Theorem: Benchmark unrestricted and memory optima

For either parity, ${\epsilon }_{\mathrm{IT}}(t)=3/4$ for $t=0,1,2$. At horizon three,

Under Assumption ass:interface,

and

Proof. Constant early prefixes give the first claim. Flagwise Bell witnesses give the unrestricted lower bounds. The complete channels

attain them.

For the even $q$-slot lower bound, the first packet must leave only classical information before the second atomic arrival. A positive effect on its occupied rail has the form $E=(\mathrm{tr}E)(I+v_{x}X+v_{y}Y+v_{z}Z)/2$, $\left|v\right|\le 1$. The conditional reference–future state is

where $A=X_2{Z}_3$, $B=Y_2{X}_3$, $C=Z_2{Y}_3$, and $AB=-C$. For joint projectors $P_{\alpha \beta }=(I+\alpha A)(I+\beta B)/4$, it equals

This is separable. Subsequent operations preserve separability. Label-mixing effects do not avoid the argument because spectral blocks are already incoherent. Decoded Bell overlap is therefore at most $p/2+(1-p)/4$.

Measuring $Z_1,X_2,Z_3$ and preparing the logical $X$ eigenstate of their negative parity on full transmission attains $p{\Delta }_X+(1-p)\mathcal{T}$. The remaining upper bounds follow from the compiled programmes below.