Isoperimetric Inequalities in Quantum Geometry

Geometric concepts are foundational in physics, from general relativity governing the largest scales of space and time to topological phases of matter revealing some of the most complex behaviors. In quantum mechanics, key quantities such as the Berry phase and quantum distance characterize the geometry of wavefunctions in Hilbert space. Here we uncover a quantum analog of the classical isoperimetric inequality, revealing a deep relationship between the Berry phase and quantum distance along closed paths in Hilbert space. For two-band systems, we establish a strong inequality that maps directly onto the classical spherical isoperimetric problem, while for general multi-band systems, we prove a weaker inequality showing that the Berry phase never exceeds the quantum distance. These results introduce new geometric principles underlying quantum mechanics — placing fundamental bounds on physical quantities in diverse systems such as Wannier function spread, quantum speed limit, electron-phonon coupling, and geometric superfluid weight — with implications for condensed matter, quantum information, and beyond.