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Majorana Signatures in the Tripartite Uncertainty Relations with Quantum Memory

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that in a two-quantum-dot device connected through a topological superconductor, the entropic uncertainty of spin measurements reaches the minimal lower bound, and that a two-step spin measurement can indirectly reveal…

desk verdict The EUR calculation is a legitimate, if modest, analytic result, but the quantum-witness section has a real error in the channel definition and the saturation claim is overstated. read the letter →

arxiv 2506.09621 v2 pith:WF2NBFAX submitted 2025-06-11 quant-ph

D. Maroulakos , C. Jasiukiewicz , A. Wal , A. Sinner , I. Weymann , T. Domański , L. Chotorlishvili This is my paper
classification quant-ph
keywords Majoranaboundstatesentropicuncertaintyrelationsquantummemorytripartiteentanglementdotstopologicalsuperconductorwitnessmeasurementinvasiveness
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 sets out to show that in a tripartite device—two quantum dots connected through a topological superconducting wire hosting Majorana boundary modes—the entropic uncertainty of spin measurements can reach the minimal lower bound allowed by quantum mechanics. It studies the quantum-memory-assisted entropic uncertainty relation for this system and shows analytically that the inequality saturates, so the measurement uncertainty equals the minimal possible value, for any coupling strength and any finite overlap between the Majorana modes. The paper also claims that the overlap between the Majorana modes tunes the uncertainty: larger overlap relative to the dot–Majorana coupling lowers the uncertainty, while weaker overlap raises it toward its maximal value. In addition, it proposes a two-step spin measurement protocol whose outcome differs from a direct measurement when Majorana modes are present, offering an indirect experimental signature.

What carries the argument

The central object is the tripartite quantum-memory-assisted entropic uncertainty relation $S(X|B)+S(Z|C)\ge\log_2(1/c)+\max\{0,\delta\}+\frac{1}{2}(S(A|B)+S(A|C))$. The calculation rides on the exactly solvable low-energy Hamiltonian of two Anderson-type quantum dots coupled to the Majorana operators $\hat\gamma_1,\hat\gamma_2$ with overlap $\varepsilon_M$, and on the resulting pure ground state $\hat\rho_{ABC}=|e_1\rangle\langle e_1|$. Substituting the derived marginals and Holevo quantities turns every term into a function of $A=\xi_+^2-\eta_+^2$, and the bound collapses into the equality $\log_2 2=1$; the quantum witness $W$, built from trace-preserving measurement maps, supplies the separate direct-versus-sequential measurement signature.

What would settle it

In a minimal Kitaev chain of coupled quantum dots operated in the strong-overlap regime $\varepsilon_M\gg\lambda$, compare the spin statistics of the second dot measured directly with those measured after first measuring the first dot: the paper predicts a difference of up to $1/4$ in the spin probabilities, so a null difference would refute the indirect-observation claim.

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Extended reading notes

Core claim

Within a specific exactly solvable parameter set (dot energies $\varepsilon=0$, couplings $\lambda_1=-\lambda_2=\sqrt{2}\,\lambda$), the paper derives closed-form expressions for all conditional entropies, mutual informations, and Holevo quantities entering the tripartite entropic uncertainty relation. It finds that both sides of the inequality reduce to the same function of $A=\xi_+^2-\eta_+^2$, so the relation collapses to the equality $\log_2 2=1$: the uncertainty of the spin measurements sits exactly at the lower bound for arbitrary $\lambda$ and any finite $\varepsilon_M$. The residual uncertainty scales as $1+\frac{\lambda^2}{4\ln(2)\,\varepsilon_M^2}\left(1+\ln\frac{4\varepsilon_M^2}{\lambda^2}\right)$ in the large-overlap limit and approaches two bits in the weak-overlap limit. For indirect observation, the quantum witness $W=\frac{1}{4}\frac{(\omega+\Delta)^2}{4\lambda^2+(\omega+\Delta)^2}$ quantifies the difference between direct and sequential spin measurements and reaches $1/4$ when $\varepsilon_M=2\omega\gg\lambda$.

Load-bearing premise

All entropy calculations start from the pure state $\hat\rho_{ABC}=|e_1\rangle\langle e_1|$, one of two degenerate ground states, and there is no argument for why the device would occupy that state rather than the other one or a superposition; if the actual ground state differs, the saturation and the monotonic decrease with overlap need not hold.

