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In interacting systems, a Majorana bound state's coupling to its environment is set by how much of the ground-state operator survives inside the coupling region.

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In interacting systems, the locality of ground-state Majorana operators, measured by fermionic partial traces, rigorously bounds how much an environment can split the ground-state degeneracy.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection Solid, honest bounds connecting ground-state Majorana locality to coupling and energy splitting; main caveat is that they are bounds on the projected low-energy theory, with virtual transitions through excited states uncontrolled. the 2 major comments →

arxiv 2510.20538 v2 pith:RLDFC2PQ submitted 2025-10-23 cond-mat.mes-hall quant-ph

Quantifying robustness and locality of Majorana bound states in interacting systems

William Samuelson , Juan Daniel Torres Luna , Sebastian Miles , A. Mert Bozkurt , Martin Leijnse , Michael Wimmer , Viktor Svensson This is my paper
classification cond-mat.mes-hall quant-ph
keywords Majorana bound statesinteracting systemsKitaev chainfermionic partial tracetopological protectionMajorana polarizationenergy splitting boundsnon-abelian braiding
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 aims to give interacting many-body systems the same guarantee that non-interacting theory gives Majorana bound states: spatial separation implies protection of the ground-state degeneracy and makes non-abelian braiding plausible. It defines Majorana operators purely from the two lowest opposite-parity states, then measures how much of each operator survives in a coupling region by a fermionic partial trace. The central inequalities state that the effective coupling of the Majorana pair to any environment is bounded by the norm of the reduced operators, and the resulting ground-state energy splitting is bounded by the same quantities times the environment size. This reduces the question 'is this interacting chain protected?' to ground-state properties computable by numerical many-body methods. A sympathetic reader cares because short quantum-dot Kitaev chains are strongly interacting, and this gives a rigorous way to assess their topological quality.

Core claim

The paper claims that for any system with two low-energy opposite-parity ground states separated by a gap, the localization of the many-body Majorana operators $\gamma$ and $\tilde{\gamma}$, measured by the Schatten norms of their fermionic partial traces $\gamma_R$ and $(i\gamma\tilde{\gamma})_R$ on the coupling region $R$, controls all low-energy consequences of coupling to an environment. Concretely, the effective odd and even couplings $F, \tilde{F}, G$ satisfy $\|F\|_p \leq \|H^o_{RB}\|_p \|\gamma_R\|_q$, and the same structure forces the associated energy splitting $\delta E$ to obey $|\delta E| \leq \sqrt{d_B}(\|(i\tilde{\gamma}\gamma)_R\|_q \|H^e_{RB}\|_p + \|\gamma_R\|_q \|H^o_{RB}\|_p + |\epsilon|)$. These are rigorous, sharp bounds; the paper illustrates them on an interacting Kitaev chain and shows that p

What carries the argument

The central object is the fermionic partial trace of the ground-state Majorana operators, $\gamma_R = \mathrm{Tr}_{\bar{R}}[\gamma]$, with $\gamma = |e\rangle\langle o| + |o\rangle\langle e|$ built from the two low-energy opposite-parity states. It measures how much of the mostly non-local many-body Majorana operator becomes accessible to a perturbation acting only in region R. Schatten-norm inequalities plus the adjoint relation between fermionic embedding and partial trace turn this locality measure into bounds on effective couplings and energy splittings; the gauge-optimized version of the bound reproduces and generalizes the Majorana polarization.

Load-bearing premise

The entire construction assumes the existence of exactly two low-energy states of opposite fermion parity, cleanly separated by an energy gap from every excited state; if a short chain has a tiny protection gap or additional low-energy states, the derived bounds stop applying.

