Non-Hermitian Quantum Geometric Tensor (NH-QGT)

• Non-Hermitian quantum geometric tensor is a biorthogonal extension of the conventional QGT tailored for systems with distinct left and right eigenstates.
• It decomposes into a real quantum metric and an imaginary Berry curvature, providing clear diagnostics for phase transitions, exceptional points, and localization phenomena.
• NH-QGT employs biorthogonal inner products and gauge-covariant frameworks to map state-space geometry, enabling practical insights into transport and dynamical behavior in non-Hermitian systems.

Searching arXiv for the cited NH-QGT literature and related recent work. The non-Hermitian quantum geometric tensor (NH-QGT) is a biorthogonal extension of the quantum geometric tensor to non-Hermitian Hamiltonians, particularly relevant when right and left eigenstates are distinct. In the formulations used across $\mathcal{PT}$-symmetric, pseudo-Hermitian, Bloch-band, and wave-packet settings, the NH-QGT unifies two geometric structures: its real part defines a quantum metric, while its imaginary or antisymmetric part defines a Berry curvature (Zhang et al., 2018, Ye et al., 2023, Hu et al., 2023). In quasi-Hermitian or unbroken $\mathcal{PT}$-symmetric regimes with real spectra, this construction recovers a Hermitian-like tensor structure once the appropriate biorthogonal inner product or metric operator is introduced (Zhang et al., 2018, Huang et al., 21 Sep 2025, Das et al., 14 Jun 2026). More generally, non-Hermiticity permits pseudo-Riemannian, complex, or degenerate geometries, and singularities of the metric identify exceptional points, spontaneous $\mathcal{PT}$-symmetry breaking, topological transitions, localization transitions, mobility edges, and many-body criticality (Ye et al., 2023, Ren et al., 2024, Das et al., 14 Jun 2026).

1. Formal definition and biorthogonal setting

For a non-Hermitian Hamiltonian $H(\lambda)$ depending on real parameters $\lambda=(\lambda^1,\lambda^2,\dots)$, the right and left eigenstates satisfy

$$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$

with biorthonormality

$$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$

A standard left-right definition is

$$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$

with $\partial_i\equiv \partial/\partial \lambda^i$ (Ye et al., 2023, Hu et al., 2023, Ren et al., 2024). Equivalently, in projector form,

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$

which makes explicit that the tensor probes variations orthogonal, in the biorthogonal sense, to the reference band (Hu et al., 2023).

In $\mathcal{PT}$0-symmetric quantum mechanics, the same object can be constructed using a $\mathcal{PT}$1-dependent inner product induced by a positive operator $\mathcal{PT}$2 satisfying

$\mathcal{PT}$3

with inner product

$\mathcal{PT}$4

If $\mathcal{PT}$5 are normalized eigenvectors of $\mathcal{PT}$6 and $\mathcal{PT}$7, then $\mathcal{PT}$8, and the extended tensor is defined by

$\mathcal{PT}$9

In that formulation, $\mathcal{PT}$0 and is independent of the particular choice of $\mathcal{PT}$1 (Zhang et al., 2018).

Recent work in the quasi-Hermitian regime reformulates the same geometry through a Dyson map $\mathcal{PT}$2, where $\mathcal{PT}$3 with $\mathcal{PT}$4. In that setting the NH-QGT is written as

$\mathcal{PT}$5

and the Dyson map becomes the central object for a gauge-covariant description of non-Hermitian geometry (Das et al., 14 Jun 2026).

2. Metric, curvature, and gauge structure

The NH-QGT is decomposed into metric and curvature components in close analogy with the Hermitian case. Several equivalent conventions appear in the literature. A common one identifies

$\mathcal{PT}$6

while in other conventions the antisymmetric part in parameter indices is emphasized directly (Ye et al., 2023, Hu et al., 2023, Ren et al., 2024). In the $\mathcal{PT}$7-symmetric construction,

$\mathcal{PT}$8

with the connection one-form

$\mathcal{PT}$9

(Zhang et al., 2018).

The gauge structure differs from the Hermitian case because left and right eigenstates transform independently but in a constrained biorthogonal manner. In the gauge-covariant framework, the transformation

$H(\lambda)$0

leaves the NH-QGT invariant owing to the projector $H(\lambda)$1 (Das et al., 14 Jun 2026). In the Dyson-map description, the connection

$H(\lambda)$2

splits into Hermitian and anti-Hermitian pieces,

$H(\lambda)$3

identified respectively as stretching and rotation components. This decomposition separates metric deformation from unitary gauge redundancy (Das et al., 14 Jun 2026).

