Isoperimetric Inequalities in Quantum Geometry

The Berry phase and quantum distance characterize the geometry of wave functions in Hilbert space. We uncover a quantum analog of the classical isoperimetric inequality, revealing a fundamental link between the Berry phase and quantum distance along closed paths. For two-band systems, we establish a strong inequality analogous to the spherical isoperimetric problem. For multiband systems, we prove a weak inequality showing that the Berry phase never exceeds the quantum distance. These results introduce new quantum geometric principles, placing fundamental bounds on physical quantities such as the Wannier function spread, quantum speed, electron-phonon coupling, and superfluid weight, with broad implications across condensed matter, quantum information, and beyond.