Vector Properties of Entanglement in a Three-Qubit System

Abstract

We suggest a dynamical vector model of entanglement in a three qubit system based on isomorphism between $su(4)$ and $so(6)$ Lie algebras. Generalizing Pl\"ucker-type description of three-qubit local invariants we introduce three pairs of real-valued $3D$ vector (denoted here as $A_{R,I}$ , $B_{R,I}$ and $C_{R,I}$). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors $A$, $B$ , $C$ under local $SU(2)$ operations is identical to $SO(3)$ evolution of single-qubit Bloch vectors of qubits $a$, $b$ and $c$ correspondingly. At the same time, general two-qubit $su(4)$ Hamiltonians incorporating $a-b$, $a-c$ and $b-c$ two-qubit coupling terms generate $SO(6)$ coupling between vectors $A$ and $B$, $A$ and $C$, and $B$ and $C$, correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors $A$, $B$, $C$ which can be controlled by single-qubit transformations. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between $W$, Greenberg-Horne-Zeilinger ($GHZ$ ) and biseparable states.