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arXiv:2607.11302

Information geometric quantification of effective privacy in quantum metrology

Luca Bianchi, Shimpei Yamaguchi, Wojciech Roga, Davide Bacco, Masahiro Takeoka

quant-phSubmitted Jul 13, 2026• Announced Jul 14, 2026• 0 citations
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Abstract

Privacy of a quantum metrological protocol concerns the extent to which single parameters can be kept inaccessible to an observer or to other users of the network. In this work, an information geometric framework is developed to quantify privacy and accessibility of functions of parameters effectively, that is, up to a finite accuracy in state discrimination. Both quantities are defined by measuring volumes in the parameter space induced by the underlying quantum states. This construction subsumes previous definitions of privacy based on the degeneracy of quantum Fisher information, naturally encompassing imperfect implementations. Using extended-GHZ states as a representative example of a quantum network scenario, privacy and accessibility are characterized by quantum correlations and accuracy, providing scaling laws depending on imperfect measurements and entanglement.

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12 pages, 4 figures

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Privacy of a quantum metrological protocol concerns the extent to which single parameters can be kept inaccessible to an observer or to other users of the network. In this work, an information geometric framework is developed to quantify privacy and accessibility of functions of parameters effectively, that is, up to a finite accuracy in state discrimination. Both quantities are defined by measuring volumes in the parameter space induced by the underlying quantum states. This construction subsumes previous definitions of privacy based on the degeneracy of quantum Fisher information, naturally encompassing imperfect implementations. Using extended-GHZ states as a representative example of a quantum network scenario, privacy and accessibility are characterized by quantum correlations and accuracy, providing scaling laws depending on imperfect measurements and entanglement.
Networks of sensors are a promising scheme to deliver the benefits of quantum technologies in coming years, offering enhanced precision and accuracy for distributed metrology through the use of large entangled states. Recent work has additionally explored the privacy of these schemes, meaning that local parameters can be kept secret while a joint function of these is estimated by the network. In this work, we use the abstract cryptography framework to relate the two proposed definitions of quasi-privacy, showing that both are composable, which enables the protocol to be securely included as a sub-routine to other schemes. We give an explicit example that estimating the mean of a set of parameters using GHZ states is composably fully secure.
Quantum sensor networks (QSN) typically encode N unknown parameters while targeting a single linear combination, rendering the N-1 remaining parameters as nuisance directions. To rigorously quantify estimation precision under such nuisances, we use the effective quantum Fisher information (EQFI) and establish a ``barrel-effect'' bottleneck: the global EQFI cannot exceed the weakest weighted local sensing capacity. To elucidate the information allocation mechanism underlying this bottleneck, we derive an exact local--global phase map that delineates how the trade-off between local and global EQFI depends dynamically on quantum correlations, and accordingly we identify concrete conditions for saturating the bottleneck bound. Notably, this geometric map uncovers a counterintuitive ``overcorrelated'' regime where excessive correlations actively degrade both local and global performance. Finally, we apply the phase map to intrinsic local privacy and identify the condition under which every local parameter is inaccessible while the desired global combination remains estimable. Overall, our work provides a principled methodology for engineering optimal network states in quantum sensing architectures.
Networks of sensors are a promising scheme to deliver the benefits of quantum technologies in coming years, offering enhanced precision and accuracy for distributed metrology through the use of large entangled states. Recent work has additionally explored the privacy of these schemes, meaning that local parameters can be kept secret while a joint function of these is estimated by the network. In this work, we use the abstract cryptography framework to relate the two proposed definitions of quasi-privacy, showing that both are composable, which enables the protocol to be securely included as a sub-routine to other schemes. We give an explicit example that estimating the mean of a set of parameters using GHZ states is composably fully secure.
The Quantum Fisher Information (QFI) metric governs a fundamental duality: it quantifies both how precisely a parameter can be estimated (metrology) and how distinguishable two quantum states are (privacy). We exploit this duality to establish a geometry-aware framework for quantum differential privacy (DP) that replaces isotropic depolarizing noise with direction-dependent noise aligned to the QFI eigenstructure of the quantum embedding. We prove six principal theorems: (1) the minimax-optimal mechanism concentrates the noise budget in the dominant QFI eigenmode, achieving ε=(Δ2/2)λmax⁡(1−cγ)\varepsilon = (Δ^2/2)λ_{\max}(1-cγ)ε=(Δ2/2)λmax​(1−cγ) with O(d/λmax⁡)O(d/λ_{\max})O(d/λmax​) advantage; (2) mixed-state QFI decomposition reveals that dephasing in the adversary's basis increases\textit{increases}increases accessible information, while misaligned-basis dephasing provides constructive privacy amplification from hardware noise; (3) a tight privacy −-− utility uncertainty relation ε⋅(1−F)≥Δ22Tr⁡(F)d\varepsilon \cdot (1 - F) \ge \frac{Δ^2}{2}\frac{\operatorname{Tr}(F)}{d}ε⋅(1−F)≥2Δ2​dTr(F)​; (4) adaptive QFI estimation converging at O(1/n)O(1/\sqrt{n})O(1/n​) yields 1.92×1.92\times1.92× tighter bounds; (5) QFI-aligned composition saturates at O(1)O(1)O(1) versus O(k)O(k)O(k) for standard composition; and (6) hardware noise can be harnessed for privacy amplification. Adversarial vulnerabilities, Wasserstein guarantees, subspace projection, and a zero-knowledge audit protocol follow as corollaries. Results are validated on Qiskit Aer GPU simulations, IBM Quantum hardware (ibm_fez, 156 qubits), and against classical DP baselines, achieving equivalent utility at ε≈0.001\varepsilon \approx 0.001ε≈0.001 versus ε≈4800\varepsilon \approx 4800ε≈4800 for classical DP.
We treat privacy in a network of quantum sensors where accessible information is limited to specific functions of the network parameters, and all other information remains private. We develop an analysis of privacy in terms of a manipulation of the quantum Fisher information matrix, and find the optimal state achieving maximum privacy in the estimation of linear combination of the unknown parameters in a network of quantum sensors. We also discuss the effect of uncorrelated noise on the privacy of the network. Moreover, we illustrate our results with an example where the goal is to estimate the average value of the unknown parameters in the network. In this example, we also introduce the notion of quasiprivacy (ε privacy), quantifying how close the state is to being private.
The distribution of quantum sensors on quantum networks is a key enabler of quantum technologies in interferometry, gravimetry, timekeeping, biological monitoring, and beyond. Yet, guaranteeing the security of these distributed sensors over noisy, insecure networks remains a formidable challenge. Previous efforts to combine quantum metrology and cryptography have encountered an apparently unavoidable tension, proposing bounds for security which are only loosely tied to the achievable measurement performance. Here we introduce a quantum remote sensing protocol that can rigorously certify privacy and integrity of the estimation. By employing offline bilateral Pauli-twirling, our approach forces the effective quantum channel into a Bell-diagonal form, independently of the attack. Surprisingly, this also preserves metrological sensitivity without introducing additional experimental overhead. Relying solely on public communication alongside an insecure quantum link, the protocol enables legitimate users to exactly quantify their estimation error relative to an eavesdropper controlling the channels. We experimentally demonstrate this framework by estimating an optical phase using entangled photons, observing that the users' precision consistently surpasses the eavesdropper's capabilities across a broad parameter regime. By unifying quantum cryptography and metrology, our results provide a practical pathway to achieve simultaneous quantum-limited precision and rigorous information security in real-world quantum networks.