Additivity Results for the Rényi-2 Entanglement of Purification

Abstract

We reformulate the Rényi entanglement of purification as a constrained minimum output Rényi entropy problem. Equivalently, for $p>1$, this formulation can be expressed in terms of a constrained maximal output Schatten $p$-norm. More precisely, for a completely positive map $\Omega :L(B^{\prime })\to L(A)$, we consider the quantity ${\upsilon }_p(\Omega )$ defined by optimizing $\Vert (\Omega \otimes {\mathrm{id}}_E)({\sigma }^{B^{\prime }E}){\Vert }_p$ over all bipartite states ${\sigma }^{B^{\prime }E}$ whose $B^{\prime }$-marginal is maximally mixed. We focus on the case $p=2$. First, we compute ${\upsilon }_2$ for the transpose-depolarizing channel and prove that it is multiplicative under tensor powers. We then establish a general multiplicativity criterion: whenever a completely positive map $N:L(B^{\prime })\to L(A)$ satisfies $N^{†}\circ N=a{\mathrm{id}}_A+b\mathrm{Tr}[\cdot ]I_d$ for some constants $a,b\ge 0$, where $N^{†}$ denotes the Hilbert-Schmidt adjoint of $N$, the quantity ${\upsilon }_2(N)$ is multiplicative under tensor powers. Examples of channels satisfying this criterion include the transpose-depolarizing channel, the depolarizing channel, and their respective complementary channels. Furthermore, we show that, for every completely positive map $\Omega$, multiplicativity of ${\upsilon }_p(\Omega )$ implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.