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arXiv:0812.3322 · Physical Review A

Freudenthal triple classification of three-qubit entanglement

L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, Hajar Ebrahim

Published Sep 22, 2009• 57 citations
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Abstract

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) $W$, 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.

DOI
10.1103/physreva.80.032326

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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.
The hierarchy of nonlocality and entanglement in multipartite systems is one of the fundamental problems in quantum physics. We study this topic in three-qubit systems considering the entanglement classification of stochastic local operations and classical communication (SLOCC). The equivalence under SLOCC divides three-qubit states into separable, biseparable, WWW, and Greenberger-Horne-Zeilinger (GHZ) classes. The WWW and GHZ are two subclasses of genuine tripartite entanglement. We adopt the family of GHZ-symmetric states as a research subject, which share the symmetries of the GHZ state and have a complete characterization of SLOCC classes. In the biseparable region (with bipartite entanglement), there exist GHZ-symmetric states that are found to be fully local. In addition, there are bilocal states in both the WWW and GHZ classes. That is, neither of the subclasses of genuine tripartite entanglement can ensure genuinely tripartite nonlocality.
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.
Many-body quantum systems can be characterized using the notions of kkk-separability and entanglement depth. A quantum state is kkk-separable if it can be expressed as a mixture of kkk entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-kkk-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, WWW-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-kkk-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.