The three kinds of three-qubit entanglement

TL;DR

This work defines a cohesive, polynomial framework for three-qubit entanglement by introducing three measures: $c_{a|bc}$ for bipartite entanglement across the $a|bc$ split, $ω$ for W-entanglement, and $τ$ for GHZ-entanglement, all grounded in the Freudenthal triple system (FTS). The authors show that these quantities form an LU-invariant set linked to fundamental invariants $I_0,I_a,I_4,I_5$, with $τ^2=4I_5$ and $ω$ related to the invariants; they prove $ω$ is an entanglement monotone under pure LOCC. A central result is the explicit entanglement ordering $0 ≤ τ ≤ ω ≤ c_{a|bc} ≤ 1$ (and $≤ n^4$ in the unnormalized case), along with extremal-state characterizations: GHZ maximizes $τ$ (=$1$ for normalized states), W maximizes $ω$ within the non-GHZ sector (=$4/(3√3)$), and the $a|bc$-class state $|0⟩_a|φ_B⟩_{bc}$ maximizes the bipartite concurrences to 1. This framework clarifies the hierarchy of three-qubit entanglement and connects algebraic invariants to potential nonlocal advantages, though the authors note open questions about operational interpretations and limitations to qubits.

Abstract

We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement, working in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three equivalence classes of entangled three-qubit states.

Figures (2)

• Figure 1: Illustration of the entanglement ordering \ref{['eq:entorder']} for the superposition $\vert \psi_x \rangle := \sqrt{1-x^2}\vert \psi_\text{W} \rangle + x\vert \psi_\text{GHZ} \rangle$. Some particular points are $\tau(\psi_x)=\omega(\psi_x)=c_{a|bc}(\psi_x)=1$ at $x=\pm1$ (GHZ state), $\tau(\psi_x)=0<\omega(\psi_x)=\sqrt{16/27}<c_{a|bc}(\psi_x)=\sqrt{8/9}$ at $x=0$ (W state), $\omega(\psi_x)=c_{a|bc}(\psi_x)=2/\sqrt{5}$ at $x=\sqrt{2/5}$, and $\tau(\psi_x)=0$ at $x=0$ and at $x=-4\sqrt{8+2\cdot2^{1/3}-3\cdot2^{2/3}}/\sqrt{155}$.
• Figure 2: Illustration of the entanglement ordering \ref{['eq:entorder']} and the maximizations of the measures for the superpositions $\vert \psi_{x\geq0} \rangle := \sqrt{1-x^2}\vert \psi_\text{W} \rangle + x\vert \psi_\text{GHZ} \rangle$ for coefficients $x\geq0$ and $\vert \psi_{x\leq0} \rangle := \sqrt{1-x^2}\vert \psi_\text{W} \rangle - x\vert \psi_{a|bc}' \rangle$ for coefficients $x\leq0$, where we use $\vert \psi_{a|bc}' \rangle := \vert 1 \rangle\otimes\bigl(\vert 01 \rangle+\vert 10 \rangle\bigr)/\sqrt{2}$, LU-equivalent with $\vert \psi_{a|bc} \rangle$, in order that the latter superposition stay in the closure of class W.