Randomized measurements for multi-parameter quantum metrology

TL;DR

This paper introduces randomized measurements based on 3-designs as an effective and practical approach for multi-parameter quantum metrology, achieving near-optimal estimation in various quantum states.

Contribution

It demonstrates that randomized measurements can perform near-optimally for multi-parameter estimation, offering a practical alternative to traditional incompatible measurement strategies.

Findings

Randomized measurements with 3-designs perform near-optimally for pure states.

They are effective for estimating multiple parameters in well-conditioned states.

Explicit examples include fidelity and Hamiltonian estimation.

Abstract

The optimal quantum measurements for estimating different unknown parameters in a parameterized quantum state are usually incompatible with each other. Traditional approaches to addressing the measurement incompatibility issue, such as the Holevo Cram\'{e}r--Rao bound, suffer from multiple difficulties towards practical applicability, as the optimal measurement strategies are usually state-dependent, difficult to implement and also take complex analyses to determine. Here we study randomized measurements as a new approach for multi-parameter quantum metrology. We show quantum measurements on single copies of quantum states given by $3$-designs perform near-optimally when estimating an arbitrary number of parameters in pure states and more generally, {approximately low-rank well-conditioned states}, whose metrological information is largely concentrated in a low-dimensional subspace. The near-optimality is also shown in estimating the maximal number of parameters for three types of mixed states that are well-conditioned on their supports. Examples of fidelity estimation and Hamiltonian estimation are explicitly provided to demonstrate the power and limitation of randomized measurements in multi-parameter quantum metrology.