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, $F_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq N$ for all unit orthogonal sensing directions $\boldsymbol{w}$ and $\boldsymbol{v}$, with equality for all equatorial states when $N=2$ and for Greenberger--Horne--Zeilinger (GHZ) states when $N\geq 2$. Heisenberg-limited precision for direction $\boldsymbol{w}$, $F_Q(\boldsymbol{w}^\top \boldsymbolθ)=N$, 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.

翻译：我们建立了具有局部相位编码的分布式量子传感器网络的量子Fisher信息（QFI）对偶性。对于任意$N$量子比特探测态（其中$N$为传感器数量），对于所有单位正交传感方向$\boldsymbol{w}$和$\boldsymbol{v}$，有$F_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq N$，当$N=2$时所有赤道态取等号，当$N\geq 2$时格林伯格-霍恩-泽林格（GHZ）态取等号。方向$\boldsymbol{w}$的海森堡极限精度$F_Q(\boldsymbol{w}^\top \boldsymbolθ)=N$饱和该界限，并同时迫使所有其他独立方向的QFI为零。这可被解释为分布式量子传感中参数隐私的条件：对传感目标达到海森堡极限精度，使得所有替代性隐私侵犯估计成为不可能。