Editorial extensions

If this is right

  • In the solved regime, spin measurements on the first quantum dot exhibit exactly the minimal entropic uncertainty, so no additional noise beyond the fundamental quantum limit is required to explain the data.
  • The Majorana overlap acts as a control knob: for $\varepsilon_M\gg\lambda$ the uncertainty sits near its floor, while for $\varepsilon_M\ll\lambda$ it rises toward two bits.
  • Sequential versus direct spin measurement on the second dot differs by up to $1/4$ in the strong-overlap limit, a concrete signature searchable in hybrid nanowire devices.
  • The predictions depend only on the dot–Majorana coupling $\lambda$ and the Majorana overlap $\varepsilon_M$, so they can be compared with transport or spectroscopy measurements without free parameters.

Reading between the lines

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

  • The paper assumes the pure ground state $|e_1\rangle$ with no mechanism selecting it over the degenerate partner $|e_2\rangle$; if a device occupies $|e_2\rangle$ or a superposition, saturation need not survive, so the universality claim is tied to that state choice.
  • Saturation of the entropic uncertainty relation is the condition for optimal guessing games, so the same device could serve as a testbed for quantum key distribution or certified randomness, applications the paper does not discuss.
  • Because the quantum witness approaches $1/4$ only when $\varepsilon_M\gg\lambda$ and vanishes in the opposite limit, the protocol could be used to calibrate the Majorana overlap in a fabricated device, not merely to detect the modes.
  • The monotonic reduction of uncertainty with overlap hints that tuning $\varepsilon_M$ could suppress measurement noise in long-distance entanglement distribution, though the paper does not model decoherence or finite temperature.
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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 studies a tripartite hybrid system in which two quantum dots are coupled through a topological superconducting nanowire hosting Majorana boundary modes. It makes two central claims. First, in Sec. IV it proposes a 'quantum witness' protocol: comparing direct and indirect spin measurements on the second dot should yield different probabilities, providing indirect evidence for Majorana modes. Second, in Sec. V it uses the tripartite quantum-memory-assisted entropic uncertainty relation (EUR) for the state rho_ABC = |e1><e1| and claims that the Majorana quasiparticles allow the measurement uncertainty to saturate the lower bound, with the bound reaching its minimal value in the strong-overlap regime. The EUR part is an algebraic consistency check on a model-specific pure state, while the quantum-witness part is intended as an experimentally testable signature.

Significance. If the quantum-witness derivation were valid, the proposed two-measurement protocol would be a useful and relatively simple experimental tool for indirect Majorana detection, and the two-parameter dependence of the EUR bound on the Majorana overlap and dot-Majorana coupling would be an appealing quantitative prediction. The entropy calculations in Sec. V are internally consistent for the assumed state: with the corrected algebra, S(X|B)+S(Z|C) equals the right-hand side of Eq. (9), so the EUR is indeed tight for that state. However, the significance is limited by the fact that the result is derived for a single chosen eigenstate |e1> with no discussion of state preparation or robustness, and the paper's strongest experimental claim in Sec. IV rests on a Kraus map that is not trace-preserving. The manuscript also contains an incorrect summary statement about Eq. (39). These issues materially affect the two headline claims, so the paper needs revision before the results can be relied upon.

