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REVIEW 2 major objections 4 minor 26 references

Tomographic Signatures of Interacting Majorana and Andreev States in Superconductor-Semiconductor Transmon Qubits

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Charge tomography distinguishes Majorana from Andreev states via quarter-integer offsets and log(4) entropy.

desk verdict Clean selection-rule derivation of quarter-charge offsets and log(4) entropy peaks for Majorana-dressed Andreev states, but the fingerprint's uniqueness against trivial odd-parity states is not established. read the letter →

arxiv 2502.07684 v1 pith:OQUCHXIW submitted 2025-02-11 cond-mat.mes-hall quant-ph

Daniel Dahan , Konstantin Yavilberg , Talya Shnaider , Elena Lupo , Malcolm R. Connolly , Eran Ginossar , Eytan Grosfeld This is my paper · ORCID
classification cond-mat.mes-hallquant-ph PACS 74.50.+r73.63.-b
keywords charge-basistomographyAndreevboundstatesMajoranazeromodestransmonqubitentanglemententropyCooper-pairboxsuperconductor-semiconductorjunctionchargeoffsets
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

By reconstructing the ground-state charge distribution of a superconductor-semiconductor transmon, the paper claims one can tell whether the junction hosts only interacting Andreev states or also Majorana fermions. Andreev states imprint peaks at integer and half-integer relative-Cooper-pair-number offsets, while Majorana-dressed states put weight in four sectors offset by 0, 1/4, 1/2, and 3/4 of a Cooper pair. The charge-entanglement entropy, obtained by tracing out the microscopic junction degrees of freedom, peaks at $\log(2)$ for a single interacting Andreev state and rises to $\log(4)$ when Majorana fermions dress it. If correct, this gives a charge-only readout that requires no microwave spectroscopy of individual junction levels.

What carries the argument

The central object is the charge-basis reduced density matrix of the junction, whose diagonal gives the probability per relative Cooper-pair number $n$ and whose von Neumann entropy measures entanglement between the condensate and the microscopic fermionic degrees of freedom. The argument is carried by selection-rule bookkeeping: total particle number $N_t=N+n_d/2$ fixes the island charge for each dot occupation, and the fermionic parity in each sector fixes the allowed quantization of $n$. Projecting the full Hamiltonian onto the Transmon plasmon ground state, valid when the dot couplings $\Gamma$, $B$, and $w$ are small compared to the plasma frequency, yields an effective two-level model for the Andreev-only case and an eight-dimensional model for the Majorana case; the quarter offsets and the $\log(4)$ peak follow from the four parity-charge sectors of the latter model.

What would settle it

Measure the ground-state charge distribution $P(n)$ of a nanowire or gatemon transmon by charge-basis tomography while sweeping the top gate $N_g$; an Andreev-only junction should show only integer and half-integer offsets with an entropy peak of $\log(2)$, whereas a Majorana-dressed junction should show the four offset sectors 0, 1/4, 1/2, 3/4 and a $\log(4)$ peak.

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Extended reading notes

Core claim

Starting from a dot-Transmon Hamiltonian, in which the weak link of a Cooper-pair box is a spin-degenerate dot representing the low-energy sector of a short semiconductor wire, the authors derive the reduced density matrix of the superconducting charge states and its diagonal, $P(n)$. Particle conservation and fermion-parity boundary conditions force the relative Cooper-pair number $n$ to lie in $\mathbb{Z}$ for an empty dot and in $\mathbb{Z}+1/2$ for a doubly occupied dot; when two non-local Majorana fermions hybridize with the dot, the allowed offsets become 0, 1/4, 1/2, and 3/4 in the four parity sectors. Near resonant gate voltages, the Andreev-only ground state is an equal superposition of empty and doubly-occupied dot states, giving von Neumann entropy $\log(2)$, while the Majorana-dressed ground state approaches an equal superposition of four parity-charge configurations, giving $\log(4)$. The $\log(4)$ peaks do not split into two independent $\log(2)$ resonances, which the authors present as a way to distinguish a Majorana-dressed Andreev state from two simultaneously resonant Andreev states.

