Precision and Privacy in Distributed Quantum Sensing: A Quantum Fisher Information Duality

Farhad Farokhi

arxiv: 2605.20765 · v1 · pith:6YZWUCBTnew · submitted 2026-05-20 · 🪐 quant-ph · cs.CR· cs.IT· math.IT

Precision and Privacy in Distributed Quantum Sensing: A Quantum Fisher Information Duality

Farhad Farokhi This is my paper

Pith reviewed 2026-05-21 05:16 UTC · model grok-4.3

classification 🪐 quant-ph cs.CRcs.ITmath.IT

keywords quantum Fisher informationdistributed quantum sensingparameter privacyHeisenberg limitGHZ statesquantum metrologyQFI dualitymulti-parameter estimation

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The pith

QFI for any two orthogonal directions on an N-qubit probe sums to at most N, so Heisenberg precision in one forces zero precision in all others.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a quantum Fisher information duality for distributed sensing with N qubits under local phase encoding. For any probe state and any pair of orthogonal unit vectors w and v, the sum F_Q(w^T θ) + F_Q(v^T θ) is bounded above by N. The bound is achieved with equality for equatorial states when N=2 and for GHZ states when N is larger. Reaching the Heisenberg limit of exactly N for one direction therefore drives the QFI to zero for every independent orthogonal direction. The authors read this saturation as the mathematical condition for parameter privacy: accurate sensing of a chosen target makes accurate estimation of all other independent parameters impossible with the same probe.

Core claim

For any N-qubit probe state the quantum Fisher information obeys F_Q(w^T θ) + F_Q(v^T θ) ≤ N for every pair of orthogonal unit vectors w and v. Equality is attained for all equatorial states at N=2 and for GHZ states at N≥2. Consequently, the Heisenberg-limited value F_Q(w^T θ)=N for one direction saturates the bound and simultaneously forces F_Q=0 for all other independent directions. This saturation is interpreted as the condition for parameter privacy in the distributed sensor network.

What carries the argument

The QFI duality bound F_Q(w^T θ) + F_Q(v^T θ) ≤ N for orthogonal unit sensing directions, which trades off metrological precision between independent linear combinations of the parameters.

Load-bearing premise

The derivation assumes an N-qubit probe state undergoing local phase encoding and restricts the sensing directions to pairwise orthogonal unit vectors.

What would settle it

Direct calculation of the two QFI values for a GHZ state of N=3 qubits along any two orthogonal directions, checking whether their sum equals exactly 3.

Figures

Figures reproduced from arXiv: 2605.20765 by Farhad Farokhi.

read the original abstract

We establish a quantum Fisher information (QFI) duality for distributed quantum sensor networks with local phase encoding. For any $N$-qubit probe state, where $N$ denotes the number of sensors, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta}) + F_Q(\boldsymbol{v}^\top \boldsymbol{\theta}) \leq N$ for all unit orthogonal sensing directions $\boldsymbol{w}$ and $\boldsymbol{v}$, with equality for all equatorial states when $N=2$ and for Greenberger--Horne--Zeilinger (GHZ) states when $N\geq 2$. Heisenberg-limited precision for direction $\boldsymbol{w}$, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta})=N$, saturates the bound and simultaneously forces zero QFI for all other independent directions. This can be interpreted as the condition for parameter privacy in distributed quantum sensing: attaining Heisenberg-limited precision for the sensing target renders all alternative privacy-intrusive estimations impossible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean, state-independent QFI bound showing that Heisenberg-limited sensing in one direction on N qubits forces zero information in any orthogonal direction.

