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Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems

Ryosuke Nogami

TL;DR

The paper extends the Watanabe--Sagawa--Ueda (WSU) estimation-based uncertainty relations to infinite-dimensional quantum systems by formulating classical and quantum estimation theory for model families with full-rank states and introducing pseudo-inverse Fisher information forms. It defines measurement errors for both bounded and unbounded observables via the pseudo-inverse of the Fisher information form and proves error--uncertainty inequalities that strengthen the original WSU bounds. The approach unifies classical and quantum CR inequalities in infinite dimensions through the spaces of operators and the SLD/RLD Fisher information forms, establishing a framework for applying estimation-based uncertainty relations to continuous observables. Limitations include restriction to full-rank states, with future work aimed at extending to boundary states, applying the framework to canonical continuous observables like position and momentum, and exploring monotonicity and disturbance-uncertainty relations in infinite dimensions.

Abstract

We extend the Watanabe--Sagawa--Ueda (WSU) uncertainty relations for measurement errors to infinite-dimensional systems. The original WSU formulation provided a definition of measurement errors with a clear physical interpretation based on quantum estimation theory, but was restricted to finite-dimensional systems, excluding important observables such as position and momentum. Using pseudo-inverse forms of positive-semidefinite forms, we develop a framework for classical and quantum estimation theory for models whose parameter space is the set of full-rank states on an infinite-dimensional Hilbert space, and derive classical and quantum Cramér--Rao inequalities. We extend the WSU measurement errors to both bounded and unbounded operators, and derive corresponding error-error uncertainty relations. The resulting uncertainty relation inequalities are stronger than the original WSU bound due to an improved derivation method. Our results provide a theoretical framework for applying estimation-based uncertainty relations to observables with continuous values in infinite-dimensional systems.

Extension of the Watanabe-Sagawa-Ueda uncertainty relation for measurement errors to infinite-dimensional systems

TL;DR

The paper extends the Watanabe--Sagawa--Ueda (WSU) estimation-based uncertainty relations to infinite-dimensional quantum systems by formulating classical and quantum estimation theory for model families with full-rank states and introducing pseudo-inverse Fisher information forms. It defines measurement errors for both bounded and unbounded observables via the pseudo-inverse of the Fisher information form and proves error--uncertainty inequalities that strengthen the original WSU bounds. The approach unifies classical and quantum CR inequalities in infinite dimensions through the spaces of operators and the SLD/RLD Fisher information forms, establishing a framework for applying estimation-based uncertainty relations to continuous observables. Limitations include restriction to full-rank states, with future work aimed at extending to boundary states, applying the framework to canonical continuous observables like position and momentum, and exploring monotonicity and disturbance-uncertainty relations in infinite dimensions.

Abstract

We extend the Watanabe--Sagawa--Ueda (WSU) uncertainty relations for measurement errors to infinite-dimensional systems. The original WSU formulation provided a definition of measurement errors with a clear physical interpretation based on quantum estimation theory, but was restricted to finite-dimensional systems, excluding important observables such as position and momentum. Using pseudo-inverse forms of positive-semidefinite forms, we develop a framework for classical and quantum estimation theory for models whose parameter space is the set of full-rank states on an infinite-dimensional Hilbert space, and derive classical and quantum Cramér--Rao inequalities. We extend the WSU measurement errors to both bounded and unbounded operators, and derive corresponding error-error uncertainty relations. The resulting uncertainty relation inequalities are stronger than the original WSU bound due to an improved derivation method. Our results provide a theoretical framework for applying estimation-based uncertainty relations to observables with continuous values in infinite-dimensional systems.
Paper Structure (22 sections, 12 theorems, 110 equations)

This paper contains 22 sections, 12 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Pseudo-inverse forms
  3. Definition
  4. Justification for the pseudo-inverse form definition
  5. Classical estimation theory for models induced by quantum measurements on full-rank states
  6. Full-rank models
  7. Fisher information form
  8. Classical Cramér--Rao inequality
  9. Quantum estimation theory for full-rank models
  10. Spaces of operators
  11. Quantum Fisher information forms
  12. Quantum Cramér--Rao inequalities
  13. Watanabe--Sagawa--Ueda uncertainty relation
  14. Expectation values, correlations, and commutators
  15. Pseudo-inverse of the SLD Fisher information form
  16. ...and 7 more sections

Key Result

Lemma 2.1

There exists a unique sesquilinear form $\mathcal{J}^{+}$ on $\mathcal{D}(\mathcal{J}^{+})$ such that $\tilde{\mathcal{J}}(\varphi) = \mathcal{J}^{+}(\varphi, \varphi)$ for all $\varphi \in \mathcal{D}(\mathcal{J}^{+})$.

Theorems & Definitions (24)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1: Classical Cramér--Rao inequality
  • proof
  • Theorem 4.1: Quantum Cramér--Rao inequalities
  • proof
  • Lemma 4.2
  • proof
  • ...and 14 more