Quantum Fisher information matrix and multiparameter estimation (1907.08037v3)

Abstract: Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cram\'{e}r-Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cram\'{e}r-Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.

• The paper introduces the Quantum Fisher Information Matrix as a pivotal tool for quantifying precision limits in multiparameter quantum estimation.
• It details methodologies such as symmetric logarithmic derivative techniques for calculating the QFIM across pure, mixed, and Gaussian quantum states.
• The study discusses practical implications in quantum metrology and beyond, emphasizing optimal measurement strategies and connections to quantum geometry via the Bures metric.

Quantum Fisher Information Matrix and Multiparameter Estimation

The paper provides a comprehensive examination of the Quantum Fisher Information Matrix (QFIM) and its role in multiparameter estimation within quantum systems. The Quantum Fisher Information (QFI) is a pivotal concept in quantum metrology due to its fundamental connection with the quantum Cramér-Rao bound (QCRB), which establishes precision limits for parameter estimation in quantum mechanics.

Overview

The QFIM extends the QFI to account for multiparameter quantum systems, which is essential for tackling complex quantum systems where multiple parameters might be estimated simultaneously. The paper highlights the mathematical formulation of the QFIM and discusses various methods of calculating it across different types of quantum states, both pure and mixed, including Gaussian states which are prevalent in continuous variable quantum information processes.

Mathematical Framework

1. Definitions and Properties:
   • The QFIM is introduced formally, with definitions aligned to the density matrix formulation in quantum mechanics.
   • It is shown that the diagonal elements of the QFIM represent the precision of estimating individual parameters and its inverse establishing the fundamental precision limits.
2. Derivation and Calculation:
   • Several mathematical techniques are reviewed for computing the QFIM, such as those relying on the symmetric logarithmic derivative (SLD). These methods are adaptable to various scenarios, including full and non-full rank density matrices.
   • The paper examines the QFIM's use in unitary processes, Gaussian states, and provides explicit computational formulas and techniques for these contexts.

Theoretical and Practical Implications

• Multiparameter Quantum Cramér-Rao Bound: The discussion extends to the QCRB's application in multiparameter estimation. Important results are derived showing conditions under which the QCRB can be attained, particularly focusing on when the optimal measurements can simultaneously satisfy the conditions for parameter estimates without bias.
• Quantum Geometry: The paper links the QFIM with the geometry of quantum mechanics, showing its relation to the Bures metric which quantifies infinitesimal distances between quantum states. Such a geometric perspective is crucial for understanding quantum state dynamics and transitions.
• Applications Beyond Quantum Metrology: The discussion broadens to the QFIM’s applicability in areas like quantum thermodynamics, quantum phase transitions, and non-Markovian dynamics. This interdisciplinary approach underscores how foundational the QFIM is to a multitude of quantum theories and its utility for elucidating the complex phenomenology observed in quantum systems.

Future Outlook

The paper identifies several open challenges in the field of quantum metrology. These include the development of practical optimal measurement strategies that are agnostic of specific parameter values and robust to noise and control imperfections. The exploration of these areas is crucial for propelling quantum estimation from theoretical frameworks to practical applications, particularly in technologies like quantum sensors and quantum computing processors.

The work serves as a detailed guide for researchers looking to harness the power of QFIM in quantum parameter estimation, offering both the theoretical underpinnings and concrete practical examples of its application. Future advancements will likely explore integrating QFIM-based techniques with emerging quantum technologies, optimizing them for greater precision and operational resilience.