What would settle it

Exact-diagonalize a small interacting Kitaev chain with a deliberately small protection gap, couple it to a quantum dot with couplings below the gap, and compare the measured ground-state splitting to the right-hand side of Eq. (14) computed from the ground-state partial traces; a violation would point to the low-energy two-state premise as the breaking step.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the reduced MBS norms are small, every local coupling to an environment is small, and the ground-state energy splitting is quantitatively bounded by the same locality measure.
  • For interacting Kitaev chains, protection of a short chain can be assessed from its two ground states alone, without diagonalizing the full spectrum.
  • The bound extends to couplings acting in multiple regions, giving quality factors Q_o and Q_e that summarize protection against odd and even perturbations.
  • In the non-interacting limit the bounds connect to the single-particle Majorana polarization and hold in the full Hilbert space, not just the low-energy sector.
  • For braiding, small ||γ_R|| and ||(iγγ̃)_R|| together with a large ||γ̃_R|| certify that three coupled copies can implement coupling-based non-abelian braiding, with undesired couplings bounded by the same locality norms.

Where Pith is reading between the lines

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

  • Extension: the same partial-trace locality measure could serve as a screening criterion for longer quantum-dot chains, predicting from ground-state data alone whether a given dot-level tuning will split the degeneracy beyond tolerance.
  • Extension: if the two-state gap premise fails in a practical device, the bounds weaken; a natural repair, hinted at but not developed in the paper, is to enlarge the low-energy subspace and re-derive the partial-trace construction for several pairs of states.
  • Extension: the bound likely also constrains dynamic processes such as quasiparticle poisoning or parity flips induced by the environment, not just static energy splittings—this is implicit in the effective-coupling formalism but not spelled out.
  • Extension: the sharpness condition H_RB = γ_R h_B suggests a direct experimental test: engineer a coupling close to this form and check whether the observed splitting approaches the bound, confirming that the locality measure is the operative quantity.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces a general framework for quantifying the locality of Majorana bound states (MBSs) in interacting systems. For any system with two low-energy opposite-parity states, it defines ground-state MBS operators via Eq. (1), and their partial traces over the complement of a region R as a locality measure. It proves Schatten-norm bounds (Eq. (8)) on the effective coupling of these MBSs to an environment, leading to energy-splitting bounds (Eq. (14)) and criteria for braiding (Sec. VII). The formalism generalizes the Majorana polarization to interacting systems (Sec. V). The results are tested on an interacting Kitaev chain, with analytical results at a sweet spot, spatial profiles of reduced MBSs, and numerical comparison of the bounds with the effective splitting when coupled to a quantum dot (Sec. VI). Full proofs are given in the appendices.

Significance. If the results hold, this is a substantial step toward rigorous quality measures for MBSs in interacting settings, directly relevant to recent quantum-dot Kitaev chain experiments. Strengths include: the proofs are self-contained and rigorous, the bounds are sharp with explicit saturation conditions, and the generalization of Majorana polarization is a useful theoretical contribution. The paper honestly notes the restriction to the low-energy subspace, but this restriction limits the advertised claim of quantifying degeneracy protection for interacting systems. The availability of code and data (Sec. IX) is a strength.

