Generalised state space geometry in Hermitian and non-Hermitian quantum systems

Kunal Pal

REVIEW 3 major objections 3 minor 95 references

Generalised state space geometry in Hermitian and non-Hermitian quantum systems

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Biorthogonal states let non-Hermitian quantum systems carry a four-part Fubini–Study geometry.

desk verdict A genuinely new biorthogonal α-deformed geometry, but the central metric claim breaks for |α|>1 and the normalization assumption is load-bearing. read the letter →

arxiv 2507.18486 v1 pith:4PDQIKFW submitted 2025-07-24 quant-ph

Kunal Pal This is my paper · ORCID

classification quant-ph MSC 53B0581Q7053Z05 PACS 03.65.-w03.65.Vz02.40.Ky

keywords quantuminformationgeometryFubini–Studymetricnon-HermitianmechanicsbiorthogonalformalismBerrycurvaturegeometrictensoralpha-connectionsnaturalgradient

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to transfer the two pillars of classical information geometry — a metric on probability distributions and a dual family of ±α connections — to the space of pure quantum states, and claims this is possible once the inner product is made biorthogonal rather than Hermitian. It constructs a one-parameter family of Fubini–Study tensors whose real symmetric part plays the role of a quantum metric and whose gauge-invariant connections satisfy a generalised ±α duality relative to that metric, with the phase of the wavefunction entering explicitly. For non-Hermitian Hamiltonians, the same machinery classifies the quantum geometric tensor into four irreducible pieces: real symmetric, imaginary antisymmetric, imaginary symmetric, and real antisymmetric, the last two being new 'flipped' tensors that vanish in the Hermitian limit. A reader should care because this gives a single geometric language for non-Hermitian state spaces and shows which part of the geometry drives natural-gradient optimisation when the cost function is complex-valued.

What carries the argument

The central object is the biorthogonal Fubini–Study tensor, built from a left state and a right state paired by the normalisation $\langle\Psi_L|\Psi_R\rangle=1$ rather than by complex conjugation. Its $\alpha$-deformed version uses the two asymmetric functions $l_1^{(\alpha)}=P^{(1-\alpha)/2}e^{i(1-\alpha)\phi}/(1-\alpha)$ and $l_2^{(-\alpha)}$, whose mutual inner product is fixed by normalisation. Expanding the biorthogonal overlap integral supplies the metric at second order and two connections at third order; subtracting gauge-dependent terms yields invariant connections. The decomposition of the tensor into four rank-two pieces — symmetric/antisymmetric times real/imaginary — carries the classification, and the identity $\mathrm{Re}[\Gamma_1^{(\alpha)}+\Gamma_2^{(-\alpha)}]=2\Gamma^{(c)}$ carries the duality claim. The same machinery produces the optimisation statement: the real symmetric part of the tensor sets the natural-gradient flow for the real part of a complex cost, and the imaginary symmetric part sets the flow for its imaginary part.

What would settle it

Take a two-level non-Hermitian Hamiltonian whose parameters trace a closed loop around an exceptional point, start from a biorthogonal pair with $\langle\Psi_L|\Psi_R\rangle=1$, and check whether the overlap remains 1 along the path; if it drifts from unity, the left–right Fubini–Study tensor, its four-piece decomposition, and the associated natural-gradient equations are not defined for that evolution, and the classification would need a different normalisation prescription.

