Non-Hermitian Quantum Geometric Tensor

• The non-Hermitian quantum geometric tensor is an extension of the standard QGT that uses distinct left and right eigenstates in a biorthogonal framework.
• It decomposes into metric and curvature sectors, with the metric encoding state distinguishability and the curvature representing geometric phase effects.
• This formulation is crucial for diagnosing criticality, exceptional points, and nonlinear responses in quantum systems, with specialized measurement and computational protocols.

Non-Hermitian quantum geometric tensor (QGT) denotes a family of extensions of the standard quantum geometric tensor to systems in which the Hamiltonian is not self-adjoint, right and left eigenstates are distinct, and the relevant geometry is therefore biorthogonal rather than purely Hilbert-space orthonormal. In the quasi-Hermitian, pseudo-Hermitian, or unbroken $\mathcal{PT}$-symmetric regimes, these constructions recover a unified picture in which a metric sector encodes distinguishability and a curvature sector encodes geometric phase; outside those regimes, Hermiticity of the tensor itself can fail, so the separation into quantum metric and Berry curvature becomes definition-dependent (Zhang et al., 2018, Huang et al., 21 Sep 2025).

1. Formal definitions and geometric decomposition

For a non-Hermitian Hamiltonian $H(\boldsymbol{\lambda})$ with right and left eigenstates

$$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$

and biorthogonality

$$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$

a standard left-right definition is

$$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$

Equivalently,

$$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$

This is the direct non-Hermitian generalization of the Hermitian QGT, with the crucial difference that both left and right eigenstates enter (Huang et al., 21 Sep 2025).

In pseudo-Hermitian systems with real spectra, the tensor admits a Hermitian-like decomposition

$$g^n_{\mu\nu} = \frac{1}{2} \left[ Q^n_{\mu\nu} + (Q^n_{\nu\mu})^* \right],\qquad F^n_{\mu\nu} = i \left[ Q^n_{\mu\nu} - (Q^n_{\nu\mu})^* \right].$$

If $Q^n_{\mu\nu}= (Q^n_{\nu\mu})^*$, this reduces to

$$Q^n_{\mu\nu} = g^n_{\mu\nu} - \frac{i}{2} F^n_{\mu\nu},\qquad g^n_{\mu\nu} = \Re Q^n_{\mu\nu},\quad F^n_{\mu\nu} = -2\,\Im Q^n_{\mu\nu}.$$

In generic non-Hermitian systems, however, $Q^n_{\mu\nu} \neq (Q^n_{\nu\mu})^*$, so the tensor is not Hermitian and the conventional metric-curvature identification becomes ambiguous (Huang et al., 21 Sep 2025).

A closely related formulation arises in unbroken $H(\boldsymbol{\lambda})$0-symmetric quantum mechanics. There the extended QGT is built from the biorthogonal pair $H(\boldsymbol{\lambda})$1, $H(\boldsymbol{\lambda})$2 and satisfies

$H(\boldsymbol{\lambda})$3

with

$H(\boldsymbol{\lambda})$4

The real part induces a metric tensor on parameter space, while the imaginary part gives a Berry curvature two-form (Zhang et al., 2018).

2. Metric operators, biorthogonality, and gauge-covariant structure

In $H(\boldsymbol{\lambda})$5-symmetric quantum mechanics, the geometric problem is complicated by the fact that the inner product itself depends on parameters. In the unbroken regime one can construct a positive definite metric operator $H(\boldsymbol{\lambda})$6 satisfying

$H(\boldsymbol{\lambda})$7

which defines the parameter-dependent inner product

$H(\boldsymbol{\lambda})$8

The left states are then induced by the metric,

$H(\boldsymbol{\lambda})$9

and obey biorthonormality and completeness. This is the natural setting for the extended QGT in $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$0-symmetric systems (Zhang et al., 2018).

