Loop-dependent entangling holonomies in localized topological quartets

Kazuki Ikeda; Yaron Oz

arxiv: 2604.11596 · v2 · submitted 2026-04-13 · ❄️ cond-mat.mes-hall · hep-th· math-ph· math.AT· math.MP· quant-ph

Loop-dependent entangling holonomies in localized topological quartets

Kazuki Ikeda , Yaron Oz This is my paper

Pith reviewed 2026-05-10 15:19 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall hep-thmath-phmath.ATmath.MPquant-ph

keywords topological quartetsentangling holonomiesBerry holonomytwo-qubit gatesBHZ modelSSH chainBBH modelspectral isolation

0 comments

The pith

Even with a local two-qubit description at each parameter point, a spectrally isolated quartet can acquire a loop holonomy outside the local U(2)⊗U(2) subgroup.

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

The paper shows that in three topological models, a four-dimensional isolated quartet admits a product structure of two qubits at every fixed point in parameter space yet can still pick up a non-local entangling operation when transported around a closed loop. Different choices of loop in the same quartet can switch the holonomy from nearly local to Ising-like or controlled-rotation type, while standard spectral invariants remain unchanged. This matters because it reveals that pointwise local reductions do not guarantee global reduction of the Berry holonomy, limiting how reliably one can treat such quartets as independent two-qubit systems under adiabatic cycling.

Core claim

A spectrally isolated quartet can admit a local two-qubit description at each point in parameter space and still acquire a loop holonomy outside the local subgroup U(2)⊗U(2). On a fixed quartet, changing only the loop moves the holonomy from almost local to entangling. In the BHZ ribbon, co-rotating and counter-rotating edge-field loops have nearly identical eigenphase data, but only the counter-rotating loop produces an Ising-like entangler. The spinful SSH chain yields a controlled-rotation example in a stable edge quartet, and the BBH model shows the same effect in a higher-order corner quartet. Standard Berry phases, Chern numbers, determinant phases, and eigenphase spectra do not detect

What carries the argument

The loop holonomy operator on the four-dimensional space of the isolated quartet, quantified by its distance to the embedded local subgroup U(2)⊗U(2).

If this is right

In the BHZ model, counter-rotating loops produce entangling holonomies while co-rotating loops stay nearly local despite matching eigenphases.
The SSH chain supplies a numerically stable example of a controlled-rotation holonomy realized on an edge quartet.
The BBH corner quartet exhibits the same loop-dependent reduction failure in a higher-order topological setting.
The subgroup-reduction problem extends to any isolated multiplet whose dimension factors as a product D = product d_alpha.
Canonical two-qubit coordinates become applicable only after the loop holonomy has been shown to lie outside the local subgroup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Parameter-space path planning must be treated as an additional design variable when engineering topological multi-qubit operations.
Similar loop dependence is likely to appear in other platforms that host isolated quartets, such as photonic lattices or superconducting qubit arrays.
Experimental tests could compare gate fidelities obtained from co-rotating versus counter-rotating adiabatic cycles on the same quartet.
The distance-to-local-subgroup diagnostic may generalize to larger product structures and provide a practical check for unwanted entanglement generation.

Load-bearing premise

The quartet stays spectrally isolated and admits a well-defined local two-qubit description at every fixed parameter point.

What would settle it

A numerical or experimental computation of the holonomy matrix for the counter-rotating edge-field loop in the BHZ ribbon that finds its distance to the nearest element of U(2)⊗U(2) is zero within numerical tolerance.

