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 improvement in measurement precision achieved by utilizing quantum resources (such as entanglement or state squeezing) compared to the best possible classical methods using separable states, i.e., states without quantum entanglement. 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.
Background Let us consider a unitary dynamics with a parameter θ {\displaystyle \theta } from initial state ϱ 0 {\displaystyle \varrho _{0}} ,
ϱ ( θ ) = exp ( − i A θ ) ϱ 0 exp ( + i A θ ) , {\displaystyle \varrho (\theta )=\exp(-iA\theta )\varrho _{0}\exp(+iA\theta ),}
the quantum Fisher information F Q {\displaystyle F_{\rm {Q}}} constrains the achievable precision in statistical estimation of the parameter θ {\displaystyle \theta } via the quantum Cramér–Rao bound as
( Δ θ ) 2 ≥ 1 m F Q [ ϱ , A ] , {\displaystyle (\Delta \theta )^{2}\geq {\frac {1}{mF_{\rm {Q}}[\varrho ,A]}},}
where m {\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 N {\displaystyle N} spin-1/2 particles
F Q [ ϱ , J z ] ≤ N {\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq N}
holds for separable states, where F Q {\displaystyle F_{\rm {Q}}} is the quantum Fisher information,
J z = ∑ n = 1 N j z ( n ) , {\displaystyle J_{z}=\sum _{n=1}^{N}j_{z}^{(n)},}
and j z ( n ) {\displaystyle j_{z}^{(n)}} is a single particle angular momentum component. Thus, the metrological gain can be characterize by
F Q [ ϱ , J z ] N . {\displaystyle {\frac {F_{\rm {Q}}[\varrho ,J_{z}]}{N}}.}
The maximum for general quantum states is given by
F Q [ ϱ , J z ] ≤ N 2 . {\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 k {\displaystyle k} ,
F Q [ ϱ , J z ] ≤ s k 2 + r 2 {\displaystyle F_{\rm {Q}}[\varrho ,J_{z}]\leq sk^{2}+r^{2}}
holds, where s = ⌊ N / k ⌋ {\displaystyle s=\lfloor N/k\rfloor } is the largest integer smaller than or equal to N / k , {\displaystyle N/k,} and r = N − s k {\displaystyle r=N-sk} is the remainder from dividing N {\displaystyle N} by k {\displaystyle k} . Hence, a higher and higher levels of multipartite entanglement is needed to achieve a better and better accuracy in parameter estimation. It is possible to obtain a weaker but simpler bound
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