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

TL;DR

This paper introduces a quantum Fisher information duality in distributed quantum sensors, linking optimal precision with privacy constraints, and characterizes states that achieve this duality.

Contribution

It establishes a duality relation for quantum Fisher information in distributed sensors and characterizes states that optimize precision while maintaining privacy.

Findings

Quantum Fisher information duality relation for distributed sensors.

Heisenberg-limited precision implies zero information for other directions.

Characterization of states (e.g., GHZ states) that saturate the bounds.

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, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta}) + F_Q(\boldsymbol{v}^\top \boldsymbol{\theta}) \leq N$ 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 $\boldsymbol{w}$, $F_Q(\boldsymbol{w}^\top \boldsymbol{\theta})=N$, 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.