Load-bearing premise

The model represents the weak link as a single spin-degenerate dot, or one low-energy channel of a short wire, with total parity fixed to $N_t=0$, so the clean 0, 1/4, 1/2, 3/4 charge sectors are a counting consequence of that single-channel, fixed-parity bookkeeping; extra channels or parity fluctuations would obscure them.

Editorial extensions

If this is right

  • A single charge-basis tomography run, sweeping the top gate, yields both $P(n)$ and the entanglement entropy from the same dataset.
  • An Andreev-only junction shows $P(n)$ confined to integer and half-integer offsets, with an entropy maximum $\log(2)$ at the dot-resonance gate voltage.
  • A Majorana-dressed junction shows all four sectors 0, 1/4, 1/2, and 3/4 in $P(n)$ near two gate voltages, with entropy peaks of $\log(4)$.
  • The $\log(4)$ peaks do not resolve into two separate $\log(2)$ peaks, distinguishing one Majorana-dressed Andreev state from two independent Andreev states.
  • Because $P(n)$ is what a charge probe reconstructs, the ground-state charge distribution becomes a classification observable for the types of excitations in the junction.

Reading between the lines

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

  • An extension the paper leaves implicit is that the same quarter-offset histogram could serve as a readout of fermionic parity: deliberately injecting a quasiparticle should smear the clean 0, 1/4, 1/2, 3/4 pattern, turning the diagnostic into a parity-error monitor.
  • A testable extension would compute $P(n)$ for a two-channel dot; if inter-channel couplings destroy the quartering and restore integer-halving, the signature may be specific to single-channel weak links rather than generic multi-channel wires.
  • The fixed-parity assumption also suggests a cross-check: comparing ground-state tomography at different total parities should shift the allowed sectors, and only the parity-resolved pattern is expected to match Tables I and II of the paper.
  • Neighboring problems could inherit the method: any hybrid system with conserved total charge and low-energy fermionic degrees of freedom may show analogous fractional offsets in the bosonic charge distribution once a parity sector is fixed.
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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

2 major / 4 minor

Summary. This paper proposes charge-basis tomography as a probe of the low-lying fermionic states in a superconductor-semiconductor transmon junction. The authors study two models: a 'dot-transmon' (DT) with even dot occupancy, yielding charge offsets of 0 and 1/2, and a 'Majorana-dot-transmon' (MDT) with an additional odd-parity sector, yielding offsets of 0, 1/4, 1/2, and 3/4. They further compute the entanglement entropy between the charge and fermionic sectors, finding Smax = log(2) for the DT and Smax = log(4) for the MDT. The selection-rule table (Table II) and the projected 8-dimensional Hamiltonian are benchmarked against exact diagonalization in Fig. 3(a). The central claim is that the quarter-integer offsets and the log(4) entropy peak are signatures that distinguish Majorana-dressed Andreev states from ordinary interacting Andreev states.

Significance. If the proposed signatures were specific to Majorana physics, the paper would provide a genuinely useful classification tool for superconducting-semiconductor qubits, complementing microwave spectroscopy. The exact parity-selection-rule bookkeeping and the analytical entropy formulas (Eqs. (7) and (14)) are valuable, and the benchmark against exact diagonalization in Fig. 3(a) gives confidence in the projected model. However, the extent to which the signatures are uniquely Majorana rather than generic odd-parity sub-gap features is the central open question and is not addressed in the current manuscript.