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Extended reading notes

Core claim

The central claim is that the Fubini–Study tensor can be consistently generalised by replacing the Hermitian inner product with a biorthogonal pairing. With the pointwise normalisation $\langle\Psi_L|\Psi_R\rangle=1$, the left–right tensor $FS^{LR}_{ij}=\langle\partial_i\Psi_L|\partial_j\Psi_R\rangle-\langle\partial_i\Psi_L|\Psi_R\rangle\langle\Psi_L|\partial_j\Psi_R\rangle$ decomposes into four gauge-invariant pieces: the real symmetric quantum metric $g^{LR}_{ij}$, the imaginary antisymmetric Berry curvature $\omega^{LR}_{ij}$, and two 'flipped' tensors $\tilde g^{LR}_{ij}$ and $\tilde\omega^{LR}_{ij}$ that are nonzero only because the left and right states are not conjugate. The same construction with $\alpha$-deformed states $l_1^{(\alpha)}$ and $l_2^{(-\alpha)}$, normalised so that $\langle l_1^{(\alpha)}|l_2^{(-\alpha)}\rangle=(1-\alpha^2)^{-1}$, yields an $\alpha$-quantum metric and an $\alpha$-Berry curvature whose real symmetric and imaginary symmetric parts govern separate natural-gradient flows. For the gauge-invariant connections built from the overlap expansion, the paper proves the duality $\mathrm{Re}[\Gamma_1^{(\alpha)}+\Gamma_2^{(-\alpha)}]=2\Gamma^{(c)}$, an exact quantum analogue of the classical $\pm\alpha$ duality, and shows that in the non-Hermitian setting the curvature of the complex Berry connection is the sum of the imaginary antisymmetric part and the flipped real antisymmetric part.

Load-bearing premise

The load-bearing premise is that the biorthogonal overlap $\langle\Psi_L(\theta)|\Psi_R(\theta)\rangle$ stays exactly equal to 1 at every point of the parameter manifold, a normalisation that evolution by a non-Hermitian Hamiltonian does not automatically preserve.

Editorial extensions

If this is right

For any non-Hermitian Hamiltonian with a biorthogonal pair of states, the quantum geometric tensor splits into four tensor pieces; the two 'flipped' pieces vanish identically in the Hermitian limit and can therefore serve as signatures of genuinely non-Hermitian geometry.
The gauge-invariant connections $\Gamma_1^{(\alpha)}$ and $\Gamma_2^{(-\alpha)}$ satisfy $\mathrm{Re}[\Gamma_1^{(\alpha)}+\Gamma_2^{(-\alpha)}]=2\Gamma^{(c)}$, recovering the classical $\pm\alpha$ duality with explicit phase contributions included.
Quantum natural-gradient descent for a complex cost splits into two generally incompatible flows, one governed by the real symmetric part of the non-Hermitian Fubini–Study tensor and one by the imaginary symmetric part.
For an associated left–right pair, imaginary-time evolution reduces to natural-gradient descent only when the generator is biorthogonally Hermitian; otherwise the two optimisation directions disagree.
The left–right, right–left, left–left, and right–right tensors are inequivalent, so fixing the inner product and normalisation determines which metric and curvature are physically relevant.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the classification holds, it provides a principled way to resolve the existing ambiguity in definitions of the quantum metric tensor for non-Hermitian systems: choose the inner product and normalisation, and the fourfold decomposition fixes the rest.
The two incompatible natural-gradient flows suggest a least-squares or Pareto-style compromise for optimising complex costs in non-Hermitian variational algorithms, a direction the paper flags as open.
Because the flipped tensors vanish for Hermitian dynamics, their magnitude could serve as a diagnostic for proximity to exceptional points or the breakdown of unitarity along a parameter trajectory.
The $\alpha$-deformed overlap construction may be the seed of a bona fide quantum $\alpha$-divergence defined from biorthogonal overlaps rather than from density-matrix monotones, which would connect this geometry to parameter-estimation bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this one. First, the biorthogonal α-deformed Fubini-Study construction is genuinely new, at least in the form presented here, and it reduces correctly in the α=0 and trivial-phase limits. Second, the central metric claim in Eq. (35) is wrong for |α|>1, because the phase covariance enters with prefactor (1−α)/(1+α), which is negative outside that interval. The paper says the phase contribution is necessarily non-negative; that is only true for −1<α<1. For a pure-phase state at α=2, g(2)_θθ is negative, so the "α-QMT" is indefinite. Since the natural-gradient optimization and the whole duality picture lean on this tensor being a metric, the unrestricted α-domain is a load-bearing flaw.

What's good: the construction is explicit and coherent. The flipped tensors in (35)-(37), the gauge-invariant connections (49)-(50), and the four-way LR/RL/LL/RR classification are all clearly laid out. The author is also honest: he flags that the bare duality (48) is not physically meaningful, that affine-coordinate interpretation is open, and that the biorthogonal normalization <ΨL|ΨR>=1 is not guaranteed by non-Hermitian dynamics. That self-awareness is real credit.

