Employing the general bilateral controlled-NOT operation and local state discrimination, the mixed state of the form ${\ensuremath{\rho}}_{d}^{(k)}{=(1/d}^{2}){\ensuremath{\sum}}_{m,n=0}^{d\ensuremath{-}1}(|{\ensuremath{\varphi}}_{\mathrm{mn}}〉〈{\ensuremath{\varphi}}_{\mathrm{mn}}|{)}^{\ensuremath{\bigotimes}k}$ is proved to be quasipure, where ${|{\ensuremath{\varphi}}_{\mathrm{mn}}〉}$ is the canonical set of mutually orthogonal maximally entangled states in $d\ifmmode\times\else\texttimes\fi{}d.$ Therefore irreversibility does not occur in the process of distillation for this family of states. Also, the distillable entanglement is calculated explicitly.
Whether entanglement with a negative partial transpose (NPT) can be undistillable is a longstanding open problem. In the canonical family of DiVincenzo \textit{et al.}, the one-copy-undistillable region was conjectured to remain undistillable at all copy numbers. We refute this: in every dimension
d≥3, one vertex of this region is two-copy distillable, via an explicit Schmidt-rank-two tight-frame witness. The separation persists on an open set, while other states in the same region are provably two-copy undistillable. Twirling our counterexample into Werner form makes it two-copy undistillable, so this standard reduction can erase finite-copy distillability.
We present an approach to a noncommutativelike phase space which allows to analyze quasifree states on the algebra of canonical anti-commutation relations (CAR) in analogy to quasifree states on the algebra of canonical commutation relations (CCR). The used mathematical tools are based on a new algebraic structure the “Grassmann algebra of canonical anticommutation relations” (GAR algebra) which is given by the twisted tensor product of a Grassmann and a CAR algebra. As a new application, the corresponding theory provides an elegant tool for calculating the fidelity of two quasifree fermionic states which is needed for the study of entanglement distillation within fermionic systems.