The quantum metrological gain is defined in the context of carrying out a metrological task using a quantum state of a multiparticle system.
Hence, the quantum metrological gain is given as the fraction of the sensitivity achieved by the state and the maximal sensitivity achieved by separable states.
The best separable state is often the trivial fully polarized state, in which all spins point into the same direction.
If the metrological gain is larger than one then the quantum state is more useful for making precise measurements than separable states.
Clearly, in this case the quantum state is also entangled.
The metrological gain is, in general, the gain in sensitivity of a quantum state compared to a product state.
[1] Metrological gains up to 100 are reported in experiments.
, the quantum Fisher information
constrains the achievable precision in statistical estimation of the parameter
For the formula, one can see that the larger the quantum Fisher information, the smaller can be the uncertainty of the parameter estimation.
spin-1/2 particles[3] holds for separable states, where
is a single particle angular momentum component.
Thus, the metrological gain can be characterize by The maximum for general quantum states is given by Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology.
Moreover, for quantum states with an entanglement depth
is the largest integer smaller than or equal to
Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation.
[4][5] It is possible to obtain a weaker but simpler bound [6] Hence, a lower bound on the entanglement depth is obtained as The situation for qudits with a dimension larger than
In this more general case, the metrological gain for a given Hamiltonian is defined as the ratio of the quantum Fisher information of a state and the maximum of the quantum Fisher information for the same Hamiltonian for separable states[7][8]
The maximum of the quantum Fisher information for separable states is given as[9] [10] [7]
denote the maximum and minimum eigenvalues of
We also define the metrological gain optimized over all local Hamiltonians as
In this case, if the local Hamitlonians are chosen to be
[11] Thus, in the case of qubits, the optimization of the gain over the local Hamiltonian can be simpler.
For qudits with a dimension larger than 2, the optimization is more complicated.
In short, we call such states metrologically useful.
all have identical lowest and highest eigenvalues, then
holds, then the state has metrologically useful genuine multipartite entanglement.
The metrological gain cannot increase if we add an ancilla to a subsystem or we provide an additional copy of the state.
[7][8] There are efficient methods to determine the metrological gain via an optimization over local Hamiltonians.
They are based on a see-saw method that iterates two steps alternatively.