Adiabatic transport in such a parameter-dependent inner product requires an additional gauge field,

$$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$1

so that the dynamics is governed by

$$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$2

The resulting Berry phase has the usual loop form $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$3, but the connection and curvature must be computed with the biorthogonal structure and the parameter-dependent metric (Zhang et al., 2018).

A more explicitly gauge-covariant formulation elevates the Dyson map $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$4 to the central geometric object. In the quasi-Hermitian regime, $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$5 maps a non-Hermitian Hamiltonian $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$6 to an equivalent Hermitian Hamiltonian $$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$7 and defines the Dyson connection

$$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$8

This connection decomposes into Hermitian and anti-Hermitian parts,

$$H(\boldsymbol{\lambda})|\phi_n^R(\boldsymbol{\lambda})\rangle = E_n(\boldsymbol{\lambda}) |\phi_n^R(\boldsymbol{\lambda})\rangle,\qquad H^\dagger(\boldsymbol{\lambda})|\phi_n^L(\boldsymbol{\lambda})\rangle = E_n^*(\boldsymbol{\lambda}) |\phi_n^L(\boldsymbol{\lambda})\rangle,$$9

identified respectively as stretching and rotation components. The stretching sector is fixed by metric deformation, whereas the rotation sector captures the unitary gauge redundancy in the Dyson map. In this framework, the non-Hermitian QGT is written in terms of gauge-covariant states built from the $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$0-covariant derivative and the orthogonal stretching fluctuation, and the geometric curvature is traced to the non-commutativity of the stretching operators at the operator level (Das et al., 14 Jun 2026).

3. Competing formulations and definitional ambiguities

A central issue in the literature is that non-Hermitian quantum geometry does not admit a unique extension once positivity, orthonormality, and Hermiticity are relaxed. One widely used option employs only normalized right eigenstates and the standard Dirac inner product,

$$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$1

With self-normal right eigenstates, the diagonal elements satisfy

$$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$2

so the non-Hermitian quantum metric equals the fidelity susceptibility of the right eigenstate. This convention is operationally effective for identifying localization transitions, mobility edges, and many-body transitions, but it does not use left eigenstates explicitly (Ren et al., 2024).

By contrast, biorthogonal formulations use both left and right states. In two-band non-Hermitian systems, several distinct prescriptions for the metric sector have been discussed: the symmetric part of the left-right QGT, its real part, and its real symmetric part. The existence of multiple definitions reflects the fact that, unlike in Hermitian geometry, the requirements “symmetric,” “real,” and “gauge-invariant” are no longer automatically equivalent (Hu et al., 2023, Ye et al., 2023).

This has produced an objective controversy rather than a merely terminological one. In exciton-polariton wavepacket dynamics and in a general two-dimensional non-Hermitian wavepacket treatment, one generalization defined using only right eigenstates and another defined using both left and right eigenstates both play a significant role in dynamics. This suggests that “the” non-Hermitian QGT is not a single universally dominant object across all observables; rather, distinct tensorial constructions become natural in different response problems (Hu et al., 2024).

4. Singular geometry, criticality, and topology

The real part of the non-Hermitian QGT is widely used as a diagnostic of criticality. In $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$3-symmetric systems, the ground-state metric tensor admits a Kubo-like form with denominators $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$4, so degeneracies or spectral coalescences can produce divergences or non-analytic behavior. In the dimerized $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$5 chain in an alternating complex field, singularities in the intensive metric tensor coincide with analytically identified quantum phase transition radii

$$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$6

while another metric component diverges as $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$7, signaling the $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$8-symmetry breaking line (Zhang et al., 2018).

A broader numerical and analytical study showed that the quantum metric of eigenstates in non-Hermitian models exactly identifies localization transitions, mobility edges, and many-body quantum phase transitions with gap closings. In that setting, peaks of the metric reproduce the known critical points in two non-Hermitian generalized Aubry-Andre models and in non-Hermitian cluster and mixed-field Ising models. The same analysis also emphasizes a limitation: transitions without gap closing are not detected by metric singularities, mirroring a familiar limitation from Hermitian fidelity-based diagnostics (Ren et al., 2024).

