Non-Hermitian Quantum Geometric Tensor

• Non-Hermitian quantum geometric tensor is a set of tensors that extend the Hermitian framework by incorporating biorthogonal left and right eigenstates and complex gauge structures.
• They distinguish between constructions like RR and LR tensors, each governing different physical observables such as anomalous velocities and energy corrections.
• Measurement protocols use nonadiabatic ramps and force operators, enabling reconstruction of tensor components in synthetic and photonic experimental platforms.

The non-Hermitian quantum geometric tensor denotes a family of geometric tensors that extend the Hermitian quantum geometric tensor to parameter-dependent non-Hermitian Hamiltonians, where left and right eigenstates differ and, in quasi-Hermitian or $\mathcal{PT}$-symmetric regimes, the physical inner product itself may depend on the parameters. In its closest analogue to the Hermitian construction, the tensor unifies a quantum metric and a Berry curvature; in more general settings, however, the tensor need not be Hermitian in its parameter indices, the metric–curvature split becomes convention-dependent, and several inequivalent constructions coexist (Zhang et al., 2018, Huang et al., 21 Sep 2025, Hu et al., 2024).

1. Hermitian benchmark and the source of non-Hermitian ambiguity

In standard quantum mechanics, for a normalized eigenstate $|\psi_n(\lambda)\rangle$, the quantum geometric tensor is usually written as

$$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$

with $$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$ and the usual Berry curvature given by the imaginary antisymmetric part. This Hermitian structure relies on a fixed inner product and on the identification of bras as Hermitian conjugates of kets (Zhang et al., 2018).

In non-Hermitian systems, the eigenproblem splits into right and left sectors,

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

with biorthogonal normalization $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$. This already removes the uniqueness of the Hermitian construction, because there is no single canonical inner product built from right states alone that remains compatible with the dynamics in full generality (Huang et al., 21 Sep 2025).

A further complication is gauge structure. For diagonalizable non-Hermitian Hamiltonians, the eigenstates carry a $GL(1,\mathbb{C})$ gauge freedom,

$$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$

rather than the $U(1)$ phase freedom familiar from Hermitian systems. As a consequence, the geometric phase is generally complex, and any non-Hermitian quantum geometric tensor must be organized so that its geometric content is well controlled under this larger gauge group (Zhang et al., 2018).

The literature therefore distinguishes several settings rather than a single universal tensor.

Setting	Representative tensor	Characteristic feature
Hermitian	$Q_{\mu\nu}^{(\mathrm{Herm})}$	Real symmetric metric, imaginary antisymmetric curvature
$|\psi_n(\lambda)\rangle$0-symmetric / pseudo-Hermitian	Biorthogonal or metric-operator QGT	Hermitian-like decomposition restored in real-spectrum regime
Generic non-Hermitian band theory	$|\psi_n(\lambda)\rangle$1, $|\psi_n(\lambda)\rangle$2	RR and LR tensors are inequivalent
Generalized biorthogonal state-space geometry	$|\psi_n(\lambda)\rangle$3	Four tensor sectors can appear

A common misconception is that the non-Hermitian quantum geometric tensor is unique once left and right eigenstates are introduced. The later literature instead treats uniqueness as a special property of Hermitian and certain pseudo-Hermitian regimes, not of generic non-Hermitian band theory (Pal, 24 Jul 2025).

2. Metric-operator and $|\psi_n(\lambda)\rangle$4-symmetric constructions

A particularly structured formulation arises in $|\psi_n(\lambda)\rangle$5-symmetric quantum mechanics. In the unbroken $|\psi_n(\lambda)\rangle$6-symmetric regime $|\psi_n(\lambda)\rangle$7, one has a real spectrum, a complete set of eigenstates, and a positive-definite metric operator $|\psi_n(\lambda)\rangle$8 satisfying

$|\psi_n(\lambda)\rangle$9

The physical inner product is then

$$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$0

and the dual vectors are defined by $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$1, yielding a biorthonormal basis $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$2 (Zhang et al., 2018).

Within this framework, Zhang, Wang, and Gong define an extended quantum geometric tensor $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$3 whose imaginary part reproduces the Berry curvature and whose real part reproduces the metric tensor,

$$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$4

The associated metric is derived from a Bures-type distance between biorthogonal pure-state projectors $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$5, while the curvature is obtained from the Berry connection $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$6 (Zhang et al., 2018).

The same real-spectrum logic reappears in pseudo-Hermitian systems. If

$$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$7

with Hermitian, invertible $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$8, and the spectrum is real, then $$Q^{(\mathrm{Herm})}_{\mu\nu} = \langle\partial_\mu\psi_n|(1-|\psi_n\rangle\langle\psi_n|)|\partial_\nu\psi_n\rangle,$$9, and the left–right tensor

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$0

becomes Hermitian in its pair of indices,

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$1

Only in that regime is the separation into a real symmetric metric and a real antisymmetric Berry curvature unambiguous: $$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$2 This restores a close parallel with the Hermitian QGT while preserving the necessity of biorthogonal left and right states (Huang et al., 21 Sep 2025).

