Refutation of the geometric tangle ansatz

Abstract

A type-4c canonical state on the Bloch-norm diagonal refutes the Benedito–Sierra geometric tangle ansatz.

Definition 1.1 (Canonical five-term three-qubit state).

$$CanonicalState=\{lambda0:\mathbb{R},lambda1:\mathbb{R},lambda2:\mathbb{R},lambda3:\mathbb{R},lambda4:\mathbb{R},phi:\mathbb{R}\}$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.CanonicalState (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The paper prints verbatim: “|ψ⟩ overset{CD}{→} |λ₀, λ⃗, λ₄; φ⟩ := [ λ₀|000⟩ + λ₁ e^{iφ}|100⟩ + λ₂|101⟩ + λ₃|110⟩ + λ₄|111⟩ ] where λⱼ ∈ [0,1] ∀ j; Σⱼ₌₀⁴ λⱼ² = 1; φ ∈ [0,π].” (arXiv v2, p. 3, Eqs. (3)–(4)). This is the six-parameter canonical state carrier used below.

Definition 1.2 (Canonical parameter conditions).

$$\forall psi\in CanonicalState,(isCanonical\left(psi\right))\iff (\left(\left(0\le lambda0\left(psi\right)\wedge lambda0\left(psi\right)\le 1\right)\wedge \left(\left(0\le lambda1\left(psi\right)\wedge lambda1\left(psi\right)\le 1\right)\wedge \left(\left(0\le lambda2\left(psi\right)\wedge lambda2\left(psi\right)\le 1\right)\wedge \left(\left(0\le lambda3\left(psi\right)\wedge lambda3\left(psi\right)\le 1\right)\wedge \left(0\le lambda4\left(psi\right)\wedge lambda4\left(psi\right)\le 1\right)\right)\right)\right)\right)\wedge \left(lambda0{\left(psi\right)}^2+lambda1{\left(psi\right)}^2+lambda2{\left(psi\right)}^2+lambda3{\left(psi\right)}^2+lambda4{\left(psi\right)}^2=1\wedge \left(0\le phi\left(psi\right)\wedge phi\left(psi\right)\le \pi \right)\right))$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isCanonical (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The five amplitudes lie in [0,1], their squares sum to one, and the phase lies in [0,π].

Definition 1.3 (Complex coefficient tensor).

$$\forall psi\in CanonicalState,amplitudes\left(psi,0,0,0\right)=lambda0\left(psi\right)\wedge \left(amplitudes\left(psi,0,0,1\right)=0\wedge \left(amplitudes\left(psi,0,1,0\right)=0\wedge \left(amplitudes\left(psi,0,1,1\right)=0\wedge \left(amplitudes\left(psi,1,0,0\right)=lambda1\left(psi\right)\cdot exp\left(phi\left(psi\right)\cdot i\right)\wedge \left(amplitudes\left(psi,1,0,1\right)=lambda2\left(psi\right)\wedge \left(amplitudes\left(psi,1,1,0\right)=lambda3\left(psi\right)\wedge amplitudes\left(psi,1,1,1\right)=lambda4\left(psi\right)\right)\right)\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.amplitudes (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The eight coefficients are listed in lexicographic qubit order A, B, C. The phase occurs only in the coefficient λ₁ exp(iφ).

Definition 1.4 (Pure-state density matrix).

$$\forall psi\in CanonicalState,\forall a\in Fin\left(2\right),\forall b\in Fin\left(2\right),\forall c\in Fin\left(2\right),\forall d\in Fin\left(2\right),\forall e\in Fin\left(2\right),\forall f\in Fin\left(2\right),jointDensity\left(psi,(a,(b,c)),(d,(e,f))\right)=amplitudes\left(psi,a,b,c\right)\cdot conj\left(amplitudes\left(psi,d,e,f\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.jointDensity (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The outer product |ψ⟩⟨ψ| has entries t_ijk conjugate(t_i’j’k’) on A × (B × C).

Definition 1.5 (The A marginal).

$$\forall psi\in CanonicalState,rhoA\left(psi\right)=partialTraceRight\left(jointDensity\left(psi\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoA (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

Tracing B and C means summing t_ijk conjugate(t_i’jk) over j,k ∈ Fin 2.

