# Characterizing non-deterministic noiseless linear amplifiers at the quantum limit

## TL;DR

This paper investigates the optimal methods for estimating the gain of non-deterministic noiseless linear amplifiers at the quantum limit, revealing that measuring the entire system yields the most information, especially when using a L"uders instrument.

## Contribution

It demonstrates that measuring the whole composite system provides the maximal information about the NLA gain, surpassing post-selection strategies, and analyzes the quantum Fisher information in detail.

## Key findings

- Measuring the entire system yields the highest quantum Fisher information.
- Post-selecting only amplified states is not optimal for gain estimation.
- Full information extraction is achieved with a L"uders instrument.

## Abstract

We address the characterization of the gain parameter of a non-deterministic noiseless linear amplifier (NLA) and compare the performances of different estimation strategies using tools from quantum estimation theory. At first, we show that, contrary to naive expectations, post-selecting only the amplified states does not offer the most accurate estimate. We then focus on minimal implementations of a NLA, i.e. those obtained by coupling the input state to a two-level system, and show that the maximal amount of information about the gain of the NLA is obtained by measuring the whole composite system. The quantum Fisher information (QFI) of this best-case scenario is analysed in some details, and compared to the QFI of the post-selected states, both for successful and unsuccessful amplification. Eventually, we show that full extraction of the available information is achieved when the non-deterministic process is implemented by a L\"uders instrument. We also analyse the precision attainable by probing NLAs by single-mode pure states and measuring the field or the number of quanta, and discuss in some details the specific cases of squeezed vacuum and coherent states.