Exceptional points provide the sharpest geometric singularities. In the biorthogonal variational quantum eigensolver framework, the metric component $$\langle \phi_i^L(\boldsymbol{\lambda}) | \phi_j^R(\boldsymbol{\lambda}) \rangle = \delta_{ij},$$9 is used as an EP diagnostic and scales as

$$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$0

while in a general gauge-covariant quasi-Hermitian treatment the quantum metric tensor exhibits a leading-order divergence $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$1 and the Berry curvature a weaker, subleading divergence $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$2 near an EP (B et al., 17 Jun 2026, Das et al., 14 Jun 2026).

Non-Hermitian topology also acquires a specifically geometric refinement. In the two-dimensional non-Hermitian $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$3-$$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$4 model, a state Chern number built from the NH-QGT Berry curvature can be $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$5 while the conventional band Chern number is zero in the same parameter region. This state-topology/band-topology discrepancy is tied to interacting and non-Hermitian effects and is one of the clearest many-body uses of the NH-QGT. In non-Hermitian SSH systems, the metric can become pseudo-Riemannian or complex, and in some phases it degenerates so that the effective dimensionality of the quantum geometry is reduced by one (B et al., 17 Jun 2026, Ye et al., 2023).

5. Adiabatic dynamics, wavepackets, and nonlinear response

The imaginary sector of the QGT governs geometric phase, but in non-Hermitian systems it also enters dynamical laws through additional gauge structures. In $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$6-symmetric adiabatic evolution, the Berry curvature determines the Berry phase through

$$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$7

yet the time dependence of the metric operator contributes through the gauge field $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$8. The resulting holonomy therefore depends not only on eigenstate transport but also on the evolution of the physical inner product itself (Zhang et al., 2018).

In two-dimensional non-Hermitian wavepacket dynamics, first-order perturbation theory shows that both the right-right and the left-right generalizations of the QGT enter the corrected semiclassical equations of motion. The anomalous Hall drift is governed by the RR Berry curvature, while the field-induced correction to the Berry phase and the anomalous Berry connection involve the LR quantum metric and LR Berry connection. This establishes that both tensorial structures are dynamically active rather than formally redundant (Hu et al., 2024).

Transport theory has extended this geometric role far beyond adiabatic transport. In line-gapped non-Hermitian systems, the quantum metric, which is a component of the QGT and takes complex values in non-Hermitian systems, generates an intrinsic nonlinear conductivity independent of the scattering time, while the complex Berry curvature induces a wavepacket-width-dependent response. The appearance of explicit wavepacket-width dependence is a distinctive non-Hermitian effect absent in Hermitian systems (Chen et al., 15 Sep 2025).

An analogous decomposition appears in nonlinear spin transport. In Floquet non-Hermitian altermagnets with a spectral line gap, the intrinsic nonlinear spin conductivity separates into quantum metric, Berry curvature, and Berry connection dipole contributions. In the studied $$Q^n_{\mu\nu}(\boldsymbol{\lambda}) := \langle \partial_\mu \phi_n^L | \partial_\nu \phi_n^R \rangle -\langle \partial_\mu \phi_n^L | \phi_n^R \rangle \langle \phi_n^L | \partial_\nu \phi_n^R \rangle.$$9-wave altermagnet, the nonlinear spin conductivity is overwhelmingly dominated by the bare quantum metric, and the optical field polarization can tune and even strictly reverse both longitudinal and transverse spin currents (Chen et al., 15 May 2026).

Non-Hermitian photonics provides a complementary example. In a square plasmonic lattice, the quantum metric is attributed to pseudospin-orbit coupling, while a non-zero Berry curvature arises exclusively from non-Hermitian effects which break the time-reversal symmetry. In that weakly non-Hermitian regime, the Hermitian-style QGT remains a good approximation to the more general biorthogonal tensor, but the physical origin of the curvature is explicitly dissipative (Cuerda et al., 2023).