A related point of comparison concerns multiple non-Hermitian generalizations. Two principal variants recur in the literature: the left-right tensor $H(\lambda)$4, built from biorthogonal states, and the right-right tensor $H(\lambda)$5, built from right eigenstates normalized in the usual Hermitian sense (Hu et al., 2023, Hu et al., 2024). The $H(\lambda)$6 tensor remains Hermitian, whereas $H(\lambda)$7 is generally non-Hermitian and can have complex metric and curvature components (Hu et al., 2024). This distinction is operational rather than merely formal, because the two tensors can control different observables in dynamical settings (Hu et al., 2023, Hu et al., 2024).

3. Relation to fidelity, adiabatic transport, and wave-packet dynamics

The quantum metric component of the NH-QGT is closely tied to infinitesimal state distinguishability. In the $H(\lambda)$8-symmetric formulation, if one regards

$H(\lambda)$9

as a pure-state density operator and defines the fidelity

$\lambda=(\lambda^1,\lambda^2,\dots)$0

then expansion to second order gives

$\lambda=(\lambda^1,\lambda^2,\dots)$1

Thus the real part of the tensor is a Bures distance element on parameter space (Zhang et al., 2018).

A related self-normalized construction shows that, even in non-Hermitian systems, the diagonal metric component can coincide with fidelity susceptibility: $\lambda=(\lambda^1,\lambda^2,\dots)$2 (Ren et al., 2024). This relation underlies the use of the metric as a diagnostic for non-Hermitian criticality.

The curvature component emerges in adiabatic evolution. In $\lambda=(\lambda^1,\lambda^2,\dots)$3-symmetric quantum mechanics, the time-dependent Schrödinger-like equation is

$\lambda=(\lambda^1,\lambda^2,\dots)$4

where

$\lambda=(\lambda^1,\lambda^2,\dots)$5

is a Hermitian gauge term guaranteeing unitary evolution with respect to $\lambda=(\lambda^1,\lambda^2,\dots)$6. Under slow variation along a closed loop $\lambda=(\lambda^1,\lambda^2,\dots)$7, the geometric phase is

$\lambda=(\lambda^1,\lambda^2,\dots)$8

and Stokes’ theorem yields $\lambda=(\lambda^1,\lambda^2,\dots)$9 (Zhang et al., 2018).

In semiclassical wave-packet dynamics, non-Hermiticity makes the distinction between $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$0 and $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$1 physically consequential. For a narrow Gaussian packet in a two-band model under external force $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$2, the center-of-mass equation contains the usual anomalous velocity term controlled by the RR Berry curvature, but also additional terms involving RR and LR connections and QGT corrections (Hu et al., 2023, Hu et al., 2024). One formulation gives

$$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$3

while an equivalent rewriting highlights $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$4 instead (Hu et al., 2023). First-order perturbation theory further shows that the RR QGT produces nonadiabatic shifts of anomalous velocity, while the LR QGT enters field-induced corrections to the intraband Berry connection and dynamical phase (Hu et al., 2024). This is one reason later work states that both right-only and biorthogonal QGTs play a significant role in non-Hermitian wave-packet dynamics (Hu et al., 2024).

4. Geometric types specific to non-Hermitian systems

In Hermitian quantum mechanics the quantum metric is positive semidefinite and thus Riemannian. Non-Hermiticity broadens this structure. In non-Hermitian Su-Schrieffer-Heeger systems, the metric built from both left and right eigenvectors correctly identifies topological phases and topological phase transitions, but the resulting geometry can become pseudo-Riemannian or complex (Ye et al., 2023). Specifically, after diagonalization the diagonal components of $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$5 can have indefinite signs, and in some non-Hermitian phases

$$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$6

so that there exists a parameter-space direction $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$7 for which

$$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$8

identically. The cited work interprets this as dimensional reduction of the quantum geometry by one (Ye et al., 2023).

The same study observes that the topological transition curves $$H(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle = E_n(\lambda)\,\bigl|\psi_n^R(\lambda)\bigr\rangle,\qquad \bigl\langle\psi_n^L(\lambda)\bigr|\,H(\lambda)=E_n(\lambda)\,\bigl\langle\psi_n^L(\lambda)\bigr|$$9 in a non-Hermitian SSH model appear exactly as the lines along which $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$0, and compares these null lines to lightlike paths in general relativity (Ye et al., 2023). This suggests an analogy between non-Hermitian phase boundaries and null structures of pseudo-Riemannian geometry, although the analogy is explicitly mathematical rather than an identification of physical spacetime.

A more systematic operator-level account is developed in the gauge-covariant framework based on the Dyson connection. There the NH-QGT can be decomposed using projected states

$$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$1

so that

$$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$2

In this formulation, the quantum metric is generally indefinite, and the non-Hermitian Berry curvature originates from the non-commutativity of the stretching components $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$3 at the operator level (Das et al., 14 Jun 2026).

