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 networks of independent sources have measured correlations above the real bound. We construct a theory on which the same conjugation fails complete positivity and still leaves every probability in the unit interval, and whose correlations are exactly those of its own conjugation-invariant subtheory in every causal structure, the bilocality scenario included. Its states are all the density matrices, its effects are the operators every partial transpose of which is again a quantum effect, and its operations are the completely PPT-preserving maps. The 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. Quantum theory admits every effect its states permit, and no theory with that property can hide a symmetry this way, so what sustains the separation measured in the network experiments is the absence of a restriction rather than any feature of conjugation. What does have a boundary is the class of theories whose correlations coincide with those of their symmetrized subtheory. 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 this theory the largest the symmetry admits and its operations the largest sectorially closed ones, so it is not one construction among several. 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.