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A discussion on separability criteria

Shengjun Wu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

The separability problem in quantum information theory was discussed and new separability criteria was searched for in order to tell whether a given quantum state is separable or entangled.Some properties of separable states as well as some separability criteria for special cases were obtained,and our discussion was focused on quantum states in a Hilbert space of 2×N or N×2.

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The separability problem in quantum information theory was discussed and new separability criteria was searched for in order to tell whether a given quantum state is separable or entangled.Some properties of separable states as well as some separability criteria for special cases were obtained,and our discussion was focused on quantum states in a Hilbert space of 2×N or N×2.
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Quantum computers have the potentials to solve certain problems faster than classical computers. In quantum computer, entanglement is one of the elements beside superposition. Recently, with the advent of quantum information theory, entanglement has become an important resource for Quantum Information and Computation. The purpose of this paper is to discuss the separability criteria and method of measurement for entanglement. This paper is aimed at viewing the method that has been proposed in previous works in bipartite and multipartite entanglement. The outcome of this paper is to classify the different method that used to measure entanglement for bipartite and multipartite cases including the advantage and disadvantage of each method.
Special types of measurements are widely used in quantum information processing. The task to detect entangled states can arise within an intermediate stage of the protocol to be performed. It is inevitable in many questions, and it is desirable to study separability criteria derived with different choices of selected states. This study aims to examine such criteria for quantum measurements assigned to projective two-designs. The considered approaches use the so-called correlation measure and the corresponding correlation matrix. Separability criteria for multipartite systems are also derived.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
Entanglement as a vital resource for information processing can be described by special properties of the quantum state. Using the well-known Weyl basis we propose a new Bloch decomposition of the quantum state and study its separability problem. This decomposition enables us to find an alternative characterization of the separability based on the correlation matrix. We show that the criterion is effective in detecting entanglement for the isotropic states, Bell-diagonal states and some PPT entangled states. We also use the Weyl operators to construct an detecting operator for quantum teleportation.
Entanglement is an important quantum resource, which can be used in quantum teleportation and quantum computation. How to judge and measure entanglement or separability has become a basic problem in quantum information theory. In this paper, by analyzing the properties of generalized ring Z[i]2n\mathbb{Z}[i]^{{2}^{n}}Z[i]2n, a new method is presented to judge the entanglement or separability of any quantum state in the discrete quantum computing model proposed by Gatti and Lacalle. Different from previous criteria based on matrices, it is relatively simple to operate in mathematical calculation. And if a quantum state is separable, it can calculate the separable mathematical expression. Taking n=2,3n=2,3n=2,3 as examples, the concrete forms of all separable states in the model are presented. It provides a new research perspective for the discrete quantum computing model.