A related projector-based pseudo-Hermitian form, used for topological bands with real spectra, is

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$3

with $$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$4. Its real part defines the non-Hermitian quantum metric and its imaginary part reproduces the biorthogonal Berry curvature (Zhu et al., 2021).

3. Competing definitions beyond the quasi-Hermitian regime

Outside the real-spectrum pseudo-Hermitian setting, two inequivalent band-geometric tensors are especially prominent. In two-band non-Hermitian systems, one may define a right–right tensor

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$5

and a left–right tensor

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$6

With the normalization $$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$7, the RR tensor yields real-valued

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$8

whereas the LR tensor is generally complex, with

$$g_{\mu\nu}^{(\mathrm{Herm})}=\Re Q^{(\mathrm{Herm})}_{\mu\nu}$$9

The Hermitian limit collapses these distinctions, but the genuinely non-Hermitian regime does not (Hu et al., 2024).

This non-uniqueness is not merely notational. One strand of the literature treats the RR tensor as the direct Fubini–Study generalization of right eigenstates, while another regards the LR tensor as the physically consistent object because it is built from the biorthogonal pairing natural to non-Hermitian evolution. A later state-space treatment sharpens the distinction by introducing a generalized biorthogonal Fubini–Study tensor

$$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$0

and decomposing it into four algebraically distinct sectors: a real symmetric metric $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$1, an imaginary antisymmetric curvature $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$2, a real antisymmetric tensor $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$3, and an imaginary symmetric tensor $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$4. In that classification, LL or RR tensors live on the standard complex projective space of a single vector space, whereas LR or RL tensors live on a different biorthogonal geometry (Pal, 24 Jul 2025).

A related SSH analysis makes the same point from a different angle. It distinguishes a metric defined by symmetrizing in parameter indices, a metric defined by taking the real part, and a metric defined by imposing both reality and symmetry. These notions coincide for Hermitian or pseudo-Hermitian systems but differ for generic non-Hermitian models, leading to pseudo-Riemannian and complex quantum geometries (Ye et al., 2023).

An important controversy concerns whether the RR metric is physically meaningful. One later wave-packet study argues that the RR quantum metric is not ill-defined but instead controls measurable second-order corrections to non-Hermitian semiclassical dynamics, while the LR metric governs different observables. The disagreement is therefore not resolved by discarding one tensor in favor of the other; it is resolved by assigning them different physical sectors (Hu et al., 2024).

4. Geometric content: phases, distances, and dynamical response

In $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$5-symmetric quantum mechanics, adiabatic transport requires a modification of the Schrödinger equation because the metric $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$6 itself depends on the parameters. Unitarity in the $$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$7-dependent inner product is preserved by a gauge field

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

so that

$$H(\lambda)|\psi_n^R(\lambda)\rangle=E_n(\lambda)|\psi_n^R(\lambda)\rangle,\qquad \langle \psi_n^L(\lambda)|H(\lambda)=\langle \psi_n^L(\lambda)|E_n(\lambda),$$9

For cyclic adiabatic evolution, the geometric phase is

$\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$0

and the curvature is $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$1. In this setting the metric measures distinguishability through fidelity, while the curvature governs the geometric phase, exactly as in Hermitian quantum mechanics but now with explicit dependence on the metric operator and biorthogonal structure (Zhang et al., 2018).

In generic non-Hermitian two-band wave-packet dynamics, the two tensors $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$2 and $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$3 enter different sectors of the semiclassical motion. The RR Berry curvature $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$4 appears directly in the first-order anomalous velocity, and the RR quantum metric $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$5 contributes to second-order corrections to the anomalous velocity and the positional shift. By contrast, the LR metric $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$6 enters field-induced corrections to the LR Berry connection and therefore to the phase and effective energy. Because the band gap $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$7 is complex, both $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$8 and $\langle\psi_m^L|\psi_n^R\rangle=\delta_{mn}$9 can influence real observables (Hu et al., 2024).

Line-gapped response theories make the same geometric split at the level of transport. For non-Hermitian electrical response, a band-renormalized left–right quantum metric

$GL(1,\mathbb{C})$0

controls the second-order band-energy shift, while the Berry curvature enters the anomalous velocity. In that framework, the intrinsic nonlinear conductivity is generated by $GL(1,\mathbb{C})$1, whereas $GL(1,\mathbb{C})$2 and $GL(1,\mathbb{C})$3 generate wave-packet-width-dependent terms that have no Hermitian analogue (Chen et al., 15 Sep 2025).

A corresponding nonlinear spin-current theory in Floquet non-Hermitian altermagnets separates the intrinsic second-order response into three gauge-invariant geometric pieces: a quantum-metric term $GL(1,\mathbb{C})$4, a Berry-curvature term $GL(1,\mathbb{C})$5, and a Berry-connection-dipole term $GL(1,\mathbb{C})$6. In the studied line-gapped Floquet phase, the total nonlinear spin conductivity tracks the geometric contribution closely, and the bare quantum metric overwhelmingly dominates the response (Chen et al., 15 May 2026).

