Quantum Metric Tensor (Fubini-Study)

Geometry of quantum state space and Berry curvature
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Quantum Geometric Tensor Qμν = ⟨∂μu|(1−|u⟩⟨u|)|∂νu⟩ unifies geometry and topology.
• Real part: Fubini-Study metric gμν — measures quantum state distinguishability.
• Imaginary part: Berry curvature Ωμν — topological Chern number source.
Left: Bloch sphere with current state (dot) and geodesic. Right: metric tensor components over parameter space.