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Qubit entanglement: A Jordan algebraic perspective?

L. Borsten

Published Aug 17, 2016• 1 citation
OpenAlex

Abstract

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 E7. 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.?Based on a talk given at 3Quantum: Algebra Geometry Information (AGMP Network),

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Invariants for multiple qubits: the case of 3 qubits

DAVID A. MEYER, Noland Wallach

Abstract The problem of quantifying entanglement in multiparticle quantum systems can be addressed using techniques from the invariant theory of Lie groups. We briefly review this theory, and then develop these techniques for application to entanglement of multiple qubits.

Chapman & Hall/CRC eBooks•Feb 14, 2002•13 citations
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 E7. 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.?Based on a talk given at 3Quantum: Algebra Geometry Information (AGMP Network),
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.
Using a generalization of Cayley's hyperdeterminant as a new measure of tripartite fermionic entanglement we obtain the SLOCC classification of three-fermion systems with six ``single-particle'' states. A special subclass of such three-fermion systems is shown to have the same properties as the well-known three-qubit ones. Our results can be presented in a unified way using Freudenthal triple systems based on cubic Jordan algebras. For systems with an arbitrary number of fermions and single-particle states we propose to use the Pl\"ucker relations as a sufficient and necessary condition of separability.