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

TL;DR

This paper investigates how nonlinear scrambling operations can improve the estimation of two phase-shift parameters in quantum systems, especially when parameters are incompatible or the model is ill-conditioned, demonstrating enhanced precision with third-order scrambling.

Contribution

It introduces the use of quadratic and cubic nonlinear scrambling to mitigate sloppiness and incompatibility in multiparameter quantum estimation, showing third-order nonlinearity's superior effectiveness.

Findings

Nonlinear scrambling reduces sloppiness and enhances estimation accuracy.

Third-order nonlinearity outperforms second-order in improving precision.

Joint estimation benefits from high nonlinear coupling, especially with coherent probes.

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.