Pith. sign in

REVIEW 3 major objections 4 minor 62 references

A privacy quantifier defined from the classical Fisher information matrix certifies that distributed quantum sensing can hide every individual parameter while estimating their average at Heisenberg-limited precision.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review

2026-08-03 07:44 UTC pith:L47ZCWIU

load-bearing objection The quantifier is a useful geometric tool, but the paper's 'no individual parameter is accessible' claim is not supported by its own CFIM, which has nonzero diagonal entries. the 3 major comments →

arxiv 2601.19206 v2 pith:L47ZCWIU submitted 2026-01-27 quant-ph

Universal Operational Privacy in Distributed Quantum Sensing

Min Namkung , Dong-Hyun Kim , Seongjin Hong , Yong-Su Kim , Su-Yong Lee , Hyang-Tag Lim This is my paper
classification quant-ph
keywords distributed quantum sensingprivacyclassical Fisher information matrixsingular information structurequantum metrologyHeisenberg limitparameter estimationentanglement
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 paper aims to establish a universal, operational privacy condition for distributed quantum sensing, defined directly from the classical Fisher information matrix (CFIM) that is experimentally measurable. The central claim is that privacy is certified by the kernel structure of the CFIM: if the matrix is singular and the hidden direction is orthogonal to the global function being estimated, then no individual parameter is exposed to untrusted servers. The paper proves the quantifier PF(w) is basis-independent, continuous under small imperfections, and robust to noise, and it demonstrates experimentally that a two-photon, four-phase network achieves PF(w)=1 with Heisenberg-scaled precision using fewer photons than parameters. This matters because it replaces idealized quantum-bound conditions with a practical, protocol-independent test applicable to any singular information structure.

Core claim

The paper's central discovery is the privacy quantifier PF(w)=1−min_{v⊥w, ||v||=1} v^T Π_F v, where Π_F projects onto the support of the CFIM. PF(w)=1 means the kernel of the CFIM contains a direction fully orthogonal to the weight vector w, so the clients' chosen global function is perfectly sensitive while all orthogonal directions—including individual parameter variations—are invisible to the servers' measurements. In the experiment, the CFIM for the two-photon state is singular with kernel spanned by [1,−1,1,−1]; for w=(1,1,1,1)/4, the minimum is zero and PF(w)=1, while the estimation variance saturates the Heisenberg bound 1/N²=1/4. The paper further shows that the condition extends to

What carries the argument

The key object is the CFIM support projector Π_F built from eigenvectors of the classical Fisher information matrix with nonzero eigenvalues. The privacy quantifier PF(w) measures how much the directions orthogonal to the target weight vector w leak into the support; it is the mechanism that turns rank deficiency into a quantitative, experimentally testable privacy guarantee.

Load-bearing premise

The guarantee rests on the premise that an untrusted server's information is fully described by the joint classical Fisher information matrix of the network's measurement record; if a server can use its own marginal outcomes or side information to estimate a local parameter, the rank-deficient CFIM may not certify privacy.

What would settle it

Measure the marginal probability distribution at a single server and compute its Fisher information with respect to its local phase; if that marginal Fisher information is nonzero and an estimator achieves nonzero precision, then the assertion that no individual parameter is accessible to an untrusted server fails.

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

Share X Bluesky LinkedIn Reddit HN

If this is right

  • Privacy certification becomes possible without implementing optimal or entangling measurements, since the CFIM is reconstructed from the actual outcome statistics.
  • The condition applies to protocols whose singular information structure has rank greater than one, going beyond rank-1 QFIM states such as GHZ states.
  • A distributed sensing protocol can use fewer photons than parameters and simultaneously achieve perfect privacy and Heisenberg-limited precision.
  • The quantifier's continuity and noise robustness mean small experimental imperfections do not destroy the privacy certificate.
  • The framework unifies distributed sensing, multiparameter estimation, and privacy verification under a single measurable criterion.

Where Pith is reading between the lines

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

  • The paper's privacy notion is statistical (Fisher-information based); a stronger adversarial model that grants servers prior knowledge or access to side channels might still allow inference, a direction the paper does not explore.
  • The quantifier could be used inversely as a design tool: engineering probe states and measurements so that the CFIM kernel contains the desired hidden directions, which may guide resource-efficient private sensor networks.
  • A natural testable extension is to apply the same CFIM-support criterion to continuous-variable or squeezed-state distributed sensing, where the CFIM can be reconstructed from homodyne outcomes.
  • Because the experimental reconstruction uses a fitted pair-sum model, an independent check would be to compute the CFIM directly from raw coincidences without imposing the sum-only structure; if the kernel persists, the privacy certificate is robust.

