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An introduction to the theory of quantum entanglement

Fei Shao -

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.

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•Sep 28, 2006•0 citations
Quantum entangled states play very important roles in quantum information processing,such as quantum teleportation,dense coding,error correction,cryptography,and computation.The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states,as well as its applications in quantum information processing.We present some basic concepts and results,including:the relationship between locality and reality in quantum mechanics and Bell inequalities,and that between Bell inequalities and the separability of quantum states;the definition of separability for pure and mixed quantum states,and some separability criteria such as the positive map approach,positive partial transpose,reduction criterion,realignment,entanglement witnesses,and the covariance matrix approach and local uncertainty relations;some entanglement measures such as the entanglement of formation,concurrence,relative entropy,negativity,tangle,entanglement of assistance,and their estimation;the change in entanglement when the physical system evolves or undergoes interaction with the environment.
Entanglement finds wide application in the development of quantum information theory. After reviewing states in classical mechanics, this chapter discusses the fundamental aspects of characterization of quantum entanglement like the definition, the detection, the classification and quantification of entanglement. The density operator is used to describe the state of a quantum system in a more general way than provided by the state vector of the Hilbert space. Multipartite entanglement theory is much complex compared to the bipartite case. One cannot generalize the results of bipartite entanglement to multipartite entanglement. The Peres criterion known as positive partial transpose criterion is an important criterion to find the separability of mixed states at least for low-dimensional systems. B.W. Terhal introduced the term entanglement witnesses for operators detecting quantum entanglement states. It is found that any entangled state can be detected by some entanglement witnesses.
Quantum entanglement plays crucial roles in quantum information processing. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. In this review, we introduce some recent results in the theory of quantum entanglement. In particular separability criteria based on the Bloch representation, covariance matrix, normal form and entanglement witness, lower bounds, subadditivity property of concurrence and tangle, fully entangled fraction related to the optimal fidelity of quantum teleportation, and entanglement distillation will be discussed in detail.
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.
All our former experience with application of quantum theory seems to say that what is predicted by quantum formalism must occur in the laboratory. But the essence of quantum formalism---entanglement, recognized by Einstein, Podolsky, Rosen, and Schr\"odinger---waited over 70years70\phantom{\rule{0.3em}{0ex}}\text{years}70years to enter laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, has potential for many quantum processes, including canonical ones: quantum cryptography, quantum teleportation, and dense coding. However, it appears that this new resource is complex and difficult to detect. Although it is usually fragile to the 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 quantification. In particular, various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, and quantum cryptography are discussed, and some interrelations are pointed out. The basic role of entanglement in quantum communication within a distant laboratory paradigm is stressed, and some peculiarities such as the irreversibility of entanglement manipulations are also discussed including its extremal form---the bound entanglement phenomenon. The basic role of entanglement witnesses in detection of entanglement is emphasized.
Abstract This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.