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arXiv:quant-ph/0607195 · Quantum Information and Computation

Separability criteria based on the Bloch representation of density matrices

Juan de Vicente

Published Sep 1, 2007• 99 citations
arXivHTMLPDF

Abstract

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

DOI
10.26421/qic7.7-5

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Quantum properties, such as entanglement and coherence, are indispensable resources in various quantum information processing tasks. However, given a quantum state, it is not a simple question to identify whether it is entangled or not. We study the entanglement of quantum systems in arbitrary dimensions using the Bloch representation of density matrix. This approach enables us to consider quantum entanglement geometrically. We first analyze the effect of local Bloch vector on determining whether a quantum state is entangled or separable, and obtain an entanglement criterion related to local Bloch vector, which is stronger than the criterion that only considers the correlation matrix. This criterion is equivalent to the optimized computable cross norm criterion in [14]. Our proof is considered from the perspective of Bloch representation, and there is no need to know the matrix relignment in advance. This criterion also be extended to the problem of the separability of tripartite quantum states. We then combine the local filtering transformation with this entanglement criterion. After applying a suitable local filtering transformation, the quantum state has at most one nonzero Bloch vector, and examples are given to illustrate that the local filtering transformation can help detect some bound states where the original entanglement criterion fails. Moreover, for a finite-dimensional quantum system in which one party is a qubit, we give an analytical form of the combined entanglement criterion.