read the original abstract

A spectrally isolated quartet can admit a local two-qubit description at each point in parameter space and still acquire a loop holonomy outside the local subgroup $\mathrm{U}(2)\otimes\mathrm{U}(2)$. We study this question in three localized topological settings, a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet. On a fixed quartet, changing only the loop can move the holonomy from almost local to entangling. In BHZ, co-rotating and counter-rotating edge-field loops have nearly the same eigenphase data, but only the counter-rotating loop yields an Ising-like entangler. SSH gives a controlled-rotation example in a numerically stable edge quartet. BBH shows the same issue in a higher-order corner quartet. Standard Berry data, including Berry phases, Chern numbers, determinant phases, and eigenphase spectra, do not separate these cases. The main diagnostic is the distance from the loop holonomy to the extracted local subgroup. Canonical two-qubit coordinates are used only after reduction failure has been identified. The quartet is the smallest setting in which this question can be tested explicitly. The same subgroup-reduction problem extends to any isolated multiplet with pointwise product type $D=\prod_{\alpha}d_\alpha$, where the relevant local subgroup is the embedded product group $G_{\mathbf d}=\mathrm{Im}[\prod_{\alpha}U(d_\alpha)\to U(D)]$. In the terminology of Ref.[arXiv:2601.13764], these examples realize loop-dependent entangling gluing.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that loop choice alone can push a topological quartet's holonomy outside the local U(2)⊗U(2) in standard models, but the procedure for extracting that local subgroup looks underspecified and risks circularity.

read the letter

The main takeaway is that a spectrally isolated quartet can look locally like a product of two qubits at every fixed parameter point yet still pick up a loop holonomy that lies outside the embedded U(2)⊗U(2) subgroup, turning it entangling. The authors demonstrate this by varying only the loop in three concrete settings: a BHZ ribbon with co-rotating versus counter-rotating edge fields, a spinful SSH edge quartet, and a BBH corner quartet. In each case the eigenphase spectra stay similar while the holonomy distance to the local subgroup jumps, and ordinary Berry phases, Chern numbers, and determinant phases fail to separate the cases. They frame the general pattern as loop-dependent entangling gluing for any isolated multiplet with product dimension structure. That observation is new relative to the usual topological invariants cited in the text, and the choice of numerically stable models plus explicit loop comparisons is a clear strength. The distance diagnostic itself is presented as the practical separator once reduction failure is spotted. The soft spot is exactly where the stress-test note flags it: the paper does not give an explicit, loop-independent procedure for pulling the local subgroup out of the isolated quartet data alone. If the extraction step leans on model-specific labels, basis choices, or an assumed tensor splitting before the distance is computed, then the claim that the loop is the sole source of entanglement becomes circular. The abstract says canonical two-qubit coordinates appear only after failure is identified, but without seeing a clean algorithmic definition from the Hamiltonian or eigenvectors, it is hard to judge whether the local description is established independently. This matters because the whole argument rests on that separation. The work is aimed at people thinking about holonomic gates in topological matter or degenerate multiplets more generally. A reader who already cares about whether Berry data suffices for classifying entangling power will find the examples useful even if the diagnostic needs tightening. It deserves a serious referee because the models are standard, the numerical claims are checkable, and the core question is well-posed, but the referee should be asked to verify the subgroup extraction step in detail. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper claims that a spectrally isolated quartet in parameter space can admit a local two-qubit description (i.e., a tensor-product structure with local subgroup U(2)⊗U(2)) at each fixed point yet still acquire a loop holonomy lying outside that subgroup. Concrete numerical examples are provided in a BHZ ribbon (co-rotating vs. counter-rotating edge-field loops yielding Ising-like entanglers), a spinful SSH chain (controlled-rotation holonomy in an edge quartet), and a BBH corner quartet. Standard Berry invariants (phases, Chern numbers, determinant phases, eigenphase spectra) fail to distinguish the cases; the sole diagnostic is the distance of the computed holonomy to the extracted local subgroup. Canonical two-qubit coordinates are introduced only after reduction failure is identified. The result is generalized to any isolated multiplet of product dimension D=∏d_α with local subgroup G_d.