major comments (3)
  1. [Sec. IV, Eqs. (6)-(8)] The quantum-witness derivation is invalid as written because the map F[rho] = sum_alpha L_alpha rho L_alpha^dagger is not trace-preserving. With L1 = sqrt(mu1)|1><0|_1 (x) |0><1|_2 and L2 = sqrt(mu2)|0><1|_1 (x) |1><0|_2, one obtains sum_alpha L_alpha^dagger L_alpha = mu1 P_{01} + mu2 P_{10}, where P_{01} = |0><0|_1 (x) |1><1|_2 and P_{10} = |1><1|_1 (x) |0><0|_2. For mu1=mu2=1/2 this is (1/2)(P_{01}+P_{10}), which is not the identity on the two-qubit space. Acting on the reduced state in Eq. (7), which contains |00><00| and |11><11| components from |Phi^->, the map annihilates those components, so Tr F[rho] < 1. Consequently the quantities P and Q in Eq. (6) are not normalized probabilities, and Eq. (8) is not a valid difference of probabilities. The claimed indirect-Majorana experimental signature is therefore unsupported by the derivation; the authors must either construct a genuine trace-preserving map with the same intended phenomenology or substantially revise the witness claim.
  2. [Sec. V, Eq. (39) and the surrounding text] The statement that 'for any A, Eq. (39) converts into equality log2 2 = 1' is not correct. For the model parameters one has A >= 0, and the right-hand side of Eq. (39) equals 1 + delta, where delta = -((1-A)/2) log2((1-A)/2) - ((1+A)/2) log2((1+A)/2). Thus Eq. (39) is indeed an equality, but the common value is 1 + delta, not 1. The value 1 is reached only in the limit A -> 1, i.e. epsilon_M >> lambda. The paper's universal claim that 'the topological Majorana quasiparticles enabled the uncertainties to reach their minimal lower bound value' for arbitrary finite overlap is therefore overstated: at finite overlap the saturated bound is 1 + delta > 1. This is a load-bearing point for the abstract and the concluding claims, and the wording of the statement after Eq. (39) should be corrected.
  3. [Sec. V, choice of state rho_ABC = |e1><e1|] All entropy and uncertainty results in Sec. V are computed for the single pure state rho_ABC = |e1><e1|. The manuscript does not state how this state is prepared or why the device should occupy it rather than another eigenstate or a mixture involving higher-energy states. Since the claimed monotonic decrease of uncertainty with increasing Majorana overlap is a property of this particular state, some justification (e.g., zero-temperature initialization in the ground state) and ideally a robustness check against small thermal admixtures or superpositions should be provided before the result is presented as a general prediction of the setup.
minor comments (4)
  1. [Eqs. (7) and (8)] The reduced state in Eq. (7) is written with |Psi^-_d>, whereas the eigenstate |e1> is defined with |Psi^+_d>. This sign discrepancy should be corrected, as it affects the subsequent application of the Kraus operators.
  2. [Eq. (24)] The expression for rho_{C|1} = rho_{C|2} contains a typo: it should be eta_+^2 |1><1|_C + xi_+^2 |0><0|_C, not eta_+^2 |1><1|_C + xi_+^2 |1><1|_C. As printed, the two terms sum to |1><1|_C, which would make H(Z:C) nonzero and inconsistent with the following line H(Z:C)=0.
  3. [Sec. V, derivations] The paper repeatedly refers to 'cumbersome calculations' and 'lengthy calculations' without showing the intermediate steps for Eqs. (19)-(39) and Eq. (8). Given the typographical issues already present, an appendix with the key algebraic steps would substantially improve verifiability.
  4. [Fig. 2 and Fig. 3] The figure captions say 'from above' with parameter lists, but the ordering of the curves is not indicated in the panels themselves. Please add labels or a legend so the reader can identify which curve corresponds to which parameter value.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EUR saturation is an exact algebraic property of the explicitly stated eigenstate, and the Majorana-overlap dependence is a derived consequence, not a fitted prediction.

full rationale

The central derivation is self-contained. The tripartite EUR in Eq. (9) is an external inequality taken from Refs. [4,50,51], not derived from the model, so checking that a particular state saturates it is a genuine calculation rather than a definitional tautology. The entropies in Section V are computed directly from the explicitly stated state rho_ABC = |e1><e1|, and Eq. (39) reduces algebraically to S(rho_X^AB) = 1 + delta by the paper's own formulas (Eqs. (26) and (40)); this equality is a property of the chosen state, not an input. The claimed dependence of uncertainty on the Majorana overlap epsilon_M follows from the algebraic expression for A = xi_+^2 - eta_+^2 and is not obtained by fitting any parameter to data. The self-citation to Ref. [14] for 'technical details' (eigenstates and basis) is not load-bearing circularity: the eigenstates are reproduced in the present text and are algebraic consequences of the stated Hamiltonian, and they do not presuppose the EUR saturation. A separate correctness concern, unrelated to circularity, is that in Section IV the map F is claimed to be trace-preserving, but for the given Kraus operators Sigma_alpha L_alpha^dagger L_alpha = mu_1 P_01 + mu_2 P_10, which is not the identity; this affects whether P and Q in Eq. (8) are normalized probabilities, but it does not make the derivation circular. Overall, no step reduces to its own input or to a fitted parameter, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The paper's central claims rest on ad hoc choices: the sign-symmetric couplings, zero dot energies, zero Coulomb repulsion, and the specific degenerate ground state |e1>. These choices are not derived from experiment or first principles, and the quantum witness result additionally assumes a Kraus map that may not be trace-preserving.

