Restricting the effects hides a nonphysical symmetry from every causal structure

Abstract: Real-amplitude quantum theory is the subtheory of quantum theory invariant under complex conjugation, and experiments in a network of independent sources have measured correlations above the real bound. A theory can nevertheless carry the same conjugation without complete positivity and still have exactly the correlations of its own conjugation-invariant subtheory in every causal structure, the bilocality scenario included. The states of that theory are all the density matrices, and its effects are the operators every partial transpose of which is again a quantum effect. Its symmetrized subtheory simulates it once each source carries a reference frame rather than each system. One map therefore receives three different verdicts in three theories, so the symmetry alone marks no boundary at all, and what sustains the separation in quantum theory is a property of quantum theory. Quantum theory admits every effect its states permit and this theory does not. What does have a boundary is the class of theories where the correlations of a theory and of its symmetrized subtheory coincide. We show that sectorial closure, meaning invariance of the effects and the operations under the symmetry acting independently on each source, suffices for the absence of a gap under any finite group, and that it cannot be weakened on the effects. Fixing unrestricted states and conjugation makes the effects of that theory the largest the symmetry admits, and its operations the largest sectorially closed ones. Locating where sectorial closure fails, for a candidate effect set built from a fixed bound entangled state, is a finite computation on a single ray of effects.