Definition 1.6 (The B marginal).

$$\forall psi\in CanonicalState,rhoB\left(psi\right)=partialTraceRight\left(partialTraceLeft\left(jointDensity\left(psi\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoB (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

First trace A, then C: the entries are the sum of t_ijk conjugate(t_ij’k) over i,k ∈ Fin 2.

Definition 1.7 (The C marginal).

$$\forall psi\in CanonicalState,rhoC\left(psi\right)=partialTraceLeft\left(partialTraceLeft\left(jointDensity\left(psi\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoC (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

First trace A, then B: the entries are the sum of t_ijk conjugate(t_ijk’) over i,j ∈ Fin 2.

Definition 1.8 (Reduced-state Bloch vectors).

$$\forall psi\in CanonicalState,blochVectors\left(psi,0\right)=bloch\left(rhoA\left(psi\right)\right)\wedge \left(blochVectors\left(psi,1\right)=bloch\left(rhoB\left(psi\right)\right)\wedge blochVectors\left(psi,2\right)=bloch\left(rhoC\left(psi\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.blochVectors (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The frozen bloch map extracts (2 Re ρ₀₁, −2 Im ρ₀₁, Re ρ₀₀ − Re ρ₁₁), the coordinates in ρ = (I + r·σ)/2. For the canonical state, A has y = 2λ₀λ₁ sin φ, while B and C have y = −2λ₁λ₃ sin φ and −2λ₁λ₂ sin φ. The coordinate identities are established inside the refutation.

Definition 1.9 (Cayley hyperdeterminant).

$$\forall t\in Fin\left(2\right)\to \left(Fin\left(2\right)\to \left(Fin\left(2\right)\to \mathbb{C}\right)\right),cayley\left(t\right)=t{\left(0,0,0\right)}^2\cdot t{\left(1,1,1\right)}^2+t{\left(0,0,1\right)}^2\cdot t{\left(1,1,0\right)}^2+t{\left(0,1,0\right)}^2\cdot t{\left(1,0,1\right)}^2+t{\left(1,0,0\right)}^2\cdot t{\left(0,1,1\right)}^2-2\cdot \left(t\left(0,0,0\right)\cdot t\left(0,0,1\right)\cdot t\left(1,1,0\right)\cdot t\left(1,1,1\right)+t\left(0,0,0\right)\cdot t\left(0,1,0\right)\cdot t\left(1,0,1\right)\cdot t\left(1,1,1\right)+t\left(0,0,0\right)\cdot t\left(1,0,0\right)\cdot t\left(0,1,1\right)\cdot t\left(1,1,1\right)+t\left(0,0,1\right)\cdot t\left(0,1,0\right)\cdot t\left(1,0,1\right)\cdot t\left(1,1,0\right)+t\left(0,0,1\right)\cdot t\left(1,0,0\right)\cdot t\left(0,1,1\right)\cdot t\left(1,1,0\right)+t\left(0,1,0\right)\cdot t\left(1,0,0\right)\cdot t\left(0,1,1\right)\cdot t\left(1,0,1\right)\right)+4\cdot \left(t\left(0,0,0\right)\cdot t\left(0,1,1\right)\cdot t\left(1,0,1\right)\cdot t\left(1,1,0\right)+t\left(0,0,1\right)\cdot t\left(0,1,0\right)\cdot t\left(1,0,0\right)\cdot t\left(1,1,1\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.cayley (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The quartic has four square-product terms, six mixed terms with coefficient −2 and two mixed terms with coefficient 4.

