# Optimal measurements for quantum fidelity between Gaussian states and its relevance to quantum metrology

## TL;DR

This paper derives the optimal measurement strategies for calculating quantum fidelity between Gaussian states, which is crucial for quantum information processing and metrology, and links these strategies to quantum parameter estimation.

## Contribution

It provides a closed-form analytical solution for optimal POVMs for Gaussian states and connects fidelity measurements with quantum parameter estimation techniques.

## Key findings

- Identified three types of optimal measurements for Gaussian states
- Derived analytical formulas for optimal POVMs between multi-mode Gaussian states
- Established equivalence between fidelity measurements and quantum parameter estimation

## Abstract

Quantum fidelity is a measure to quantify the closeness of two quantum states. In an operational sense, it is defined as the minimal overlap between the probability distributions of measurement outcomes and the minimum is taken over all possible positive-operator valued measures (POVMs). Quantum fidelity has been investigated in various scientific fields, but the identification of associated optimal measurements has often been overlooked despite its great importance for practical purposes. We find here the optimal POVMs for quantum fidelity between multi-mode Gaussian states in a closed analytical form. Our general finding is specified for selected single-mode Gaussian states of particular interest and we identify three types of optimal measurements: a number-resolving detection, a projection on the eigenbasis of operator $\hat{x}\hat{p}+\hat{p}\hat{x}$, and a quadrature detection, each of which is applied to distinct types of single-mode Gaussian states. We also show the equivalence between optimal measurements for quantum fidelity and those for quantum parameter estimation, enabling one to easily find the optimal measurements for displacement, phase, squeezing, and loss parameter estimations using Gaussian states.