eigenvalue.
Parent problems
- Special case of Do bound entangled states with negative partial transpose exist? · open · from revision 1 The parent was revised since revision 1: it now states another question.
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