This is an unreviewed preprint posted to arXiv. It examines real-amplitude quantum theory—the subtheory of standard quantum mechanics left invariant under complex conjugation—which has recently been probed experimentally through network Bell-type experiments that measure correlations exceeding what real-amplitude theory can produce. The paper's central claim is that this experimental gap does not trace back to the symmetry itself, but to a more subtle structural feature of quantum theory.
The authors construct an alternative theory that also carries conjugation symmetry but lacks complete positivity. Despite this difference, they argue it reproduces exactly the same correlations as its own conjugation-invariant subtheory in every causal structure, including the bilocality scenario used in recent experiments. The key move is restricting the allowed effects: rather than permitting every effect the state space allows, this theory permits only operators whose partial transposes remain valid quantum effects. Because a single map receives three different verdicts in three distinct theories, the authors conclude that the symmetry alone marks no physical boundary.
The paper then characterizes precisely which class of theories does exhibit a gap between a theory and its symmetrized subtheory. The authors introduce sectorial closure—invariance of effects and operations under the symmetry acting independently on each source—and prove it is sufficient to eliminate any such gap under any finite group, and that the condition cannot be weakened on the effects. They also show that checking whether sectorial closure fails for a candidate effect set built from a fixed bound-entangled state reduces to a finite computation on a single ray of effects, suggesting a concrete route to probing these boundaries.