On Strong Superadditivity of the Entanglement of Formation

Abstract

We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation $E_F$ and the conjugate function $E^{\ast }$ of the entanglement function $E(ρ)=S(\trace_Aρ)$. We then consider the conjectured strong superadditivity of the entanglement of formation $E_F(\rho )\ge E_F({\rho }_I)+E_F({\rho }_{II})$, where ${\rho }_I$ and ${\rho }_{II}$ are the reductions of $\rho$ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of $E^{\ast }$. As an application, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for all non-trace-preserving quantum channels, when purity is measured by Schatten $p$-norms for $p$ tending to 1.

11 pages; refs added, explanatory improvements

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