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

TL;DR

This paper investigates three-qubit graph and hypergraph states to identify conditions for achieving the Heisenberg limit in phase measurement, highlighting the roles of entanglement and symmetry.

Contribution

It demonstrates that certain three-qubit graph and hypergraph states can reach the Heisenberg limit, with specific states showing optimal entanglement and measurement properties.

Findings

GHZ and symmetric graph states attain the Heisenberg limit.

Graph states exhibit measurement parameter invariance in RMQFI.

States violate Bell's inequality with $F_Q > N$.

Abstract

We study the reciprocal of the mean quantum Fisher information (RMQFI), $\chi^2$ for general three qubit states, having graph and hypergraph states as special cases, for identifying genuine multi party entanglement characterized by $\chi^2 <1$. 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.