5. Criticality, exceptional structures, and non-Bloch geometry

The real part of the non-Hermitian quantum geometric tensor is often used as a detector of singular spectral structure. In the $GL(1,\mathbb{C})$7-symmetric extended-QGT framework, the tensor is well defined only in the unbroken regime where the spectrum is real and the biorthonormal basis is complete. Approaching $GL(1,\mathbb{C})$8-symmetry-breaking points or exceptional points, energy levels coalesce, denominators involving energy gaps vanish, and the metric tensor becomes singular. In the dimerized $GL(1,\mathbb{C})$9-symmetric XY chain, the metric detects both conventional quantum phase transitions and spontaneous $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$0-symmetry breaking (Zhang et al., 2018).

For non-Hermitian SSH systems, the left–right metric identifies the full topological phase diagram, whereas LL or RR constructions each miss part of the transition structure. The resulting geometry is no longer purely Riemannian: introducing non-Hermiticity leads to pseudo-Riemannian and complex geometries, and in some phases the metric degenerates so that the effective dimensionality of the quantum geometry is reduced by one. That analysis explicitly draws a correspondence between topological phase-transition curves and lightlike paths of a pseudo-Riemannian metric (Ye et al., 2023).

Boundary sensitivity introduces another layer of geometric distinction. In the non-Hermitian skin effect, the localization length scale is encoded in the quantum metric defined solely from right eigenstates, but not in the biorthogonal quantum metric. In the Hatano–Nelson model, the RR metric captures the skin localization scale, whereas the LR metric remains blind to it because the left and right exponential envelopes cancel in the biorthogonal product. In non-Bloch SSH models, both metrics show power-law divergences at gapless points that depend on boundary conditions, and cusps of the generalized Brillouin zone are signaled by discontinuities in the quantum metrics (Imura et al., 11 Apr 2026).

A gauge-covariant Dyson-map formulation of quasi-Hermitian systems places these singularities into a hierarchy. There the Dyson connection is decomposed into Hermitian stretching and anti-Hermitian rotation components, and the non-Hermitian geometric curvature is traced to the non-commutativity of the stretching components. Near a general two-level exceptional point, the quantum metric tensor exhibits a leading-order divergence

$$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$1

while the Berry curvature shows a weaker, subleading divergence

$$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$2

with $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$3 the displacement from the exceptional point along parameter axis $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$4 (Das et al., 14 Jun 2026).

6. Measurement strategies and experimental realizations

Direct reconstruction of the non-Hermitian quantum geometric tensor has moved from formal possibility to explicit protocol. In pseudo-Hermitian systems with real spectra, two quantum-simulation schemes recover all components of the left–right QGT. One uses generalized expectation values of the energy-fluctuation operator $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$5 between left- and right-evolved states; the other uses generalized force operators $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$6 together with evolutions under $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$7 and $$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$8. Both rely on controlled nonadiabatic parameter ramps and generalized expectation values of the form

$$|\phi_j\rangle \to f_j(R)|\phi_j\rangle,\qquad |\chi_j\rangle \to f_j(R)^{-1}|\chi_j\rangle,$$9

which can be accessed by a controlled-swap circuit (Huang et al., 21 Sep 2025).

Pseudo-Hermitian topological phases built from $U(1)$0-deformed matrices provide a complementary experimental route. In that setting the full non-Hermitian QGT, together with Abelian and non-Abelian tensor Berry connections, can be extracted in synthetic matter by modulation spectroscopy after a Hermitian dilation of the pseudo-Hermitian Hamiltonian. The same framework emphasizes that pseudo-Hermitian systems can share topological invariants with Hermitian models while retaining different band geometries encoded in the metric sector of the QGT (Zhu et al., 2021).

Driven-dissipative photonic systems supply experimentally accessible band geometries beyond the pseudo-Hermitian regime. In a non-Hermitian exciton-polariton platform, generalized QGT components can be reconstructed from measured pseudospins once the left eigenstates are also constructed experimentally. That work reviews RR and LR definitions side by side and shows that generalized QGT components leave clear signatures in wave-packet dynamics, where anomalous Hall drift arises from both non-Hermitian Berry curvature and Berry connection (Hu et al., 2023).

A weakly non-Hermitian plasmonic lattice demonstrates a more limited but instructive regime. There the standard Hermitian formulas were used as a good approximation because the non-Hermiticity is weak, bands remain non-degenerate, and the Berry curvature is several orders of magnitude smaller than the metric. Even so, the local Berry curvature arises exclusively from non-Hermitian effects which break time-reversal symmetry, whereas the quantum metric originates from a pseudospin-orbit coupling given by the polarization and directional dependence of the radiation (Cuerda et al., 2023).

Taken together, these developments suggest a clear present picture. In real-spectrum quasi-Hermitian and $U(1)$1-symmetric regimes, the non-Hermitian quantum geometric tensor can closely parallel the Hermitian object. In generic non-Hermitian systems with complex spectra, however, the field consists of multiple inequivalent tensors—RR, LR, and generalized biorthogonal Fubini–Study constructions—whose metric and curvature sectors govern different observables, singularities, and transport phenomena.