# Information geometry of Gaussian channels

## TL;DR

This paper introduces a new information-geometric metric for Gaussian channels based on the Bures-Fisher metric, providing bounds, properties, and computational methods for channel estimation using Gaussian states.

## Contribution

It defines a novel Riemannian metric for Gaussian channels, analyzes its properties, and develops algorithms for optimal channel estimation with Gaussian probe states.

## Key findings

- The metric bounds quantum Fisher information for Gaussian channels.
- Optimal probe states are always pure and have limited ancillary modes.
- Provides explicit formulas and algorithms for multiparametric quantum Fisher information.

## Abstract

We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated from distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a byproduct, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications: It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulae for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states, and provide the optimal observables for the estimation of the channel parameters (e.g. bath couplings, squeezing, and temperature).