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The likelihood operator and Fisher information in quantum probability

Kalyan B. Sinha, Ritabrata Sengupta

TL;DR

The paper addresses how to define and analyze quantum likelihood operators (QLO) and their relationship to quantum Fisher information (QFI) across several logarithmic-derivative (LD) models. It shows that commutativity between a parametrised state $\rho_\theta$ and its derivative is not guaranteed and can significantly affect QFI and CR bounds, motivating a range of LD definitions including Boltzmann-von Neumann (BvN) LD and the standard symmetric logarithmic derivative (SLD). The authors derive finite- and infinite-dimensional CR bounds for multiple LDs, establish equivalences under specific conditions, and provide explicit calculations for two-level systems and infinite-dimensional coherent-state models to illustrate how eigenvalue and eigenvector dependence shapes the neo-classical and quantum contributions to QFI. They also connect QFI with quantum relative entropy in the BvN framework and extend the theory to infinite dimensions via quadratic forms, highlighting when QFI exists and how it can be computed in physically relevant cases. The results offer a structured framework for choosing LD models and for understanding the information geometry of quantum parameter estimation in both finite and infinite-dimensional settings.

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 $ρ_θ$ 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.

The likelihood operator and Fisher information in quantum probability

TL;DR

The paper addresses how to define and analyze quantum likelihood operators (QLO) and their relationship to quantum Fisher information (QFI) across several logarithmic-derivative (LD) models. It shows that commutativity between a parametrised state and its derivative is not guaranteed and can significantly affect QFI and CR bounds, motivating a range of LD definitions including Boltzmann-von Neumann (BvN) LD and the standard symmetric logarithmic derivative (SLD). The authors derive finite- and infinite-dimensional CR bounds for multiple LDs, establish equivalences under specific conditions, and provide explicit calculations for two-level systems and infinite-dimensional coherent-state models to illustrate how eigenvalue and eigenvector dependence shapes the neo-classical and quantum contributions to QFI. They also connect QFI with quantum relative entropy in the BvN framework and extend the theory to infinite dimensions via quadratic forms, highlighting when QFI exists and how it can be computed in physically relevant cases. The results offer a structured framework for choosing LD models and for understanding the information geometry of quantum parameter estimation in both finite and infinite-dimensional settings.

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 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.

Paper Structure

This paper contains 15 sections, 12 theorems, 106 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Revisiting classical theory
  3. Quantum case
  4. Boltzmann-von Neumann Logarithmic Derivative operator (or BvN-LD for short)
  5. Other Logarithmic Derivative Operators (LD)
  6. Various models of LDs
  7. Cramér-Rao bound for finite dimension
  8. Quantum CR bound for LD$_1$
  9. CR-bound for LD$_2$:
  10. CR bound for LD$_3$
  11. Quantum CR bound for BvNLD ($\dim \mathcal{H} < \infty$).
  12. LD and QFI for infinite dimensional case
  13. Quantum two-level systems
  14. Two level system with $\theta$ dependent eigenvalues and eigenvectors
  15. 2-level system with $\theta$-independent eigenvalues and $\theta$-dependent eigenvectors

Key Result

Lemma 3.3

Let the family $\{\rho_\theta\}$ of states satisfy the Assumption 3.2. Then

Theorems & Definitions (33)

  • Remark 2.1
  • Example 3.1
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • proof
  • ...and 23 more