Soft spots, in proportion. The positivity issue is the big one; it needs either a stated and proved restriction to |α|<1 or a different argument. The biorthogonal normalization is indeed load-bearing, and the author acknowledges it but doesn't resolve it; for a paper that claims a geometry of non-Hermitian state spaces, this is a real limitation, not a footnote. Several long formulas, (27)-(30) and (52)-(53), are stated without derivation; that's fine for a first pass but should be filled in. Finally, the α-FS tensor is derived from an α-QFI defined precisely so the duality (51) holds by construction, so the depth of that result is modest.

Who is this for? People working on quantum information geometry, natural gradient methods, or non-Hermitian QGT. It's a useful starting point, but it needs a careful referee. I'd send it to review, with the expectation of significant revision. My own verdict is conditional, not accept.

Referee Report

3 major / 3 minor

Summary. The paper develops a one-parameter (α) deformation of the Fubini–Study tensor for pure quantum states, using two biorthogonal families l1(α) and l2(−α) with the normalization ⟨l1(α)|l2(−α)⟩ = 1/(1−α²). It derives a four-component decomposition — a real symmetric α-QMT, an imaginary antisymmetric α-Berry curvature, and two 'flipped' tensors — and constructs gauge-invariant connections Γ1 and Γ2 that are claimed to satisfy a generalized ±α duality (Eqs. (48), (51), (54)). The same construction is then transferred to non-Hermitian systems, where LR/RL/LL/RR Fubini–Study tensors are classified and quantum natural-gradient optimization for complex-valued cost functions is discussed. The α=0 limit and the trivial-phase limit are checked against the standard QMT and the classical α-connections.

Significance. If the technical issues identified below are resolved, the framework would be a useful phenomenological extension of quantum information geometry: it gives explicit formulas for a deformed metric, Berry curvature, and connections in a unified biorthogonal language, and the four-way tensor classification usefully organizes several inequivalent definitions that exist in the non-Hermitian literature. The manuscript is candid about the lack of gauge invariance of the bare connections and about the role of normalization, and the limiting checks are correct as far as they go. However, the load-bearing question of whether the α-deformed object is actually a metric is not settled for the α domain advertised, and the biorthogonal normalization assumption is stated but not justified as a dynamically preserved condition. The paper would also be materially strengthened by a concrete physical example showing which of the four tensor components are nonvanishing.

major comments (3)

[Section III B 1, Eq. (35)] Eq. (35) defines g^{(α)}_{ij} = (1/4) E[∂_i l_θ ∂_j l_θ] + ((1−α)/(1+α))(E[∂_i φ ∂_j φ] − E[∂_i φ] E[∂_j φ]). The phase covariance is positive semidefinite, but its coefficient is negative for |α|>1, so the statement immediately after Eq. (35) that the phase contribution is 'necessarily non-negative' is false outside the interval −1<α<1. For a pure-phase state at α=2, g^{(2)}_{θθ} is negative, so the object is not a metric, and the steepest-descent reading of Eqs. (57) and (66) is not defined. Please either restrict α to the interval where positivity holds and prove that restriction, or explicitly present the sign-indefinite case as a pseudo-Riemannian/indefinite structure rather than a quantum metric.
[Section IV A, Eq. (59) and Section III B] The LR (and RL) construction, the four-component decomposition, and the natural-gradient equations (66) all depend on the pointwise biorthogonal normalization ⟨Ψ_L|Ψ_R⟩ = 1, together with the analogous α-normalization ⟨l1(α)|l2(−α)⟩ = 1/(1−α²). The text itself acknowledges immediately after Eq. (59) that this normalization is not guaranteed for evolution generated by a non-Hermitian Hamiltonian, and Appendix A supplies constraints only for a single-parameter family with commuting generators. The classification should therefore be stated as conditional on the state trajectory remaining in the normalization class, or the paper should prove invariance of that class for the physical evolutions considered. Without this, the claimed systematic classification of non-Hermitian QGTs is narrower than the presentation suggests.
[Section III B 4, Eq. (42)] The 'derivation' of the α-FS tensor from the α-QFI is definitional, because ρ^{(α)} = |l2(−α)⟩⟨l1(α)| and the trace normalization were chosen so that (1−α²)² Tr[ρ^{(α)} ∂_i ρ^{(α)} ∂_j ρ^{(α)}] reduces to the overlap expression (31). The construction is internally consistent, but the claim that the object is obtained from a Fisher-information-like quantity requires at least one property of a genuine quantum Fisher information (for example, monotonicity under stochastic maps or a Cramér–Rao-type bound), or an explicit statement that this is only a formal analogue.