Definition 1.10 (The five J invariants).

$$\forall psi\in CanonicalState,jInvariants\left(psi,0\right)={\left|lambda1\left(psi\right)\cdot lambda4\left(psi\right)\cdot exp\left(phi\left(psi\right)\cdot i\right)-lambda2\left(psi\right)\cdot lambda3\left(psi\right)\right|}^2\wedge \left(jInvariants\left(psi,1\right)=lambda0{\left(psi\right)}^2\cdot lambda2{\left(psi\right)}^2\wedge \left(jInvariants\left(psi,2\right)=lambda0{\left(psi\right)}^2\cdot lambda3{\left(psi\right)}^2\wedge \left(jInvariants\left(psi,3\right)=lambda0{\left(psi\right)}^2\cdot lambda4{\left(psi\right)}^2\wedge jInvariants\left(psi,4\right)=lambda0{\left(psi\right)}^2\cdot \left({\left|lambda1\left(psi\right)\cdot lambda4\left(psi\right)\cdot exp\left(phi\left(psi\right)\cdot i\right)-lambda2\left(psi\right)\cdot lambda3\left(psi\right)\right|}^2+lambda2{\left(psi\right)}^2\cdot lambda3{\left(psi\right)}^2-lambda1{\left(psi\right)}^2\cdot lambda4{\left(psi\right)}^2\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.jInvariants (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

Entry k is J_(k+1). These are the Acín invariants cited by the source: J₁ = |λ₁λ₄ exp(iφ) − λ₂λ₃|², J₂ = λ₀²λ₂², J₃ = λ₀²λ₃², J₄ = λ₀²λ₄² and J₅ = λ₀²(J₁ + λ₂²λ₃² − λ₁²λ₄²).

Definition 1.11 (The GHZ-class condition).

$$\forall psi\in CanonicalState,(isGHZ\left(psi\right))\iff (lambda0\left(psi\right)\cdot lambda4\left(psi\right)\ne 0)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isGHZ (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The GHZ class condition is λ₀ λ₄ ≠ 0.

Definition 1.12 (The type-5 exclusion).

$$\forall psi\in CanonicalState,(isType5\left(psi\right))\iff (lambda0\left(psi\right)\ne 0\wedge \left(lambda1\left(psi\right)\ne 0\wedge \left(lambda2\left(psi\right)\ne 0\wedge \left(lambda3\left(psi\right)\ne 0\wedge \left(lambda4\left(psi\right)\ne 0\wedge \left(\forall k\in Fin\left(5\right),jInvariants\left(psi,k\right)\ne 0\right)\right)\right)\right)\right))$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isType5 (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

Type 5 requires every λⱼ and every Jₖ to be nonzero. The witness has λ₁ = 0, so the full predicate excludes it without evaluating the J invariants.

Definition 1.13 (Bloch-norm coordinates).

$$\forall psi\in CanonicalState,blochLengths\left(psi\right)=(sqrt\left(\sum _{i:Fin\left(3\right)}(blochVectors{\left(psi,0,i\right)}^2)\right),sqrt\left(\sum _{i:Fin\left(3\right)}(blochVectors{\left(psi,1,i\right)}^2)\right),sqrt\left(\sum _{i:Fin\left(3\right)}(blochVectors{\left(psi,2,i\right)}^2)\right))$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.blochLengths (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The three coordinates are the Euclidean lengths of the Bloch vectors obtained from the literal reduced density matrices. Each finite sum is the squared Euclidean norm.

Definition 1.14 (Squared Bloch-vector norm).

$$\forall rA\in \mathbb{R},\forall rB\in \mathbb{R},\forall rC\in \mathbb{R},normSquared\left((rA,rB,rC)\right)=r{A}^2+r{B}^2+r{C}^2$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.normSquared (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The squared norm of (r_A,r_B,r_C) is the sum of the three squared Bloch lengths.

Definition 1.15 (The main diagonal line).

$$V_{line}:Set\left(EuclideanSpace\left(\mathbb{R},Fin\left(3\right)\right)\right)=Set.range\left((t:\mathbb{R})\mapsto WithLp.toLp\left(2,![t,t,t]\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.V_line (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The main diagonal is the range of t ↦ WithLp.toLp 2 ![t,t,t] in EuclideanSpace ℝ (Fin 3).

Definition 1.16 (The Euclidean Bloch-length vector).

$$\forall rA\in \mathbb{R},\forall rB\in \mathbb{R},\forall rC\in \mathbb{R},euclideanVector\left((rA,rB,rC)\right):EuclideanSpace\left(\mathbb{R},Fin\left(3\right)\right)=WithLp.toLp\left(2,![rA,rB,rC]\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.euclideanVector (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The real triple (r_A,r_B,r_C) is embedded as WithLp.toLp 2 ![r_A,r_B,r_C] in EuclideanSpace ℝ (Fin 3), which carries the Euclidean metric.

