Calculating the quantum Fisher information via the truncated Wigner method

Abstract

In this work, we propose new methods of parameter estimation using stochastic sampling quantum phase-space simulations. We show that it is possible to compute the quantum Fisher information (QFI) from semiclassical stochastic samples using the Truncated Wigner Approximation (TWA). This method extends the class of quantum systems whose fundamental sensitivity limit can be computed efficiently to any system that can be modelled using the TWA, allowing the analysis of more meteorologically useful quantum states. We illustrate this approach with examples, including a system that evolves outside the spin-squeezing regime, where the method of moments fails.

Figures (5)

• Figure 1: A small subsample of individual trajectories (white dots) overlaid with the analytic form of the Wigner function for (a) an initial vacuum state, (b) the state at time $t_1$, and (c) the time-reversed state after encoding. The arrows indicate the derivative of the trajectories w.r.t. to $\omega$, or 'flow'. The state preparation breaks the rotational symmetry of the state, so the state is changed by the encoding, so the QFI becomes non-zero. When the trajectories from (b) are mapped to $t=0$, we can compute this non-zero QFI directly from the trajectories, as the flow has gained a component in the direction of the initial gradient.
• Figure 2: QFI calculated from Eq. (\ref{['eq:TWA_QFI_EXP']}) (blue line) compared to the analytic solution Eq. (\ref{['FQ_opo_anal']}) (red circles). 1 million trajectories were used. Parameters: $\alpha_0=10$, $\theta=0$.
• Figure 3: (a) Populations of cavity and pump modes during state preparation, following Eq. (\ref{['undepeletedTrajectories']}). Blue line: $N_a = \langle \hat{a}^{\dagger}\xspace(t_1)\hat{a}\xspace(t_1)\rangle$, red dashed line $2N_b = 2\langle\hat{b}^{\dagger}\xspace(t_1)\hat{b}\xspace(t_1)\rangle$, black dotted line: $N_a + 2N_b$. (b) QFI calculated from Eq. (\ref{['eq:TWA_QFI_EXP']}) (blue line), QFI calculated from variance $4 \mathrm{Var}(\hat{a}^{\dagger}\xspace(t_1)\hat{a}\xspace(t_1))$ (red circles), QFI contribution from the $\partial_\omega \alpha$ terms (black dashed line). All three traces were evaluated via the TW method, using 1 million trajectories. Parameters: $\alpha_0=10$, $\beta_0 = \sqrt{1000}$, $\theta=0$.
• Figure 4: $F_Q$ calculated from the trajectory method (blue soild line), compared to the exact solution from the Schrodinger equation (black dashed line), and from the solution to Eq. (\ref{['FP_Kerr']}), both with (green stars) and without (red circles) the inclusion of the third-order derivative terms. The equivalent sensitivity metric from the MoM estimator $1/(\delta x_0)^2 = 1/\mathrm{Var}(\hat{X})$, calculated via exaction solution from the Schrodinger equation, is shown with the orange dot-dashed line. An initial coherent state $|\alpha_0\rangle$ with $\alpha_0 = 4$ was used.
• Figure 5: $W(\alpha,t)$ with (left column) and without (middle column) the inclusion of the third-order derivative terms, compared to a subset of stochastic trajectories (right column), at $t=0$ (top row), $\chi t=0.03$ (middle row), and $\chi t = 0.07$ (bottom row). The high-frequency fringes in the bottom left frame are the result of negativity (indicated by red) in the Wigner function. An initial coherent state $|\alpha_0\rangle$ with $\alpha_0 = 4$ was used.