Direct measurement of quantum Fisher information

In the adiabatic perturbation theory, Berry curvature is related to the generalized force, and the quantum metric tensor is linked with energy fluctuation. While the former is tested with numerous numerical results and experimental realizations, the latter is less considered. Quantum Fisher information, the key to quantum precision measurement, is a four times quantum metric tensor. It is difficult to relate the quantum Fisher information with some physical observable. One interesting candidate is the square of the symmetric logarithmic derivative, which is usually tough to obtain, both theoretically and experimentally. The adiabatic perturbation theory enlightens us to measure the energy fluctuation to directly extract the quantum Fisher information. In this article we first adopt an alternative way to derive the link of energy fluctuation to the quantum Fisher information. Then we numerically testify to the direct extraction of the quantum Fisher information based on adiabatic perturbation in two-level systems and simulate the experimental realization in a nitrogen-vacancy center with experimentally practical parameters. Statistical models such as the transverse-field Ising model and Heisenberg spin chains are also discussed to compare with the analytical result and show the level crossing, respectively. Our discussion will provide a practical scheme to measure the quantum Fisher information and will also be a benefit to quantum precision measurement and the understanding of the quantum Fisher information.

Figure 1

Illustration of the extraction of the QFI (matrix). The initial state is prepared to be the ground state of the initial Hamiltonian. (The green block represents the population in each level and it is located in the ground state, obviously.) Then the parameter(s) of the Hamiltonian is (are) slowly changed with vanishing switching-on velocity. The populations in higher levels are still small but not zero due to quasiadiabaticity. When the Hamiltonian is evolved into the interested one, the energy fluctuation is measured in the final state of the system. The ratio between the expectation to the square of the final velocity of parameter(s) gives the QFI (matrix).

Figure 2

Measurement of the QFI with respect to $\theta$, $F_{\theta }$, of the ground state of a two-level system. The numerical result is obtained by the dynamical simulation using the software packages developed in Refs. [40, 41].

Figure 3

(a) Sum of elements of the QFIM with respect to $x\left(0\right)$ when the final Hamiltonian's ground state is estimated. (b) Diagonal elements of the QFIM evaluated using Eq. (18) and off-diagonal elements of the QFIM obtained using half of the difference between the sum of all QFIM elements and all the diagonal QFIM elements.

Figure 4

(a) Energy splitting first due to the zero-field splitting $D$ and then the external field $B$. (b) Demonstration of the protocol of measuring QFI in NV center.

Figure 5

QFI of $\theta$ and $\varphi$ in a rotating frame interacting two-qubit model, which is experimentally measured in [9]. The result using our method with the same parameters is exactly coincident with that of [9]. We list the actual parameters here: $A_x=2.79\mathrm{MHz}$, $A_z=11.832\mathrm{MHz}$, ${\mathrm{\Omega }}_{\mathrm{mw}}=2.13\mathrm{MHz}$, ${\gamma }_n{B}_{\left|\right|}=1.07\times 749.32$ kHz, and $\varphi =$ 0.

Figure 6

QFI of the ground state of transverse-field Ising model with respect to the field strength. The dot is the measured quantum Fisher information using our method, while the line is the analytic solution using Eq. (28). In our case, the positive parity is used since the energy is lower in the situation of positive parity.

Figure 7

(a) Level crossing of the ground state of the Heisenberg spin chain. In the considered region of the coupling strength, level crossing occurs twice at $J=-0.5$ and $J=-0.25$, respectively. (b) QFI with respect to the field strength of the ground state of the Heisenberg spin chain. We obtain the QFI using our method. It is apparent that the QFI also shows two abrupt changes at the corresponding values of $J$, which depicts the level crossing.