The qutrit Weyl–Bloch one-norm

Abstract

The two-qutrit Weyl–Bloch entrywise one-norm is bounded by 25 on separable states.

Definition 1.1 (The qutrit Weyl operators).

$$\forall k\in \mathrm{Fin}\left(3\right),\forall l\in \mathrm{Fin}\left(3\right),\forall j\in \mathrm{Fin}\left(3\right),\forall c\in \mathrm{Fin}\left(3\right),W\left(k,l,j,c\right)=\mathrm{if}\left(c=j+l,\mathrm{omega}{\left(\right)}^{\mathrm{val}\left(j\right)\cdot \mathrm{val}\left(k\right)},0\right)$$

Formalization. D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.W (✓ std3).

Citation. Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr (2026). Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits. DOI: 10.1103/z7jn-4try. URL: https://arxiv.org/abs/2508.18393v2.

Commentary.

For row j and column c, W(k,l) has the cubic phase omega to j times k when c equals j+l modulo three, and is zero otherwise. omega denotes D5.S3.QuantumContext.HesseSicCertificate.omega: the complex exponential of two pi times the imaginary unit divided by three.

Definition 1.2 (The Weyl–Bloch coefficient).

$$\forall rho\in \mathrm{Matrix}\left(\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Complex}\left(\right)\right),\forall i\in \mathrm{Fin}\left(3\right),\forall j\in \mathrm{Fin}\left(3\right),\forall k\in \mathrm{Fin}\left(3\right),\forall l\in \mathrm{Fin}\left(3\right),\mathrm{bloch}\left(rho,i,j,k,l\right)=\mathrm{trace}\left(rho\cdot \mathrm{conjTranspose}\left(\mathrm{kronecker}\left(W\left(i,j\right),W\left(k,l\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.bloch (✓ std3).

Citation. Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr (2026). Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits. DOI: 10.1103/z7jn-4try. URL: https://arxiv.org/abs/2508.18393v2.

Commentary.

The coefficient is the trace pairing with the conjugate transpose of a tensor product of two Weyl operators.

Definition 1.3 (The entrywise one-norm).

$$\forall rho\in \mathrm{Matrix}\left(\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Complex}\left(\right)\right),l1\left(rho\right)=\sum _{i:\mathrm{Fin}\left(3\right)}(\sum _{j:\mathrm{Fin}\left(3\right)}(\sum _{k:\mathrm{Fin}\left(3\right)}(\sum _{l:\mathrm{Fin}\left(3\right)}(\left|\mathrm{bloch}\left(rho,i,j,k,l\right)\right|))))$$

Formalization. D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.l1 (✓ std3).

Citation. Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr (2026). Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits. DOI: 10.1103/z7jn-4try. URL: https://arxiv.org/abs/2508.18393v2.

Commentary.

The quantity l1 is the sum of the complex norms of all 81 Weyl–Bloch coefficients.

Definition 1.4 (The sharp separable bound and an entangled violation).

$$(\mathrm{claim}\left(\right))\iff ((\forall rho\in \mathrm{Matrix}\left(\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Complex}\left(\right)\right),(\mathrm{IsSeparable}\left(rho\right))\Rightarrow (l1\left(rho\right)\le 25))\wedge ((\exists rho\in \mathrm{Matrix}\left(\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right),\mathrm{Complex}\left(\right)\right),(\mathrm{IsSeparable}\left(rho\right))\wedge (l1\left(rho\right)=25))\wedge (\exists psi\in \mathrm{Prod}\left(\mathrm{Fin}\left(3\right),\mathrm{Fin}\left(3\right)\right)\to \mathrm{Complex}\left(\right),(\mathrm{dotProduct}\left(\mathrm{star}\left(psi\right),psi\right)=1)\wedge (25<l1\left(\mathrm{vecMulVec}\left(psi,\mathrm{star}\left(psi\right)\right)\right)))))$$

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

Citation. Tobias C. Sutter, Christopher Popp and Beatrix C. Hiesmayr (2026). Group-Theoretic Perspective on the PPT and Realignment Criteria in the Magic Simplex for Bipartite Qutrits. DOI: 10.1103/z7jn-4try. URL: https://arxiv.org/abs/2508.18393v2.

Commentary.

Every separable state has l1 at most 25, a separable state attains 25, and a normalized pure two-qutrit vector produces a Weyl–Bloch one-norm strictly above 25.

Theorem 1.5 (The sharp separable bound).

$$\mathrm{claim}\left(\right)$$

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

Resolves. Problems/sutter-popp-hiesmayr-2025-qutrit-weyl-bloch-norm-bound (proved) by D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.result.

Source. Repository-derived.

Commentary.

For every qutrit density matrix, the squared Weyl coefficients sum to at most three. The identity coefficient is one, leaving a squared sum at most two over the other eight coefficients. Cauchy–Schwarz bounds their sum of moduli by four. The local one-norm is therefore at most five.

Weyl coefficients factor on tensor products. The product one-norm is at most twenty-five, and convexity preserves this bound for every finite separable mixture.

The vector (0,1,-1)/sqrt(2) gives a qutrit density matrix with local one-norm five; its tensor square attains twenty-five. The normalized two-qutrit vector (2|00>+|02>+|11>-|20>)/sqrt(7) has one-norm (169+2sqrt(13))/7, which exceeds twenty-five. Thus the separable threshold is sharp and its strict violation detects entanglement.

References

• Truth anchor: D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.W
• Truth anchor: D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.bloch
• Truth anchor: D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.claim
• Truth anchor: D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.l1
• Truth anchor: D5/S3/Quantum/Entanglement/QutritWeylBlochNormSeparableBound.result
• Dependency: D5/S3/Quantum/Entanglement/GHZMeasureBiseparableBound
• Dependency: D5/S3/Quantum/Information/SLDSpectralEnergy
• Dependency: D5/S3/Quantum/QuantumChannels/IptPptSquaredEquivalence
• Dependency: D5/S3/QuantumContext/HesseSicCertificate