Quantum Geometrical Tensor

• Quantum Geometrical Tensor is a geometric object that embodies both quantum metrics (state distances) and curvature (Berry curvature) on parameterized state manifolds.
• It uses methods like perturbation theory and the N-bein formalism to quantify nonadiabatic transitions and inter-state mixing.
• Recent extensions to multi-state and mixed-state systems reveal new observables such as quantum torsion and state-to-state correlations.

The quantum geometrical tensor (QGT) is a central geometric object that encodes both the metric and curvature properties of quantum state manifolds parameterized by external controls or system parameters. It provides a unified framework for quantifying quantum distances, phase holonomies, and nonadiabatic state mixing, with broad utility in quantum physics, condensed matter theory, quantum information, and quantum materials. Recent developments generalize the QGT to multi-state and mixed-state settings, uncovering richer geometric structures and enabling new diagnostic and computational techniques.

1. Mathematical Definition and Structure

For a non-degenerate quantum state |n(λ)⟩, parameterized by a set {λ^i} in control or momentum space, the QGT is given by

$$Q_{ij}^{(n)} = \langle \partial_i n | (1 - |n\rangle\langle n|) | \partial_j n\rangle,$$

with ∂_i ≡ ∂/∂λ^i. The QGT admits a decomposition into its symmetric (real) and antisymmetric (imaginary) parts,

$$Q_{ij}^{(n)} = Y_{ij}^{(n)} + i\,\Omega_{ij}^{(n)},$$

where

• $Y_{ij}^{(n)} = \operatorname{Re} Q_{ij}^{(n)}$ supplies a Riemannian metric (the quantum metric, or Fubini–Study metric),
• $\Omega_{ij}^{(n)} = \operatorname{Im} Q_{ij}^{(n)}$ encodes the local Berry curvature.

For a parameter-dependent Hamiltonian H(λ) with eigenstates |ϕ_n(λ)⟩, the QGT can be computed via perturbation theory as

$$Q_{ij} = \sum_{m \neq n} \frac{\langle \varphi_n | \partial_i H | \varphi_m \rangle \langle \varphi_m | \partial_j H | \varphi_n \rangle}{(E_n - E_m)^2}.$$

This singularly emphasizes the importance of near-degeneracies, as the denominator vanishes when energy levels cross.

Typically, the real part of the QGT is directly linked to fidelity susceptibility and is sensitive to quantum phase transitions, while the imaginary part underlies geometric phase effects (1012.1337).

2. N-bein Generalization and the Two-State Quantum Geometric Tensor

The N-bein formalism generalizes the standard (single-state) QGT, drawing on the analogy with vielbeins (orthonormal frames) in Cartan geometry. The derivative ∂_i|n(λ)⟩ can always be expanded as

$$|\partial_i n\rangle = |n\rangle \langle n|\partial_i n\rangle + \sum_{m\neq n} |m\rangle \langle m|\partial_i n\rangle.$$

Defining the N-bein as $e_i^{(n)}{}_m = i\langle m|\partial_i n\rangle$, the QGT is reconstructed as

$$Q_{ij}^{(n)} = \sum_{m \neq n} e_i^{(n)*}{}_m e_j^{(n)}{}_m,$$

so the N-bein serves as a “square root” of the QGT.

Advancing further, the two-state quantum geometric tensor is introduced as

$$M_{ij}^{(n,m)} = \langle \partial_i m | P^{(n,m)} | \partial_j n \rangle,$$

where the projector $$P^{(n,m)} = 1 - |n\rangle\langle n| - |m\rangle\langle m|$$ excludes both |n⟩ and |m⟩, quantifying the amplitude for going from state |n⟩ to |m⟩ via two consecutive infinitesimal parameter variations. Its symmetric part $\mathcal{G}_{ij}^{(n,m)}$ generalizes the quantum metric to a two-state context (measuring correlations in their response to parameter changes), while the antisymmetric part $$T_{ij}^{(n,m)} = i(M_{ij}^{(n,m)} - M_{ji}^{(n,m)})$$ is interpreted as a quantum torsion, encoding the (non)commutativity of parameter variations as reflected in state transitions (Romero et al., 27 Jun 2024).

3. Connections, Covariant Derivatives, and Curvature

A connection distinct from the standard Berry connection is defined on the parameter space via

$$\Gamma_i^{(n,m)} = A_i^{(n)} - A_i^{(m)}, \qquad A_i^{(n)} = i \langle n | \partial_i n \rangle,$$

which governs relative parallel transport between two quantum states. Under a gauge transformation, it transforms Abelianly with the relative phase α_{nm}.

