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The quantum metrological gain is defined in the context of carrying out a metrological task using a quantum state of a multiparticle system. It is the sensitivity of parameter estimation using the state compared to what can be reached using separable states, i.e., states without quantum entanglement. 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.^[2]

Let us consider a unitary dynamics with a parameter ${\displaystyle \theta }$ from initial state ${\displaystyle \varrho _{0}}$,

${\displaystyle \varrho (\theta )=\exp(-iA\theta )\varrho _{0}\exp(+iA\theta ),}$

the quantum Fisher information ${\displaystyle F_{\rm {Q}}}$ constrains the achievable precision in statistical estimation of the parameter ${\displaystyle \theta }$ via the quantum Cramér–Rao bound as

${\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,A]}},}$

where ${\displaystyle m}$ is the number of independent repetitions. For the formula, one can see that the larger the quantum Fisher information, the smaller can be the uncertainty of the parameter estimation.

For a multiparticle system of ${\displaystyle N}$ spin-1/2 particles^[3]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N}$

holds for separable states, where ${\displaystyle F_{\rm {Q}}}$ is the quantum Fisher information,

${\displaystyle J_{z}=\sum _{n=1}^{N}j_{z}^{(n)},}$

and ${\displaystyle j_{z}^{(n)}}$ is a single particle angular momentum component. Thus, the metrological gain can be characterize by

${\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}.}$

The maximum for general quantum states is given by

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N^{2}.}$

Hence, quantum entanglement is needed to reach the maximum precision in quantum metrology. Moreover, for quantum states with an entanglement depth ${\displaystyle k}$,

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq sk^{2}+r^{2}}$

holds, where ${\displaystyle s=\lfloor N/k\rfloor }$ is the largest integer smaller than or equal to ${\displaystyle N/k,}$ and ${\displaystyle r=N-sk}$ is the remainder from dividing ${\displaystyle N}$ by ${\displaystyle k}$. 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]

${\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq Nk.}$

Hence, a lower bound on the entanglement depth is obtained as

${\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}\leq k.}$

The situation for qudits with a dimension larger than ${\displaystyle d=2}$ is more complicated. 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]

${\displaystyle g_{\mathcal {H}}(\varrho )={\frac {{\mathcal {F}}_{Q}[\varrho ,{\mathcal {H}}]}{{\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})}},}$

where the Hamiltonian is

${\displaystyle {\mathcal {H}}=h_{1}+h_{2}+...+h_{N},}$

and ${\displaystyle h_{n}}$ acts on the nth spin. The maximum of the quantum Fisher information for separable states is given as^{[9] [10] [7]}

${\displaystyle {\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})=\sum _{n=1}^{N}[\lambda _{\max }(h_{n})-\lambda _{\min }(h_{n})]^{2},}$

where ${\displaystyle \lambda _{\max }(X)}$ and ${\displaystyle \lambda _{\min }(X)}$ denote the maximum and minimum eigenvalues of ${\displaystyle X,}$ respectively.

We also define the metrological gain optimized over all local Hamiltonians as

${\displaystyle g(\varrho )=\max _{\mathcal {H}}g_{\mathcal {H}}(\varrho ).}$

The case of qubits is special. In this case, if the local Hamitlonians are chosen to be

${\displaystyle h_{n}=\sum _{l=x,y,z}c_{l,n}\sigma _{l},}$

where ${\displaystyle c_{l,n}}$ are real numbers, and ${\displaystyle |{\vec {c}}_{n}|=1,}$ then

${\displaystyle {\mathcal {F}}_{Q}^{({\rm {sep}})}({\mathcal {H}})=4N}$,

independently from the concrete values of ${\displaystyle c_{l,n}}$.^[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.

If the gain larger than one

${\displaystyle g(\varrho )>1,}$

then the state is entangled, and it is more useful metrologically than separable states. In short, we call such states metrologically useful. If ${\displaystyle h_{n}}$ all have identical lowest and highest eigenvalues, then

${\displaystyle g(\varrho )>k-1}$

implies metrologically useful ${\displaystyle k}$-partite entanglement. If for the gain^[8]

${\displaystyle g(\varrho )>N-1}$

holds, then the state has metrologically useful genuine multipartite entanglement.^[7] In general, for quantum states ${\displaystyle g(\varrho )\leq N}$ holds.

The metrological gain cannot increase if we add an ancilla to a subsystem or we provide an additional copy of the state.^[7][8] The metrological gain ${\displaystyle g(\varrho )}$ is convex in the quantum 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.^[7]