Quantum network sensing shows potential to enhance the estimation precision for functions of spatially distributed parameters beyond the shot noise limit. The key resource required for this task is possibly multi-partite quantum entanglement. Such protocols can also provide privacy, preventing sensitive information from leaking to unauthorized parties. The photonic entanglement is the most natural for this task; however, distributing it over long distances presents significant difficulties, mainly because of unavoidable loss in communication channels. The resource efficiency is also fundamental, both for precise network sensing and for private sensing In this research, we analyze a quantum network sensing protocol based on a recently proposed, efficient GHZ state distribution scheme. In comparison to conventional methods based on entanglement distribution, our protocol shows the decreasing loss-induced estimation error of certain functions of distributed parameters including their arbitrary linear combinations. Moreover, we consider a scenario in which one person can estimate linear combinations of distributed parameters without violating privacy of the other users. We show that an additional parameter can be used to hide the information about the target parameter from everyone, except the person who controls the parameter.
Quantum network sensing shows potential to enhance the estimation precision for functions of spatially distributed parameters beyond the shot noise limit. The key resource required for this task is possibly multi-partite quantum entanglement. Such protocols can also provide privacy, preventing sensitive information from leaking to unauthorized parties. The photonic entanglement is the most natural for this task; however, distributing it over long distances presents significant difficulties, mainly because of unavoidable loss in communication channels. The resource efficiency is also fundamental, both for precise network sensing and for private sensing In this research, we analyze a quantum network sensing protocol based on a recently proposed, efficient GHZ state distribution scheme. In comparison to conventional methods based on entanglement distribution, our protocol shows the decreasing loss-induced estimation error of certain functions of distributed parameters including their arbitrary linear combinations. Moreover, we consider a scenario in which one person can estimate linear combinations of distributed parameters without violating privacy of the other users. We show that an additional parameter can be used to hide the information about the target parameter from everyone, except the person who controls the parameter.
Distributed quantum sensing enables the estimation of multiple parameters encoded in spatially separated probes. While traditional quantum sensing is often focused on estimating a single parameter with maximum precision, distributed quantum sensing seeks to estimate some function of multiple parameters that are only locally accessible for each party involved. In such settings, it is natural to not want to give away more information than is necessary. To address this, we use the concept of privacy with respect to a function, ensuring that only information about the target function is available to all the parties, and no other information. We define a measure of privacy (essentially how close we are to this condition being satisfied) and show it satisfies a set of naturally desirable properties of such a measure. Using this privacy measure, we identify and construct entangled resource states that ensure privacy for a given function under different resource distributions and encoding dynamics, characterized by Hamiltonian evolution. For separable and parallel Hamiltonians, we prove that the GHZ state is the only private state for certain linear functions, with the minimum amount of required resources, up to SLOCC. Recognizing the vulnerability of this state to particle loss, we create families of private states, that remain robust even against loss of qubits, by incorporating additional resources. We then extend our findings to different resource distribution scenarios and Hamiltonians, resulting in a comprehensive set of private and robust states for distributed quantum estimation. These results advance the understanding of privacy and robustness in multi-parameter quantum sensing.
Entangling distant quantum sensors is a key application of future quantum networks, allowing for the estimation of global functions of local parameters, with precision that is not possible with stand-alone, individual, quantum sensors. However, with this advantage comes the risk of information leakage over the network via possible malicious parties. It is often particularly important that local parameters remain unknown and that only the global function is shared across the network. Recently, the notion of privacy has been introduced in this context, which ensures that only the agreed function of parameters is shared over the network, even when malicious parties control the network itself, whilst maintaining the estimation advantage. In this work, we introduce a noise-robust protocol for private quantum networked sensing, which we realise using a high-fidelity Greenberger-Horne-Zeilinger (GHZ) state source. We further run a comparative analysis and simulate attacks using three different quantum states, a four-qubit GHZ state, two Bell pairs, and a fully separable state. Our results highlight the clear advantage of GHZ states in maintaining high precision, accuracy, and privacy for the distributed estimation task.
Quantum sensing is a promising technology capable of demonstrating clear advantage over comparable classical techniques for precise measurement. One application of quantum sensing is in function estimation, which can be done using a network of entangled quantum sensors, allowing for measurements with greater optimal sensitivity than unentangled sensing protocols. In cases where quantum sensor networks will be used to measure data that should remain private (e.g., biomedical data), it is imperative that these protocols include a privacy mechanism to hide sensitive information. In this work, we show that entangled sensor networks are vulnerable to certain privacy-violating attacks. To mitigate these attacks, we introduce secure sensing protocols endowed with differential privacy. We reconcile differential privacy with retaining Heisenberg-limited scaling, and introduce several protocols achieving varying balances between the two. We show that our main protocol, an n-node network sensing protocol that injects noise directly into the sensing Hamiltonian, exhibits a tradeoff between the desirable O(1/n2) Heisenberg scaling of the mean-squared error of the function estimate and the level of privacy attainable. Under assumptions on the network (a common source of randomness and a constant fraction of honest parties), we show that this protocol is locally implementable and achieves (O(1),δ)-differential privacy for arbitrarily small δ while retaining Heisenberg scaling of the mean-squared error. We prove that our protocols are resilient to attacks by broad classes of classical and quantum adversaries, and find advantages in the privacy-utility tradeoff when using quantum techniques.
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.
Abstract Quantum networks can enhance both security and privacy conditions for multi-user communication, delegated computation, and distributed sensing tasks. In distributed quantum sensing, it is often desirable to extract only global information from a network of sensors while keeping individual sensor values hidden. Private parameter estimation (PPE), a recently proposed protocol, formalises this requirement by defining a notion of privacy that bounds information leakage about local parameters while allowing estimation of a global function. Here, we present a proof-of-principle implementation of PPE using a three-photon Greenberger–Horne–Zeilinger state distributed across three sensors and verified via stabiliser measurements. We demonstrate Heisenberg-limited scaling for the global parameter while suppressing local metrological information by up to three orders of magnitude, establishing both the privacy and precision performance of the protocol. This work, which integrates privacy in distributed quantum sensing, marks a crucial step towards developing advanced quantum-secure-and-private protocols in complex quantum networks.
Private distributed quantum sensing aims to estimate an authorized collective parameter while preventing independent estimation of individual local parameters. Here, we analytically characterize Gaussian quantum networks satisfying this perfect local privacy condition. Using a graph representation of the Gaussian pairing matrix, we show that two-mode squeezed vacuum states constitute the essential building blocks of privacy-preserving Gaussian states. We then analytically derive the optimal sensitivity under perfect local privacy and determine a Gaussian probe that achieves Heisenberg scaling with respect to both the photon number and the number of sensing modes. Using local homodyne measurements and maximum-likelihood estimation, we numerically verify quantum-enhanced sensitivity beyond the shot-noise limit while preserving perfect local privacy. We further show that the local-privacy condition is preserved under arbitrary phase-independent quantum channels, including optical loss. Our results provide a general framework for constructing and optimizing privacy-preserving continuous-variable quantum sensing networks.