Optimal witnessing of the quantum Fisher information with few measurements

We show how to verify the metrological usefulness of quantum states based on the expectation values of an arbitrarily chosen set of observables. In particular, we estimate the quantum Fisher information as a figure of merit of metrological usefulness. Our approach gives a tight lower bound on the quantum Fisher information for the given incomplete information. We apply our method to the results of various multiparticle quantum states prepared in experiments with photons and trapped ions, as well as to spin-squeezed states and Dicke states realized in cold gases. Our approach can be used for detecting and quantifying metrologically useful entanglement in very large systems, based on a few operator expectation values. We also gain new insights into the difference between metrological useful multipartite entanglement and entanglement in general.

Figure 1

(a) Fidelity vs lower bound on the quantum Fisher information for GHZ states of $N$ qubits. The quantum Fisher information is 0 if the fidelity is less than $0.5.$ (b) The same, but for Dicke states with $N=6$ (solid line) and $N=40$ (dashed line).

Figure 2

(a) Optimal lower bound on the quantum Fisher information ${\mathcal{F}}_{\mathrm{Q}}[\varrho ,J_y]$ based on collective measurements for spin-squeezing with $N=4.$ The mean spin points in the $z$ direction. Below the dashed line we have ${\mathcal{F}}_{\mathrm{Q}}[\varrho ,J_y]/N>1.$ For the description of points P, D, M, and C, see the text. (b) Lower bound on ${\mathcal{F}}_{\mathrm{Q}}[\varrho ,J_y]$ for $〈J_z〉=1.5$ and ${\left(\mathrm{\Delta }{J}_x\right)}^2=0.567,$ as a function of $\langle J_x^4\rangle .$ The corresponding point in (a) is denoted by a cross. Dashed horizontal line: Lower bound without constraining $\langle J_x^4\rangle .$ Dotted horizontal line: Lower bound for states in the symmetric subspace. As shown, an additional constraint or assuming symmetry improves the bound.

Figure 3

Optimal lower bound on the quantum Fisher information for symmetric states close to Dicke states for $N=6.$

Figure 4

The lower bound on ${\mathcal{F}}_{\mathrm{Q}}[\varrho ,J_y],$ denoted $\mathcal{B}\left(F_{\mathrm{Dicke}}\right),$ for various particle numbers, for $F_{\mathrm{Dicke}}=0.2$ (diamonds), $F_{\mathrm{Dicke}}=0.5$ (circles), and $F_{\mathrm{Dicke}}=0.7$ (triangles). For $F_{\mathrm{Dicke}}=0.2$ the calculated values times 10 are shown, for better visibility.

Figure 5

Behavior of the bound in Eq. (2) for points at the boundary of physical states. The relative difference with respect to the optimal lower bound is plotted for $N=4$ (solid line), $N=20$ (dashed line), and $N=1000$ (dotted line).

Figure 6

Lower bound on the quantum Fisher information based on $\langle J_z\rangle$ and ${\left(\mathrm{\Delta }{J}_x\right)}^2$ obtained for different particle numbers, making calculations in the symmetric subspace. $N=2300$ corresponds to the spin-squeezing experiment in Ref. [7]. (a) Almost fully polarized spin-squeezed state. Even for a moderate $N^{\prime }$, the bound is practically identical to the right-hand side of Eq. (18). (b) Spin-squeezed state that is not fully polarized. For large $N^{\prime },$ the bound converges to the right-hand side of Eq. (18), represented by the dashed line. In both panels, circles correspond to the results of our calculations, which are connected by straight lines to guide the eye.

Figure 7

Quantum Fisher information extrapolated to $N=7900$ from calculations with different particle numbers $N^{\prime }$ in an experiment creating Dicke states. Circles correspond to the results of our calculations, which are connected by straight lines to guide the eye.