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

3 major / 4 minor

Summary. The paper proposes a universal operational privacy quantifier for distributed quantum sensing, defined as PF(w)=1−min_{v⊥w,||v||=1} vᵀΠ_F v (Eq. 1), where Π_F is the projector onto the support of the classical Fisher information matrix (CFIM). It claims that PF(w)=1 certifies that no information about individual parameters is accessible to untrusted parties, and it experimentally demonstrates a four-phase, two-photon protocol whose CFIM (Eq. 5) is singular with kernel spanned by [1,−1,1,−1], yielding PF(w)=1 for w=(1,1,1,1)/4 together with Heisenberg-scaled uncertainty.

Significance. The geometric quantifier PF(w) is mathematically coherent and could be a useful measure of how much of the parameter space is hidden by a singular CFIM. The experiment is substantial: it implements a two-photon distributed sensing network, measures sixteen coincidence probabilities with high visibility, and reconstructs a singular CFIM in agreement with the ideal model. However, the central privacy claim is not supported. PF(w)=1 only certifies the existence of a direction orthogonal to w in the CFIM kernel; it does not imply that individual parameter directions e_i are unobservable. In the reported CFIM, the diagonal entries F_ii=1/2 are strictly positive, so the joint outcome record contains Fisher information about each individual phase. The claimed 'universal operational privacy' therefore does not follow from the presented mathematics unless a specific adversary model—such as a single server seeing only its own marginal outcome—is explicitly adopted and analyzed. As stated, the paper overreaches its evidence.

major comments (3)
  1. [Eq. (1), Eq. (5), and Abstract] The condition PF(w)=1 occurs when the kernel of the full CFIM contains a vector v orthogonal to w. In the experimental CFIM Eq. (5), the kernel is spanned by [1,−1,1,−1], which contains no elementary direction e_i. Meanwhile the diagonal elements F_11=F_22=F_33=F_44=1/2 are nonzero, so each individual phase is estimable from the joint outcome record with finite Fisher information. The abstract's claim that 'no information about individual parameters is accessible' is therefore not entailed by PF(w)=1. To support this claim, the manuscript must either restate the guarantee as privacy of a specific linear combination, or introduce and analyze an explicit adversary model (e.g., a single server whose marginal CFIM is zero).
  2. [Eq. (4) and experimental section (Fig. 4)] The privacy structure is effectively assumed by the probability model. Eq. (4) postulates that every two-photon coincidence probability depends only on φ_μ+φ_ν, so the per-pair CFIM is automatically rank-one and the global kernel necessarily contains the alternating vector [1,−1,1,−1]. Fitting data to this pair-sum-only model cannot independently verify that the kernel exists; it only checks consistency with an input assumption. To claim experimental verification of privacy, the authors should analyze the model-free CFIM or provide a test that could in principle detect a non-singular information structure.
  3. [Eq. (2) and 'universal/protocol-independent' wording] The CFIM is by construction dependent on the chosen measurement (POVM), so calling the condition 'protocol-independent' is misleading. The text says Eq. (2) verifies continuity, but no derivation is given in the main text and the Supplemental Material was not available for review. If the universality claim rests on this continuity property, the proof should be stated or the claim should be weakened to 'valid for the class of protocols whose CFIM is singular as specified.'
minor comments (4)
  1. [References] Reference [3] contains a typo: 'A VS Quantum Sci.' should be 'AVS Quantum Sci.' Reference [54] has a doubled 'and' in the author list.
  2. [Fig. 4 caption] The caption states that 'the horizontal and vertical axes denote eigenvalues and the corresponding vector components, respectively.' This is unclear: the horizontal axis appears to label eigenvalues, while the vertical axis shows components, but the layout of three subpanels should be explained more explicitly.
  3. [Notation] The symbol Π_F is introduced as the projector onto the support space of the CFIM, but the text does not explicitly define the support space for a rank-deficient matrix. A one-sentence definition (range of F, or span of eigenvectors with nonzero eigenvalue) would improve accessibility.
  4. [General] The paper repeatedly refers to 'Supplemental Material I-B' and 'I-D' for key derivations (e.g., the claim that kernel structure prevents exposure of any local parameter). Since these derivations are central to the privacy claim, at least their statements should be included or summarized in the main text.

