We suggest a dynamical vector model of entanglement in a three-qubit system based on isomorphism between su(4) and so(6) Lie algebras. This model allows one to write an evolution equation for three-qubit entanglement parameters under an arbitrary pairwise qubit coupling. Generalizing a Plücker-type description of three-qubit local invariants we introduce three pairs of real-valued three-dimensional vectors (denoted here as ${\mathbf{A}}_{R,I}$, ${\mathbf{B}}_{R,I}$, and ${\mathbf{C}}_{R,I}$). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$ 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 $a\text{−}b$, $a-c$, and $b\text{−}c$ two-qubit coupling terms generate SO(6) coupling between vectors $\mathbf{A}$ and $\mathbf{B}$, $\mathbf{A}$ and $\mathbf{C}$, and $\mathbf{B}$ and $\mathbf{C}$ correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between $W$, Greenberg-Horne-Zeilinger, and biseparable states.

• Received 31 March 2020
• Accepted 28 July 2020

DOI:https://doi.org/10.1103/PhysRevA.102.032401