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arXiv:quant-ph/0408094

Measures of entanglement in quantum mechanics

Geir Ove Myhr

Published Aug 14, 2004• 6 citations
arXivHTMLPDF

Abstract

I give an overview of some of the most used measures of entanglement. To make the presentation self-contained, a number of concepts from quantum information theory are first explained. Then the structure of bipartite entanglement is studied qualitatively, before a number of bipartite entanglement measures are described, both for pure and mixed states. Results from the study of multipartite systems and continuous variable systems are briefly discussed.

DOI
10.48550/arxiv.quant-ph/0408094

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I give an overview of some of the most used measures of entanglement. To make the presentation self-contained, a number of concepts from quantum information theory are first explained. Then the structure of bipartite entanglement is studied qualitatively, before a number of bipartite entanglement measures are described, both for pure and mixed states. Results from the study of multipartite systems and continuous variable systems are briefly discussed.
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This article presents the basis of a theory of entanglement. We begin with a classical theory of entangled discrete measures. Then, we treat quantum mechanics and discuss the statistics of bounded operators on a Hilbert space in terms of context coefficients. Finally, we combine both topics to develop a general theory of entanglement for quantum states. A measure of entanglement called the entanglement number is introduced. Although this number is related to entanglement robustness, its motivation is not the same and there are some differences. The present article only involves bipartite systems and we leave the study of multipartite systems for later work. Quanta 2020; 9: 7–15.
Characterizing entanglement, including quantifying and distribution of entanglement, which lies at heart of the quantum resource theory, have been investigated extensively ever since Bennett \etal proposed three seminal measures of entanglement in 1996. Up to now, there are numerous measures of entanglement that have been proposed from different point of view and plenty of monogamy relations have been explored which make the distribution of entanglement became more and more clear. While this is relatively easy in the case of pure states, it is much more intricate for the case of mixed quantum states especially with higher dimension and more particles in the system. We present here an overview of the theory along this line. We outline most of the results in this field historically and focus on the finite-dimensional systems. In particular we emphasize the point of view that (i) which yardsticks haven been applied in quantifying entanglement and its distribution, (ii) what are the substantive characteristics and interrelations of these measures and their monogamy relations mathematically by comparing, and (iii) which concepts should be improved or revised and how they were developed accordingly.
Quantum information and computation is one of the new fields of physics and information science. Entanglement is one of the most significant resources in quantum information processing. The most basic quantitative measure of entanglement, particularly for mixed states of a bipartite system is the entanglement of formation. This entanglement of formation is used in the concurrence, another measurement of entanglement. Here I have briefly reviewed the entanglement of formation and the concurrence of bipartite systems. I have discussed that for a three qubit pure state, if A, B and C are the three qubits, then sum of the squared concurrence between A and B and the squared concurrence between A and C never exceeds the squared concurrence between A and BC pair. The residual entanglement, obtained from the aforementioned inequality, is also discussed here. Keywords- Quantum information, entanglement, qubit.
Quantum information theory and quantum computing are theoritical basis of quantum computers. Thanks to entanglement, quantum mechanical systems are provisioned to realize many information processing problems faster than classical counterparts. For example, Shor’s factorization algorithm, Grover’s search algorithm, quantum Fourrier transformation, etc. [1]. Entanglement, is the theoretical basis providing the expected speedups. It can be view in bipartite or multipartite forms. In order to quantify entanglement, some measures are defined. On the other hand, a general and accepted criterion, which can measure the amount of entanglement of multilateral systems, has not yet been found. However, since it must be used in many information processing tasks, the production and processing of multilateral quantum entangled systems is at the top of the hot topics of recent years [5-11]. Much of the work in the basic quantum technologies, such as quantum cryptography, communications, and computers, requires multi-partite entangled systems such as GHZ, W [22,30]. It can be suggested that the quantum entanglement criteria reflects the different properties of the systems. Many recent research has been done in entanglement and its related disciplines like entanglement measures and majorization, etc. [62-71]. In this work, we make an overview of recent research on the topic entanglement monotones/measures with an analitical approach.