Circularity Check

3 steps flagged

The experimental 'demonstration' of PF(w)=1 is forced by the pair-sum-only fitting model and the quantifier's definition; the further claim that this hides individual parameters is not supported by the global CFIM.

specific steps
  1. self definitional [Eq. (1) and following paragraph]
    "PF(w) = 1−min_{v⊥w,∥v∥2=1} vTΠFv ... In the extreme case where the kernel of the CFIM contains directions fully orthogonal to w, the minimum in Eq. (1) vanishes and PF(w) = 1, meaning that both desired privacy and sensitivity are simultaneously achieved by the clients."

    PF(w)=1 is defined as the existence of a CFIM kernel direction orthogonal to w. The paper then treats PF(w)=1 as the operational certificate of privacy. Therefore the statement that the demonstrated protocol is private is a restatement of the defining condition of the quantifier rather than an independent prediction.

  2. fitted input called prediction [Eq. (4) and experimental CFIM reconstruction (Eq. (5))]
    "p[(xµ,xν)=(±,±)|φ] = 1/16 [1+V^{±±}_{µν} cos(φµ+φν)] ... Notably, each probability depends only on the sum φµ+φν ... These data are used to reconstruct the experimental CFIM, which closely reproduces the ideal matrix in Eq. (5)."

    The CFIM used to certify privacy is reconstructed by fitting data to a pair-sum-only model. This functional form forces the singular structure: the shift [1,-1,1,-1] leaves every φµ+φν invariant, so the CFIM kernel and hence PF(w)=1 are built into the fit. Only the visibilities are free; the rank deficiency is an input, not an empirical finding.

  3. self definitional [Paragraph after Eq. (1): 'Importantly, the proposed privacy condition clarifies...']
    "Importantly, the proposed privacy condition clarifies that not only the exposure of all local parameters but also the partial exposure of any individual parameter is prevented."

    This equates the global rank-deficiency condition PF(w)=1 with absence of information about individual parameters. But PF(w)=1 only requires some kernel direction v⊥w; it does not require individual directions e_i to lie in the kernel. In the paper's own Eq. (5), F_{ii}=1/2, so the joint outcomes carry Fisher information about each φ_i. The privacy conclusion is thus an interpretation imposed on the quantifier, not derived from it.

full rationale

The paper defines a genuine mathematical quantifier PF(w) and correctly computes its value from the CFIM. The circularity enters at the experimental-validation level: the CFIM is reconstructed from Eq. (4), which restricts every two-photon probability to a function of φµ+φν only. That restriction alone forces the kernel direction [1,-1,1,-1] and therefore PF(w)=1 for w=(1,1,1,1)/4. The headline 'satisfies the universal privacy condition' is consequently an algebraic consequence of the fitting ansatz, not an independent result extracted from the data. Additionally, the semantic claim that PF=1 protects individual parameters is not supported by the global CFIM, whose diagonal entries Eq. (5) show individual phases remain Fisher-informative; a local-server marginal model would be needed but is not analyzed. Self-citations (e.g., [25], [46]) are present but not the load-bearing source of circularity. Score 6 reflects that the central demonstration reduces by construction, while the framework itself is not vacuous and could assign PF<1 to other protocols.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central claim rests on the assumption that information-theoretic privacy is fully captured by the classical Fisher information matrix of a fixed measurement. That assumption is load-bearing and unproven. The experimental demonstration additionally depends on a fitted model whose singular structure is assumed. No new physical entities are introduced; the privacy quantifier is a mathematical functional, not an invented physical object.

