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REVIEW 3 major objections 4 minor 21 references

Protocol for detecting the nonlocality of the multi-Majorana Systems

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A parity-only entanglement witness can certify genuine Majorana nonlocality in a six-Majorana system, distinguishing MZM pairs from trivial Andreev states.

desk verdict A parity-only witness with a clean analytical detection rate, but the entanglement-elimination step that makes the zero-false-positive claim work is not implementable as written. read the letter →

arxiv 2506.17003 v1 pith:3BWY3PAR submitted 2025-06-20 quant-ph

Bai-Ting Liu , Peng Qian , Zhan Cao , Dong E. Liu This is my paper · ORCID
classification quant-ph MSC 81P4081P68
keywords MajoranazeromodesAndreevboundstatesentanglementwitnessparitymeasurementtopologicalquantumcomputingnonlocalityquasiparticlepoisoningdots
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

This paper tries to establish that the nonlocal character of Majorana zero modes can be certified using an entanglement witness built only from two-site parity measurements, without full state tomography. The protocol first applies local projections to remove bipartite correlations from trivial states, then measures the parity of every pair spanning the two subsystems and forms the linear combination $R = \sum a_{ij} d_{ij}$. For a six-site toy model, the witness is shown to have negative expectation on roughly 18% of the odd-parity MZM subspace and nonnegative expectation on separable ABS states, so a negative $R$ certifies MZM pairing. This matters because trivial Andreev bound states mimic zero-bias conductance peaks and can obscure the nonlocal correlations that topological qubits rely on.

What carries the argument

The load-bearing object is the parity-only witness $W = I - \sum_{\langle i,j\rangle} a_{ij} p_{ij}$, where $p_{ij} = i\gamma_i\gamma_j$ for MZM pairs and, in the ABS case, the measured quantity is the occupation of an effective fermionic mode $b^\dagger_{ij,\mathrm{ABS}}b_{ij,\mathrm{ABS}}$ formed from the two sites and the tunneling amplitudes. The witness parameters are constrained by a candidate condition, $\operatorname{Tr}[W(\rho_1\otimes\rho_2)] \ge 1 - \sum_{ij} a_{ij}(1 + \sqrt{2} + T_{ij}) \ge 0$, derived from a Cauchy-Schwarz bound over product states; this guarantees nonnegative values on separable ABS states. The MZM detection rate is then computed analytically by parameterizing the four-dimensional odd-parity MZM state space and integrating the condition $\operatorname{Tr}(W\rho_{\mathrm{odd}}) < 0$, giving the ratio $(\pi^2 - 2\pi)/(2\pi^2)$.

What would settle it

Prepare a known six-site Andreev bound state with inter-subsystem entanglement, apply the paper's entanglement-elimination projection, measure $R$ with $m = 1$, $a_{52} = -1$, $\cos\theta = -1$; a single ABS instance with $R < 0$ would refute the zero-false-positive claim.

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Extended reading notes

Core claim

The central claim is that a single operator, $W = I - \sum_{\langle i,j\rangle} a_{ij} i\gamma_i\gamma_j$ for MZMs, with the same parameters used for ABSs, separates the two classes in a bipartite six-site system. After an entanglement-elimination step that local projections are assumed to make ABS states separable while preserving Majorana pairing, the witness value $R = \operatorname{Tr}(W\rho)$ is nonnegative for any ABS state that is a product across the bipartition, but for MZM-paired states the negative region has volume fraction $(\pi^2 - 2\pi)/(2\pi^2) \approx 0.1817$ in the odd-parity subspace for parameters $m = 1$, $a_{52} = -1$, $\cos\theta = -1$. Repeating the projection-and-measurement cycle on fresh copies turns this into a detection protocol: the first negative $R$ certifies nonlocal MZM pairing. The same calculation gives the same detection fraction in the even-parity subspace when $a_{52} = +1$, and numerical simulations show the detection rate decreases with quasiparticle poisoning strength.

Load-bearing premise

The protocol assumes that the local projection step completely removes bipartite entanglement from Andreev bound states while leaving Majorana pairing intact, and the paper does not specify the exact projective operation that achieves this.

