Determine whether the stabilizer entanglement inequality holds for graph states

Determine whether the inequality E_{sg}(\ket{G}) \geq E_{ss}(\ket{G}) holds for graph states, where E_{sg} is the stabilizer geometric measure and E_{ss} is the stabilizer Schmidt measure.

Background

The paper defines E_{sg} as the stabilizer-product-state analogue of the geometric measure of entanglement and E_{ss} as the logarithm of the minimum number of stabilizer product states required to decompose a stabilizer state. It proves the inequality E_{sg}(\ket{G}) \geq E_{ss}(\ket{G}) for the graph-state setting under discussion by constructing an orthogonal stabilizer product-state decomposition from an optimal nearest stabilizer product state.

The authors emphasize that the corresponding inequality between the ordinary geometric and Schmidt measures does not hold in general. They explicitly leave unresolved whether the stabilizer analogue E_{sg}(\ket{G}) \geq E_{ss}(\ket{G}) holds for all graph states.

References

This establishes the relation $E_{sg}(\ket{G}) \geq E_{ss}(\ket{G})$ between the stabilizer product state counterparts that does not hold generally between the geometric and Schmidt measure and, to our knowledge, is unknown to hold even when inputs are restricted to graph states.

— Complexity and Applications of Nearest Stabilizer Product State Problems  (2610.02037 - Grier et al., 1 Oct 2026) in Appendix, Section “Measures of entanglement”