Significance. If the central claim holds, the work shows that holonomy in isolated multiplets can be loop-dependent and entangling in a manner invisible to pointwise local descriptions or conventional Berry data. The explicit numerical demonstrations in three standard models (BHZ, SSH, BBH) together with the distance-based diagnostic and the generalization to arbitrary product-type multiplets constitute a concrete, falsifiable advance in the study of topological holonomies. The observation that only the loop choice, not the pointwise structure, produces the entangling effect is a useful clarification for systems where local tensor structure is assumed.

major comments (2)

[Diagnostic construction (following abstract statement of the distance metric)] The manuscript states that the local subgroup is extracted from the isolated quartet data at each fixed parameter point and that canonical coordinates are introduced only after reduction failure is identified, yet no explicit algorithm, optimization procedure, or mathematical definition is supplied for performing this extraction from the 4D subspace alone (independent of any pre-chosen basis labeling or model-specific splitting). This extraction step is load-bearing for the central claim that the local two-qubit description exists independently of the loop; without it, attribution of entangling holonomy solely to loop choice cannot be verified as non-circular.
[BHZ ribbon results] In the BHZ ribbon example, the claim that co-rotating and counter-rotating loops produce nearly identical eigenphase spectra but qualitatively different holonomies (one local, one Ising-like entangler) rests on the numerical stability of the distance diagnostic; however, no quantitative error bounds, convergence checks with respect to discretization, or comparison against an independent verification of the local subgroup are reported, leaving open whether the observed difference survives controlled numerical perturbations.

minor comments (2)

[Generalization paragraph] Notation for the embedded product group G_d is introduced without an explicit embedding map or reference to the prior work (arXiv:2601.13764) that presumably defines it; a short appendix recalling the construction would improve readability.
[Figure captions] Figure captions for the SSH and BBH examples should explicitly state the parameter values at which the quartets are extracted and the loop paths used, to allow direct reproduction of the distance values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight the need for greater explicitness in our diagnostic procedure and additional numerical validation, both of which strengthen the presentation. We address each major comment in turn below and have revised the manuscript to incorporate the requested clarifications and checks.

read point-by-point responses

Referee: The manuscript states that the local subgroup is extracted from the isolated quartet data at each fixed parameter point and that canonical coordinates are introduced only after reduction failure is identified, yet no explicit algorithm, optimization procedure, or mathematical definition is supplied for performing this extraction from the 4D subspace alone (independent of any pre-chosen basis labeling or model-specific splitting). This extraction step is load-bearing for the central claim that the local two-qubit description exists independently of the loop; without it, attribution of entangling holonomy solely to loop choice cannot be verified as non-circular.

Authors: We agree that the original manuscript did not supply a sufficiently explicit, model-independent procedure for extracting the local subgroup from the 4D data. In the revised version we have added a new subsection (II.C) that defines the extraction mathematically and algorithmically. At each fixed parameter point the local subgroup is the image of an embedding U(2)⊗U(2)↪U(4) realized by a unitary change-of-basis matrix V∈U(4) that identifies the 4D space with ℂ²⊗ℂ². The optimal V is obtained by a numerical minimization of the Frobenius distance between each computed holonomy operator H and the nearest element of the candidate local subgroup: min_V inf_{U1,U2∈U(2)}‖H−V(U1⊗U2)V†‖_F. The optimization is performed over the Grassmannian of possible tensor-product splittings (parameterized by the coset SU(4)/[SU(2)×SU(2)×U(1)]), using a gradient-descent routine initialized from random SU(4) elements and converged to machine precision. Pseudocode, convergence tolerances, and verification on the three model examples are now included. This procedure uses only the isolated 4D subspace and the holonomy data; no pre-chosen basis labels or model-specific splittings enter the extraction. With this addition the attribution of entangling holonomy to loop choice is manifestly non-circular. revision: yes
Referee: In the BHZ ribbon example, the claim that co-rotating and counter-rotating loops produce nearly identical eigenphase spectra but qualitatively different holonomies (one local, one Ising-like entangler) rests on the numerical stability of the distance diagnostic; however, no quantitative error bounds, convergence checks with respect to discretization, or comparison against an independent verification of the local subgroup are reported, leaving open whether the observed difference survives controlled numerical perturbations.

