The likelihood operator and Fisher information in quantum probability

TL;DR

This paper investigates the quantum likelihood operator and its relationship with quantum Fisher information, highlighting the importance of non-commutativity in parametrized quantum states and analyzing specific examples.

Contribution

It reveals the limitations of existing approaches assuming commutativity and explores the implications for quantum Fisher information in non-commutative cases.

Findings

Non-commutativity affects quantum Fisher information calculations.

Detailed analysis of two-level quantum systems.

Examination of coherent states in infinite-dimensional systems.

Abstract

We study the problem of Quantum Likelihood Operators (LO) and their connection with quantum Fisher information (QFI). It is observed that the present approaches to this problem tacitly assume commutativity of parametrised density matrix $\rho_\theta$ and its derivative, which, in general, need not be true, and this has nontrivial consequences in QFI. As examples, we discuss the parametrised two-level system exhaustively, and, as a further example, the one-mode coherent states of an infinite-dimensional system.