# Quantum Semiparametric Estimation

## TL;DR

This paper develops a quantum semiparametric estimation theory that provides simple bounds for high-dimensional quantum systems with limited prior assumptions, applicable to practical quantum measurement scenarios.

## Contribution

It introduces a new framework for quantum semiparametric estimation that overcomes high dimensionality and limited prior knowledge, linking bounds to Holevo's quantum Cramér-Rao bound.

## Key findings

- Provides analytic bounds for high-dimensional quantum estimation problems.
- Relates bounds to Holevo's quantum Cramér-Rao bound for asymptotic attainability.
- Applicable to quantum state properties like fidelity, purity, and entropy.

## Abstract

In the study of quantum limits to parameter estimation, the high dimensionality of the density operator and that of the unknown parameters have long been two of the most difficult challenges. Here we propose a theory of quantum semiparametric estimation that can circumvent both challenges and produce simple analytic bounds for a class of problems in which the dimensions are arbitrarily high, few prior assumptions about the density operator are made, but only a finite number of the unknown parameters are of interest. We also relate our bounds to Holevo's version of the quantum Cram\'er-Rao bound, so that they can inherit the asymptotic attainability of the latter in many cases of interest. The theory is especially relevant to the estimation of a parameter that can be expressed as a function of the density operator, such as the expectation value of an observable, the fidelity to a pure state, the purity, or the von Neumann entropy. Potential applications include quantum state characterization for many-body systems, optical imaging, and interferometry, where full tomography of the quantum state is often infeasible and only a few select properties of the system are of interest.