Can antisymmetry become an operational resource?

Explicit variants

1. Alternative restrictions

   Repeat the question with geometry, bounded adaptive depth, or a restricted preparation library as the charged resource. Each choice defines a different resource theory and must have its own free states and operations.

2. Monotones of the actual STP process

   On a fixed closed register, characterize functions $M$ satisfying $$\sum_o p_o M(\rho_o)\le M(\rho)$$ for every allowed pair measurement. Separate conserved sector weights from quantities sensitive to the multiplicity state, and ask whether any such quantity bounds a stated conversion cost. An average monotone need not decrease in each selected branch.

3. Information encoded in antisymmetry

   At fixed total spin, compare states with the same total singlet weight but different pair probabilities, multiplicity correlations, or coherence. Which distinctions predict advantage for a stated conversion or inference task? Determine which candidate information is operationally accessible under the chosen real invariant instruments.

4. Subsystem-spin martingales and constraints

   For outcome history $h_t$ on a fixed register, let $p_t(S,j)=\operatorname{Tr}(\Pi_{S,j}\rho_{h_t})$. Which functions of the overlapping distributions $p_t(S,j)$ are conserved, martingales, or supermartingales under every allowed next pair measurement? Include functions of multiplicity data when the marginal distributions are insufficient. A bound on a stopped protocol must state its stopping and integrability assumptions.