bibkey: benedito2025visualizing authors: A. Benedito; G. Sierra year: 2025 title: “Visualizing Three-Qubit Entanglement” doi: null url: https://arxiv.org/abs/2505.23638v2 claim: “The authors state the canonical five-term decomposition, the Bloch-norm coordinates of the one-qubit reduced states, the tangle tau(psi) = 4 |Hdet(t_ijk)| = 4 lambda_0^2 lambda_4^2, and conjecture a geometric ansatz for normalized GHZ states excluding type 5.” strata_touched:

• D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation license: citation-only triage: anchor

Visualizing Three-Qubit Entanglement

A. Benedito and G. Sierra, Visualizing Three-Qubit Entanglement, arXiv:2505.23638v2 (2025), published as Entropy 27(8), 800. The canonical coefficient tensor, finite-sum one-qubit partial traces, Pauli-coordinate Bloch lengths, full type-5 classification and Cayley three-tangle define the quantities in the geometric ansatz.

The paper prints the canonical decomposition (arXiv v2, p. 3, Eqs. (3)–(4)):

“$|\psi ⟩\stackrel{\text{CD}}{\to }|{\lambda }_0,\vec{\lambda },{\lambda }_4;\phi ⟩:=[{\lambda }_0|000⟩+{\lambda }_1{e}^{i\phi }|100⟩+{\lambda }_2|101⟩+{\lambda }_3|110⟩+{\lambda }_4|111⟩]$ where ${\lambda }_j\in [0,1]\forall j;{\sum }_{j=0}^4{\lambda }_j^2=1;\phi \in [0,\pi ]$.”

For a one-qubit reduced state it writes $\rho =\frac{1}{2}(1+\vec{r}\cdot \vec{\sigma })$ and collects the three Bloch lengths in $\vec{r}=(r_A,r_B,r_C)$. It gives (arXiv v2, p. 5, Eq. (10))

“$\tau (\psi )=4\left|\text{Hdet}(t_{ijk})\right|=4{\lambda }_0^2{\lambda }_4^2$.”

The GHZ class is characterized by ${\lambda }_0{\lambda }_4\ne 0$. Its type 4c states have ${\lambda }_1=0$, while type 5 requires all ${\lambda }_j\ne 0$ and all $J_k\ne 0$. The geometric ansatz is printed as (arXiv v2, p. 7, Eq. (15))

“$\tau \left(\vec{r}\right)=1-\frac{{|\vec{r}|}^2}{3}-d\left(\vec{r},V_{\text{line}}\right)\cdot \mathcal{F}\left(\vec{r}\right)$ where $|\psi ⟩\in \text{GHZ}$ excluding type 5 and $\mathcal{F}(\vec{r})\ge 0$.”

The classification refers to the Acín invariants from A. Acín et al., Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States, Phys. Rev. Lett. 85, 1560–1563 (2000), arXiv:quant-ph/0003050, Eq. (23). With ${\mu }_j={\lambda }_j^2$ and $\Delta =|{\lambda }_1{\lambda }_4{e}^{i\phi }-{\lambda }_2{\lambda }_3{|}^2$, they are $J_1=\Delta$, $J_2={\mu }_0{\mu }_2$, $J_3={\mu }_0{\mu }_3$, $J_4={\mu }_0{\mu }_4$ and $J_5={\mu }_0(\Delta +{\mu }_2{\mu }_3-{\mu }_1{\mu }_4)$. The full type-5 predicate requires all five amplitudes and all five invariants to be nonzero.

The source defines the distance in Appendix B, before the displayed distance formula:

“the distance from that point to the straight line spanned by the main diagonal is the length of the vector connecting it to a point on the line such that this vector is perpendicular to the line.”

The formal diagonal is V_line : Set (EuclideanSpace ℝ (Fin 3)) := Set.range (fun t : ℝ => WithLp.toLp 2 ![t,t,t]). The embedding is euclideanVector r := WithLp.toLp 2 ![r.1,r.2.1,r.2.2], and distanceToDiagonal r := Metric.infDist (euclideanVector r) V_line is the infimum of the Euclidean distances to that line.

The literal density matrix has entries $t_{ijk}\overline{t_{i^{\prime }{j}^{\prime }{k}^{\prime }}}$. Its finite-sum partial traces give ${\rho }_A$, ${\rho }_B$ and ${\rho }_C$. The Bloch vectors use $(2\operatorname{Re}ρ_{01}, -2\operatorname{Im}ρ_{01}, \operatorname{Re}ρ_{00} - \operatorname{Re}ρ_{11})$. The three-tangle is $4|\mathrm{Hdet}(t)|$, using Cayley’s full quartic; its canonical specialization $4{\lambda }_0^2{\lambda }_4^2$ follows by substitution.

The formalized counterexample is the type 4c state $\frac{1}{2}(|000⟩+|101⟩+|110⟩+|111⟩)$. Its reduced states have Bloch lengths $(1/2,1/2,1/2)$, so it lies on the diagonal and the distance term vanishes; its tangle is $1/4$, whereas the ansatz gives $3/4$.

Verified locator

https://arxiv.org/abs/quant-ph/0003050, Eq. (23), supplies the cited J invariants.

https://arxiv.org/abs/2505.23638v2, pp. 3, 5 and 7, Eqs. (3)–(4), (10) and (15).