The abort-aware ideal output and conditional marginals are states

The real state is nonnegative because its orthogonal classical blocks are nonnegative by [F1]. Its trace is the sum of their traces by [F5], which is one by [F1], so it is a density operator by [F3]. If $p_{\mathrm{a}\mathrm{c}\mathrm{c}}>0$, dividing each accepted block by $p_{\mathrm{a}\mathrm{c}\mathrm{c}}$ preserves nonnegativity and makes the trace sum equal to one; hence ${\rho }_{ABTE}^{\mathrm{\top }}$ is a density operator.

In the explicit sums of [F2], each $ATE$ block is $p_{\mathrm{a}\mathrm{c}\mathrm{c}}^{-1}{\sum }_b{\omega }_E^{a,b,t}$ and each $TE$ block is $p_{\mathrm{a}\mathrm{c}\mathrm{c}}^{-1}{\sum }_{a,b}{\omega }_E^{a,b,t}$. These are nonnegative finite sums. By [F5], their traces sum to $p_{\mathrm{a}\mathrm{c}\mathrm{c}}^{-1}{p}_{\mathrm{a}\mathrm{c}\mathrm{c}}=1$, so both marginals are density operators by [F3]. The states ${\tau }_A$ and ${\tau }_{AB}$ are diagonal with $m$ entries equal to $1\mathrm{/}m$ and all other entries zero, so each is nonnegative with trace one. In the product register, ${\tau }_A\otimes {\rho }_{TE}^{\mathrm{\top }}$ has one block $m^{-1}{\rho }_{TE}^{\mathrm{\top }}$ per key, hence is nonnegative and has trace $m(m^{-1})=1$.

The ideal accepted blocks occur only at key pairs $(a,a)$ and equal $m^{-1}{\sum }_{b,c}{\omega }_E^{b,c,t}$ for each $a,t$; they are nonnegative and their total trace is $p_{\mathrm{a}\mathrm{c}\mathrm{c}}$ by [F5]. Its abort blocks are the original nonnegative blocks with total trace $1-p_{\mathrm{a}\mathrm{c}\mathrm{c}}$ by [F1]–[F2]. These lie in mutually orthogonal classical slots, so the full ideal state is nonnegative and has trace $p_{\mathrm{a}\mathrm{c}\mathrm{c}}+1-p_{\mathrm{a}\mathrm{c}\mathrm{c}}=1$, hence is a density operator by [F3]. If $p_{\mathrm{a}\mathrm{c}\mathrm{c}}=0$, nonnegativity makes each accepted block's trace nonnegative; their sum is zero, so each trace is zero. By [F6], each such block has only nonnegative eigenvalues summing to zero and is therefore zero. The accepted ideal block is also zero, so the ideal output equals the real abort-only state.