6. Measurement protocols, variational algorithms, and platforms

Direct access to the NH-QGT has become an experimental and algorithmic problem in its own right. In pseudo-Hermitian systems with real spectra, two complete and independent measurement schemes have been proposed. One extracts the full QGT from generalized expectation values of the energy fluctuation operator $$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$0 measured between two nonadiabatically prepared states; the other reconstructs Berry curvature and quantum metric separately from generalized-force measurements. Both use generalized expectation values of the form

$$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$1

and a controlled-swap circuit is presented as the universal building block for these measurements (Huang et al., 21 Sep 2025).

Exciton-polariton systems provide a different route based on pseudospin tomography. The right-right QGT can be reconstructed from polarization-resolved pseudospins exactly as in Hermitian two-band systems. For the left-right QGT, a two-band biorthogonal constraint allows the left pseudospin to be inferred from measured right pseudospins, because the left pseudospin of one band is antipodal to the right pseudospin of the other band. This yields an experimentally motivated protocol for reconstructing both RR and LR quantum geometry in a non-Hermitian hybrid photonic system (Hu et al., 2023).

On the computational side, the Biorthogonal Variational Quantum Eigensolver uses independent ansätze for left and right eigenstates and includes an NH-QGT readout after variational convergence. In that workflow the tensor is estimated via a biorthogonal parameter-shift rule, with circuit count scaling $$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$2, identical to the Hermitian cost. The same framework uses the NH-QGT metric to locate exceptional points and the biorthogonal Berry curvature to compute a state Chern number in interacting many-body models (B et al., 17 Jun 2026).

7. Bounds, limitations, and broader outlook

Recent work has established non-Hermitian geometric bounds that constrain mixed left-right tensors by symmetric right-right and left-left tensors together with connection-difference terms. For a band $$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$3, the mixed tensor obeys

$$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$4

and related inequalities bound non-Hermitian Chern numbers and optical weights. In two-band cases, the trace of the optical weight is directly expressed through the right-basis quantum metric and a complex spectral angle factor, showing that QGT geometry constrains experimentally accessible response functions even in open-system settings (Matraszek et al., 29 Dec 2025).

These constructions are regime-sensitive. Pseudo-Hermiticity with real spectra, unbroken $$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$5 symmetry, or quasi-Hermiticity restores a well-controlled geometric framework with positive-definite or at least physically meaningful metric operators. In more generic non-Hermitian systems, positivity and completeness can fail, Berry phases and curvatures can become fully complex, and the geometric picture may be more subtle or partially ill-defined. The reliance on nondegenerate bands and the adiabatic perturbative regime is also explicit in several dynamical and measurement proposals (Huang et al., 21 Sep 2025, Zhang et al., 2018).

A plausible implication is that non-Hermitian quantum geometry is moving toward a layered rather than a monolithic formulation. Pure-state biorthogonal QGTs, gauge-covariant quasi-Hermitian tensors, response-renormalized band metrics, and mixed-state bundle constructions each organize part of the same problem. The mixed-state $$Q^n_{\mu\nu} = \sum_{m \neq n} \langle \partial_\mu \phi_n^L | \phi_m^R \rangle \, \langle \phi_m^L | \partial_\nu \phi_n^R \rangle.$$6 QGT has been presented as conceptually close to what is needed in non-Hermitian geometry because one replaces orthonormal eigenbases by biorthogonal ones and the usual inner product by a modified one, but this remains a structural bridge rather than a completed non-Hermitian theory (Wang et al., 2024).

In this sense, the non-Hermitian QGT is best understood not as a single universally fixed tensor, but as the geometric core of biorthogonal quantum theory: a framework in which metric deformation, Berry curvature, exceptional-point singularity, fidelity susceptibility, nonlinear response, and many-body state topology are treated as different projections of the same non-unitary parameter-space structure.