major comments (2)
  1. [Section 3, Eq. (9) and Table II] The paper claims (Abstract and Conclusions) that quarter-integer offsets in P(n) are a signature of Majorana fermions, but the comparison DT model in Section 2 is restricted to even dot occupations (see the sentence 'We can further simplify the model by focusing on an even nd' preceding Eq. (3)) and therefore contains no single-electron transfer processes. The sectors n ∈ Z ± 1/4 in Table II originate from the operators in Eq. (9) that move one electron between an island and the dot; the phase factors e^{-iφ/4} and e^{+iφ/4} are kinematic consequences of the half-integer change in n_L or n_R, not of the Majorana algebra. An ordinary, topologically trivial single-level dot in its odd-occupation doublet has the same single-electron tunneling vertices with amplitude t instead of wγ, and would therefore produce the same 0, 1/4, 1/2, 3/4 charge sectors. The paper establishes the forward direction (MDT implies quarter offsets) but does not establish the converse, so the proposed Majorana discriminator is not proven. Please add a calculation for a trivial odd-parity Andreev doublet (e.g., a spin-degenerate dot with single-electron tunneling t_L, t_R and pairing Γ) and show whether it reproduces the offset pattern and the entropy values; if it does, the conclusions need to be revised to attribute the signatures to odd-parity sub-gap states rather than specifically to Majorana fermions.
  2. [Section 3 and Conclusions] The claim that the entanglement entropy peaks at log(4) for a Majorana-dressed Andreev state, and that these peaks do not split into two log(2) peaks in contrast to two resonant Andreev states, is not supported by a comparison calculation. A trivial odd-parity doublet with two degenerate spin channels would also allow a rank-4 reduced charge density matrix and hence log(4) at resonance, so the entropy magnitude alone is not a unique Majorana fingerprint. Moreover, the undemonstrated statement about two simultaneously resonant Andreev states appears only in the Conclusions and would need a concrete model (e.g., two dots or two levels) to be substantiated. Please either supply the comparison calculation or qualify the claim.
minor comments (4)
  1. [Supplementary Material, Eqs. (S12) and (S15)] The set '{(o,e),(o,e)}' appears in both equations; this should be '{(o,e),(e,o)}' for consistency with Table II and Eq. (S17).
  2. [Eq. (11)] The occupation numbers m_L and m_R are used without definition; please state explicitly that m_L = 0 for parity e and 1 for parity o, and similarly for m_R.
  3. [Section 2, harmonic approximation] The paper states that the harmonic Gaussian projection is valid for Γ and B below the transmon plasma frequency, but it does not quantify the error introduced by neglecting the periodic (or anti-periodic) boundary conditions of the transmon wavefunctions; a brief estimate of this error for the parameters used in Figs. 2 and 3 would help the reader assess the approximation.
  4. [Fig. 2(a)] The color scale in the density plot of P(n) is not defined; please add a colorbar or state the normalization used.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the quarter-offset and entropy signatures are derived from the stated model Hamiltonians, not fitted or definitionally equivalent to the inputs; minor self-citations do not carry the derivation.

full rationale

The central predictions follow from explicit model assumptions rather than from a fitted parameter renamed as a prediction. For the DT model, the integer/half-integer charge offsets in P(n) are determined by the parity bookkeeping in Table I (n ∈ Z for the empty dot and n ∈ Z+1/2 for the doubly occupied dot), and the entropy maximum Smax = log(2) is obtained by minimizing the projected two-level Hamiltonian (4), with the resonance condition Ng,max given by Eq. (8). For the MDT model, the quarter-integer offsets 0, 1/4, 1/2, 3/4 are derived from the selection rule in Eq. (S11) and Table II, which are consequences of fermionic parity conservation and the phase factors in the Majorana-dot coupling Eq. (9); the entropy peak Smax = log(4) arises from the four-state superposition in Eq. (12), and the peak positions Ng,1 and Ng,2 are found analytically by setting Γ = w = t0 = 0 in the projected Hamiltonian. None of these steps is an input that has been renamed as an output. The paper does rely on self-citations: Eq. (9) is adopted from the authors' earlier work [7], and the tomography premise cites the same group's unpublished preprint [18]. These are model and feasibility inputs, however; the specific signatures are new derived consequences, so the self-citations are not load-bearing in the circularity sense. The skeptic concern that a trivial odd-parity Andreev doublet might also produce the same 0, 1/4, 1/2, 3/4 pattern is a question of model completeness and specificity of the discriminant, not a circularity of the derivation, because the paper's derivation from its stated single-channel MDT Hamiltonian is self-contained.