Editorial extensions

If this is right

  • If the zero-false-positive property survives the entanglement-elimination step, a single negative $R$ from nine inter-subsystem parity measurements is enough to certify MZM pairing without state tomography.
  • Because the witness is built from the same tunnelling-based parity measurements already demonstrated on tetron-like devices, the protocol can be added to existing nanowire setups without new measurement hardware.
  • The analytic detection rate of about 18% per run means that repeating the protocol on fresh projected copies raises the cumulative probability of detecting a true MZM state, while ABS states never turn negative.
  • The sign of $a_{52}$ selects the parity sector; quasiparticle poisoning transfers states between sectors, which is why the simulated detection rate drops and approaches zero near $p = 0.6$.
  • For a mixed MZM-ABS state, negativity of the reduced witness still indicates that at least some MZM pairing is present.

Reading between the lines

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

  • One testable extension is to simulate the entanglement-elimination projection explicitly on finite-size nanowires with overlapping Majorana wavefunctions; the paper's own caveat suggests residual ABS entanglement could produce false positives, and the size of that effect is not quantified.
  • The 18% figure is a uniform-volume average over the MZM state space; a real device prepares specific fusion states, so the practical detection rate could be higher or lower depending on where those states sit in the negative region.
  • The same construction, an operator linear in inter-subsystem parities constrained by a product-state bound, could be adapted to other anyonic encodings or to larger arrays, where the pairing graph may admit more favorable witness parameters.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a protocol to distinguish Majorana zero modes (MZMs) from trivial Andreev bound states (ABSs) in a six-site system, using an entanglement witness constructed from parity measurements. The central claims are a zero false-positive rate for ABS states and an approximate 18% detection probability for MZM-paired states, together with robustness under quasiparticle poisoning. The algebraic derivation of the witness value and the phase-space volume calculation are analytic, and the noise analysis is numerical.

Significance. If the protocol were realizable, it would offer a quantum-information-based discriminator for MZMs, complementing conductance measurements that cannot easily tell MZMs from ABSs. The witness construction and the explicit calculation of the negative-subspace fraction are useful contributions, and the numerical study of quasiparticle poisoning is a sensible addition. However, the experimental implementability claim and the zero-false-positive guarantee rest on an underspecified 'entanglement elimination' step, and the detection-rate calculation is not reconciled with that step.

major comments (3)
  1. [Section IV A and Fig. 2(a)] The 'entanglement elimination' step is not a well-defined physical operation. The only implementation indicated, a single-site QD measurement, cannot projectively measure a single Majorana mode: with t2=0 in Eq. (12), the energy shift in Eq. (13) is independent of p12, so the QD readout carries no parity information. Moreover, the local Majorana operators γ1, γ2, γ3 in subsystem A anticommute, so a simultaneous projective measurement onto their eigenspaces is impossible. The paper must specify an explicit POVM or other operation and prove that it renders ABS states separable across the A|B partition while preserving the MZM-paired form used in the witness calculation.
  2. [Section VI A] The paper acknowledges that if the projection leaves residual ABS entanglement, the witness can turn negative for an ABS state, i.e., a false positive. Since no concrete projection operator is given, the claimed zero-false-positive guarantee is an unverified assumption rather than a derived result. This is load-bearing for the central claim of the protocol.
  3. [Sections IV A and V] The projection step and the 18% detection-rate calculation are mutually inconsistent. A projective measurement of all sites in subsystem A, as described in the Figure 2 caption, would leave the post-measurement state as a product across the A|B partition; for a state with fixed total parity, all crossing-parity expectations ⟨iγ_iγ_j⟩ would then vanish, giving R=0 rather than R<0. The calculation in Section V assumes the projected state still has the nonlocal paired form of Eq. (7), but the paper does not explain how a disentangling projection of A can produce that form.
minor comments (4)
  1. [Section V, Eq. (41)] The negative-subspace volume is written as π^2 − π/2, but the quoted ratio (π^2 − 2π)/(2π^2) = 0.1817 corresponds to a numerator of π^2 − 2π. Please correct the intermediate expression.
  2. [Appendix C, Eq. (C7)] The final bound should be 2|t*_1 t_2|, not 2|t*_1 t_2|^2.
  3. [Appendix C, Eq. (C10)] The right-hand side '(√2 + Tij)' omits the leading '1+' that appears in the preceding expression.
  4. [Abstract] There is a typo: 'in the resence of quasiparticle contamination' should read 'presence'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 18% detection rate is a geometric volume calculation for fixed witness parameters, with no fitted input relabeled as a prediction.

