# Qubit-channel metrology with very noisy initial states

## TL;DR

This paper investigates how to optimize the estimation of a qubit channel parameter using very noisy initial states, revealing that correlated input states can significantly improve accuracy for unital channels, but not for non-unital channels.

## Contribution

It demonstrates that correlated input states enhance estimation accuracy for unital channels with low purity, while showing no gain for a broad class of non-unital channels.

## Key findings

- Correlated states improve estimation accuracy for unital channels by a factor between n-1 and n.
- No gain is observed for non-unital channels with low purity.
- The advantage depends on the channel's unital or non-unital nature.

## Abstract

We consider an arbitrary qubit channel depending on a single parameter, which is to be estimated by a physical process. Using the quantum Fisher information per channel invocation to quantify the estimation accuracy, we consider various estimation protocols when the available initial states are mixed with very low purity. We compare a protocol using a single channel invocation on one out of $n$ qubits prepared in a particular correlated input state to the optimal protocol using uncorrelated input states, with the same initial state purity. We show that, to lowest order in initial-state purity, for a unital channel this correlated state protocol enhances the estimation accuracy by a factor between $n-1$ and $n.$ We also show that to lowest order in initial-state purity, a broad class of non-unital channels yields no gain regardless of the input state.