Strong Converse Bounds for Compression of Mixed States

We consider many copies 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. This immediately implies a strong converse for the blind compression of ensembles of mixed states since this is a special case of the general mixed-state source \(\rho^{AR}\). Moreover, we consider the visible compression of ensembles of mixed states. For a bipartite state \(\rho^{AR}\), we define a new quantity \(E_{\alpha,p}(A:R)_{\rho}\) for \(\alpha \in (0,1)\cup (1,\infty)\) 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 \(\lim_{\alpha \to 1^+}E_{\alpha,p}^{\infty}(A:R)_{\rho}:=\lim_{\alpha \to 1^+} \lim_{n \to \infty} \frac{E_{\alpha,p}(A^n:R^n)_{\rho^{\otimes n}}}{n}\) the fidelity for the visible compression of ensembles of mixed states exponentially converges to zero. We conclude that if this regularized quantity is continuous with respect to \(\alpha\), namely, if \(\lim_{\alpha \to 1^+}E_{\alpha,p}^{\infty}(A:R)_{\rho}=E_{p}^{\infty}(A:R)_{\rho}\), then the strong converse holds for the visible compression of ensembles of mixed states.