🤖 AI Summary
This work addresses the inherent tension between estimation precision and parameter privacy in distributed quantum sensing, where achieving Heisenberg-limited accuracy along a target direction may inadvertently leak information about orthogonal, irrelevant parameters. The authors establish a dual inequality for the quantum Fisher information in orthogonal directions, revealing a fundamental trade-off: for any $N$-qubit probe state, $F_Q(\mathbf{w}^\top\boldsymbol{\theta}) + F_Q(\mathbf{v}^\top\boldsymbol{\theta}) \leq N$, with equality attained by the GHZ state for $N \geq 2$. This result demonstrates that Heisenberg-limited sensitivity in a desired direction can be achieved while completely suppressing information leakage in any orthogonal direction, thereby enabling intrinsically private sensing. The analysis leverages many-body quantum state characterization, a local phase encoding model, and information-theoretic bounds under directional orthogonality constraints.
📝 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, $F_Q(\boldsymbol{w}^\top \boldsymbolθ) + F_Q(\boldsymbol{v}^\top \boldsymbolθ) \leq N$ 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.