read the letter

The main point is a duality bound: for any N-qubit probe and any two orthogonal unit vectors w and v, F_Q(w^T θ) + F_Q(v^T θ) ≤ N. Equality holds for equatorial states at N=2 and for GHZ states at any N. Reaching the Heisenberg limit in one direction therefore sets the QFI to zero in the orthogonal one, which the authors read as automatic parameter privacy in a distributed sensor network. The bound follows from the operator relation on the effective generators A = w·G and B = v·G, where the sum of squares is controlled by the rank-2 projector from the orthogonal directions. This is state-independent and saturates on GHZ states when the plane includes the all-ones vector. That part is straightforward and checks out from the standard properties of qubit phase encoding. What is actually new is the explicit pairing of the inequality with the equality cases plus the privacy interpretation for sensing networks; it is not just a restatement of earlier QFI bounds. The paper keeps the assumptions explicit and avoids overclaiming. The soft spots are modest. The privacy reading is a direct consequence rather than an independent mechanism, so its novelty sits mostly in the framing. The result is limited to orthogonal directions and local encoding, which is fine but means it does not automatically extend to correlated noise or non-orthogonal parameters that appear in some real networks. No load-bearing gaps or circularity show up in the central argument. This is for people working on multi-parameter quantum metrology who care about precision-privacy trade-offs. A reader who wants a simple, usable bound to plug into protocol design will find it worth their time. It shows clear engagement with the literature and the math holds up, so it deserves a serious referee. I would send it for peer review.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes a quantum Fisher information duality for distributed quantum sensing with local phase encoding on N-qubit probe states. It proves that for any such state and any pair of orthogonal unit vectors w and v, F_Q(w^T θ) + F_Q(v^T θ) ≤ N, with equality for equatorial states at N=2 and for GHZ states at N≥2. Achieving the Heisenberg limit F_Q(w^T θ)=N in one direction is shown to force zero QFI in all orthogonal directions, which the authors interpret as a condition for parameter privacy in sensor networks.

Significance. If the central bound holds, the result provides a state-independent fundamental limit on the trade-off between precision in orthogonal sensing directions. This has clear implications for multi-parameter quantum metrology and for privacy-preserving protocols in distributed quantum sensing, where entanglement (e.g., via GHZ states) can be used to saturate the bound and enforce privacy. The operator-inequality origin of the bound and its saturation conditions are strengths that offer concrete, falsifiable predictions for experiments.

major comments (1)

The section deriving the main bound (around the operator inequality A² + B² ≤ (N/4)I for generators A = w·G and B = v·G): while the rank-2 projector argument on M = wwᵀ + vvᵀ correctly yields the state-independent bound via σᵀMσ ≤ N, the manuscript should explicitly verify that this extends to the quantum Fisher information for mixed states, where the SLD-based definition of F_Q may require additional steps beyond the pure-state variance bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment, and recommendation for minor revision. We address the major comment below.

read point-by-point responses

Referee: The section deriving the main bound (around the operator inequality A² + B² ≤ (N/4)I for generators A = w·G and B = v·G): while the rank-2 projector argument on M = wwᵀ + vvᵀ correctly yields the state-independent bound via σᵀMσ ≤ N, the manuscript should explicitly verify that this extends to the quantum Fisher information for mixed states, where the SLD-based definition of F_Q may require additional steps beyond the pure-state variance bound.

Authors: We thank the referee for this helpful suggestion. The operator inequality A² + B² ≤ (N/4)I is state-independent. For any state ρ (pure or mixed), the QFI satisfies F_Q(ρ, H) ≤ 4 Var_ρ(H) for generator H. Thus F_Q(wᵀθ) + F_Q(vᵀθ) ≤ 4[Var(w·G) + Var(v·G)] ≤ N follows directly from the variance bound obtained via the rank-2 projector on M. We will add an explicit paragraph clarifying this extension to mixed states via the general QFI-variance inequality in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The claimed QFI duality bound follows directly from the standard definition of the quantum Fisher information for pure qubit states (F_Q = 4 Var(A) where A = w · G) combined with the algebraic operator inequality A² + B² ≤ (N/4) I. This inequality is obtained because M = w wᵀ + v vᵀ is a rank-2 projector, so for any eigenvalue vector σ with components ±1/2 the quadratic form σᵀ M σ ≤ N holds independently of the probe state. Equality saturation for GHZ states (when the plane includes the all-ones vector) and equatorial states (N=2) is a direct consequence of this state-independent bound, without any fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation therefore reduces to elementary properties of Pauli operators and QFI variance and remains fully self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of quantum Fisher information for phase estimation and the modeling choice of local phase encoding on an N-qubit register; no free parameters or new entities are introduced.