minor comments (3)

[Section III B 3] In the paragraph after Eq. (39), 'the moralisation is maintained throughout the evolution' should read 'the normalisation is maintained'; the typo makes the sentence difficult to parse.
[Section III A, Eqs. (27)–(30)] The lengthy expressions (27)–(30) are central objects of the first construction but are presented without derivation or reference to an appendix. Please provide the intermediate steps, or at least describe the structure of the calculation, so that the formulas can be checked without rederiving the whole expansion.
[General] There are numerous typographical spacing errors ('di fferent', 'a ffect', 'e ffect') and inconsistent notation for symmetrization ('sym' versus explicit (ij) indices). A careful proofread is needed before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the α-FS tensor, QFI consistency check, and gauge-invariant connections are explicit constructions whose formal duality is acknowledged in the paper, not hidden predictions.

full rationale

The derivation chain is self-contained and the apparent self-reference is openly disclosed rather than used as evidence. Section III B defines the biorthogonal functions l1(α), l2(−α) in Eq. (32), fixes the normalization in Eq. (33), and then constructs the α-FS tensor in Eq. (31); Eq. (35) is a direct expansion of that definition. The modified QFI in Eq. (42) is introduced after the FS tensor has already been defined, with the prefactor and the choice ρ(α)=|l2(−α)⟩⟨l1(α)| selected so that the trace reproduces Eq. (31); the paper presents this as a consistency check, not as an independent first-principles prediction. Likewise, the gauge-invariant connections (49)-(50) are obtained by adding gauge-restoring terms to the bare connections (43)-(44), and the duality identity (51) is an algebraic consequence of that construction. The paper itself flags the limitation: in Section III B 5 it states that the bare identity is 'not meaningful to interpret this as a quantum generalisation of such duality,' and in Section III B 6 it says that 'whether the apparent duality (51) can be interpreted as the existence of affine coordinates like the classical case remains to be seen.' The four-tensor classification in Section IV is a decomposition of the LR-FS tensor into real/imaginary symmetric/antisymmetric parts, again a definitional taxonomy tied to the chosen normalization, and the paper emphasizes that the different normalizations are context-dependent. There are no fitted parameters, no load-bearing self-citations (the only self-citation, [16], appears in a review-style literature list), and no uniqueness theorem imported from the author's own prior work to force the construction. The possible non-positivity of Eq. (35) for |α|>1 is a mathematical correctness concern rather than a circularity, and it does not change the circularity verdict.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

All substantive results are derived from the chosen biorthogonal inner product and normalization. There are no fitted constants, so the free-parameter list is empty. The main axioms are smoothness and integrability of the wavefunction and its deformed powers, pointwise biorthogonal normalization, and invertibility of the metric in optimization applications. The invented alpha-density matrix and BSLD are formal devices that carry no independent evidence outside the paper.