full rationale

The paper's central quantitative claim, the approximate 18% detection rate, is a self-contained calculation: the witness parameters m=1, a52=-1, cos(theta)=-1 are chosen explicitly, and the negative-subspace volume is computed from the parameterized MZM state space (Eqs. 33-42) after verifying that these parameters satisfy the witness-candidate inequality (Eq. 26). No parameter is fitted to data, and no quantity measured from MZM states is inserted back into the derivation. The measurement model is imported from the external references [6] and [53], not from the present authors' prior work, so there is no load-bearing self-citation chain. The zero-false-positive claim for ABS states is conditional on the Section IV A 'entanglement elimination' assumption that local operations render ABS states separable; this is an unproven physical premise and is explicitly acknowledged in Section VI A as a possible source of false positives when residual ABS entanglement remains. That unsupported premise is a physical-viability and correctness concern, but it is not a circular reduction of the target result to the input: the witness condition, the separability assumption, and the volume calculation are distinct statements, and the paper does not redefine the outcome in terms of the assumption. Overall, no step in the derivation is equivalent by construction to its own input.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on standard MZM algebra and the positivity of the ABS number operator. The nonstandard assumption is the availability and effect of the entanglement-elimination projection. No new particles, forces, or dimensions are introduced.

free parameters (1)
  • witness parameters {a14, a16, a34, a36, a52} = a14=-1, a16=0, a34=0, a36=-1, a52=-1 (m=1, cosθ=-1)
    Chosen by hand to give a simple numerical example; the paper does not optimize them, and the 18% detection rate depends on this choice.
assumptions (6)
  • standard math Majorana operators satisfy γ_i=γ_i† and {γ_i,γ_j}=2δ_ij
    Eq. (1): the algebraic backbone for all parity operators and witness expressions.
  • domain assumption The six-site system is modeled by three MZM pairs with fixed total parity, giving a 4-dimensional subspace
    Section II A, Eq. (4): the odd-parity (and even-parity) MZM state space used for the detection rate calculation.
  • domain assumption ABS are local fermionic modes, and b†_{ij,ABS}b_{ij,ABS} is a positive operator with eigenvalues 0 and 1
    Section II B, Eqs. (5) and (18): the zero-false-positive bound in Section III D relies on this positivity.
  • ad hoc to paper There exists a local projection that renders any ABS state separable across the bipartition while leaving MZM-paired states in the paired subspace
    Section IV A: asserted without a concrete measurement; Section VI A concedes residual entanglement would break the classifier.
  • domain assumption Parity measurements across arbitrary cross-subsystem pairs are available via the QD tunneling scheme
    Section II C, Eqs. (12)-(15): the readout model is imported from Ref. 6.
  • standard math The chosen witness parameters satisfy the ABS EW candidate condition
    The chosen a_ij are all non-positive, making W_A=I+positive operator, so the condition in Eq. (26) holds trivially.

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Cite this review

Pith. "Pith review of Protocol for detecting the nonlocality of the multi-Majorana Systems." pith.science (2026). https://pith.science/paper/3BWY3PAR

@misc{pith2026250617003,
  author       = {Pith},
  title        = {Pith review of: Protocol for detecting the nonlocality of the multi-Majorana Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3BWY3PAR}},
  note         = {Machine review of arXiv:2506.17003}
}
read the original abstract

Majorana zero modes (MZMs) are non-Abelian quasiparticles with the potential to serve as topological qubits for fault-tolerant quantum computing due to their ability to encode quantum information nonlocally. In multi-Majorana systems configured into two separated subsystems, nontrivial quantum correlations persist, but the presence of trivial Andreev bound states (ABSs) can obscure this nonlocality if MZM preparation fails. To address this, we propose a protocol using an entanglement witness based solely on parity measurements to distinguish the nonlocal characteristics of MZM systems. Our framework, which is experimentally implementable, achieves a detection probability of approximately 18% in a 6-site system and demonstrates robustness under environmental noise, albeit with a reduced detection rate in the resence of quasiparticle contamination.

Figures

Figures reproduced from arXiv: 2506.17003 by the authors.

