Extend the strong quantum isoperimetric inequality beyond two bands

Prove that for any closed loop of normalized pure states in the complex projective space CP^{M−1} with M > 2, the strong quantum isoperimetric inequality (|γ_B| − π)^2 + d_FS^2 ≥ π^2 holds, where d_FS denotes the total Fubini–Study length of the loop and γ_B denotes the Berry phase of the loop taken in the principal branch (−π, π].

Background

The paper derives a strong quantum isoperimetric inequality for two-band systems by mapping CP^1 to a sphere of radius 1/2 and importing the classical spherical isoperimetric inequality. In this setting, the total Fubini–Study length d_FS of a closed loop on the Bloch sphere and its Berry phase γ_B satisfy (|γ_B| − π)^2 + d_FS^2 ≥ π^2, with equality on circles of constant polar angle.

For multi-band systems (M ≥ 2), the authors establish a weak inequality d_FS ≥ γ_B and show in the Supplementary Materials that the equality case of the strong inequality holds for circles embedded in CP^{M−1} (since CP^1 ⊂ CP^{M−1}) and that circles are extremal for the Berry phase under infinitesimal variations. Based on these observations, they conjecture that the full strong inequality extends to CP^{M−1} for M > 2.