assumptions (5)

domain assumption Wavefunction normalization and smoothness: Psi(x;theta) is normalized to unity and P, phi are smooth and differentiable on the parameter manifold.
Invoked throughout Sections I and II to define the FS tensor and metric connection from the polar decomposition of the wavefunction.
domain assumption Square-integrability of deformed states l1(alpha) and l2(-alpha) for the alpha values used.
Section III B notes that the polynomial combinations must lie in the Hilbert space of square-integrable functions, but no explicit constraints on alpha are stated.
domain assumption Biorthogonal normalization <Psi_L|Psi_R>=1 holds at every point of the parameter manifold.
Section IV A, eq. (59), is used to obtain the gauge-invariant LR-FS tensor and the natural gradient equations.
ad hoc to paper The alpha-density matrix rho^(alpha)=|l2(-alpha)><l1(alpha)| has trace 1/(1-alpha^2) and is used as the basis for a QFI-like derivation.
Defined in Section III B 4. It is not a physical density matrix, and its trace is fixed by the chosen normalization to reproduce the FS tensor.
domain assumption Metric invertibility for natural gradient equations.
Equations (55)-(57), (66), and (75) solve for delta_theta using the inverse metric; the paper mentions pseudo-inverses but relies on generic invertibility.

invented entities (2)

alpha-density matrix rho^(alpha)
purpose: To derive the alpha-FS tensor from a modified quantum Fisher information expression.
Non-Hermitian operator defined as |l2(-alpha)><l1(alpha)| whose trace is set to 1/(1-alpha^2); it is not the physical density matrix and has no external falsifiable handle.
Biorthogonal symmetric logarithmic derivative (BSLD) L^LR_theta
purpose: Formally defined to write a biorthogonal quantum Fisher information equal to four times the LR-FS tensor.
Introduced speculatively in Section IV A; no equation of motion or estimation protocol is given.

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Cite this review

Pith. "Pith review of Generalised state space geometry in Hermitian and non-Hermitian quantum systems." pith.science (2026). https://pith.science/paper/4PDQIKFW

@misc{pith2026250718486,
  author       = {Pith},
  title        = {Pith review of: Generalised state space geometry in Hermitian and non-Hermitian quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4PDQIKFW}},
  note         = {Machine review of arXiv:2507.18486}
}

read the original abstract

One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher--Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states -- the Fubini-Study tensor, which is preserved under the unitary evolution. In this work, we explore how modifying the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini--Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.

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Works this paper leans on

95 extracted references · 53 canonical work pages

[1]

Derivation from QFI In this subsection, we ask: if it is possible to obtain the form of this Fubini-Study structure, and subsequently the α- quan- tum metric tensor from (modified version of) quantum Fisher information, as was done in [25] for the case ofα = 0. It turns out that, due to the lack of normalisation (for the two func- tions l(α), l(−α)) it is...
[2]

Berry curvature Even though we have decomposed the non-Hermitian FS tensor, where the real, symmetric part can be ascribed to the α-QMT 6, the significance of the rest of the parts is not en- tirely evident at the moment. In particular, it can be seen that the antisymmetric field strength of theα-gauge fields, namely, ˜Ai = i⟨l1(α)|∂ jl2(−α)⟩ is, ˜Fi j = ...
[3]

(27) This is one of the simplest generalised versions of the QMT, which we will call theα- quantum-metric tensor (QAMT)

Then, using the form of the generalised tensor structure (25), we can obtain the generalised metric on this space given by the real and symmetric part, the explicit form of which can be written down as g(α) i j = 1 4Ep h cos (2αϕ)∂ilθ∂ jlθ i +Ep h cos (2αϕ)∂iϕ∂ jϕ i − 2 (α2− 1) Ep h cos (2αϕ)∂iϕ i Ep h cos (2αϕ)∂ jϕ i + Ep h cos (2αϕ)∂ jlθ i Ep h cos (2αϕ...
[4]

Components of the α-FS tensor Let us first explicitly extract the form of the four compo- nents of the α-FS tensor. The real and the symmetric tensor, which we call theα-QMT, is given in terms of the probability distribution function and the phase of the position space of the wave function as g(α) i j = 1 4Ep h ∂ilθ∂ jlθ i + (1−α) (1 +α) Ep h ∂iϕ∂ jϕ i −E...
[5]

(39) Here we have assumed that the two sets of generators and the initial states are not related for general α , 0, and are equal only for α = 0