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

No numbers are fitted to data. The plotted Hamiltonian parameters (EC=E'_C, EC/EJ=7.14, Gamma/E'_C=0.01, B/E'_C=0.75, w/E'_C=0.086, ng=1/10) are representative inputs, not adjustments to reproduce a target result. The Majorana modes are assumed from prior topological-superconductivity models rather than newly introduced, so no invented entities are added by this work.

assumptions (5)
  • domain assumption The weak link is a single spin-degenerate quantum dot (or the low-energy sector of a short wire) with charging energy E'_C, Zeeman splitting B, and proximity pairing Gamma; only empty, singly, and doubly occupied dot states are retained.
    This single-channel structure generates the charge-offset selection rules in Tables I and II. Extra dot orbitals or channels would add or smear the n offsets, so the signature claim is conditional on this description.
  • domain assumption Total particle number is conserved, the total parity sector Nt=0 is selected, and the relative Cooper-pair number n plus the dot and Majorana parities determine the allowed charge lattices.
    The integer, half-integer, and quarter-integer offsets are direct consequences of these counting rules; a different parity sector or parity flips would change the predicted pattern.
  • domain assumption The transmon is treated in the harmonic oscillator approximation for its ground state, with a Gaussian wavefunction, neglected periodicity, and integration limits extended to infinity.
    This projection is used to obtain the analytic two-level and eight-level Hamiltonians and the entropy formulas in Eqs. (4), (10), (7), and (14). It is valid when Gamma and B are below the plasma frequency, and outside that regime quantitative values shift.
  • domain assumption The Majorana modes near the weak link, gamma_2 and gamma-bar_3, hybridize with the dot, while the remote Majoranas gamma-bar_1 and gamma_4 do not hybridize with the dot.
    Footnote [26] states this assumption explicitly. If remote-end hybridization is non-negligible, the eight-state basis and the quarter-offset picture require modification.
  • domain assumption The dot-island coupling Gamma is symmetric on both sides of the junction.
    Footnote [25] acknowledges that asymmetric coupling couples the transmon ground and first excited states and is not treated here. This is a material simplification for real devices.

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Cite this review

Pith. "Pith review of Tomographic Signatures of Interacting Majorana and Andreev States in Superconductor-Semiconductor Transmon Qubits." pith.science (2026). https://pith.science/paper/OQUCHXIW

@misc{pith2026250207684,
  author       = {Pith},
  title        = {Pith review of: Tomographic Signatures of Interacting Majorana and Andreev States in Superconductor-Semiconductor Transmon Qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OQUCHXIW}},
  note         = {Machine review of arXiv:2502.07684}
}
read the original abstract

Semiconductor-based Josephson junctions embedded within a Cooper-pair-box can host complex many-body states, such as interacting Andreev states and potentially other quasi-particles of topological origin. Here, we study the insights that could be revealed from a tomographic reconstruction of the Cooper-pair charge distribution of the junction prepared in its ground state. We posit that interacting and topological states can be identified from distinct signatures within the probability distribution of the charge states. Furthermore, the comprehensive dataset provides direct access to information theory metrics elucidating the entanglement between the charge sector of the superconductor and the microscopic degrees of freedom in the junction. We demonstrate how these metrics serve to further classify differences between the types of excitations in the junction.

Figures

Figures reproduced from arXiv: 2502.07684 by the authors.

Figure 1
Figure 1. FIG. 1. Quantum circuit describing the qubit considered. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Probability density [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Spectrum characteristics of the Majorana-DT. (a) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Reference graph

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    We assume negligible hybridization of the dot operators and the nearby γ2 and ¯γ3 with the Majorana fermions ¯γ1 and γ4 at the nanowire’s remote ends. SUPPLEMENT AR Y MA TERIAL T racing out degrees of freedom This section explains how we trace out degrees of free- dom in our m...

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