Corollary 5: Renyi monotonicity from one half through infinity

For all positive natural numbers $a,b,c,d$, every order $\alpha\in[0,\infty]$ with $\alpha\ge \tfrac12$, and all unit vectors $\psi\in\mathbb C^a\otimes\mathbb C^b$ and $\phi\in\mathbb C^c\otimes\mathbb C^d$, suppose $L$ is a map from operators on the first space to operators on the second that is a channel, preserves the separable positive cone, and satisfies $L(|\psi\rangle\langle\psi|)=|\phi\rangle\langle\phi|$. Here being a channel means that there is a finite family of matrices $E_j$ such that $L(X)=\sum_j E_jXE_j^\dagger$ for every $X$ and $\sum_jE_j^\dagger E_j=I$; the non-entangling hypothesis additionally requires a finite Kraus representation, without a normalization requirement on that representation, and preservation of every finite nonnegative sum of tensor products of positive semidefinite trace-one matrices, including the empty sum $0$. Then $H_\alpha(\phi)\le H_\alpha(\psi)$. To define this quantity for a vector $v\in\mathbb C^u\otimes\mathbb C^v$, let $\lambda_0,\ldots,\lambda_{u-1}$ be the eigenvalues, in nonincreasing order and with multiplicities, of the matrix with entries $\sum_j v_{ij}\overline{v_{i'j}}$, and let $r$ be the number of strictly positive $\lambda_i$. The definition is $H_\infty(v)=-\log_2\lambda_0$, $H_0(v)=\log_2r$, $H_1(v)=-\sum_i\lambda_i\log_2\lambda_i$, and, for any other finite order $t$, $H_t(v)=\log_2(\sum_i\lambda_i^t)/(1-t)$. The order-zero case is excluded by the hypothesis, while order one, infinite order, and zero eigenvalues are included; the real logarithm at zero is assigned the value $0$.