Quantum Differential Geometry from the Quantum Geometric Tensor Basis

Vinicius Ferreira & Professor Vincent Rodgers

Abstract: Metrics and curvature are central to differential geometry (DG), which describes smooth spaces and how vectors and curves behave within them. In quantum mechanics, the quantum metric has drawn growing attention for its role in diverse phenomena while Berry curvature has been extensively explored. Then, we develop a geometric construction inspired by classical DG that has the gauge-invariant Quantum Geometric Tensor (QGT), which encodes quantum metric and Berry curvature in a single object, as its Hermitian metric in projective Hilbert space. Although DG has already been used to describe the entire Hilbert space, what remains unanswered is what further geometric structures can be extracted from the full QGT. We explore this question through a geometric construction using families of pure quantum states that depend on real parameters. We use Berry-covariant derivatives to separate the local phase direction from state-changing tangent directions. These tangent directions allow us to construct a basis that naturally has the Quantum Geometric Tensor as its Hermitian metric, a step-by-step construction intimately related to the classical differential geometry approach on real manifolds. The resulting connection defines parallel transport, velocity, acceleration, and optimal curves in that space, with interesting geometric and physical properties. We call this framework Quantum Differential Geometry. Qubit and qutrit quantum states connect the geometry to energy fluctuations, survival probabilities, and geometric phases, enabling different platforms (e.g., NV centers) for probing geometrical objects. QDG also applies to the time-dependent variational principle, where a restricted family can be approximated to computationally-expensive quantum evolutions.

To participate in this event virtually via Zoom, go to https://uiowa.zoom.us/j/99570315915.