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REVIEW 3 major objections 5 minor 26 references

Shot noise from which-path detection in a chiral Majorana interferometer

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Which-path detection converts deterministic charge transfer in a chiral Majorana interferometer into binomial 2e shot noise, with a Fano factor that diverges as the detector becomes perfect.

desk verdict Tight FCS calculation showing which-path detection can produce divergent shot noise in a Majorana interferometer, but the Fano factor has a factor-4 error and the measurement channel is assumed rather than derived. read the letter →

arxiv 2505.23670 v1 pith:AGXOIJTK submitted 2025-05-29 cond-mat.mes-hall cond-mat.supr-con

C.W.J.Beenakker This is my paper
classification cond-mat.mes-hallcond-mat.supr-con PACS 73.23.-b72.70.+m
keywords fullcountingstatisticsMajoranainterferometerwhich-pathdetectionshotnoiseFanofactorbinomialweakmeasurementtopologicalsuperconductor
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 asks what happens when a chiral Majorana interferometer — a device in which a Dirac mode splits into two Majorana edge modes that recombine after encircling vortices — is watched by its environment. Without any coupling, the transferred charge is deterministic: each injected electron leaves as an electron or a hole according to the vortex parity. The paper argues that once occupation measurements leak which path the Majorana fermion took, charge transfer acquires a stochastic component of size 2e, with binomial full counting statistics. The mean charge shrinks while the variance grows, so the Fano factor — shot noise divided by conductance — diverges as the detection probability approaches unity. A sympathetic reader cares because this turns decoherence itself into a measurable noise signature unique to the Majorana interferometer.

What carries the argument

The carrying object is a monitored quantum channel: unitary scattering operators alternate with weak measurements of the occupation of each Majorana mode, represented by Kraus operators $\hat{P}_{+,n}=\delta_n\hat{I}+\varepsilon_n a_n^\dagger a_n$ and their empty-mode counterparts, chosen so that $\hat{P}_{+,n}^2+\hat{P}_{-,n}^2=\hat{I}$. Each measurement outcome enters a determinant formula for the moment generating function, obtained by tracing products of Gaussian operators, and particle-hole symmetry imposes $S(-E)=S^*(E)$. At low voltage the factor $\cos(2k\delta L)\to 1$, so all energy channels contribute identically and the sum over energies collapses to the simple binomial form.

What would settle it

A concrete check is to compute or measure the transferred-charge distribution through a chiral Majorana interferometer as a function of detector coupling; if at fixed $N_{\rm in}$ the variance-to-mean ratio deviates from $T(1-T)/(1-2T)$, with $T=\tfrac12(p_1+p_2-p_1p_2)$ obtained from independently calibrated detection probabilities, the binomial claim is falsified.

Watch

Extended reading notes

Core claim

The central discovery is that a weak which-path measurement of Majorana-mode occupancy converts the noiseless, deterministic charge transfer of the Z2 interferometer into a binomial random process. The cumulant generating function is $C(\xi)=N_{\rm in}\ln[\cosh\xi+s_v(1-p_1)(1-p_2)\sinh\xi]$, equivalent to a deterministic transfer of $s_v$ electrons plus random transfers of $-2s_v$ with probability $T=\tfrac12(p_1+p_2-p_1p_2)$. Consequently the variance is ${\rm Var}\,Q=4N_{\rm in}T(1-T)$, while the mean is $s_v N_{\rm in}(1-2T)$, giving a Fano factor $F=T(1-T)/(1-2T)$, which diverges as $T\to\tfrac12$. In the full-detection limit the distribution becomes independent of vortex parity, $P(Q)=2^{-N_{\rm in}}\binom{N_{\rm in}}{(N_{\rm in}-Q)/2}$.

Load-bearing premise

The environment is modelled as a weak measurement of Majorana-mode occupation with free probabilities $p_n$, and the paper does not derive these probabilities from a microscopic Hamiltonian; if the real coupling measures a different operator, both the binomial form and the Fano divergence could change.

Editorial extensions

If this is right

  • At full which-path detection $T=\tfrac12$, the transferred-charge distribution is $P(Q)=2^{-N_{\rm in}}\binom{N_{\rm in}}{(N_{\rm in}-Q)/2}$, independent of the vortex parity $s_v$, so the measurement erases the interferometer's parity memory.
  • The Fano factor $F=T(1-T)/(1-2T)$ grows without bound as $T\to\tfrac12$, so even a weak leak of path information produces strongly super-Poissonian shot noise.
  • The binomial form survives decoherence in this monitored-channel treatment, in contrast to dephasing-probe models of electron interferometers where the counting statistics generally deviates from binomial.
  • Because the stochastic charge is $2e$, the shot noise directly reflects the superconducting condensate absorbing the missing charge when the electron exits as a hole.

