Universal Operational Privacy in Distributed Quantum Sensing

Technical Summary: Universal Operational Privacy in Distributed Quantum Sensing

Problem Statement
Distributed quantum sensing networks enable multiple clients to estimate spatially distributed parameters using shared quantum resources. A critical, yet under-explored, requirement in these networks is privacy: clients must estimate global functions of parameters without revealing individual local parameter values to untrusted servers or eavesdroppers. Existing privacy conditions rely on the rank deficiency of the Quantum Fisher Information Matrix (QFIM), specifically assuming idealized rank-1 structures (e.g., Greenberger–Horne–Zeilinger states). However, these conditions face a fundamental mismatch with realistic experimental scenarios. In practice, optimal measurements required to reach the QFIM are often infeasible due to experimental constraints or incompatible Hamiltonians. Consequently, the information actually accessible to untrusted parties is determined by the Classical Fisher Information Matrix (CFIM), which may be singular with a rank greater than one. Current frameworks fail to provide a universal, operational privacy criterion applicable to these realistic, experimentally accessible CFIMs.

Methodology
The authors propose a universal operational privacy framework formulated directly in terms of the experimentally accessible CFIM, denoted as $F$. The core of this methodology is the introduction of a privacy quantifier, $P_F(w)$, defined for a weight vector $w$ representing the global linear function $f(\phi) = w^T\phi$ to be estimated:

$$P_F(w) = 1 - \min_{v \perp w, \|v\|^2=1} v^T \Pi_F v$$

where $\Pi_F$ is the projector onto the support space of the CFIM (spanned by eigenvectors with non-zero eigenvalues).

• Geometric Interpretation: If the CFIM is non-singular, $\Pi_F$ is the identity, yielding $P_F(w) = 0$ (no privacy). If the CFIM is singular, the support space is a strict subspace of the parameter space. Privacy emerges ($P_F(w) > 0$) when directions orthogonal to the estimation vector $w$ lie within the kernel of the CFIM, rendering individual parameters inaccessible to server measurements.
• Properties: The quantifier is basis-independent, continuous under small experimental imperfections, and robust against parameter-independent noise. It satisfies a continuity relation where deviations from the ideal CFIM result in second-order changes to the privacy quantifier.

Experimental Implementation and Results
To validate the framework, the authors implemented a distributed quantum sensing network using entangled photon pairs.

• Setup: A polarization-entangled Bell state was generated in a Sagnac interferometer and distributed via a beam splitter network to prepare a two-photon entangled state $|\Psi^2_4\rangle$ shared among four spatial modes.
• Protocol: The state was sent through 3-km optical fiber links to four servers. Each server applied local phase encoding ($\phi_1, \phi_2, \phi_3, \phi_4$) and performed projective measurements in the $\hat{\sigma}_x$ basis.
• Outcome: The experiment measured two-photon coincidence probabilities, which were found to depend only on the sum of phase pairs ($\phi_\mu + \phi_\nu$), preventing the inference of individual phases.
• Performance:
  • The experimentally reconstructed CFIM was singular, with a kernel spanned by the vector $[1, -1, 1, -1]$.
  • For the global weight vector $w = (1, 1, 1, 1)/4$, the privacy quantifier achieved $P_F(w) = 1$, indicating perfect privacy.
  • Simultaneously, the protocol achieved Heisenberg-limited precision ($1/N^2 = 1/4$) using only two photons to estimate four parameters.
  • The experimental CFIM closely matched the theoretical prediction, with an average visibility of $V = 0.968 \pm 0.003$.

Key Contributions

1. Universal Operational Framework: The paper introduces a privacy condition based on the CFIM rather than the QFIM, making it applicable to arbitrary protocols with singular information structures of rank greater than one.
2. Protocol-Independent Criterion: The proposed condition ensures that no information about individual parameters is accessible to untrusted parties, regardless of the specific estimation strategy or measurement configuration.
3. Experimental Demonstration: The work provides the first experimental demonstration of a distributed sensing protocol that simultaneously satisfies a universal privacy condition and achieves Heisenberg-limited precision using fewer photons than the number of estimated parameters.

Significance and Claims
The authors claim that their results establish universal operational constraints governing privacy in distributed quantum sensing networks. By shifting the focus from idealized quantum bounds to experimentally accessible quantities, the framework provides a foundation for practical, privacy-preserving quantum sensing beyond full-rank regimes. The work unifies distributed sensing, multiparameter estimation with singular CFIMs, and experimental privacy verification, offering a general basis for scalable quantum sensing networks across various platforms, including photonic systems, trapped ions, and superconducting circuits. The paper emphasizes that privacy and quantum-enhanced precision can be simultaneously realized in practical networks, addressing a fundamental gap between theoretical privacy conditions and operational reality.