free parameters (1)
  • two-photon interference visibilities V^{±±}_{μν} and V^{±∓}_{μν} = V = 0.968 ± 0.003 (average over the sixteen outcomes)
    These visibility parameters are fitted to the measured coincidence probabilities in Eq. (4) and set the scale of the reconstructed CFIM. The rank-deficiency (kernel) of the CFIM is independent of V as long as V>0, but the numerical 'agreement' with theory in Fig. 4 derives from this fitted model.
axioms (4)
  • domain assumption Privacy is equivalent to rank deficiency of the classical Fisher information matrix of the actual measurement.
    PF(w) is defined via the support projector of F, and the paper reads the operational guarantee 'no information about individual parameters' off the CFIM kernel. This equivalence is asserted, not derived from an adversary model; it appears immediately after Eq. (1) and in the abstract.
  • domain assumption The untrusted server/eavesdropper is limited to the specified measurement record and has no side information.
    The CFIM is computed for the implemented σx coincidence measurements. The claim that individual parameters are hidden depends on the adversary not choosing a different POVM, not combining the data with prior knowledge, and not exploiting global state correlations. This restriction is never stated explicitly.
  • standard math Zero Fisher information along a direction on the parameter space is sufficient to conclude the corresponding parameter combination is not accessible.
    The privacy argument uses local CFIM singularity to claim non-identifiability. For smooth models this is valid over an open set only if the singularity persists there—the paper says this persistence is demonstrated in Supplemental Material II—but the main text shows only two reconstructed points.
  • ad hoc to paper The coincidence probability model of Eq. (4), with each outcome depending only on φμ+φν, correctly describes the implemented experiment.
    The experimental privacy demonstration reconstructs the CFIM from this assumed model. If real imperfections produced additional individual-phase dependence, the fitted-reconstruction procedure could miss it, and the singular structure would be an artifact of the model rather than an independent observation.

reviewed 2026-08-03 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Universal Operational Privacy in Distributed Quantum Sensing." pith.science (2026). https://pith.science/paper/L47ZCWIU

@misc{pith2026260119206,
  author       = {Pith},
  title        = {Pith review of: Universal Operational Privacy in Distributed Quantum Sensing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L47ZCWIU}},
  note         = {Machine review of arXiv:2601.19206}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Privacy is a fundamental requirement in distributed quantum sensor networks, where multiple clients estimate spatially distributed parameters using shared quantum resources while interacting with potentially untrusted servers. Despite its importance, existing privacy conditions rely on idealized quantum bounds and do not fully capture the operational constraints imposed by realistic measurements. Here, we introduce a universal operational privacy framework for distributed quantum sensing, formulated in terms of the experimentally accessible Fisher information matrix and applicable to arbitrary protocols characterized by singular information structures. The proposed condition provides a protocol-independent criterion, ensuring that no information about individual parameters is accessible to untrusted parties. We further experimentally demonstrate that a distributed quantum sensing protocol employing fewer photons than the number of estimated parameters simultaneously satisfies the universal privacy condition and achieves Heisenberg-limited precision. Our results establish universal operational constraints governing privacy in distributed quantum sensor networks and provide a foundation for practical, privacy-preserving quantum sensing beyond full-rank regimes.

Figures

Figures reproduced from arXiv: 2601.19206 by Dong-Hyun Kim, Hyang-Tag Lim, Min Namkung, Seongjin Hong, Su-Yong Lee, Yong-Su Kim.

Figure 1
Figure 1. Figure 1: FIG. 1. Conceptual schematic of a private distributed quan [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Experimental implementation of a private distributed quantum sensing network. A polarization-entangled Bell state is [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Experimentally measured two-photon outcome probabilities for the distributed quantum sensing protocol. The sixteen [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Experimental verification of privacy in distributed quantum sensing via the singular CFIM. Eigenvalues and eigenvectors [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 4 linked inside Pith

  1. [1]

    Quantum limits in optical interferometry,Prog

    Demkowicz-Dobrzanski, R., Jayzyna, M., & Kolodyn- ski, J. Quantum limits in optical interferometry,Prog. Opt.60, 345 (2015)

    2015

  2. [2]

    After a dual wavelength half waveplate (DWH) set at 45 ◦, the state is transformed into|Φ +⟩= (|HH⟩+|V V⟩)/ √

  3. [3]

    The state |Ψ2 4⟩is distributed to the servers through 3-km optical fiber spools

    A beam splitter network (BSN) com- posed of two 50/50 fiber beam splitters (FBSs) then dis- tributes the photons across four spatial modes, preparing the two-photon entangled state [25] |Ψ2 4⟩= 1 2 |Φ2 1,2⟩+|Φ 2 2,3⟩+|Φ 2 3,4⟩+|Φ 2 4,1⟩ ,(3) where|Φ 2 µ,ν⟩= (|H µHν⟩+|V µVν⟩)/ √ 2 denotes a Bell state shared between theµth andνth servers. The state |Ψ2 4⟩i...

