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arXiv:1905.07678 · Journal of Physics A Mathematical and Theoretical

On the convex cones arising from classifications of partial entanglement in the three qubit system*

Kyung Hoon Han, Seung-Hyeok Kye

Published Nov 8, 2019• 6 citations
arXivHTMLPDF

Abstract

Abstract In order to classify partial entanglement of multi-partite states, it is natural to consider the convex hulls, intersections and differences of basic convex cones obtained from partially separable states with respect to partitions of systems. In this paper, we consider convex cones consisting of -shaped three qubit states arising in this way. The class of -shaped states includes important classes like Greenberger–Horne–Zeilinger diagonal states. We find all the extreme rays of those convex cones to exhibit corresponding partially separable states. We also give characterizations for those cones which give rise to necessary criteria in terms of diagonal and anti-diagonal entries for general three qubit states.

DOI
10.1088/1751-8121/ab5593

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Abstract We characterize the separability of three qubit GHZ diagonal states in terms of entries. This enables us to check separability of GHZ diagonal states without decomposition into the sum of pure product states. In the course of discussion, we show that the necessary criterion of Gühne (2011 Entanglement criteria and full separability of multi-qubit quantum states Phys. Lett. A 375 406–10 ) for (full) separability of three qubit GHZ diagonal states is sufficient with a simpler formula. The main tool is to use entanglement witnesses which are tri-partite Choi matrices of positive bi-linear maps.
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.
We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial separability classes in the tripartite case contrary to a former 1+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.
We give separability criteria for general multi-qubit states in terms of diagonal and anti-diagonal entries. We define two numbers which are obtained from diagonal and anti-diagonal entries, respectively, and compare them to get criteria. They give rise to characterizations of separability when all the entries are zero except for diagonal and anti-diagonal, like Greenberger-Horne-Zeilinger diagonal states. The criteria is strong enough to get nonzero volume of entanglement with positive partial transposes.
We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases and find three-qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7.