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arXiv:1311.4149 · Journal of Physics Conference Series

3-qubit entanglement: A Jordan algebraic perspective

L Borsten

Published Sep 10, 2014• 0 citations
arXivHTMLPDF

Abstract

We review work classifying the physically distinct forms of 3-qubit entanglement using the elegant framework of Jordan algebras, Freudenthal triple systems and groups of type E_7. While this framework is, in the first instance, specific to three qubits, it is shown here how the essential features may be naturally generalised to an arbitrary number of qubits.

DOI
10.1088/1742-6596/532/1/012003

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We review work classifying the physically distinct forms of 3-qubit entanglement using the elegant framework of Jordan algebras, Freudenthal triple systems and groups of type E_7. While this framework is, in the first instance, specific to three qubits, it is shown here how the essential features may be naturally generalised to an arbitrary number of qubits.
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 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 present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen. 34 10243). The Hilbert space of the three-qubit system is the 15-dimensional sphere S 15 , which allows for a natural (last) Hopf fibration with S 8 as base and S 7 as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.