  4. [4]

    Szczykulska, T

    M. Szczykulska, T. Baumgratz, and A. Datta, Multi- parameter quantum metrology, Adv. Phys. X1, 621 (2016)

    2016

  5. [5]

    Polino, M

    E. Polino, M. Valeri, N. Spagnolo, and F. Sciarrino, Pho- tonic quantum metrology, A VS Quantum Sci.2, 024703 (2020)

    2020

  6. [6]

    Barbieri, I

    M. Barbieri, I. Gianani, A. Z. Goldberg, and L. L. S´ anchez-Soto, Quantum multiphase estimation, Contemp. Phys.65, 112–124 (2024)

    2024

  7. [7]

    Dorneret al., Optimal quantum phase estimation, Phys

    U. Dorneret al., Optimal quantum phase estimation, Phys. Rev. Lett.102, 040403 (2009)

    2009

  8. [8]

    Demkowicz-Dobrzanskiet al., Quantum phase estima- tion with lossy interferometers, Phys

    R. Demkowicz-Dobrzanskiet al., Quantum phase estima- tion with lossy interferometers, Phys. Rev. A80, 013825 (2009)

    2009

  9. [9]

    H. Lee, P. Kok, and J. P. Dowling, A quantum Rosetta stone for interferometry, J. Mod. Opt.49, 2325–2338 (2002)

    2002

  10. [10]

    P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walm- sley, Quantum enhanced multiple phase estimation, Phys. Rev. Lett.111, 070403 (2013)

    2013

  11. [11]

    Yue, Y.-R

    J.-D. Yue, Y.-R. Zhang, and H. Fan, Quantum-enhanced metrology for multiple phase estimation with noise, Sci. Rep.4, 5933 (2014)

    2014

  12. [12]

    Rocciaet al., Multiparameter approach to quantum phase estimation with limited visibility, Optica5, 1171 (2018)

    E. Rocciaet al., Multiparameter approach to quantum phase estimation with limited visibility, Optica5, 1171 (2018)

    2018

  13. [13]

    ur Rehmanet al., Optimal strategy for multiple-phase 6 estimation under practical measurement with multimode NOON states, Phys

    J. ur Rehmanet al., Optimal strategy for multiple-phase 6 estimation under practical measurement with multimode NOON states, Phys. Rev. A106, 032612 (2022)

    2022

  14. [14]

    Namkunget al., Optimal multiple-phase estimation with multi-mode NOON states against photon loss, New J

    M. Namkunget al., Optimal multiple-phase estimation with multi-mode NOON states against photon loss, New J. Phys.26, 073028 (2024)

    2024

  15. [15]

    Namkung, D.-H

    M. Namkung, D.-H. Kim, S. Hong, C. Lee, and H.- T. Lim, Persistent quantum advantage with definite photon-number states in lossy multiple-phase estimation, Phys. Rev. A112, 013714 (2025)

    2025

  16. [16]

    Polinoet al., Experimental multiphase estimation on a chip, Optica6, 288 (2019)

    E. Polinoet al., Experimental multiphase estimation on a chip, Optica6, 288 (2019)

    2019

  17. [17]

    Kacprowicz, R

    M. Kacprowicz, R. Demkowicz-Dobrzanski, W. Wasilewski, K. Banaszek, and I. A. Walmsley, Experimental quantum-enhanced estimation of a lossy phase shift, Nat. Photon.4, 357 (2010)

    2010

  18. [18]

    Honget al., Quantum enhanced multiple-phase esti- mation with multi-mode N00N states, Nat

    S. Honget al., Quantum enhanced multiple-phase esti- mation with multi-mode N00N states, Nat. Commun.12, 5211 (2021)

    2021

  19. [19]

    Honget al., Practical sensitivity bound for multiple phase estimation with multi-mode N00N states, Laser Photon

    S. Honget al., Practical sensitivity bound for multiple phase estimation with multi-mode N00N states, Laser Photon. Rev.16, 2100682 (2022)

    2022

  20. [20]

    Gessner, L

    M. Gessner, L. Pezz` e, and A. Smerzi, Sensitivity bounds for multiparameter quantum metrology, Phys. Rev. Lett. 121, 130503 (2018)

    2018

  21. [21]

    T. J. Proctor, P. A. Knott, and J. A. Dunningham, Mul- tiparameter estimation in networked quantum sensors, Phys. Rev. Lett.120, 080501 (2018)

    2018

  22. [22]