major comments (2)
  1. [Sec. IV and VI.C (Eq. (14), Fig. 3)] The energy-splitting bound Eq. (14) is proven for eigenstates of the effective low-energy Hamiltonian H_eff, not for eigenstates of the full Hamiltonian H = H_S + H_RB + H_B. The exact splitting differs by virtual transitions through excited states of S, of order ||H_RB||^2/Δ, and no bound on this leakage is provided. The paper states this limitation in Sec. IV, but the abstract claims the result 'quantifies the protection of the energy degeneracy' for interacting systems. Fig. 3 projects onto H_eff before diagonalizing, so the numerical test does not validate the bound for the full model. I recommend adding an estimate of the projection error (e.g., in terms of the gap Δ and coupling norm) or explicitly restricting all claims to the low-energy effective theory and adjusting the abstract/title accordingly.
  2. [Sec. VII (Eqs. (38)-(39))] The conclusion that three copies of the system can implement non-abelian braiding rests on the assumption that the coupling-based protocol of Refs. [16,61,62] works if unwanted couplings are small. The paper bounds these couplings but does not prove that the bounds are sufficient for adiabatic non-abelian braiding in an interacting system. The effective Hamiltonian Eq. (38) has the same form as the low-energy non-interacting model, so the conclusion is plausible, but the logical dependency should be stated explicitly, e.g., 'provided the conditions of Refs. [16,61,62] are met.'
minor comments (4)
  1. [Sec. VI.C, Fig. 3 caption] The solid curve is computed after projecting H onto H_eff; the caption and text should state this explicitly to avoid giving the impression that the exact splitting of the full chain-dot Hamiltonian is bounded. The text mentions the projection, but the figure caption alone is ambiguous.
  2. [Sec. V.A.1, Eq. (23)] The statement that Eq. (23) 'puts the use of Eq. (23) on a firm footing' could be more precise: it bounds the energy splitting only for odd couplings that are sums of single-mode fermion tunnelings; many-body odd couplings require the general Q_o of Eq. (17a).
  3. [Eq. (9)] The Schatten norm definition appears garbled as 'Tr(p A†A p)^{1/p}'; it should read (Tr[(A†A)^{p/2}])^{1/p}.
  4. [Sec. II, Eq. (3)] The ground-state MBSs γ and γ̃ are defined on the two-dimensional ground-state subspace. For the partial trace in Eq. (3) to be meaningful, state explicitly that they are extended to the full Hilbert space by acting as zero on excited states.

Circularity Check

0 steps flagged

No material circularity: the central bounds are derived from the definitions via partial-trace/Hölder inequalities; self-citations are contextual, not load-bearing.

full rationale

The derivation chain is self-contained rather than circular. The central objects—the ground-state MBS operators γ, γ~ and their partial traces γ_R—are defined directly from the two low-energy opposite-parity ground states in Sec. II, and the coupling bounds in Eq. (8) follow from the adjointness of the fermionic embedding and Hölder's inequality in App. A, with no fitted parameters or target result inserted. The energy-splitting bound Eq. (14) is obtained by commuting γ with the explicitly constructed effective Hamiltonian and then applying the already-proved coupling bounds in App. B. The only caveat is explicitly stated: the bounds 'only hold in the low-energy subspace—they can be violated if the coupling is comparable to the excitation gap in S' (Sec. IV). This is a stated limitation of the projection, not a circular step. Self-citations, such as Ref. [38] for local distinguishability and for a special-case bound, are contextual and are independently generalized in the present proofs; no load-bearing premise rests on an unverified self-citation. The numerical check in Fig. 3 is honest about projecting H → H_eff and about λ = t/100 being much smaller than the excitation gap. There is no fitted input renamed as a prediction and no known result merely relabeled.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The theoretical bounds have no fitted parameters. The model parameters in the illustrative Kitaev-chain example are hand-chosen inputs, not data fits. The main load-bearing assumptions are the two-state low-energy sector, locality of the coupling region, and weak coupling relative to the excitation gap.

free parameters (2)
  • λ (overall chain-dot coupling in Sec. VI C) = t/100
    Hand-picked small coupling for the quantum-dot example (Eq. 34) to stay in the weak-coupling regime; not fitted to data and not required for the general bounds.
  • φ (tunneling/pairing mixing angle in Sec. VI C) = π/6
    Hand-picked to parametrize t_c and Δ_c in Eq. (34); illustrative only, no effect on the central theorems.
axioms (5)
  • domain assumption Existence of exactly two low-energy states |e⟩, |o⟩ of opposite fermion parity separated from all excited states by an energy gap.
    Sec. II, first paragraph: the ground-state MBSs γ, ˜γ are defined on this sector; if more states become low-energy or the gap closes, the bounds in Secs. III–IV do not apply.
  • domain assumption The system–environment coupling H_RB acts only in a fixed subregion R of the system, with no direct long-range couplings outside R.
    Sec. III: all bounds use the reduced operators γ_R and (iγ˜γ)_R; couplings delocalized over the full system would evade the localization measure.
  • domain assumption The coupling obeys ||H_RB|| below the excitation gap of S for the low-energy effective description.
    Sec. IV: 'These bounds only hold in the low-energy subspace—they can be violated if the coupling is comparable to the excitation gap in S.'
  • standard math Standard Schatten-norm inequalities, fermionic partial-trace adjointness and contractivity (Refs. [49,67,69]).
    Used throughout Appendices A and B in the proofs of the coupling and energy-splitting bounds.
  • domain assumption Coupling-based braiding protocols with tunable desired couplings and small undesired couplings realize non-abelian braiding (prior work Refs. [16,61,62]).
    Sec. VII: the paper bounds the unwanted couplings but does not re-derive the adiabatic braiding theorem; it inherits the sufficiency condition from cited protocols.

