Quantum Geometric Inequality and Its Classical Wave Verification

The study of the geometric properties of quantum states in Hilbert space-particularly through the lens of the quantum geometric tensor (QGT)-has profoundly advanced the fields of condensed matter physics and materials science. The real and imaginary parts of the QGT, the quantum metric and Berry curvature, characterize the distance and phase variation of two adjacent quantum states, respectively. Here, we unveil a fundamental global inequality rooted in these two local quantities between the Fubini-Study quantum distance (d_{FS}) and the Berry phase (ϕ_{B}), d_{FS}≥ϕ_{B}, for any closed momentum path. Interestingly, the equality occurs when the closed path is mapped to a great circle on the Bloch sphere, i.e., d_{FS}=ϕ_{B}=π for a topologically nontrivial path, qualifying quantum distance as an alternative probe for nontrivial band topology. Experimentally, using acoustic metamaterials, we measure the full QGT of concrete models and provide compelling evidence for this quantum geometric inequality. Our findings shed new light on the geometric characteristics of quantum matter.