Induced fluctuations The physical significance of this α- QMT and theα- Berry curvature for a given quantum system can be illuminated by studying the case, where we assume the two independent states|l1(α)⟩ and|l2(−α)⟩ are constructed from a set of fixed, mutually biorthogonal states on the Hilbert space so that we can write |l1(α)⟩ = eiskAk 1(α)|l0 1(α)⟩,...
[6]

(41) Importantly, note that this form of the α-density matrices is not Hermitian ρ(α) , (ρ(α))†

α- Fubini-Study tensor from theα-quantum Fisher information To obtain the form of the tensor structure, let us define for the pure quantum states Ψ, a generalised version of the stan- dard density matrix ρ, which we call α-density matrix given by 7 ρ(α) =|l2(−α)⟩⟨ l1(α)| . (41) Importantly, note that this form of the α-density matrices is not Hermitian ρ(...
[7]

Expansion of the overlap integrals and the non-metric Connections In this section, we will do a systematic expansion of the Provost-Vallee-like overlap integral for the non-Hermitian in- ner product (34) and extract the metric and the connection co- efficients in a similar way as is done in the classical formu- lation of information geometry [1]. As was e...
[8]

Gauge-invariant connections It is straightforward to show that, even though neither con- nections (43) and (44) are invariant under gauge transforma- tions|l1(α)⟩→ eiβ|l1(α)⟩ and|l2(−α)⟩→ eiβ|l2(−α)⟩, the following combinations are Γ1(α) i j,k =⟨∂i∂ jl1(α)|∂kl2(−α)⟩− (1−α2) ⟨∂i∂ jl1(α)|l2(−α)⟩⟨ l1(α)|∂kl2(−α)⟩ + 2⟨∂(il1(α)|l2(−α)⟩⟨∂ j)l1(α)|∂kl2(−α)⟩ +2(1...

Show all 95 references

[9]

Decomposition of the α-connections Since both the±α-connections obtained in the earlier sec- tions are already symmetric in the first two indices, in this section we will focus on the real and imaginary part of these gauge-invariant connections and their relation with the metr...
[10]

(65) This is essentially the antisymmetric part of FS LR i j , i.e

Non-Hermitian extension of Berry curvature To understand the role of each component, let us first note that, for the complex-valued Berry connection in this case ˜Ai = i⟨ΨL|∂ jΨR⟩, the curvature of this complex 2-form is ˜FLR =∂i ˜A j−∂ j ˜Ai = i ⟨∂iΨL|∂ jΨR⟩−⟨ ∂ jΨL|∂iΨR⟩ . (...
[11]

Optimisation problem and the LR (RL) QMT As was discussed in section III C, the quantum natural gra- dient descent techniques uses the natural metric on the quan- tum state space to optimise a given cost, which in the quantum variational eigensolver problem is the expectation ...
[12]

LR (RL) connections and their duality In this section, we will briefly discuss the connections in- duced on the parameter manifold from the non-Hermitian in- ner product structure for α = 0. Following similar expansion as we did in the section III B 5, we can obtain the two ra...
[13]

Optimisation problem and the RR (LL) QMT The difference between the complex-valued LR (RL)-QMT and the LL(RR)-QMT can probably be best illustrated by con- sidering the natural gradient optimisation problem, where in this case the cost is tailored for right-right or left-left s...
[14]

LL (RR) connections For the sake of completeness, we also briefly mention the connections built solely from LL (or RR) overlap integral, DII (θ,θ′) = ⟨ΨI(θ +δθ)− ΨI(θ)|ΨI(θ +δθ)− ΨI(θ)⟩, which will of the form, after restoring gauge invariance, ΓII i j,k =⟨∂i∂ jΨI|∂kΨI⟩− ⟨∂i∂ ...
[15]

Since we are considering the evolu- tion of a right state|ΨR⟩ and the related associated state⟨ΨL| [54], the generator is biorthogonal-Hermitian H# R = HR

LR (or RL) imaginary-time evolution Let us consider the following imaginary-time evolution of the biorthogonal right state and the corresponding associated state combinations generated respectively byHL and HR, such that the time-evolved states are |ΨR(τ;θ)⟩ = e−HRδτ|ΨR(τ = 0;...
[16]