Figure 1
Figure 1. FIG. 1. The 6 sites are labeled as 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The example design of six MZMs. The blue wires are topological superconductor wires, which consists a wire of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The graphic illustration of the definition of an entan [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The graphical representation of the MZM detection protocol. The input of the protocol is the prepared initial state [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The performance of the witness with quasiparticle poisoning. (a) is the result for 10000 random odd-parity MZM [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

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Reference graph

Works this paper leans on

21 extracted references · 20 canonical work pages

  1. [1]

    Each wire, a composite of a spin-orbit coupled semiconductor and a supercon- ducting layer, enters a TS phase under an external magnetic field

    In our setup, there are several topological supercon- ductor (TS) wires hosting Majorana Zero Modes (MZMs) at their ends. Each wire, a composite of a spin-orbit coupled semiconductor and a supercon- ducting layer, enters a TS phase under an external magnetic field. These six wires are interconnected at one end by a superconducting bulk, enabling the poten...

    work page

  2. [2]

    The design of the full T-junction structure for TS illustrated in 45 is avoided to en- sure that the external magnetic field aligns with all the wires

    It should be noted that the TS wires are parallel to each other. The design of the full T-junction structure for TS illustrated in 45 is avoided to en- sure that the external magnetic field aligns with all the wires. This arrangement facilitates the simul- taneous induction of a TS phase across all wires through a strategically oriented magnetic field

    work page

  3. [3]

    The TS wires are deliberately fabricated to be lengthy, such thatLW≫ξ, where ξ represents the coherence length within the TS framework. This design is important as it significantly reduce the ground state degeneracy splitting, attributable to the overlap of the MZM wave functions, and is quantitatively expressed as∼ exp(−LW/ξ)43

    work page

  4. [4]

    However, an increase in the length of the TS wires (LW ) inversely impactsEC, dimin- ishing the charging energy

    The charging energy, denoted by EC, is instrumen- tal in preventing quasiparticle events and exponen- tially safeguarding the parity of MZMs, following exp(−EC/T ). However, an increase in the length of the TS wires (LW ) inversely impactsEC, dimin- ishing the charging energy. Therefore, it is crucial to establish an optimal balance between EC and LW . Th...

    work page

  5. [5]

    Moreover, with the help of the two QDs in the middle, all projective mea- surements over a certain subset of the six sites are possible

    According to 6, the structure of the six MZMs forms a hexon design of a qubit. Moreover, with the help of the two QDs in the middle, all projective mea- surements over a certain subset of the six sites are possible. Each point in the center of two MZMs has two gates connected to the neighboring MZM sites, and the two QDs are also coupled together by a gat...

    work page

  6. [6]

    It must yield nonnegative mean values when ap- plied to all separable states, i.e., ⟨ψA|⟨ϕB|W|ψA⟩|ϕB⟩ ⩾ 0 (19)

    work page

  7. [7]

    It must possess at least one negative eigenvalue, indicating its ability to detect entanglement. If an operator W satisfies only the first condition, it is classified as an entanglement witness (EW) candidate.41 In other words, a necessary condition for the sepa- rability of a state ρAB is: Tr( WρAB)≥ 0, and a suf- ficient condition for ρAB to be an entan...

    work page

  8. [8]

    The CHSH inequality can be violated by entangled bipartite states

    And n = (sinθL cosϕL, sinθL sinϕL, cosθL) is a unit vector. The CHSH inequality can be violated by entangled bipartite states. Follow this inequality, we can construct an opera- torW that satisfies the entanglement witness conditions: WCHSH =I− X i,j∈{1,2} (−1)k(i,j) D ˆLi ˆRj E (22) To measure the CHSH witness, the measurement needed for the witness are ...

    work page

Show all 21 references
  1. [9]

    And similarly for the right counterpart the Pauli operators are {I,−iγ4γ6,−iγ6γ2,−iγ2γ4}/ √

  2. [10]

    However, due to the different physical nature of MZMs and ABSs, the same measurement operations yield dis- tinct outcomes

    The Pauli operators are MZM parity measurements, which can be measured by the tunneling method described in Section II C. However, due to the different physical nature of MZMs and ABSs, the same measurement operations yield dis- tinct outcomes. In the case of local ABSs, the d...

  3. [11]

    This outcome follows directly from the conservation of total parity

    Further analy- sis (detailed in Section V) reveals that, for MZM states 10 within the odd-parity manifold (a result that also applies to the even-parity manifold), only five of the nine expec- tation values are nonzero. This outcome follows directly from the conservation of to...

  4. [12]

    The definition of an EW In a review paper 55, the witness is defined as an op- erator W∈ L(HAB) is an EW if and only if it is block- positive but not positive. Block-positive: A Hermitian operator W∈ L(HAB) is block-positive if for all product states in L(HAB), W is positive: ...