Reading between the lines

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

  • If the claim is right, any unavoidable environmental coupling in a realistic Majorana interferometer — not just a deliberate detector — would produce $2e$ shot noise; a microscopic model of the coupling, which the paper does not provide, would be needed to predict the effective probabilities $p_n$.
  • The divergence of the Fano factor coincides with the average current going to zero, so practical detection of the divergence requires resolving increasingly rare transfer events; sensitivity, not integration time, will set the measurement limit.
  • The same monitored-channel counting method, applied to other charge or parity observables or to interferometers with more than two modes, could predict analogous noise signatures; those extensions are not made in the paper.
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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 / 5 minor

Summary. The manuscript derives the full counting statistics of charge transfer in a chiral Majorana interferometer when which-path information leaks to the environment. The author models the environment as a monitored quantum channel: weak measurements of the occupation numbers of the two Majorana arms, described by the POVM in Eq. (2.2), are interspersed with unitary scattering. At zero temperature and low bias, the cumulant generating function is claimed to be Eq. (3.4), i.e., a deterministic transfer of s_v electron charges plus a stochastic transfer of -2s_v charges with binomial probability T=(p1+p2-p1p2)/2. This implies a Fano factor that diverges as the which-path detection probability approaches unity, turning the otherwise noiseless interferometer into a source of 2e shot noise. The paper also recovers the known deterministic limit p1=p2=0 and closes by acknowledging that a microscopic model of the environment is needed to assess how effectively which-path detection is realized in a realistic geometry.

Significance. If the result holds, this is a conceptually striking and clean prediction: which-path detection alone converts a noiseless Majorana interferometer into a source of 2e shot noise with binomial statistics and an unbounded Fano factor. Strengths of the paper include its self-contained use of standard fermionic trace-determinant techniques, the explicit reproduction of the deterministic no-measurement limit, and the fact that the two model parameters p1,p2 are not fitted to force the binomial result. The prediction is falsifiable in principle through conductance and shot-noise measurements. However, the central claim rests on an assumed occupation-measurement POVM, and one displayed formula contains an internal factor-of-4 inconsistency. Once those points are addressed, this would be a valuable contribution to the theory of Majorana interferometry and monitored quantum transport.

major comments (3)
  1. [III, Eq. (3.10)] The Fano factor stated in Eq. (3.10) is inconsistent with Eqs. (3.8) and (3.9). Using x=(1-p1)(1-p2)=1-2T, Eq. (3.8) gives |Q|=Nin x and Eq. (3.9) gives VarQ=Nin(1-x^2), so F=VarQ/|Q|=(1-x^2)/x=4T(1-T)/(1-2T). The expression T(1-T)/(1-2T) in Eq. (3.10) is smaller by a factor of 4. Since Fig. 3 is computed from Eq. (3.10), the plotted Fano factor is also off by a factor of 4. The divergence as T→1/2 is unaffected, but this is one of the two main quantitative predictions of the paper and must be corrected.
  2. [III, Eq. (3.3)] The central result of the paper, Eq. (3.3), is obtained from Eq. (2.13) without showing the determinant evaluation. This step is load-bearing: all subsequent results, including the binomial form Eq. (3.4), the variance Eq. (3.9), and the Fano factor, follow from Eq. (3.3). Please provide the calculation, at least in an appendix or supplement, with particular attention to the sums over s1,s2 and the way the product (1-p1)(1-p2) emerges from the coefficients c_{s1}^2 c_{s2}^2.
  3. [IV (and II, Eq. (2.2))] The physical relevance of the result depends on whether a realistic environment in the proposed geometry realizes the occupation-number POVM of Eq. (2.2). The manuscript explicitly states in Sec. IV that a microscopic model is needed. This is not a cosmetic caveat: the interfering arms are neutral chiral Majorana modes, so a generic electromagnetic environment couples to charge or current densities, which need not be the occupation numbers n_1,n_2. If the environment measures a different fermion bilinear, the trace-determinant step leading to Eq. (3.3) changes and the binomial form, as well as the divergent Fano factor, are not guaranteed. Please either supply a microscopic derivation of the POVM for a concrete environment, or explicitly present the result as conditional on the assumed coupling and state clearly what physical observable would need to be measured to realize the predicted effect.
minor comments (5)
  1. [III, Eq. (3.4a)-(3.5)] The quantity N_in is used in Eq. (3.4a) but only defined in Eq. (3.5); please define it before first use or add a forward reference.
  2. [III, Eq. (3.6)] The binomial distribution in Eq. (3.6) should state explicitly that N_in is an integer number of injected electrons at this point, since the earlier large-time limit in Eq. (3.5) treats it as a continuous variable.
  3. [III, Eq. (3.7)] For T=1/2, Eq. (3.7) should specify that P(Q) vanishes unless Q has the same parity as N_in; otherwise the support of the distribution is not fully defined.
  4. [Fig. 3] After correcting Eq. (3.10), the curve in Fig. 3 must be recomputed; the caption should also state the corrected formula.
  5. [II, Eq. (2.2)] The paper calls p_n=epsilon_n^2 the probability of which-path detection, but the outcome probabilities of the POVM in Eq. (2.2) for an occupied mode are (delta_n+epsilon_n)^2 and delta_n^2. A sentence explaining the operational meaning of p_n would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the binomial full counting statistics is derived from an explicitly stated weak-measurement POVM, with free parameters p1,p2, and the self-cited method is restated in the paper.