RR (or LL) imaginary-time evolution Let us now consider the imaginary-time evolution of only the right-right (or the left-left) state combinations |ΨR⟩,⟨ΨR| (or|ΨL⟩,⟨ΨL|) generated respectively by HR and H† R (or HL and H† L), such that the time-evolved states are |ΨR(τ;θ)⟩ = ...
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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Derivation from QFI In this subsection, we ask: if it is possible to obtain the form of this Fubini-Study structure, and subsequently the α- quan- tum metric tensor from (modified version of) quantum Fisher information, as was done in [25] for the case ofα = 0. It turns out that, due to the lack of normalisation (for the two func- tions l(α), l(−α)) it is...

[2] [2]

Berry curvature Even though we have decomposed the non-Hermitian FS tensor, where the real, symmetric part can be ascribed to the α-QMT 6, the significance of the rest of the parts is not en- tirely evident at the moment. In particular, it can be seen that the antisymmetric field strength of theα-gauge fields, namely, ˜Ai = i⟨l1(α)|∂ jl2(−α)⟩ is, ˜Fi j = ...

[3] [3]

(27) This is one of the simplest generalised versions of the QMT, which we will call theα- quantum-metric tensor (QAMT)

Then, using the form of the generalised tensor structure (25), we can obtain the generalised metric on this space given by the real and symmetric part, the explicit form of which can be written down as g(α) i j = 1 4Ep h cos (2αϕ)∂ilθ∂ jlθ i +Ep h cos (2αϕ)∂iϕ∂ jϕ i − 2 (α2− 1) Ep h cos (2αϕ)∂iϕ i Ep h cos (2αϕ)∂ jϕ i + Ep h cos (2αϕ)∂ jlθ i Ep h cos (2αϕ...

[4] [4]

Components of the α-FS tensor Let us first explicitly extract the form of the four compo- nents of the α-FS tensor. The real and the symmetric tensor, which we call theα-QMT, is given in terms of the probability distribution function and the phase of the position space of the wave function as g(α) i j = 1 4Ep h ∂ilθ∂ jlθ i + (1−α) (1 +α) Ep h ∂iϕ∂ jϕ i −E...

[5] [5]

(39) Here we have assumed that the two sets of generators and the initial states are not related for general α , 0, and are equal only for α = 0

Induced fluctuations The physical significance of this α- QMT and theα- Berry curvature for a given quantum system can be illuminated by studying the case, where we assume the two independent states|l1(α)⟩ and|l2(−α)⟩ are constructed from a set of fixed, mutually biorthogonal states on the Hilbert space so that we can write |l1(α)⟩ = eiskAk 1(α)|l0 1(α)⟩,...

[6] [6]

(41) Importantly, note that this form of the α-density matrices is not Hermitian ρ(α) , (ρ(α))†

α- Fubini-Study tensor from theα-quantum Fisher information To obtain the form of the tensor structure, let us define for the pure quantum states Ψ, a generalised version of the stan- dard density matrix ρ, which we call α-density matrix given by 7 ρ(α) =|l2(−α)⟩⟨ l1(α)| . (41) Importantly, note that this form of the α-density matrices is not Hermitian ρ(...

[7] [7]

Expansion of the overlap integrals and the non-metric Connections In this section, we will do a systematic expansion of the Provost-Vallee-like overlap integral for the non-Hermitian in- ner product (34) and extract the metric and the connection co- efficients in a similar way as is done in the classical formu- lation of information geometry [1]. As was e...

[8] [8]

Gauge-invariant connections It is straightforward to show that, even though neither con- nections (43) and (44) are invariant under gauge transforma- tions|l1(α)⟩→ eiβ|l1(α)⟩ and|l2(−α)⟩→ eiβ|l2(−α)⟩, the following combinations are Γ1(α) i j,k =⟨∂i∂ jl1(α)|∂kl2(−α)⟩− (1−α2) ⟨∂i∂ jl1(α)|l2(−α)⟩⟨ l1(α)|∂kl2(−α)⟩ + 2⟨∂(il1(α)|l2(−α)⟩⟨∂ j)l1(α)|∂kl2(−α)⟩ +2(1...