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Quantum entanglement

Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, Karol Horodecki

TL;DR

The paper surveys the multifaceted role of quantum entanglement as a resource in quantum information science. It weaves together foundational concepts (entanglement structure, Bell nonlocality, and entropic criteria) with operational frameworks (LOCC, PPT, and positive maps), detailing both detection and manipulation methods including entanglement witnesses, distillation, and activation. It highlights the divergence between entanglement and nonlocality, the emergence of bound entanglement, and the rich geometry of entangled states, emphasizing practical implications for cryptography, communication complexity, and quantum computation. Collectively, the work outlines a comprehensive theory linking fundamental quantum correlations to actionable protocols and tasks across bipartite and multipartite settings.

Abstract

All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schrödinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized.

Quantum entanglement

TL;DR

The paper surveys the multifaceted role of quantum entanglement as a resource in quantum information science. It weaves together foundational concepts (entanglement structure, Bell nonlocality, and entropic criteria) with operational frameworks (LOCC, PPT, and positive maps), detailing both detection and manipulation methods including entanglement witnesses, distillation, and activation. It highlights the divergence between entanglement and nonlocality, the emergence of bound entanglement, and the rich geometry of entangled states, emphasizing practical implications for cryptography, communication complexity, and quantum computation. Collectively, the work outlines a comprehensive theory linking fundamental quantum correlations to actionable protocols and tasks across bipartite and multipartite settings.

Abstract

All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schrödinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized.

Paper Structure

This paper contains 61 sections, 116 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Entanglement as a quantum property of compound systems
  3. Pioneering effects based on entanglement
  4. Pioneering effects based on entanglement
  5. Quantum key distribution based on entanglement
  6. Quantum dense coding
  7. Quantum teleportation
  8. Entanglement swapping
  9. Beating classical communication complexity bounds with entanglement
  10. Correlation manifestations of entanglement: Bell inequalities.
  11. Bell theorem: CHSH inequality.
  12. The optimal CHSH inequality for $2\times2$ systems
  13. Nonlocality of quantum states and LHV model
  14. Pure states
  15. Mixed states
  16. ...and 46 more sections

Figures (3)

  • Figure 1: The concept of state merging: before and after.
  • Figure 2: The line represents hyperplane corresponding to the entanglement witness $W$. All states located to the left of the hyperplane or belonging to it (in particular all separable states) provide nonnegative mean value of the witness, i.e. ${\tr}(W\varrho_{sep}) \ge 0$ while those located to the right are entangled states detected by the witness.
  • Figure 3: Schematic representation of the set of all states with example of entanglement witness and its optimization