The literature also contains sector-resolved generalizations beyond the standard LR tensor. In the non-Hermitian Zeeman QGT, the tensor is generically non-Hermitian and splits into normal and anomalous sectors, producing an imaginary symmetric metric-like tensor and a real antisymmetric curvature-like tensor with no counterpart in the standard Hermitian QGT (Cui et al., 9 Apr 2026). This is not the same object as the usual NH-QGT of band geometry, but it illustrates how non-Hermitian geometry can support additional symmetry-resolved structures.

5. Criticality, exceptional points, and phase transitions

A central application of the NH-QGT is the detection of critical points through singularities of the metric. In $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$4-symmetric systems, when the ground state undergoes a level crossing with an excited state or coalesces at a $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$5-breaking exceptional point, the ground-state metric

$$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$6

diverges because $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$7 in the denominator (Zhang et al., 2018). The same work states that the metric diverges both at conventional quantum phase transition points and at spontaneous $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$8-breaking points (Zhang et al., 2018).

Near exceptional points, several works emphasize distinct singular mechanisms. In the biorthogonal picture, derivatives diverge as left and right eigenvectors coalesce and $$\bigl\langle \psi_n^L(\lambda)\bigm|\psi_m^R(\lambda)\bigr\rangle=\delta_{nm}.$$9 (Ren et al., 2024). In the gauge-covariant framework, the metric deformation operator and stretching sector dominate the leading divergence, and for a generic second-order exceptional point one finds

$$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$0

with $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$1 denoting displacement from the exceptional point along parameter axis $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$2 (Das et al., 14 Jun 2026). The B-VQE study likewise reports

$$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$3

as an operational signature used to locate exceptional points (B et al., 17 Jun 2026).

The scope of NH-QGT diagnostics extends well beyond isolated exceptional-point physics. In non-Hermitian generalized Aubry-André models, the quantum metric exactly identifies localization transitions and mobility edges (Ren et al., 2024). In a non-Hermitian cluster Ising model, peaks of $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$4 and $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$5 coincide with real or imaginary gap closings and with changes in the string order $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$6 or staggered magnetization $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$7 (Ren et al., 2024). In a non-Hermitian mixed-field Ising model, the real-to-complex transition of the ground-state energy is accompanied by a divergence of $$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$8 (Ren et al., 2024). The same study further reports finite-size scaling at single-particle localization transitions,

$$Q_{ij}(\lambda)\equiv \bigl\langle \partial_i \psi^L \bigm| \partial_j \psi^R \bigr\rangle -\bigl\langle \partial_i \psi^L \bigm|\psi^R \bigr\rangle \bigl\langle \psi^L \bigm|\partial_j \psi^R \bigr\rangle,$$9

with measured $\partial_i\equiv \partial/\partial \lambda^i$0 for one non-Hermitian generalized Aubry-André model and $\partial_i\equiv \partial/\partial \lambda^i$1 for another (Ren et al., 2024).

These results counter a possible misconception that NH-QGT singularities are limited to topological band touchings or exceptional degeneracies. The cited evidence shows that localization transitions, mobility-edge crossings, and many-body gap-closing transitions can also be encoded in the non-Hermitian metric (Ren et al., 2024).

6. Representative models and experimental or computational access

Several model systems illustrate the range of NH-QGT phenomena.

In the dimerized XY chain with alternating complex field,

$\partial_i\equiv \partial/\partial \lambda^i$2

one obtains four bands $\partial_i\equiv \partial/\partial \lambda^i$3, with unbroken $\partial_i\equiv \partial/\partial \lambda^i$4 regime $\partial_i\equiv \partial/\partial \lambda^i$5 (Zhang et al., 2018). The ground-state metric per site $\partial_i\equiv \partial/\partial \lambda^i$6 diverges at two circular quantum phase transition loci in the anisotropic case, over a finite critical region in the pseudo-isotropic case, and at the $\partial_i\equiv \partial/\partial \lambda^i$7-breaking threshold $\partial_i\equiv \partial/\partial \lambda^i$8 (Zhang et al., 2018).

In non-Hermitian SSH models, the metric identifies all topological phase transitions only when both left and right eigenvectors are used (Ye et al., 2023). For real non-reciprocal hoppings, one finds four transition lines

$\partial_i\equiv \partial/\partial \lambda^i$9

and in the non-Hermitian topological regions the metric is pseudo-Riemannian and satisfies $$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$0 (Ye et al., 2023). For complex non-reciprocal hoppings, the transition lines become

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$1

with the same pseudo-Riemannian and degenerate features (Ye et al., 2023).

