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

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, $F_Q({\mathbf{w}}^{\top }\hat{\mathbf{I}}¸)+F_Q({\mathbf{v}}^{\top }\hat{\mathbf{I}}¸)\le N$ for all unit orthogonal sensing directions $\mathbf{w}$ and $\mathbf{v}$, 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 $\mathbf{w}$, $F_Q({\mathbf{w}}^{\top }\hat{\mathbf{I}}¸)=N$, 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.