axioms (2)

domain assumption Local phase encoding on each of the N qubits
The setup models each sensor experiencing an independent phase shift θ_i.
standard math Standard definition and properties of the quantum Fisher information
The bound is derived using the usual QFI as the figure of merit for estimation precision.

pith-pipeline@v0.9.0 · 5697 in / 1371 out tokens · 44370 ms · 2026-05-21T05:16:09.827012+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Theorem 1.For any two unit vectors ˆw⊥ ˆv: FQ( ˆw⊤θ) +F Q(ˆv⊤θ)≤N.(7) Equality holds with equatorial probe states forN= 2

Now, we are ready to present the main result of this letter. Theorem 1.For any two unit vectors ˆw⊥ ˆv: FQ( ˆw⊤θ) +F Q(ˆv⊤θ)≤N.(7) Equality holds with equatorial probe states forN= 2. Proof.We first show that Tr(F Q(θ))≤N. Settingi=j in (5) gives [FQ(θ)]ii =⟨(σ (i) z )2⟩ − ⟨(σ(i) z )⟩2 ≤ ⟨(σ (i) z )2⟩= ⟨I⟩= 1. Therefore, Tr(F Q(θ)) = PN i=1[FQ(θ)]ii ≤N. P...

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R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Multi-parameter estimation beyond quantum Fisher in- formation, J. Phys. A53, 363001 (2020)

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

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Farokhi, Maximal information leakage from quantum encoding of classical data, Phys

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Farokhi, Barycentric and pairwise R´ enyi quantum leakage with application to privacy-utility trade-off, Proc

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C. Hirche, C. Rouz´ e, and D. Stilck Fran¸ ca, Quantum differential privacy: an information theory perspective, IEEE Trans. Inf. Theory69, 5771 (2023)

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Cheng, N

H.-C. Cheng, N. Datta, N. Liu, T. Nuradha, R. Salz- mann, and M. M. Wilde, An invitation to the sample complexity of quantum hypothesis testing, npj Quantum Information11, 94 (2025)

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[1] [1]

Theorem 1.For any two unit vectors ˆw⊥ ˆv: FQ( ˆw⊤θ) +F Q(ˆv⊤θ)≤N.(7) Equality holds with equatorial probe states forN= 2

Now, we are ready to present the main result of this letter. Theorem 1.For any two unit vectors ˆw⊥ ˆv: FQ( ˆw⊤θ) +F Q(ˆv⊤θ)≤N.(7) Equality holds with equatorial probe states forN= 2. Proof.We first show that Tr(F Q(θ))≤N. Settingi=j in (5) gives [FQ(θ)]ii =⟨(σ (i) z )2⟩ − ⟨(σ(i) z )⟩2 ≤ ⟨(σ (i) z )2⟩= ⟨I⟩= 1. Therefore, Tr(F Q(θ)) = PN i=1[FQ(θ)]ii ≤N. P...

work page

[2] [2]

Every equatorial two-qubit state lies exactly on the conservation-law lineF Q( ˆw⊤θ) +F Q(ˆv⊤θ) = 2 (The- orem 1,N= 2 equality); none lies strictly inside it. Three states from the Bell-state family cos(ϕ)|Φ +⟩+ sin(ϕ)|Ψ +⟩are highlighted (arrow: direction of increasingϕ): the Bell state |Φ+⟩(red,ϕ= 0) at the co-optimal corner (2,0), where Heisenberg-limi...

work page

[3] [3]

Komar, E

P. Komar, E. M. Kessler, M. Bishof, L. Jiang, A. S. Sørensen, J. Ye, and M. D. Lukin, A quantum network of clocks, Nat. Phys.10, 582 (2014)

work page 2014

[4] [4]

J. F. Barry, M. J. Turner, J. M. Schloss, D. R. Glenn, Y. Song, M. D. Lukin, H. Park, and R. L. Walsworth, Optical magnetic detection of single-neuron action potentials using quantum defects in diamond, Proc. Natl. Acad. Sci.113, 14133 (2016)

work page 2016

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Stray, A

B. Stray, A. Lamb, A. Kaushik, J. Vovrosh, A. Rodgers, J. Winch, F. Hayati, D. Boddice, A. Stabrawa, A. Nigge- baum,et al., Quantum sensing for gravity cartography, Nature602, 590 (2022)

work page 2022

[6] [6]

