Phase estimation with limited coherence

TL;DR

This paper explores the fundamental limits of quantum phase estimation based on the coherence of the probe, revealing how the estimation variance scales with coherence and the probe size, and identifying optimal states.

Contribution

It derives the minimum estimation variance for pure states based on coherence, characterizes the optimal states, and shows the variance scales as a Heisenberg-like relation independent of system dimension.

Findings

Pure states are optimal only if coherence scales linearly with system size.

Optimal state rank increases as coherence decreases, eventually becoming full-rank.

Estimation variance scales as $V(C) \\sim a_n/C^2$, with $a_n$ approaching $\\pi^2/3$ as system size grows.

Abstract

We investigate the ultimate precision limits for quantum phase estimation in terms of the coherence, $C$, of the probe. For pure states, we give the minimum estimation variance attainable, $V(C)$, and the optimal state, in the asymptotic limit when the probe system size, $n$, is large. We prove that pure states are optimal only if $C$ scales as $n$ with a sufficiently large proportionality factor, and that the rank of the optimal state increases with decreasing $C$, eventually becoming full-rank. We show that the variance exhibits a Heisenberg-like scaling, $V(C) \sim a_n/C^2$, where $a_n$ decreases to $\pi^2/3$ as $n$ increases, leading to a dimension-independent relation.