Strong Converse Bounds for Compression of Mixed States

In this paper, we study strong converse properties for both visible and blind compression of mixed states. The optimal rate of a visible compression scheme is obtained in terms of the entanglement of purification, whose additivity remains unknown so far. For a variation of extendible states, we prove that the entanglement of purification is additive and apply this to obtain a “pretty strong” converse bound for the blind and visible compression of such states. Namely, when the rate decreases below the optimal rate, the error exhibits a discontinuous jump from 0 to at least $\frac{1}{3\sqrt{2}}$. To deal with the visible case for general states, we define a new quantity $E_{\alpha ,p}(A:R{)}_{\rho }$ for a bipartite state ${\rho }^{AR}$ and $\alpha \in (0,1)\cup (1,\mathrm{\infty })$ as the $\alpha$-Rényi generalization of the entanglement of purification $E_p(A:R{)}_{\rho }$. For $\alpha =1$, we define $E_{1,p}(A:R{)}_{\rho }:=E_p(A:R{)}_{\rho }$. We show that for any rate below the regularization ${\lim }_{\alpha \to 1^{+}}{E}_{\alpha ,p}^{\mathrm{\infty }}(A:R{)}_{\rho }:={\lim }_{\alpha \to 1^{+}}{\lim }_{n\to \mathrm{\infty }}\frac{E_{\alpha ,p}(A^n:R^n{)}_{{\rho }^{\otimes n}}}{n}$ the fidelity for the visible compression exponentially converges to zero. Moreover, we consider blind compression of a general mixed-state source ${\rho }^{AR}$ shared between an encoder and an inaccessible reference system $R$. We obtain a strong converse bound for the compression of this source by assuming that the decoder is a super-unital channel. This immediately implies a strong converse for the blind compression of ensembles of mixed states, by assuming a super-unital decoder, as this is a special case of the general mixed-state source ${\rho }^{AR}$ where the reference system $R$ has a classical structure.