# Quantum multi-parameter estimation with generalized balanced multi-mode NOON-like states

## TL;DR

This paper investigates multi-parameter quantum estimation using generalized balanced multi-mode states, deriving bounds and comparing different entangled states to identify optimal configurations for phase imaging.

## Contribution

It introduces a generalized class of multi-mode entangled states for quantum metrology, derives the quantum Cramer-Rao bound for these states, and compares their performance.

## Key findings

- Squeezed vacuum states outperform other entangled states at fixed photon number.
- Balanced entangled states can outperform unbalanced states in certain scenarios.
- An experimental setup is proposed to produce states that surpass NOON states in quantum metrology.

## Abstract

The simultaneous multi-parameter estimation problem using a class of multi-mode entangled states is investigated in this paper. Specifically, the problem of optical phase imaging is considered and the quantum probe is taken to be a balanced coherent superposition of components with an arbitrary quantum state in one mode and vacuum states in all the other modes, which is a generalization of the multi-mode NOON state. The analytical form for the quantum Cramer-Rao bound (QCRB) is presented, which shows the performance by providing a lower bound of the estimation uncertainty. It is shown that the NOON state has the worst performance among those in the class of the entangled states considered. We also analyze in detail four different scenarios, which are the NOON state, the entangled coherent state, the entangled squeezed coherent state, and the entangled squeezed vacuum state. From the comparison among these four states, we find that when the mean photon number is fixed, the squeezed vacuum state has the smallest QCRB, followed by the squeezed coherent state, entangled coherent state, and NOON state. We also illustrate that the balanced entangled state can perform better than a more generalized unbalanced form studied in previous works for certain scenarios. Finally, we give an experimental setup for producing a two-mode entangled state that can beat the NOON state in quantum metrology.