    W. Ge, K. Jacobs, Z. Eldredge, A. V. Gorshkov, and M. Foss-Feig, Distributed quantum metrology with linear networks and separable inputs, Phys. Rev. Lett.121, 043604 (2018)

    2018

  23. [23]

    Zhang and Q

    Z. Zhang and Q. Zhuang, Distributed quantum sensing, Quantum Sci. Technol.6, 043001 (2021)

    2021

  24. [24]

    Liuet al., Distributed quantum phase estimation with entangled photons, Nat

    L.-Z. Liuet al., Distributed quantum phase estimation with entangled photons, Nat. Photon.15, 137 (2021)

    2021

  25. [25]

    Zhaoet al., Field demonstration of distributed quantum sensing without post-selection, Phys

    S.-R. Zhaoet al., Field demonstration of distributed quantum sensing without post-selection, Phys. Rev. X 11, 031009 (2021)

    2021

  26. [26]

    Malitesta, A

    M. Malitesta, A. Smerzi, and L. Pezz` e, Distributed quan- tum sensing with squeezed-vacuum light in a configurable array of Mach–Zehnder interferometers, Phys. Rev. A 108, 032621 (2023)

    2023

  27. [27]

    Kimet al., Distributed quantum sensing of mul- tiple phases with fewer photons, Nat

    D.-H. Kimet al., Distributed quantum sensing of mul- tiple phases with fewer photons, Nat. Commun.15, 266 (2024)

    2024

  28. [28]

    Pezz` e and A

    L. Pezz` e and A. Smerzi, Distributed quantum multi- parameter estimation with optimal local measurements, arXiv:2405.18404

    Pith/arXiv arXiv

  29. [29]

    Kimet al., Distributed quantum sensing with multimode N00N states, Phys

    D.-H. Kimet al., Distributed quantum sensing with multimode N00N states, Phys. Rev. Lett.135, 050802 (2025)

    2025

  30. [30]

    Albarelli, M

    F. Albarelli, M. Barbieri, M. G. Genoni, and I. Gianani, A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imag- ing, Phys. Lett. A384, 126311 (2020)

    2020

  31. [31]

    Montenegroet al., Review: quantum metrology and sensing with many-body physics, Phys

    V. Montenegroet al., Review: quantum metrology and sensing with many-body physics, Phys. Rep.1134, 1–62 (2025)

    2025

  32. [32]

    Demkowicz-Dobrzanski, K

    R. Demkowicz-Dobrzanski, K. Banaszek, and R. Schn- abel, Fundamental quantum interferometry bound for the squeezed-light-enhanced gravitational-wave detector GEO600, Phys. Rev. A88, 041802(R) (2013)

    2013

  33. [33]

    R. K. Gosalia and R. Malaney, Quantum-enhanced clock synchronization using prior statistical information, Quantum Sci. Technol.10, 025022 (2025)

    2025

  34. [34]

    Huang, C

    Z. Huang, C. Lupo, and P. Kok, Quantum-limited esti- mation of range and velocity, PRX Quantum2, 030303 (2021)

    2021

  35. [35]

    Huanget al., Experimental blind quantum com- puting for a classical client, Phys

    H.-L. Huanget al., Experimental blind quantum com- puting for a classical client, Phys. Rev. Lett.119, 050503 (2017)

    2017

  36. [36]

    Polacchiet al., Multi-client distributed blind quantum computation with the Qline architecture, Nat

    B. Polacchiet al., Multi-client distributed blind quantum computation with the Qline architecture, Nat. Commun. 14, 7743 (2023)

    2023

  37. [37]

    Weiet al., Universal distributed blind quantum computing with solid-state qubits, Science388, 6746 (2025)

    Y.-C. Weiet al., Universal distributed blind quantum computing with solid-state qubits, Science388, 6746 (2025)

    2025

  38. [38]

    Shettel, M

    N. Shettel, M. Hassani, and D. Markham, Private network parameter estimation with quantum sensors, arXiv:2207.14450

    Pith/arXiv arXiv

  39. [39]

    de Jonget al., Anonymous and private parameter esti- mation in networks of quantum sensors, Phys

    J. de Jonget al., Anonymous and private parameter esti- mation in networks of quantum sensors, Phys. Rev. Ap- plied24, 054053 (2025)

    2025

  40. [40]

    Hassani, S

    M. Hassani, S. Scheiner, M. G. A. Paris, and D. Markham, Privacy in networks of quantum sensors, Phys. Rev. Lett.134, 030802 (2025)

