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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Separable states and separability criteria

Shengjun Wu, J. Anandan

Published Aug 5, 2003• 0 citations
Publisher

Abstract

A quantum state which can be written as a convex sum of product states is a separable state; such a convex sum allows for a local description. There are many different local descriptions for a given separable state, and some new results about the minimal number of product states in the convex sum wil be given in this paper. Any quantum state is either entangled or separable. However it is very difficult to tell whether a given state is separable or entangled. This difficulty has not been solved yet although a lot of separability criteria have already be proposed. The various separability criteria will also be discussed.

DOI
10.1117/12.492907

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A quantum state which can be written as a convex sum of product states is a separable state; such a convex sum allows for a local description. There are many different local descriptions for a given separable state, and some new results about the minimal number of product states in the convex sum wil be given in this paper. Any quantum state is either entangled or separable. However it is very difficult to tell whether a given state is separable or entangled. This difficulty has not been solved yet although a lot of separability criteria have already be proposed. The various separability criteria will also be discussed.
We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.
When developing and operating quantum computers, it is necessary to be able to check quantum registers for the presence of quantum entanglement between its subsystems, due to which quantum computers gain computational advantage in solving many problems over classical computers. The state in which a quantum system consists of two or more subsystems that are not entangled with each other is called separable.In this article, we provide an overview of various existing approaches to solving the problem of determining separability in a quantum system. To do this, we provide an overview of works on this topic, describing both established and new approaches, highlighting directions in which new possibilities for solving the problem can be explored. We provide a definition of the problem of checking a system for separability, based on which we describe several applicable separability criteria. We provide a model-theoretic approach to solving the separability problem, describing a quantum system as a blurry model.
We extend the classification of mixed states of quantum systems composed of arbitrary numbers of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+11+18+11+18+1 partial separability classes in the tripartite case contrary to a former 1+8+11+8+11+8+1. Then we give necessary and sufficient criteria for these classes, which make it possible to determine to which class a mixed state belongs. These criteria are given by convex roof extensions of functions defined on pure states. In the special case of three-qubit systems, we define a different set of such functions with the help of the Freudenthal triple system approach of three-qubit entanglement.