Gaps for weakly nonphysical non-sectorially-closed symmetries

Determine whether a weakly nonphysical symmetry whose effect set is not sectorially closed can produce a symmetrized–nonsymmetrized causal compatibility gap.

Background

Sectorial closure is proved sufficient for equality of the correlation sets of a theory and its symmetrized subtheory, and the paper proves that the corresponding condition cannot be weakened on effects when the encoding must work for every measurement. Nevertheless, the paper constructs candidate effect sets contained in the maximal sectorially closed effect set but not themselves sectorially closed.

For such candidates, weak nonphysicality is not automatic because the operation class must also preserve the relevant dual cone under conjugated operations. Even when weak nonphysicality is supplied, shrinking the effect set changes both the original and symmetrized correlation sets, so it is unresolved whether one can obtain a gap.

References

Whether a weakly nonphysical symmetry that is not sectorially closed can produce a gap is left open.

— Restricting the effects hides a nonphysical symmetry from every causal structure  (2609.05322 - Kam, 4 Sep 2026) in Section 7, subsection “How far the encoding reaches”; Section 8, Discussion

Whether any condition of this kind is necessary we do not know.

— Restricting the effects hides a nonphysical symmetry from every causal structure  (2609.05322 - Kam, 4 Sep 2026) in Section 8, Discussion

A gap for $E_{1}$ would therefore have to come from four-party correlations whose entanglement is confined entirely to the $2$:$2$ cuts, with no outer party holding anything negative under partial transposition against the other three. We do not know of a mechanism of that shape.

— Restricting the effects hides a nonphysical symmetry from every causal structure  (2609.05322 - Kam, 4 Sep 2026) in Section 8, Discussion; Appendix “The effect set $E_{1}$”

Is fermionic quantum theory sectorially closed under the parity symmetry? It too fails tomographic locality, and unlike the theories considered here it is not a foil, so an answer would say something about a theory people take seriously rather than about a constructed one.

— Restricting the effects hides a nonphysical symmetry from every causal structure  (2609.05322 - Kam, 4 Sep 2026) in Section 8, Discussion