Lemma 10: universal primal upper bound, corrected from its proof

Let $\lambda=(\lambda_1,\ldots,\lambda_d)$ be a finite probability vector: every entry is nonnegative and the entries sum to one. Write $\rho_\lambda=\sum_{i,j}\sqrt{\lambda_i\lambda_j}|ii\rangle\langle jj|$. All logarithms have base two. A PPT operation is a completely positive, trace-preserving map $L$ such that $\Gamma_{\rm out}L\Gamma_{\rm in}$ is completely positive for arbitrary finite ancillary systems; $\Gamma$ transposes Bob in the fixed product basis. Assume in addition that all entries of $\lambda$ are strictly positive, so $d$ is its Schmidt rank. For $K\ge2$, let $T(K;\lambda)$ be the minimum of the Schmidt-coordinate primal SDP (20), defined independently of operational reachability. Its real variables $s_{ij}$ for $j\le i$ and $a_{ij}$ for $j<i$ obey $s_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K+1)$ and $a_{ij}\ge\sqrt{\lambda_i\lambda_j}/(K-1)$, and the matrix $\sum_{i\ge j}s_{ij}\sigma_{ij}^{\Gamma}+\sum_{i>j}a_{ij}\alpha_{ij}^{\Gamma}$ is positive semidefinite. The objective is $\sum s_{ij}+\sum a_{ij}$. Here $\sigma_{ii}=|ii\rangle\langle ii|$, while the off-diagonal symmetric and antisymmetric projectors are $(|ij\rangle\pm|ji\rangle)(\langle ij|\pm\langle ji|)/2$. For a finite order $t$, $S_t$ is the Rényi entropy, with $S_0$ the log support size, $S_1=-\sum_i\lambda_i\log_2\lambda_i$ (using $0\log_2 0=0$), and $S_t=(1-t)^{-1}\log_2\sum_i\lambda_i^t$ otherwise. The separate infinity order is $S_\infty=-\log_2\max_i\lambda_i$. The proof-supported universal upper bound is

This bound controls asymptotic and catalytic preparation costs.

Source correction. Printed equation (25) subtracts one in the numerator. The proof explicitly gives the bound displayed here; the printed stronger bound is false for the allowed scalar spectrum $(1)$.