Authors: We concur that quantitative error analysis is required to substantiate the numerical claims. In the revised manuscript we have added Appendix C containing systematic convergence tests. The parameter loops were discretized with N=50,100,200,500 points; the underlying BHZ ribbon was sampled with k-point densities increased by successive factors of two. For each resolution we recomputed the holonomy operators via the product of short-time propagators and evaluated the distance diagnostic. The counter-rotating loop distance converges to 0.87±0.02 (well outside the local subgroup), while the co-rotating distance remains below 0.04±0.01. Error bars are obtained from finite-difference variations under ±1% perturbations of the loop path and from the difference between fourth- and sixth-order Runge–Kutta integrators. As an independent cross-check we also extracted the local subgroup by computing the commutant of the holonomy set with the edge-field operators that define the ribbon geometry; the two extraction methods agree to within 0.03 in the distance metric. These controlled tests confirm that the qualitative distinction between the two loops is robust under discretization and numerical perturbations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via model examples and direct distance diagnostic

full rationale

The paper states that a spectrally isolated quartet admits a local two-qubit description at each fixed parameter point as a premise, then demonstrates via explicit BHZ, SSH, and BBH calculations that loop choice alone can produce holonomy outside U(2)⊗U(2) while standard Berry invariants cannot distinguish the cases. The main diagnostic is presented as the distance from the computed holonomy to the extracted local subgroup, applied only after reduction failure is identified, with canonical coordinates introduced post-failure. The reference to arXiv:2601.13764 supplies terminology for 'loop-dependent entangling gluing' but is not invoked to justify the existence of the local description or the distance metric itself. No equation or step reduces the entangling diagnosis to a fitted parameter, self-citation chain, or presupposed factorization by construction; the quartet examples serve as independent, numerically verifiable tests. The derivation chain therefore remains self-contained against external model benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard assumptions of topological band theory and adiabatic evolution.

pith-pipeline@v0.9.0 · 5592 in / 1238 out tokens · 40695 ms · 2026-05-10T15:19:58.886036+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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Ikeda,Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras,arXiv e-prints(2026) arXiv:2601.13764 [2601.13764]

K. Ikeda,Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras,arXiv e-prints(2026) arXiv:2601.13764 [2601.13764]

work page arXiv 2026

[2] [2]

Berry,Quantal phase factors accompanying adiabatic changes,Proc

M.V. Berry,Quantal phase factors accompanying adiabatic changes,Proc. R. Soc. Lond. A 392(1984) 45

work page 1984

[3] [3]

Wilczek and A

F. Wilczek and A. Zee,Appearance of gauge structure in simple dynamical systems,Phys. Rev. Lett.52(1984) 2111

work page 1984

[4] [4]

Zanardi and M

P. Zanardi and M. Rasetti,Holonomic quantum computation,Phys. Lett. A264(1999) 94

work page 1999

[5] [5]

Sj¨ oqvist, D.M

E. Sj¨ oqvist, D.M. Tong, L.M. Andersson, B. Hessmo, M. Johansson and K. Singh, Non-adiabatic holonomic quantum computation,New J. Phys.14(2012) 103035

work page 2012

[6] [6]

Yu, X.-L

R. Yu, X.-L. Qi, A. Bernevig, Z. Fang and X. Dai,An equivalent expression ofZ 2 topological invariant for band insulators using the non-abelian Berry connection,Phys. Rev. B84(2011) 075119

work page 2011

[7] [7]

Soluyanov and D

A.A. Soluyanov and D. Vanderbilt,Computing topological invariants without inversion symmetry,Phys. Rev. B83(2011) 235401

work page 2011

[8] [8]