  5. [13]

    More precisely, we have the Lemma: Lemma 1 56: For any entangled state ρ∈H AB, there exists a Hermitian W such that Tr(Wρ ) and Tr( Wσ ) for all separable σ

    Banach separation theorem: The completeness of witnesses For any entangled states, there exist an EW separate it fram the convex set of product states. More precisely, we have the Lemma: Lemma 1 56: For any entangled state ρ∈H AB, there exists a Hermitian W such that Tr(Wρ ) a...

  6. [14]

    However the discussion in 46 is not very clear

    Optimal EW and the formation of EW The form of EW used in the paper comes from 46. However the discussion in 46 is not very clear. Here the optimization conditions for any EW, as well as a closely related form of EW, which is based on realignment cri- terion, would be discusse...

  7. [15]

    Then we get: ⟨α⊗β|W|α⊗β⟩ = 1− X k akbk≥ 0

    With Cauchy-Schwarz inequality,P kakbk≤ 1. Then we get: ⟨α⊗β|W|α⊗β⟩ = 1− X k akbk≥ 0. (A13) For an entangled state that can be detected by this EW, the trace would be negative: Tr(Wρ ) = 1− X k λk < 0 (A14) So W defined in Eq. (A10) is an EW. Appendix B: The Perturbation Calcu...

  8. [16]

    (12) H (1,2) tunn,MZM =−ie−iϕ/2 2 t1f† 1γ1 +t2f† 1γ2 + H.c

    The Majorana zero mode tunneling For the Majorana zero mode state, the tunneling term in the Hamiltonian is the same as Eq. (12) H (1,2) tunn,MZM =−ie−iϕ/2 2 t1f† 1γ1 +t2f† 1γ2 + H.c. (B2) According to the perturbation theory, the second order perturbation of the ground state ...

  9. [17]

    (B5) Now we consider the initial state as |NS = 0⟩⊗| nf = 0⟩, and the state after tunneling is|NS =−1⟩⊗|nf = 1⟩

    iγ1γ2 4 [EC (1− 2Ng) +ϵ0−ϵ1] =ECN 2 g +ϵ1−|t1|2 +|t2|2 + ip12 (t∗ 1t2−t1t∗ 2) 4 [EC (1− 2Ng) +ϵ0−ϵ1] . (B5) Now we consider the initial state as |NS = 0⟩⊗| nf = 0⟩, and the state after tunneling is|NS =−1⟩⊗|nf = 1⟩. The total energy with perturbation can be obtained in the sam...

  10. [18]

    The tunneling term is the same as Eq

    The fermionic state tunneling Now we consider the fermionic case, which is a special case of ABS. The tunneling term is the same as Eq. (14) H (1,2) tunn,F =−ie−iϕ/2 2 t1f† 1c1 +t2f† 1c2 + H.c. (B7) For the ground state |NS = 0⟩⊗| nf = 1⟩ and the excited state|NS = 1⟩⊗| nf = 0...

  11. [19]

    (B13) Note that the only difference between the ABS tunnel- ing Hamiltonian and the fermion tunneling Hamiltonian Eq

    The Andreev bound state tunneling As for the Andreev bound state, the tunneling term in the total Hamiltonian is H (1,2) tunn,ABS =− ie−iϕ/2 2 [t1f† 1(|u1|c1 +|v1|c† 1) +t2f† 1(|u2|c2 +|v2|c† 2)] + H.c. (B13) Note that the only difference between the ABS tunnel- ing Hamiltonia...

  12. [20]

    We denote trace on site 1 by ⟨·⟩1 and trace on site 2 ⟨·⟩2

    The witness candidate condition of fermionic witness The result in Section II C as well as in Appendix B 2 shows that two site measurement in fermionic state is b† 12b12, with b† 12 = t∗ 1c† 1 +t∗ 2c† 2q |t1|2 +|t2|2 (C1) The measurement outcome on a pair of sites, for ex- amp...

  13. [21]

    Roadmap to fault tolerant quantum computation using topological qubit arrays,

    The witness candidate condition of ABS witness The fermion state is a special case of Andreev bound state. For the case of ABS, the tunneling of particle and hole on each site can have different strength. The operators measured in the witness would be in terms of b† ij,ABSbij,...

    2000 arXiv

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