Definition 1.17 (Distance to the main diagonal).

$$\forall r\in Prod\left(\mathbb{R},Prod\left(\mathbb{R},\mathbb{R}\right)\right),distanceToDiagonal\left(r\right)=Metric.infDist\left(euclideanVector\left(r\right),V_{line}\right)$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.distanceToDiagonal (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The paper defines d(r⃗,V_line) as the Euclidean distance from r⃗ to the diagonal line. Metric.infDist takes the infimum of the Euclidean distances from euclideanVector(r) to points of V_line.

Definition 1.18 (Canonical three-tangle).

$$\forall psi\in CanonicalState,tangle\left(psi\right)=4\cdot \left|cayley\left(amplitudes\left(psi\right)\right)\right|$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.tangle (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The paper prints verbatim: “τ(ψ) = 4 |Hdet(t_ijk)| = 4 λ₀² λ₄².” (arXiv v2, p. 5, Eq. (10)). The definition uses the full Cayley quartic of the coefficient tensor. On this canonical family the quartic is λ₀²λ₄²; hence the three-tangle is 4λ₀²λ₄².

Definition 1.19 (The Benedito–Sierra geometric ansatz).

$$(claim)\iff (\exists F\in Prod\left(\mathbb{R},Prod\left(\mathbb{R},\mathbb{R}\right)\right)\to \mathbb{R},\left(\forall r\in Prod\left(\mathbb{R},Prod\left(\mathbb{R},\mathbb{R}\right)\right),0\le F\left(r\right)\right)\wedge \left(\forall psi\in CanonicalState,(isCanonical\left(psi\right))\Rightarrow ((isGHZ\left(psi\right))\Rightarrow ((\neg isType5\left(psi\right))\Rightarrow (tangle\left(psi\right)=1-\frac{normSquared\left(blochLengths\left(psi\right)\right)}{3}-distanceToDiagonal\left(blochLengths\left(psi\right)\right)\cdot F\left(blochLengths\left(psi\right)\right))))\right))$$

Formalization. D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.claim (✓ std3).

Citation. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

The paper states verbatim: “τ ( r⃗ ) = 1 − |r⃗|²/3 − d( r⃗, V_line ) · 𝓕(r⃗) where |ψ⟩ ∈ GHZ excluding type 5 and 𝓕(r⃗) ≥ 0.” (arXiv v2, p. 7, Eq. (15)). The formal encoding quantifies a nonnegative real function over all normalized canonical states in the GHZ class that are not type 5, with the one-qubit Bloch lengths and diagonal distance defined above.

Theorem 1.20 (A type-4c diagonal counterexample).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.result (✓ std3). ∎

Resolves. Problems/benedito-sierra-2025-geometric-tangle-ansatz-refutation (refuted) by D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.result.

Source. Repository-derived.

Acknowledgement. A. Benedito; G. Sierra (2025). Visualizing Three-Qubit Entanglement. URL: https://arxiv.org/abs/2505.23638v2.

Commentary.

For ψ = (|000⟩ + |101⟩ + |110⟩ + |111⟩)/2, the canonical parameters are (1/2,0,1/2,1/2,1/2;0). Its reduced states have Bloch lengths (1/2,1/2,1/2), so the distance to V_line is zero. The ansatz therefore gives 3/4, while τ = 4(1/2)²(1/2)² = 1/4. Hence no nonnegative F can satisfy the universal claim.

References

• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.CanonicalState
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.V_line
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.amplitudes
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.blochLengths
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.blochVectors
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.cayley
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.claim
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.distanceToDiagonal
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.euclideanVector
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isCanonical
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isGHZ
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.isType5
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.jInvariants
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.jointDensity
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.normSquared
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.result
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoA
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoB
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.rhoC
• Truth anchor: D5/S3/Quantum/Entanglement/ThreeQubitGeometricTangleRefutation.tangle
• Dependency: D5/S3/Quantum/Information/ActualPureQubitGeometry
• Dependency: D5/S3/Quantum/Information/PartialTraceMutualInformation