C. L. Degen, F. Reinhard, and P. Cappellaro, Quantum sensing, Rev. Mod. Phys.89, 035002 (2017)

work page 2017

[7] [7]

Giovannetti, S

V. Giovannetti, S. Lloyd, and L. Maccone, Quantum- enhanced measurements: beating the standard quantum limit, Phys. Rev. Lett.96, 010401 (2006)

work page 2006

[8] [8]

Eldredge, M

Z. Eldredge, M. Foss-Feig, J. A. Gross, S. L. Rolston, and A. V. Gorshkov, Optimal and secure measurement protocols for quantum sensor networks, Phys. Rev. A97, 042337 (2018)

work page 2018

[9] [9]

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

work page 2018

[10] [10]

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

work page 2018

[11] [11]

W. N. Price and I. G. Cohen, Privacy in the age of med- ical big data, Nature Medicine25, 37 (2019)

work page 2019

[12] [12]

Farokhi and H

F. Farokhi and H. Sandberg, Ensuring privacy with con- strained additive noise by minimizing Fisher information, Automatica99, 275 (2019)

work page 2019

[13] [13]

I. Issa, A. B. Wagner, and S. Kamath, An operational ap- proach to information leakage, IEEE Trans. Inf. Theory 66, 1625 (2020)

work page 2020

[14] [14]

K. Wei, J. Li, M. Ding, C. Ma, H. H. Yang, F. Farokhi, S. Jin, T. Q. S. Quek, and H. V. Poor, Federated learn- ing with differential privacy: algorithms and performance analysis, IEEE Trans. Inf. Forensics Secur.15, 3454 (2020)

work page 2020

[15] [15]

S. Ragy, M. Jarzyna, and R. Demkowicz-Dobrza´ nski, Compatibility in multiparameter quantum metrology, Phys. Rev. A94, 052108 (2016)

work page 2016

[16] [16]

Demkowicz-Dobrza´ nski, W

R. Demkowicz-Dobrza´ nski, W. G´ orecki, and M. Gut ¸˘ a, Multi-parameter estimation beyond quantum Fisher in- formation, J. Phys. A53, 363001 (2020)

work page 2020

[17] [17]

Gessner, L

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

work page 2018

[18] [18]

Farokhi, Maximal information leakage from quantum encoding of classical data, Phys

F. Farokhi, Maximal information leakage from quantum encoding of classical data, Phys. Rev. A109, 022608 (2024)

work page 2024

[19] [19]

Farokhi, Barycentric and pairwise R´ enyi quantum leakage with application to privacy-utility trade-off, Proc

F. Farokhi, Barycentric and pairwise R´ enyi quantum leakage with application to privacy-utility trade-off, Proc. R. Soc. A480, 20240319 (2024)

work page 2024

[20] [20]

Hirche, C

C. Hirche, C. Rouz´ e, and D. Stilck Fran¸ ca, Quantum differential privacy: an information theory perspective, IEEE Trans. Inf. Theory69, 5771 (2023)

work page 2023

[21] [21]

Farokhi, Sample complexity bounds for scalar param- eter estimation under quantum differential privacy, IEEE Control Syst

F. Farokhi, Sample complexity bounds for scalar param- eter estimation under quantum differential privacy, IEEE Control Syst. Lett.9, 240 (2025)

work page 2025

[22] [22]

Cheng, N

H.-C. Cheng, N. Datta, N. Liu, T. Nuradha, R. Salz- mann, and M. M. Wilde, An invitation to the sample complexity of quantum hypothesis testing, npj Quantum Information11, 94 (2025)

work page 2025

[23] [23]

J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum Fisher information matrix and multiparameter estima- tion, J. Phys. A53, 023001 (2020)

work page 2020

[24] [24]

C. W. Helstrom, Minimum mean-squared error of esti- mates in quantum statistics, Physics Letters A25, 101 (1967)

work page 1967

[25] [25]

Kifer and A

D. Kifer and A. Machanavajjhala, No free lunch in data privacy, inProceedings of the 2011 ACM SIGMOD Inter- national Conference on Management of Data(2011) pp. 193–204

work page 2011