The covariant derivative of the N-bein is

$$D_i e_j^{(n)}{}_m = \partial_i e_j^{(n)}{}_m + i \Gamma_i^{(n,m)} e_j^{(n)}{}_m.$$

The torsion two-form is defined as

$$T_{ij}^{(n,m)} = D_i e_j^{(n)}{}_m - D_j e_i^{(n)}{}_m,$$

coinciding with the antisymmetric part of M_{ij}^{(n,m)}. This torsion measures the failure of parameter variations to commute in taking |n⟩ to |m⟩—a nonzero torsion signals path dependence in parameter space (Romero et al., 27 Jun 2024).

The curvature associated with the connection $\Gamma_i^{(n,m)}$ is

$$R_{ij}^{(n,m)} = \partial_i \Gamma_j^{(n,m)} - \partial_j \Gamma_i^{(n,m)} = F_{ij}^{(n)} - F_{ij}^{(m)},$$

where F_{ij}^{(n)} is the Berry curvature for |n⟩; curvature thus quantifies the difference in geometric phase accumulation between the two levels under parameter cycling.

4. Physical and Geometric Significance

The N-bein and its associated tensors quantify the nonadiabatic response of a quantum system to parameter changes. A nonzero N-bein indicates “leakage” from the instantaneous eigenstate; thus, the QGT (and its generalizations) naturally capture fidelity susceptibilities and the (local) metric structure of quantum state space.

The torsion term T_{ij}^{(n,m)} is nontrivial whenever the order of parameter variation matters in taking the system between two nearby eigenstates, establishing a geometric obstruction akin to torsion in differential geometry. The new geometric invariants constructed from these tensors, such as

$$N_{ijkl}^{(\Xi)} = \operatorname{Re} \Xi_{ij} \operatorname{Re} \Xi_{kl} + \operatorname{Im} \Xi_{ij} \operatorname{Im} \Xi_{kl},$$

are gauge-independent and serve as new physical observables, quantifying inter-state quantum correlations inaccessible in the standard (single-state) QGT formalism (Romero et al., 27 Jun 2024).

The symmetric part $\mathcal{G}_{ij}^{(n,m)}$ generalizes quantum distance to state pairs, while the torsion and curvature capture the “geometry of transport” on Hilbert space, sensitive to the global ordering of physical processes.

5. Practical Computation and Examples

Two representative examples illustrate the practicality of this extended framework:

Harmonic Oscillator with a Linear Term

• Hamiltonian: $\hat{H} = \frac{1}{2}(q^2 + Z p^2) + W q$.
• The N-beins are nonvanishing for levels n ↔ n ± 1 and n ↔ n ± 2, reflecting the selection rules on parameter-induced transitions.
• The associated QGT splits into a contribution from the linear term (W) and a standard oscillator contribution (Z).
• Nonzero torsion occurs for transitions linking adjacent levels, indicating noncommutativity in parameter-driven transitions.

Generalized Oscillator with Linear and Cross Terms

• Hamiltonian: $$\hat{H} = \frac{1}{2}(q^2 + Y(qp+pq) + Z p^2) + W q$$.
• For this more complex system, the N-bein, quantum metric, Berry curvatures, and torsion explicitly depend on W, Y, Z, and the “frequency” parameter ω.
• Scalar invariants built from these geometric objects peak for low-lying states and decrease for higher states, which suggests that state-to-state geometric correlations are most relevant in the infrared sector.

6. Differential Forms and Cartan Structure

The formalism admits a natural differential form expression. The connection, torsion, and curvature introduced here satisfy generalized Cartan structure equations, with the torsion two-form playing a central role in capturing noncommutativity of parameter flows. The construction satisfies the analog of Bianchi identities, underscoring its consistency as a geometric framework.

These structures admit further generalizations and physical interpretation, including the interpretation of geometric tensors as describing generalized quantum state transport and their potential relevance for the analysis of entanglement, correlation, and quantum control protocols in parameter-rich quantum systems.

7. Implications and Outlook

The N-bein approach and multi-state QGT generalize the traditional quantum metric and Berry curvature to encode not only local quantum state geometry but also higher-order, state-to-state quantum correlations and nontrivial geometry induced by inter-level transitions. The construction is naturally adapted to gauge-covariant settings and provides a toolkit for quantifying controllability, adiabaticity violations, and the feasibility of state transformations in quantum systems.

The identification of “quantum torsion” and noncommutative transport points to new observables and geometric invariants, which, as demonstrated in the harmonic and generalized oscillator exemplars, distinguish physical effects inaccessible by other means. This generalization bridges the gap between geometric quantum mechanics and classical geometric frameworks (Cartan geometry, vielbein/tetrad approaches), with applications across quantum many-body theory, condensed matter, and quantum information (Romero et al., 27 Jun 2024).

A plausible implication is that these geometric invariants, constructed from N-beins, could serve as sensitive diagnostics for quantum phase transitions, decoherence pathways, or quantum control strategies where multi-level effects and nonadiabatic phenomena become prominent. Experimental access to such quantities would further illuminate the geometric structure underlying quantum dynamics in complex systems.