Introduction To Quantum-state Estimation

Yong Siah Teo

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814678865
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Quantum-state estimation is an important field in quantum information theory that deals with the characterization of states of affairs for quantum sources. This book begins with background formalism in estimation theory to establish the necessary prerequisites. This basic understanding allows us to explore popular likelihood- and entropy-related estimation schemes that are suitable for an introductory survey on the subject. Discussions on practical aspects of quantum-state estimation ensue, with emphasis on the evaluation of tomographic performances for estimation schemes, experimental realizations of quantum measurements and detection of single-mode multi-photon sources. Finally, the concepts of phase-space distribution functions, which compatibly describe these multi-photon sources, are introduced to bridge the gap between discrete and continuous quantum degrees of freedom.

This book is intended to serve as an instructive and self-contained medium for advanced undergraduate and postgraduate students to grasp the basics of quantum-state estimation. Any reader with a solid foundation in quantum mechanics, linear algebra and calculus would be able to follow the book comfortably.

Quantum-state estimation is an important field in quantum information theory that deals with the characterization of states of affairs for quantum sources. This book begins with background formalism in estimation theory to establish the necessary prerequisites. This basic understanding allows us to explore popular likelihood- and entropy-related estimation schemes that are suitable for an introductory survey on the subject. Discussions on practical aspects of quantum-state estimation ensue, with emphasis on the evaluation of tomographic performances for estimation schemes, experimental realizations of quantum measurements and detection of single-mode multi-photon sources. Finally, the concepts of phase-space distribution functions, which compatibly describe these multi-photon sources, are introduced to bridge the gap between discrete and continuous quantum degrees of freedom.

This book is intended to serve as an instructive and self-contained medium for advanced undergraduate and postgraduate students to grasp the basics of quantum-state estimation. Any reader with a solid foundation in quantum mechanics, linear algebra and calculus would be able to follow the book comfortably.

Readership: Advanced undergraduate and graduate students as well as researchers interested in quantum information, estimation theory, and quantum optics.
Key Features:

• This is a relatively new topic with a paucity of literature in the market
• This book is a set of lecture notes that is more suitable for graduate-module teaching, covering the necessary basic elements of this subject in a pedagogical manner before introducing more advanced notions in the later chapters
• A solution manual for the problems in the book is currently in the works

Chapter 1

Preliminaries of Quantum-State Estimation

1.1Overview of quantum-state estimation

Quantum-state preparation is the first important step for any protocol that makes use of a source of quantum systems. For instance, a quantum-state teleportation protocol that is carried out with optical equipment requires a source that produces two photons in a maximally-entangled quantum state; that is, each photon in every photon pair possesses maximum quantum correlation with each other. In order to verify the integrity of the quantum state that appropriately describes the source prepared, one carries out quantum-state tomography on the source. Measurements are performed on a collection of identical copies of quantum systems (electrons, photons, etc.) that are prepared by the source. These measurements are generically described by a set of positive operators {Π_j ≥ 0} that compose a probability-operator measurement (POM), where ∑_j Π_j = 1. Here, we remind the reader that Π_j ≥ 0 means that 〈 |Π_j| 〉 ≥ 0 for any | 〉.

A familiar special case would be the mutually exclusive measurement outcomes in quantum mechanics. The corresponding Π_j = |j〉; 〈j| are prototypically orthonormal pure states that form a complete set — Π_j Π_k = δ_j,k Π_j and ∑_j Π_j = ∑ |j〉 〈j| = 1 —, so that repeated always measurements on the same quantum system, if physically possible,* always result in the same outcome for any Π_j. Such measurements are routinely implemented in practice. For the electron-spin measurement in a Stern–Gerlach^† experiment, either the “up spin” or the “down spin” outcome is measured at any given instance through a path deflection by a spatially inhomogeneous magnet. For the photon-polarization measurement, either the “horizontal” or “vertical” polarization is detected with a partially polarizing beam splitter (see Chapter 4) and a photodetector.

The POM description therefore generalizes these mutually exclusive measurements to measurements with outcomes that are not necessarily mutually exclusive. We shall postpone the discussion on realizing such generalized measurements to Chapter 4. For now, we shall take this general description for granted. A reason for this generalization will become clear very soon.

The measurement data obtained are used to infer the identity of the source. Such a procedure of state inference is known as quantum-state estimation. In quantum mechanics, a quantum state encodes all information the observer has about a source after the measurement has been performed. This means that all quantum-mechanical probabilities for any future measurement...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface
• Acknowledgments
• Acronyms
• Symbols and Notations
• 1. Preliminaries of Quantum-State Estimation
• 2. Informationally Complete Estimation
• 3. Informationally Incomplete Estimation
• 4. Practical Aspects of State Estimation
• 5. Quasi-Probability Distributions
• Hints to All Problems
• Sample Solutions to All Problems
• Appendix: Squeezed Coherent States
• Index