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QUANTUM ENTANGLED STATES

Zheng Y. Zhou

Published Jan 1, 2000• 2 citations
OpenAlex

Abstract

Entanglement is one of the intriguing characteristics of quantum mechanics. We review the history of quantum entangled states and the main advances in the measure of entanglement, the distinguishing of entangled states from separable states and the classification of entangled states.

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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 is one of most peculiar characteristic of Quantum Mechanics (probably the other most mentioned is the particle tunneling through a barrier) because it entails correlations that supersedes classical correlations. It is conceived as the genuine quantum characteristic by many researchers and quite recently it has emerged as a physical resource to produce non-classical task, like quantum teleportation, quantum cryptography and quantum information. Entanglement could be distributed (efficient quantum communication is equivalent to efficient entanglement distribution (Plenio & Virmani, 2007)), concentrated (given an amount of entanglement, equal to the full content of entanglement in n pairs of identical entangled pure state systems, it is possible, using local operations on each system, to concentrate the total amount of entanglement into a smaller number m, m < n, of maximally pure entangled state system (Bennett, et al.; 1996)) and used to perform many quantum information tasks useful to overcome technical restrictions present on classical communication. In this chapter, we review the entanglement of quantum mechanical systems. First, in Section 2, we give a characterization of entanglement in terms of its special features as resource and its mathematical structure. Then, in section 3, we review some of the Bell inequalities. In Sections 4 and 5, we review some of the most important entanglement measures published for two and three entangled systems. After that, in Section 6, we give a characterization of quantum gates and operators by its entanglement power, i. e. the amount of entanglement that they can produce when acting on an arbitrary state. Finally, in Section 7, we briefly review the experimental detection of entanglement.
Quantum entanglements, describing truly quantum couplings, are studied and classified for discrete compound states. We show that classical-quantum correspondences such as quantum encodings can be treated as d-entanglements leading to a special class of separable compound states. The mutual information for the d-compound and for q-compound (entangled) states leads to two different types of entropies for a given quantum state. The first one is the von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the second one is the dimensional entropy, which is achieved at the standard entanglement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-conditional entropy and q-capacity of a quantum noiseless channel, defined as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semiquantum couplings (d-entanglements, or encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra. The entropic measure for essential entanglement is introduced.