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Cite this review

Pith. "Pith review of Quantifying robustness and locality of Majorana bound states in interacting systems." pith.science (2026). https://pith.science/paper/RLDFC2PQ

@misc{pith2026251020538,
  author       = {Pith},
  title        = {Pith review of: Quantifying robustness and locality of Majorana bound states in interacting systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLDFC2PQ}},
  note         = {Machine review of arXiv:2510.20538}
}
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read the original abstract

Protecting qubits from perturbations is a central challenge in quantum computing. Topological superconductors with separated Majorana bound states (MBSs) provide a strong form of protection that only depends on the locality of perturbations. While the link between MBS separation, robust degeneracy, and protected braiding is well understood in non-interacting systems, recent experimental progress in short quantum-dot-based Kitaev chains highlights the need to establish these connections rigorously for interacting systems. We do this by defining MBSs from many-body ground states and show how their locality constrains their coupling to an environment. This, in turn, quantifies the protection of the energy degeneracy and the feasibility of non-abelian braiding.

Figures

Figures reproduced from arXiv: 2510.20538 by A. Mert Bozkurt, Juan Daniel Torres Luna, Martin Leijnse, Michael Wimmer, Sebastian Miles, Viktor Svensson, William Samuelson.

Figure 1
Figure 1. Figure 1: FIG. 1. We show that the separation of MBSs implies energy [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quality measures and locality of ground-state MBSs [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The ground-state energy splitting [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. When coupling multiple systems, each with a low [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. In the gauge that maximizes Majorana separation [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

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Cited by 2 Pith papers

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    Microscopic treatment of the hybrid segment in mesoscopic Kitaev chains shows that Andreev bound state parity crossings define optimal sweet spots for localized Majoranas with large gaps.

  2. Optimal Majoranas in Mesoscopic Kitaev Chains

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    For quantum-dot Kitaev chains, the optimal Majorana sweet-spot—well-localized Majoranas together with a large excitation gap—occurs at the parity-crossing of the spin-split Andreev bound state in the mesoscopic superc...

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    The energy difference between |e⟩|ψ⟩ and |o⟩|ψ⟩, where |ψ⟩ is any state in the environ- ment is δE ≡ ⟨ψ|⟨e|Heff|e⟩|ψ⟩ − ⟨ψ|⟨o|Heff|o⟩|ψ⟩ = ⟨ψ|⟨o|[γ, Heff]|e⟩|ψ⟩

    Uncoupled eigenstates A simpler bound is obtained by considering the origi- nal uncoupled eigenstates. The energy difference between |e⟩|ψ⟩ and |o⟩|ψ⟩, where |ψ⟩ is any state in the environ- ment is δE ≡ ⟨ψ|⟨e|Heff|e⟩|ψ⟩ − ⟨ψ|⟨o|Heff|o⟩|ψ⟩ = ⟨ψ|⟨o|[γ, Heff]|e⟩|ψ⟩. (B20) Together with Eqs. (8) and (12) it is straightforward to show that |δE − ε| ≤ ∥(iγ˜γ)R...

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.