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.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.

quant-pharXiv:2605.20765May 21, 20260 citations

Achieving Heisenberg limit in the phase measurement through three-qubit graph states

Subhasish Bag, Sarkar, Ramita, Prasanta Kumar Panigrahi

We study the reciprocal of the mean quantum Fisher information (RMQFI), $χ^2$ for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterized by $χ^2 <1$ . We demonstrate that the most symmetric graph state and the GHZ state have the lowest RMQFI values leading to the highest statistical speed showing that both these states attain the Heisenberg limit in phase sensitivity. Unlike the GHZ state, graph states have the same RMQFI values for measurement through different parameters, a property shared by the hypergraph states. Three qubit graph and hypergraph states can violate Bell's inequality as $F_Q > N$ . Both the GHZ state and the most symmetric graph state have the highest concurrence equalling 3 and the maximum QFI values.We study the reciprocal of the mean quantum Fisher information (RMQFI), $χ^2$ for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterized by $χ^2 <1$. We demonstrate that the most symmetric graph state and the GHZ state have the lowest RMQFI values leading to the highest statistical speed showing that both these states attain the Heisenberg limit in phase sensitivity. Unlike the GHZ state, graph states have the same RMQFI values for measurement through different parameters, a property shared by the hypergraph states. Three qubit graph and hypergraph states can violate Bell's inequality as $F_Q > N$. Both the GHZ state and the most symmetric graph state have the highest concurrence equalling 3 and the maximum QFI values.

arXiv:2208.07772Aug 16, 20220 citations

Preserving Heisenberg-Limited Metrological Information during Storage via Correlated-Noise Correction

Hang Xu, Xue-Ke Song, Jingzheng Huang +2 more

Quantum error correction has become an indispensable tool for restoring Heisenberg-limited precision in noisy quantum metrology. Existing protocols, however, almost exclusively focus on correcting noise during the signal-encoding stage and implicitly assume that the probe is measured immediately after sensing. In many quantum information processing tasks, the encoded probe must instead be stored before subsequent quantum operations, during which environmental noise can significantly degrade the accumulated metrological information. Here, we propose a correlated-noise correction (CNC) protocol for protecting quantum probes during the storage stage. By correlating probe errors with auxiliary qubits through fixed two-body entangling gates, memory errors are converted into measurable syndromes that are extracted only once after storage. We show that the protocol naturally extends from single-qubit to multi-qubit probes and protects the stored quantum Fisher information against dephasing, bit-flip, and amplitude-damping noise. Furthermore, we demonstrate that preserving the quantum Fisher information does not necessarily require restoring the entire quantum state when the probe is measured immediately after storage, whereas full state recovery becomes essential for subsequent rounds of quantum signal processing. Our results establish correlated-noise correction as a practical framework for protecting metrological information during quantum memory and provide a useful building block for sensing-enabled quantum information processing.Quantum error correction has become an indispensable tool for restoring Heisenberg-limited precision in noisy quantum metrology. Existing protocols, however, almost exclusively focus on correcting noise during the signal-encoding stage and implicitly assume that the probe is measured immediately after sensing. In many quantum information processing tasks, the encoded probe must instead be stored before subsequent quantum operations, during which environmental noise can significantly degrade the accumulated metrological information. Here, we propose a correlated-noise correction (CNC) protocol for protecting quantum probes during the storage stage. By correlating probe errors with auxiliary qubits through fixed two-body entangling gates, memory errors are converted into measurable syndromes that are extracted only once after storage. We show that the protocol naturally extends from single-qubit to multi-qubit probes and protects the stored quantum Fisher information against dephasing, bit-flip, and amplitude-damping noise. Furthermore, we demonstrate that preserving the quantum Fisher information does not necessarily require restoring the entire quantum state when the probe is measured immediately after storage, whereas full state recovery becomes essential for subsequent rounds of quantum signal processing. Our results establish correlated-noise correction as a practical framework for protecting metrological information during quantum memory and provide a useful building block for sensing-enabled quantum information processing.

quant-pharXiv:2608.08130Aug 11, 20260 citations

Quantum Fisher information for a single qubit system

S. Abdel‐Khalek, K. Berrada, A.‐S. F. Obada

The European Physical Journal DMar 1, 201236 citations

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.

We study the reciprocal of the mean quantum Fisher information (RMQFI),

χ^2

for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterized by

χ^2 <1

. We demonstrate that the most symmetric graph state and the GHZ state have the lowest RMQFI values leading to the highest statistical speed showing that both these states attain the Heisenberg limit in phase sensitivity. Unlike the GHZ state, graph states have the same RMQFI values for measurement through different parameters, a property shared by the hypergraph states. Three qubit graph and hypergraph states can violate Bell's inequality as

F_Q > N

. Both the GHZ state and the most symmetric graph state have the highest concurrence equalling 3 and the maximum QFI values.