Quadratic and cubic scrambling in the estimation of two successive phase-shifts

TL;DR

The paper tackles sloppiness and incompatibility in multiparameter quantum metrology for two successive phase shifts in a bosonic mode. It proposes nonlinear scrambling between the two encodings, considering $m=2$ (quadratic) and $m=3$ (cubic) forms, and analyzes both coherent and squeezed-vacuum probes. The results show that scrambling reduces sloppiness, enhances parameter compatibility, and improves joint estimation precision, with third-order scrambling generally superior; a threshold in nonlinear coupling governs when joint estimation surpasses stepwise strategies. By deriving and comparing quantum bounds $C_Q$, $C_H$, and $C_T$ and evaluating the quantum Fisher information matrix and mean Uhlmann curvature, the work demonstrates a practical route to mitigating sloppiness via active encoding control, with potential impact for continuous-variable quantum sensing.

Abstract

Multiparameter quantum estimation becomes challenging when the parameters are incompatible, i.e., when their respective symmetric logarithmic derivatives do not commute, or when the model is sloppy, meaning that the quantum probe depends only on combinations of parameters leading to a degenerate or ill-conditioned Fisher information matrix. In this work, we explore the use of scrambling operations between parameter encoding to overcome sloppiness. We consider a bosonic model with two phase-shift parameters and analyze the performance of second- and third-order nonlinear scrambling using two classes of probe states: squeezed vacuum states and coherent states. Our results demonstrate that nonlinear scrambling mitigates sloppiness, increases compatibility, and improves overall estimation precision. We find third-order nonlinearity to be more effective than second-order under both fixed-probe and fixed-energy constraints. Furthermore, by comparing joint estimation to a stepwise estimation strategy, we show that a threshold for nonlinear coupling exists. For coherent probes, joint estimation outperforms the stepwise strategy if the nonlinearity is sufficiently large, while for squeezed probes, this advantage is observed specifically with third-order nonlinearity.

Figures (3)

• Figure 1: The sloppy model considered in this paper involves encoding the model parameters $\phi_1$ and $\phi_2$ via the unitary operations $\hat{V}_1$ and $\hat{V}_2$. To remove sloppiness, we introduce a scrambling operation, represented by the intermediate unitary transformation $\hat{U}$.
• Figure 2: Plots of sloppiness $S$ versus scrambling strength $\gamma$: (a) Squeezed vacuum probe with cubic scrambling; (b) Squeezed vacuum probe with quadratic scrambling; (c) Coherent probe with cubic scrambling; (d) Coherent probe with quadratic scrambling.
• Figure 3: The bounds $C_{\rm Step}^{1}$, $C_{\rm Step}^{2}$, $C_{Q}$ and $C_{T}$ as functions of the scrambling strength $\gamma$ for probes with $\bar{n}=1$: (a) Squeezed vacuum with cubic scrambling; (b) Squeezed vacuum with quadratic scrambling; (c) Coherent state with cubic scrambling; (d) Coherent state with quadratic scrambling. For both probes and limited nonlinearity, the best strategy is step-wise estimation, with cubic scrambling being more effective than quadratic. When squeezed probes and cubic scrambling are used, joint estimation becomes superior to step-wise estimation once the nonlinearity passes a certain threshold. For quadratic scrambling, this cannot be proved (see text). For coherent probes, a second threshold on the nonlinearity appears, above which joint estimation outperforms the step-wise, while in the intermediate nonlinearity regime this cannot be proved.