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

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, $F_Q({\mathit{w}}^{\mathrm{\top }}\mathit{\theta })+F_Q({\mathit{v}}^{\mathrm{\top }}\mathit{\theta })\le N$ for all unit orthogonal sensing directions $\mathit{w}$ and $\mathit{v}$, with equality for all equatorial states when $N=2$ and for Greenberger–Horne–Zeilinger (GHZ) states when $N\ge 2$. Heisenberg-limited precision for direction $\mathit{w}$, $F_Q({\mathit{w}}^{\mathrm{\top }}\mathit{\theta })=N$, 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.