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arXiv:1705.01377 · The European Physical Journal D

Distinguishing different classes of entanglement of three-qubit pure states

Chandan Datta, Satyabrata Adhikari, Arpan Das, Pankaj Agrawal

Published Sep 1, 2018• 14 citations
arXivHTMLPDF
DOI
10.1140/epjd/e2018-90199-2

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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.
Employing the Pauli matrices, we have constructed a few operators, which can be used to distinguish six inequivalent classes of entanglement under SLOCC (stochastic local operation and classical communication) for three-qubit pure states. Motivated by the form of Mermin's operator, these operators have very simple structure and most importantly they can be implemented in an experiment to distinguish the types of entanglement present in a state. We have shown that the measurement of only one operator is sufficient to show that whether the tangle of the state in question is zero or not zero. Thus we are able to detect whether the given state is of GHZ-type or not. Also it is shown that it is possible to detect and classify W-type states or biseparable states or separable states by performing a small number of measurements. Furthermore, we discuss the teleportation scheme of Lee et.al. \cite{lee} and have shown that the partial tangle and hence teleportation fidelity can be computed experimentally. We have also shown that the partial tangle and hence teleportation fidelity can be used to classify genuine entangled state, biseparable state and separable state.
We explore nonlocality of three-qubit pure symmetric states shared between Alice, Bob and Charlie using the Clauser-Horne-Shimony-Holt (CHSH) inequality. We show that the reduced two-qubit states, extracted from an arbitrary pure entangled symmetric three-qubit state, do not violate the CHSH inequality and hence they are CHSH-local . However, when Alice and Bob perform the CHSH test, after conditioning over measurement results of Charlie, nonlocality of the state is revealed. It is also shown that two different classes of three-qubit pure symmetric states, consisting of two and three distinct constituent spinors (qubits) respectively, can be distinguished through the conditional CHSH non-locality test.