Quantum Geometric Tensor Overview

• Quantum Geometric Tensor is a fundamental object that unifies the Fubini–Study metric and Berry curvature to measure quantum distance and phase evolution.
• It decomposes into a real symmetric metric and an imaginary curvature form, underpinning fidelity susceptibility and topological responses in quantum systems.
• Its derivation from the Fubini–Study metric offers a clear framework for analyzing quantum phase transitions and evolution in both adiabatic and non-adiabatic regimes.

The quantum geometric tensor (QGT) is the foundational geometric object in quantum mechanics that unifies a Riemannian metric structure (quantum distance) and a symplectic structure (Berry curvature) on the projective Hilbert space of quantum states. It arises as the gauge-invariant bilinear form measuring both the infinitesimal distinguishability of quantum states and the geometric phase accumulated under cyclic adiabatic evolution. The QGT is realized as the Fubini–Study metric on complex projective space, and decomposes into a real symmetric metric tensor and an imaginary antisymmetric curvature two-form. Its dual roles are pivotal for quantum phase transitions, topological phenomena, and the geometric theory of quantum evolution (Cheng, 2010).

1. Formal Definition and Gauge Structure

Given a normalized quantum state $|\psi(X)\rangle$ that depends smoothly on a set of real parameters %%%%1%%%%, the quantum geometric tensor is defined by

$$Q_{\mu\nu}(X) = \langle \partial_\mu \psi | \partial_\nu \psi \rangle - \langle \partial_\mu \psi | \psi \rangle \langle \psi | \partial_\nu \psi \rangle$$

where $$\partial_\mu \equiv \frac{\partial}{\partial X^\mu}$$.

This subtraction ensures U(1) gauge invariance under $|\psi\rangle \to e^{i\alpha(X)}|\psi\rangle$, by projecting out the local phase and restricting to the complex rays in Hilbert space (projective Hilbert space). In explicit terms, $Q_{\mu\nu}=Y_{\mu\nu}-B_\mu B_\nu$, where $$Y_{\mu\nu}=\langle \partial_\mu \psi | \partial_\nu \psi \rangle$$ and $B_\mu(X)=i\langle \psi | \partial_\mu \psi \rangle$ is the Berry connection.

2. Emergence from the Fubini–Study Metric

The QGT emerges naturally from the infinitesimal distance in projective Hilbert space: $$ds^2 = 1 - |\langle \psi(X) | \psi(X+dX) \rangle|^2$$ Expanding to second order in $dX$ yields the line element

$ds^2 = g_{\mu\nu}(X)\, dX^\mu dX^\nu$

where

$g_{\mu\nu}(X) = \mathrm{Re}\, Q_{\mu\nu}(X)$

This is the Fubini–Study metric, which is the canonical Riemannian structure on complex projective space $\mathrm{CP}^{n-1}$, quantifying the minimal quantum distance between rays.

3. Decomposition: Metric and Berry Curvature

Because the inner product is Hermitian, $Q_{\mu\nu}=Q^*_{\nu\mu}$, allowing decomposition:

• Quantum metric:

$g_{\mu\nu} = \mathrm{Re}\, Q_{\mu\nu}$

symmetric, measures quantum state distinguishability.

• Berry curvature:

$F_{\mu\nu} = -2\,\mathrm{Im}\, Q_{\mu\nu}$

antisymmetric, acts as the curvature of the U(1) fiber bundle and yields the geometric phase.

The QGT can thus be written as

$Q_{\mu\nu} = g_{\mu\nu} - \frac{i}{2} F_{\mu\nu}$

and underlies the geometric structure of parameter-dependent quantum systems.

4. Physical Interpretation and Manifestation

• Quantum metric ($g_{\mu\nu}$): The geodesic length $\int \sqrt{g_{\mu\nu} dX^\mu dX^\nu}$ gives the minimal quantum distance between two rays, central to fidelity susceptibility in quantum phase transitions and providing a physically meaningful metric on parameter space.
• Berry curvature ($F_{\mu\nu}$): Its flux through parameter loops determines the Berry (geometric) phase:

$$\gamma_{\mathrm{Berry}} = \int F_{\mu\nu} dX^\mu \wedge dX^\nu$$

governing topological responses.

5. Paradigmatic Example: Two-Level System

For a spin-$\frac{1}{2}$ particle in a magnetic field of fixed magnitude $B$ but varying direction (spherical angles $\theta$, $\phi$), the instantaneous ground state is

$$|\psi(\theta,\phi)\rangle = \cos(\theta/2)e^{-i\phi/2}|\uparrow\rangle + \sin(\theta/2)e^{i\phi/2}|\downarrow\rangle$$

The QGT in these coordinates yields: $$g_{\theta\theta} = \frac{1}{4}, \quad g_{\phi\phi} = \frac{1}{4} \sin^2\theta, \quad g_{\theta\phi}=0$$

$F_{\theta\phi} = \frac{1}{2}\sin\theta$

The metric coincides with the round metric on the Bloch sphere of radius $1/2$, and $F$ represents the curvature of a monopole field, aligning geometric phase accumulation with the solid angle subtended by the trajectory.

6. Adiabatic and Non-Adiabatic Regimes

• Adiabatic: For a single nondegenerate eigenstate, $$Q_{\mu\nu} = \sum_{m \neq n} \frac{\langle \psi_n | \partial_\mu H | \psi_m \rangle \langle \psi_m | \partial_\nu H | \psi_n \rangle}{(E_n - E_m)^2}$$. The real part encodes fidelity susceptibility, imaginary part the Berry curvature.
• Non-Adiabatic: For general time-dependent states, $Q_{\mu\nu}$ calculated on $|\psi(t)\rangle$ yields the quantum speed via the Anandan–Aharonov relation:

$\frac{ds}{dt} = \frac{2\Delta E}{\hbar}$

relating the rate of change of quantum distance to energy fluctuations, with $Q_{\mu\nu}$ capturing nonadiabatic state evolution.

7. Geometric Visualization and Topology

The QGT encapsulates quantum geometry on the Bloch sphere for two-level systems; geodesics track optimal quantum transitions, while Berry curvature visualizes the field of a magnetic monopole, underpinning phase holonomy. Its decomposition provides a unified geometric language for quantum distinguishability and topological invariants.

8. Summary and Applications

The quantum geometric tensor, $Q_{\mu\nu}$, is a central object in geometric quantum mechanics, furnishing:

• The real, symmetric Fubini–Study metric ($g_{\mu\nu}$) for quantum state distances.
• The imaginary, antisymmetric Berry curvature ($F_{\mu\nu}$) governing phase and topological responses.

This tensor admits transparent derivation, clear decomposition, and direct computation in simple models, serving as a pedagogical and practical tool for fidelity susceptibility, quantum criticality, geometric phases, and quantum speed limits (Cheng, 2010). Its geometric unification of metric and holonomy deepens the conceptual understanding of quantum phase, distance, and evolution in parameter space.