Alexandradinata, X

A. Alexandradinata, X. Dai and B.A. Bernevig,Wilson-loop characterization of inversion-symmetric topological insulators,Phys. Rev. B89(2014) 155114

work page 2014

[9] [9]

Gresch, G

D. Gresch, G. Aut` es, O.V. Yazyev, M. Troyer, D. Vanderbilt, B.A. Bernevig et al.,Z2Pack: Numerical implementation of hybrid Wannier centers for identifying topological materials, Phys. Rev. B95(2017) 075146

work page 2017

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Bradlyn and M

B. Bradlyn and M. Iraola,Lecture notes on Berry phases and topology,SciPost Phys. Lect. Notes51(2022) 51

work page 2022

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Sugawa, F

S. Sugawa, F. Salces-Carcoba, Y. Yue, A. Putra and I.B. Spielman,Wilson loop and Wilczek–Zee phase from a non-abelian gauge field,npj Quantum Inf.7(2021) 144

work page 2021

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Tyner, S

A.C. Tyner, S. Sur, Q. Zhou, D. Puggioni, P. Darancet, J.M. Rondinelli et al.,In-plane Wilson loop for measurement of quantized non-abelian Berry flux,Phys. Rev. B109(2024) 195149

work page 2024

[13] [13]

Bernevig, T.L

B.A. Bernevig, T.L. Hughes and S.-C. Zhang,Quantum spin Hall effect and topological phase transition in HgTe quantum wells,Science314(2006) 1757

work page 2006

[14] [14]

Hasan and C.L

M.Z. Hasan and C.L. Kane,Colloquium: Topological insulators,Rev. Mod. Phys.82(2010) 3045

work page 2010

[15] [15]

Qi and S.-C

X.-L. Qi and S.-C. Zhang,Topological insulators and superconductors,Rev. Mod. Phys.83 (2011) 1057

work page 2011

[16] [16]

Benalcazar, B.A

W.A. Benalcazar, B.A. Bernevig and T.L. Hughes,Quantized electric multipole insulators, Science357(2017) 61

work page 2017

[17] [17]

Schindler, A.M

F. Schindler, A.M. Cook, M.G. Vergniory, Z. Wang, S.S.P. Parkin, B.A. Bernevig et al., Higher-order topological insulators,Sci. Adv.4(2018) eaat0346

work page 2018

[18] [18]

Fukui, Y

T. Fukui, Y. Hatsugai and H. Suzuki,Chern numbers in discretized Brillouin zone: Efficient method of computing (spin) Hall conductances,J. Phys. Soc. Jpn.74(2005) 1674

work page 2005

[19] [19]

Leroux, K

F. Leroux, K. Pandey, R. Rehbi, F. Chevy, C. Miniatura, B. Gr´ emaud et al.,Non-abelian adiabatic geometric transformations in a cold strontium gas,Nat. Commun.9(2018) 3580. – 29 –

work page 2018

[20] [20]

James, P.G

D.F.V. James, P.G. Kwiat, W.J. Munro and A.G. White,Measurement of qubits,Phys. Rev. A 64(2001) 052312

work page 2001

[21] [21]

Chuang and M.A

I.L. Chuang and M.A. Nielsen,Prescription for experimental determination of the dynamics of a quantum black box,J. Mod. Opt.44(1997) 2455

work page 1997

[22] [22]

Zanardi, C

P. Zanardi, C. Zalka and L. Faoro,Entangling power of quantum evolutions,Phys. Rev. A62 (2000) 030301

work page 2000

[23] [23]

Makhlin,Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations,Quantum Inf

Y. Makhlin,Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum computations,Quantum Inf. Process.1(2002) 243

work page 2002

[24] [24]

Zhang, J

J. Zhang, J. Vala, S. Sastry and K.B. Whaley,Geometric theory of nonlocal two-qubit operations,Phys. Rev. A67(2003) 042313. – 30 –

work page 2003