free parameters (4)
  • λ (QD-Majorana coupling) = not fitted
    Physical coupling strength; the result is expressed in terms of λ, but the sign symmetry λ_1=-λ_2=√2 λ is an ad hoc choice that makes the model exactly solvable.
  • ω = ε_M/2 (Majorana overlap) = not fitted
    Overlap between Majorana modes; sets the scale for the universal bound. The paper finds uncertainty decreases with ω/λ.
  • ε_iσ (QD on-site energies) = 0
    Chosen to be zero to enable analytic diagonalization; not justified physically.
  • U (Coulomb repulsion) = 0
    Coulomb interaction is set to zero, a strong simplification; the paper notes that including U would require numerics.
assumptions (4)
  • standard math Tripartite entropic uncertainty relation (Eq. 9) from Ref. [4]
    The improved EUR with quantum memory is a known theorem; the paper uses it as the framework.
  • domain assumption Low-energy model Hamiltonian (Eq. 5) for QDs coupled to Majorana modes
    The model is standard for Majorana-dot hybrids, but its validity for the proposed experiment is assumed.
  • ad hoc to paper The system is prepared in the pure state |e1>, one of two degenerate ground states
    No physical mechanism is given for selecting |e1> over |e2> or any superposition; all results depend on this choice.
  • ad hoc to paper Kraus map F in Section IV is trace-preserving with the given L1, L2
    The map is stated without proving trace preservation on the support of the states; this underlies the quantum witness result.

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Cite this review

Pith. "Pith review of Majorana Signatures in the Tripartite Uncertainty Relations with Quantum Memory." pith.science (2026). https://pith.science/paper/WF2NBFAX

@misc{pith2026250609621,
  author       = {Pith},
  title        = {Pith review of: Majorana Signatures in the Tripartite Uncertainty Relations with Quantum Memory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WF2NBFAX}},
  note         = {Machine review of arXiv:2506.09621}
}
read the original abstract

Quantumness imposes a fundamental limit on measurement accuracy. The paradigmatic cases are Heisenberg's uncertainty relation in the original formulation, Robertson's formulation, and improved uncertainty relations. However, the more universal measures are given in terms of quantum entropies. Uncertainties of measurements done on one quantum system correlated with another quantum system constitute a more intriguing question. Quantum correlations can influence the lower bound of uncertainties, and the reason for this is the quantum memory. In this article, we study uncertainties of measurements performed on one quantum dot correlated with the second one through the superconductor, hosting the Majorana boundary modes. We prove that the Majorana quasiparticles allow the uncertainties to reach the minimal possible lower bound. By rigorous theoretical considerations, we obtain the result of experimental relevance expressed in terms of only two parameters: the overlap between Majorana modes and their coupling strength with the quantum dots. We show that the overlap between Majorana modes reduces quantum uncertainties, which is a general result of fundamental importance. We also propose the protocol to measure spins in both quantum dots, consecutively, and demonstrate that the result of the second measurement would depend on the presence of Majorana quasiparticles. This could serve as an indirect tool for their empirical observation, which is of importance for the ongoing discussions concerning unambiguous detection of the Majorana quasiparticles in nanoscopic hybrid structures.

Figures

Figures reproduced from arXiv: 2506.09621 by the authors.

Figure 1
Figure 1. FIG. 1. The tripartite system, consisting of two quantum [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Entropy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Holevo quantity [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Entanglement dynamics in minimal Kitaev chains

    quant-ph 2025-07 reject novelty 5.0 of 10

    Two- and three-site Kitaev chains can dynamically generate maximally entangled two-qubit and GHZ-type three-qubit states, while a pure W state is forbidden by parity conservation.

Reference graph

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