How hard is predicting singlet transmission under STP?

Explicit variants

1. Cost of transporting a singlet correlation

   Fix an interaction graph, an initial supply of singlets and maximally mixed qubits, an untouched reference $R$ initially paired with an input qubit, and an output qubit $B$. For a history $h$, let $c(h)$ count all pair measurements, or only measurements crossing a specified spatial cut. For a target fidelity $F>1/2$, define the minimum expected cost over adaptive protocols satisfying $\operatorname{Tr}(P_s^{RB}\rho_{\mathrm{out}})\ge F$, where the output averages over all histories. Require termination and state whether fresh ancillas, discarding, and record-dependent output registers are permitted. Bound this cost using geometry and the multiplicity-space correlations created during the protocol. This is an optimization problem; distinguish it from predicting the output of a supplied controller. For fixed-total-spin comparisons, use closed registers and compatible initial and target states.