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\(boldsymbol{w}^→p boldsymbolθ\) + F_Q\(boldsymbol{v}^→p boldsymbolθ\) 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 Ngeq 2. Heisenberg-limited precision for direction boldsymbol{w}, F_Q\(boldsymbol{w}^→p boldsymbolθ\)=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.