Holevo Cram\'er-Rao bound: How close can we get without entangling measurements?

TL;DR

This paper investigates the potential precision gains from entangling measurements in multi-parameter quantum metrology, showing that while maximum enhancement can be linear in the number of parameters, it is limited by the qudit dimension.

Contribution

The work establishes bounds on the collective quantum enhancement achievable with entangling measurements, revealing limitations and proposing a conjecture on the maximum enhancement.

Findings

Maximum enhancement can be a factor of n for n parameters

Enhancement is linear in the qudit dimension

Bound is not tight for large n

Abstract

In multi-parameter quantum metrology, the resource of entanglement can lead to an increase in efficiency of the estimation process. Entanglement can be used in the state preparation stage, or the measurement stage, or both, to harness this advantage; here we focus on the role of entangling measurements. Specifically, entangling or collective measurements over multiple identical copies of a probe state are known to be superior to measuring each probe individually, but the extent of this improvement is an open problem. It is also known that such entangling measurements, though resource-intensive, are required to attain the ultimate limits in multi-parameter quantum metrology and quantum information processing tasks. In this work we investigate the maximum precision improvement that collective quantum measurements can offer over individual measurements for estimating parameters of qudit states, calling this the 'collective quantum enhancement'. We show that, whereas the maximum enhancement can, in principle, be a factor of $n$ for estimating $n$ parameters, this bound is not tight for large $n$. Instead, our results prove an enhancement linear in dimension of the qudit is possible using collective measurements and lead us to conjecture that this is the maximum collective quantum enhancement in any local estimation scenario.