Additivity Results for the Rényi-2 Entanglement of Purification

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 $Ω:L(B')\to L(A)$, we consider the quantity $\upsilon_p(Ω)$ defined by optimizing $\|(Ω\otimes \mathrm{id}_E)(σ^{B'E})\|_p$ over all bipartite states $σ^{B'E}$ whose $B'$-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 $N:L(B')\to L(A)$ satisfies $N^{\dagger} \mathbin{\circ} N=a\,\mathrm{id}_A+b\,\mathrm{Tr}[\cdot]\,I_d$ for some constants $a,b\ge 0$, where $N^{\dagger}$ 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 $Ω$, multiplicativity of $\upsilon_p(Ω)$ implies multiplicativity for its complementary map. This yields the corresponding additivity statements for the associated Rényi-2 entanglement of purification.