full rationale

The paper's central result, Eqs. (3.4) and (3.10), is a direct mathematical consequence of the monitored quantum channel defined in Eqs. (2.2), (2.6), and (2.13). The POVM projectors P_{±,n} are given explicitly, the Gaussian representation (2.11) is stated, and Klich's trace-determinant relation (2.12) is an external standard result; no fitted parameter is used to enforce the binomial form. The parameters p1 and p2 enter as free model parameters describing the assumed weak measurement, and the Fano-factor divergence as T→1/2 is a parametric prediction of that model, not a restatement of an input. The self-citation of Ref. [13] is methodological and transparent: the paper reproduces the essential channel and determinant formulas rather than importing them as an unverified black box. The acknowledged limitation in Sec. IV, that 'One would need a microscopic model of this coupling,' concerns the physical realization of the POVM, not circularity; the derivation is conditional on the stated measurement model, and the paper says so. The deterministic benchmark p1=p2=0 recovers noiseless Z2 charge transfer, and the standard cumulant identities (3.8)–(3.9) are used. An arithmetic inconsistency exists in Eq. (3.10) (the cumulants give a factor 4 larger Fano factor), but that is a numerical error, not a circular step. No self-definitional reduction, fitted-input-as-prediction, or uniqueness imported from the authors' prior work is present.

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

The central claim rests on two free phenomenological detection probabilities and on the assumption that the environment acts as a Gaussian weak measurement of occupation. No new particles or forces are introduced.

free parameters (2)
  • p1 = not fitted
    Which-path detection probability in Majorana arm 1, introduced phenomenologically; the divergence claim uses p1 tending to 1.
  • p2 = not fitted
    Which-path detection probability in Majorana arm 2, introduced phenomenologically; enters through T = (p1+p2-p1p2)/2.
assumptions (5)
  • standard math Klich trace-determinant formula for Gaussian fermionic operators
    Used in Eq. (2.12) to evaluate the moment generating function.
  • domain assumption Fermionic anticommutation relations and particle-hole symmetry S(-E)=S*(E)
    Eqs. (2.3) and (3.1) define the Majorana mode algebra and constrain the scattering matrix.
  • domain assumption Weak measurement projectors of Eq. (2.2) form a completely positive trace-preserving channel
    The monitored-quantum-channel model of Ref. 13 is assumed to describe which-path detection; no microscopic derivation is given.
  • domain assumption Junctions treated as point scatterers evaluated at E=0, with scattering matrices of SO(2)
    Used in Eq. (3.2) to write Sin and Sout in terms of Pauli matrices.
  • domain assumption Zero temperature and low voltage V << hbar vF/(e delta L), so cos(2k delta L) tends to 1
    Used after Eq. (3.3) to obtain the energy-independent binomial form (3.4).

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Cite this review

Pith. "Pith review of Shot noise from which-path detection in a chiral Majorana interferometer." pith.science (2026). https://pith.science/paper/AGXOIJTK

@misc{pith2026250523670,
  author       = {Pith},
  title        = {Pith review of: Shot noise from which-path detection in a chiral Majorana interferometer},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGXOIJTK}},
  note         = {Machine review of arXiv:2505.23670}
}
read the original abstract

We calculate the full counting statistics of charge transfer in a chiral Majorana interferometer - a setup where a Dirac mode (an electron-hole mode) is split into two Majorana modes that encircle a number of h/2e vortices in a topological superconductor. Without any coupling to the environment it is known that the low-energy charge transfer is deterministic: An electron is transferred either as an electron or as a hole, dependent on the parity of the vortex number. We show that a stochastic contribution appears if which-path information leaks into the environment, producing the shot noise of random 2e charge transfers with binomial statistics. The Fano factor (dimensionless ratio of shot noise power and conductance) increases without bound as the which-path detection probability tends to unity.

Figures

Figures reproduced from arXiv: 2505.23670 by the authors.

Figure 1
Figure 1. FIG. 1. Layout of the [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Scattering geometry corresponding to Fig. 1. The [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Fano factor as a function of [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

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