    2025

  41. [41]

    Bugalho, M

    L. Bugalho, M. Hassani, Y. Omar, and D. Markham, Pri- vate and robust states for distributed quantum sensing, Quantum9, 1596 (2025)

    2025

  42. [42]

    Junioret al., Privacy in continuous-variable dis- tributed quantum sensing, arXiv:2509.12338

    A. Junioret al., Privacy in continuous-variable dis- tributed quantum sensing, arXiv:2509.12338

    arXiv

  43. [43]

    Hoet al., Quantum-private distributed sensing, arXiv:2410.00970 (2024)

    J. Hoet al., Quantum-private distributed sensing, arXiv:2410.00970 (2024)

    arXiv 2024

  44. [44]

    Alushi and R

    U. Alushi and R. D. Candia, Privacy in distributed quantum sensing with Gaussian quantum networks, arXiv:2509.22103

    Pith/arXiv arXiv

  45. [45]

    A. T. Rezakhani, M. Hassani, and S. Alipour, Continuity of the quantum Fisher information, Phys. Rev. A100, 032317 (2019)

    2019

  46. [46]

    Y. Yang, V. Montenegro, and A. Bayat, Overcoming quantum metrology singularity through sequential mea- surements, Phys. Rev. Lett.135, 010401 (2025)

    2025

  47. [47]

    Wang and G

    J. Wang and G. S. Agarwal, Exact quantum Fisher matrix results for distributed phases using multiphoton polarization Greenberger–Horne–Zeilinger states, Phys. Rev. A111, 012414 (2024)

    2024

  48. [48]

    Namkung, C

    M. Namkung, C. Lee, and H.-T. Lim, Unified strategy for non-invertible Fisher information matrix in quantum metrology, Phys. Lett. A564, 131075 (2025)

    2025

  49. [49]

    Pezz` e and A

    L. Pezz` e and A. Smerzi, Quantum theory of phase esti- mation, arXiv:1411.5164

    Pith/arXiv arXiv

  50. [50]

    C. W. Helstrom,Quantum detection and estimation the- ory(Academic Press, 1969)

    1969

  51. [51]

    Cram´ er,Mathematical methods of statistics(Prince- ton University Press, 1946)

    H. Cram´ er,Mathematical methods of statistics(Prince- ton University Press, 1946)

    1946

  52. [52]

    C. R. Rao,Linear statistical inference and its applications (Wiley, New York, 1973)

    1973

  53. [53]

    J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estima- tion, J. Phys. A: Math. Theor.53, 023001 (2020)

    2020

  54. [54]

    M. G. A. Paris, Quantum estimation for quantum tech- nology, Int. J. Quantum Inf.7, 125–137 (2009)

    2009

  55. [55]

    Penrose, A generalized inverse for matrices, Proc

    R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc.51, 406 (1955)

    1955

  56. [56]

    Li and and P.-C

    Y.-H. Li and and P.-C. Yeh, An Interpretation of the Moore-Penrose Generalized Inverse of a Singular Fisher Information Matrix,IEEE Trans. Sign. Proc.60, 5532 (2012). 7

    2012

  57. [57]

    Fujiwara and H

    A. Fujiwara and H. Nagaoka, Quantum Fisher metric and estimation for pure state models, Phys. Lett. A201, 119–124 (1995)

    1995

  58. [58]

    M. G. Genoniet al., Optimal estimation of joint param- eters in phase space, Phys. Rev. A87, 012107 (2013)

    2013

  59. [59]

    Vaneph, T

    C. Vaneph, T. Tufarelli, and M. G. Genoni, Quantum estimation of a two-phase spin rotation, Quantum Meas. Quantum Metrol.1, 12–20 (2013)

    2013

  60. [60]

    See Supplemental Material for the theoretical details of the universal privacy condition and the corresponding experimental details

  61. [61]

    Bateet al., Experimental distributed quantum sensing in a noisy environment, Phys

    J. Bateet al., Experimental distributed quantum sensing in a noisy environment, Phys. Rev. Lett.135, 220801 (2025)

    2025

  62. [62]

    Zhanget al., Distributed multi-parameter quantum metrology with a superconducting quantum network, arXiv:2412.18398 (2025)

    J. Zhanget al., Distributed multi-parameter quantum metrology with a superconducting quantum network, arXiv:2412.18398 (2025)

    arXiv 2025

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.