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Journal of Physics Conference Series

Global and Bipartite Entanglement for Three-Qubit System Local Unitary Classes

S Luna-Hernández

Published Feb 1, 2023• 2 citations
Publisher

Abstract

Abstract Entanglement of multipartite quantum systems is an important resource in the field of quantum information processing. Using a measure of global entanglement and the notion of concurrence, the properties of entanglement for a three-qubit system are studied and classified in terms of local unitary classes. The connection of such classification with the three-qubit entanglement polytope is also discussed.

DOI
10.1088/1742-6596/2448/1/012020

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International Journal of Advanced Computer Science and Applications•Jan 1, 2019•6 citations
Abstract Entanglement of multipartite quantum systems is an important resource in the field of quantum information processing. Using a measure of global entanglement and the notion of concurrence, the properties of entanglement for a three-qubit system are studied and classified in terms of local unitary classes. The connection of such classification with the three-qubit entanglement polytope is also discussed.
In this study, local unitary (LU) properties of two- and three-qubit quantum systems are studied. Specifically, the Schmidt decomposition approach to LU classification for two qubits is re-examined using a more general approach, i.e. higher order singular value decomposition (HOSVD). Later, HOSVD is used to classify three-qubit pure states by LU operations. We found that due to HOSVD, it is possible to characterize the entanglement classes of three qubits by the eigenvalues distribution of its one-particle reduced density matrices. This finding generalized the similar classification results in the literature and it is hoped that this work will fill in the gap of LU classification from bipartite to multipartite quantum states by using HOSVD.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.
Several entanglement measures are used to define equivalence classes in the set of hypergraph states of three qubits. Our classifications reveal that (i) under local unitary transformations, hypergraph states of three qubits are split into six classes and only one of them is not equivalent to any graph state; (ii) under stochastic local operations with classical communication, for the single-copy case hypergraph states of three qubits, partitioned into five classes which cannot be converted into a WWW state, are equivalent to graph states; and (iii) when bipartite entanglement in three qubits is considered, hypergraph states of three qubits are split into five classes and only one of them has the same entangled graph as the WWW state.
We investigate the equivalence of quantum states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of non-generic three-qubit mixed states. It is shown that two such states in this class are locally equivalent if and only if all these invariants have equal values for them. PACS numbers: 03.67.-a, 02.20.Hj, 03.65.-w Key words: Invariant, quantum state, local unitary transformation Quantum entangled states are playing very important roles in quantum information processing and quantum computation [1]. As the properties of entanglement of a multipartite quantum system remain invariant under local unitary transformations on the subsystems, the entanglement can be characterized by all the invariants of the local unitary transformations. The trace norms of realigned or partial transposed density matrices in entanglement measure and separability criteria are some of these invariants [2]. Invariants are also relevant in the discussions of Bell inequalities [3, 4] and teleportation, etc. [5]. A complete
Entanglement is a physical phenomenon that links a pair, or a set of particles that correlates with each other, regardless of the distance between them. Recent researches conducted on entanglement are mostly focused on measurement and classification in multiqubit systems. Classification of two qubits will only distinguish the quantum state as either separable or entangled, and it can be done by measurement. Meanwhile, in a three-qubit system, it becomes more complex because of the structure of the three qubits itself. It is not sufficient to do measurement because the states are divided into three types, including fully separable state, biseparable state, and genuine entangled state. Therefore, the classification is needed to distinguish the type of states in the three-qubit system. This study aims to classify the entanglement of three-qubit pure states using a combination model of special unitary groups, SU(2) and SU(4), by changing the angle of selected parameters in SU(4) and acting on the separable pure state. The matrix represents SU(2) is 2×2 matrix while matrix for SU(4) is 4×4 matrix. Hence, the combination of SU(2) and SU(4) represent 8×8 matrix. This classification uses the von Neumann entropy and three tangle measurements to classify the class, respectively. The results of this study have indicated that the three-qubit pure state has been successfully classified into different classes, namely, A-B-C, A-BC, C-AB, GHZ, and W, with A-B-C being a fully separable state, A-BC and C-AB are biseparable states, and GHZ and W are genuine entangled states. The results show that this model can change separable pure states to other entanglement class after transformation is done.