Entanglement of purification for Werner states: canonical purification and nonadditivity

Abstract

We prove that the von Neumann entanglement of purification is nonadditive. For two-qubit Werner states $W(f)$ with singlet weight $1/64\le f\le1/4$, we show that canonical purification is globally optimal: its entanglement, also known as reflected entropy, gives the exact entanglement of purification. In contrast, the regularized entanglement of purification lies strictly below the reflected entropy for $0<f<f_\ast$, where $f_\ast\approx0.017$. Together, these results establish nonadditivity throughout $1/64\le f<f_\ast$. Our canonical-optimality proof generalizes the proof in [1], which established R\'enyi nonadditivity for $0\le\alpha<1$ using classical two-qubit states. We thus resolve the additivity question at $\alpha=1$. Remarkably, nonadditivity is again witnessed by separable states and no entanglement in the underlying states is required.