In exciton-polariton systems, the generalized QGT components can be reconstructed from experimental observables (Hu et al., 2023). Angle- and polarization-resolved spectroscopy yields the Stokes intensities $$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$2, from which one forms the right-eigenstate pseudospin components

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$3

Using the relation

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$4

for a two-band system, one reconstructs the left pseudospin, forms the complex biorthogonal pseudospin

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$5

extracts complex Bloch angles $$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$6, and then obtains $$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$7 and $$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$8 from two-band formulas (Hu et al., 2023). This provides a direct route to imaging both RR and LR geometries in a photonic platform.

On the algorithmic side, direct measurement protocols have been proposed for pseudo-Hermitian systems with real spectra. One study develops two schemes based on generalized expectation values

$$Q^{LR}_{n,ij}\equiv \langle\partial_i \psi_n^L | \Pi_n | \partial_j \psi_n^R\rangle, \qquad \Pi_n=1-|\psi_n^R\rangle\langle\psi_n^L|,$$9

between two nonadiabatically evolved states, allowing extraction of the full QGT via either generalized energy fluctuations or generalized forces (Huang et al., 21 Sep 2025). Another study introduces B-VQE, which employs independent variational circuits for left and right eigenstates and reads out the NH-QGT on NISQ hardware through generalized parameter-shift overlaps and Hadamard-test-style circuits (B et al., 17 Jun 2026). These approaches are specifically formulated for pseudo-Hermitian or quasi-Hermitian settings with real spectra.

7. Transport, topology, and broader implications

The NH-QGT is not solely a diagnostic of state-space geometry; it also appears in transport and response theory. In systems with a spectral line gap, the non-Hermitian QGT governs nonlinear electrical responses (Chen et al., 15 Sep 2025). In the narrow-wavepacket limit, the second-order DC conductivity contains a scattering-time-independent term controlled by the band-renormalized non-Hermitian quantum metric $\mathcal{PT}$00, defining an intrinsic nonlinear conductivity (Chen et al., 15 Sep 2025). For finite wavepacket width $\mathcal{PT}$01, additional nonlinear terms arise that depend on the imaginary part of the Berry curvature, leading to $\mathcal{PT}$02 and $\mathcal{PT}$03 corrections absent in Hermitian systems (Chen et al., 15 Sep 2025). This suggests that non-Hermitian transport depends on geometric data beyond the Hermitian metric-curvature pair in its usual form.

In plasmonic lattices, a biorthogonal QGT analysis shows that a non-zero local Berry curvature can arise even in a square lattice without magnetic field (Cuerda et al., 2023). There the effective two-band Hamiltonian contains a real pseudospin-orbit coupling term responsible for the quantum metric and an imaginary non-Hermitian term, $\mathcal{PT}$04, arising from radiative and dissipative loss differences between TE and TM modes (Cuerda et al., 2023). The work attributes the non-zero Berry curvature exclusively to non-Hermitian effects which break time-reversal symmetry, while the quantum metric originates from pseudospin-orbit coupling (Cuerda et al., 2023).

Several recent works also link NH-QGT structures to topological invariants and their bounds. In pseudo-Hermitian settings with real spectra, the Berry curvature obtained from the NH-QGT yields Chern numbers in the usual way (Huang et al., 21 Sep 2025). In interacting non-Hermitian many-body systems, the B-VQE framework distinguishes state-topological and band-topological signatures and computes a state Chern number from the biorthogonal Berry curvature (B et al., 17 Jun 2026). A separate line of work proves geometric bounds on non-Hermitian QGTs and response functions, including a Chern-number bound involving RR and LL metric blocks and connection differences (Matraszek et al., 29 Dec 2025).

A recurring misconception is that a single non-Hermitian QGT universally controls all observables. The available formulations suggest a more differentiated picture. In exciton-polariton and wave-packet dynamics, $\mathcal{PT}$05 is associated with fidelity-susceptibility scaling near phase transitions, while transverse drifts appear to couple to $\mathcal{PT}$06 or to equivalent expressions involving LR curvature plus Berry-connection gradients (Hu et al., 2023, Hu et al., 2024). This suggests that non-Hermitian geometry is intrinsically multi-representational: the physically relevant tensor depends on the observable, the normalization convention, and whether the problem is formulated in biorthogonal, right-normalized, pseudo-Hermitian, or gauge-covariant language.

Across these formulations, the common principle remains stable. The NH-QGT is the object that organizes distance, curvature, adiabatic phase, critical singularity, and, in several settings, measurable dynamical or transport response for non-Hermitian quantum states (Zhang et al., 2018, Ye et al., 2023, Hu et al., 2023).