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Journal of Physics Conference Series

The Freudenthal triple classification of 3-qubit entanglement andSTUblack hole entropy

L. Borsten

Published Dec 31, 2013• 2 citations
Publisher

Abstract

We show that the 3-qubit entanglement classification is naturally encoded by that elegant algebraic structure, well known from supergravity, the Freudenthal triple system . In particular, we show that the six entanglement classes: (1) Separable A-B-C , (2a) Biseparable A-BC , (2b) Biseparable B-CA , (2c) Biseparable C-AB , (3) W and (4) Greenberger-Horne-Zeilinger correspond respectively to the Freudenthal triple ranks 1, 2a, 2b, 2c, 3, 4. Based on work done in collaboration with D. Dahanayake, M. J. Duff, H. Ebrahim and W. Rubens.

DOI
10.1088/1742-6596/462/1/012005

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We show that the 3-qubit entanglement classification is naturally encoded by that elegant algebraic structure, well known from supergravity, the Freudenthal triple system . In particular, we show that the six entanglement classes: (1) Separable A-B-C , (2a) Biseparable A-BC , (2b) Biseparable B-CA , (2c) Biseparable C-AB , (3) W and (4) Greenberger-Horne-Zeilinger correspond respectively to the Freudenthal triple ranks 1, 2a, 2b, 2c, 3, 4. Based on work done in collaboration with D. Dahanayake, M. J. Duff, H. Ebrahim and W. Rubens.
We show that the three-qubit entanglement classes, (0) null, (1) separable $A\text{\ensuremath{-}}B\text{\ensuremath{-}}C$, (2a) biseparable $A\text{\ensuremath{-}}BC$, (2b) biseparable $B\text{\ensuremath{-}}CA$, (2c) biseparable $C\text{\ensuremath{-}}AB$, (3) WWW, and (4) Greenberger-Horne-Zeilinger, correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3, and 4 of a Freudenthal triple system defined over the Jordan algebra $\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}\ensuremath{\bigoplus}\mathbb{C}$. We also compute the corresponding stochastic local operations and classical communication orbits.
We introduce a classification of mixed three-qubit states, in which we define the classes of separable, biseparable, W, and Greenberger-Horne-Zeilinger states. These classes are successively embedded into each other. We show that contrary to pure W-type states, the mixed W class is not of measure zero. We construct witness operators that detect the class of a mixed state. We discuss the conjecture that all entangled states with positive partial transpose (PPTES) belong to the W class. Finally, we present a new family of PPTES "edge" states with maximal ranks.
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.