Enhancing distributed sensing with imperfect error correction

Entanglement has shown promise in enhancing information processing tasks in a sensor network via distributed quantum sensing protocols. As noise is ubiquitous in sensor networks, error correction schemes based on Gottesman-Kitaev-Preskill (GKP) states are required to enhance the performance, as shown in [Q. Zhuang, J. Preskill, and L. Jiang, New J. Phys. 22, 022001 (2020)1367-263010.1088/1367-2630/ab7257] assuming homogeneous noise among sensors and perfect GKP states. Here, we extend the analyses of performance enhancement to finite-squeezed GKP states in a heterogeneous noise model. To begin with, we study different concatenation schemes of GKP-two-mode-squeezing codes. While traditional sequential concatenation schemes in previous works do improve the suppression of noise, we propose a balanced concatenation scheme that outperforms the sequential schemes in the presence of finite GKP squeezing. We then apply these results to two specific tasks in distributed quantum sensing (parameter estimation and hypothesis testing) to understand the trade-off between imperfect squeezing and performance. For the first task, we consider an energy-constrained scenario and provide an optimal way to distribute the energy of the finite-squeezed GKP states among the sensors. For the later task, we show that the error probability can still be drastically lowered via concatenation of realistic finite-squeezed GKP codes.