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Cambridge University Press eBooks

Mixed-State Entanglement

Gilad Gour

Published Apr 10, 2025• 0 citations
Publisher

Abstract

Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.

DOI
10.1017/9781009560870.013

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Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.
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A complete chapter on quantum entanglement measures and detection of entanglement is given because there are a wide variety of entanglement measures for pure and mixed states which are used very often. The interesting feature of these routines is the fact that, these operations can be done between any random set of qubits not necessarily contiguous qubits. Key recipes to perform operations like partial tracing and partial transpose are disscussed in detail. After that, it was decided to devote several routines to calculate the entanglement measures like concurrence and its variants, von Neumann entropy, block entropy, Renyi entropy, Q measure, negativity and log negativity, entanglement spectrum and residual entanglement (three tangle). Last but not least, certain entanglement detection procedures like Positive Partial Transpose (the Peres-Horodecki criteria) and the reduction criteria are also explained via examples.
We simulate quantum spin systems and measure entanglement using circuits tailored for near-term quantum computers. Traditional tools like entanglement entropy are limited to pure states and require full state tomography, making them impractical on current hardware. Instead, we employ the novel approach, Positive Partial Transpose (PPT) criterion to efficiently detect pairwise entanglement from two-spin reduced density matrices, applicable to both pure and mixed states. This method enables scalable entanglement detection, providing a practical route to study quantum correlations, phase transitions, and benchmark quantum devices.
This chapter introduces the notion of entanglement, methods of detection, and various types of two atomic ensemble entangled states. Bipartite The bipartite system constitutes is the simplest and most straightforward way of understanding entanglement, where there is a well-developed theory of detecting and quantifying it. We discuss how von Neumann entropy and negativity are two simple and powerful quantifiers of entanglement in pure and mixed systems, respectively. In the absence of a way to topographically reconstruct the density matrix, other correlation-based entanglement criteria are important practically, since they only involve the measurement of some key correlations. Various approaches to this are introduced, such as the Duan--Giedke--Cirac--Zoller criterion, Hillery--Zubairy criteria, entanglement witness approaches, and covariance matrices. Finally, two types of squeezed states for two atomic ensembles are introduced, : namely the one-axis two-spin and two-axis two-spin squeezed states are introduced.
Chapter 12 provides an in-depth exploration of pure-state entanglement. It begins with a clear definition of quantum entanglement for pure states, emphasizing its critical role in quantum computing and communication. The chapter highlights various strategies for entanglement manipulation, encompassing deterministic, stochastic, and approximate methods. Quantification of bipartite entanglement is a key focus, with emphasis on entropy of entanglement and the Ky Fan norm-based entanglement monotones. Additionally, the chapter delves into entanglement catalysis and embezzlement of entanglement, presenting them as a nuanced nonintuitive phenomena that underscore the challenges of entanglement preservation during quantum operations. A notable aspect of this chapter is its connection between entanglement theory and the theory of majorization discussed in Chapter 4. Through a comprehensive treatment of these topics, the chapter equips readers with a robust understanding of the intricacies of pure-state entanglement theory.
The aim of this chapter is to introduce the reader to various aspects of quantum entanglement. A detailed summary of the jargon of important mathematical notions, and concepts used in entanglement theory is provided. Various methods of entanglement generation are discussed followed by an introduction to the methods of detection and quantification of quantum entanglement in terms of various witnesses and measures. A brief account of some well-known applications of quantum entanglement is presented. Finally, a set of problems with solutions is provided, illustrating various important concepts discussed in the chapter.