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Optimality of the coefficient in the entanglement-breaking norm inequality

Abstract

For every dimension d ≥ 2, every Hermitian traceless basis λ₁, …, λ_{d²−1} with tr(λᵢλⱼ) = 2δᵢⱼ and every real B > d(d − 1)/2, some entanglement-breaking channel has Bloch data (A, c) with ‖A‖_*² + B|c|² > (d − 1)²; hence d(d − 1)/2 is the largest coefficient of |c|² for which the norm inequality for entanglement-breaking channels can hold.

Definition 1.1 (Generalised Bloch bases).

Formalization. D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.IsBlochBasis (✓ std3).

Citation. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

A family λ of d² − 1 complex d × d matrices, indexed by Fin(d² − 1) with d² − 1 computed in the natural numbers, is a generalised Bloch basis when every member is Hermitian and traceless and tr(λᵢλⱼ) equals 2 for i = j and 0 otherwise; ite(p, x, y) denotes x when p holds and y otherwise. Together with the identity such a family spans the d × d matrices, and every density matrix is I/d + ½ Σᵢ rᵢλᵢ with real rᵢ = tr(λᵢρ). For d = 2 the Pauli matrices are an example and for general d the generalised Gell-Mann matrices.

Definition 1.2 (Entanglement-breaking maps in Holevo form).

Formalization. D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.IsEntanglementBreaking (✓ std3).

Citation. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

A linear map φ on complex d × d matrices is entanglement breaking when it has a Holevo form φ(X) = Σ_a tr(E_a X) σ_a with finitely many positive semidefinite E_a summing to the identity and positive semidefinite σ_a of trace 1. MatrixMap is the frozen type of linear maps between matrix spaces of D5/S3/Quantum/Foundation/FiniteKrausChannel, and IsPOVM is the frozen predicate of D5/S3/Quantum/QuantumChannels/ConcealmentKernelNecessityRefutation: every E_a is positive semidefinite and Σ_a E_a = I. A map of this form is completely positive and trace preserving, and (id ⊗ φ)(ρ) is separable for every state ρ.

Definition 1.3 (The linear part of the Bloch representation).

Formalization. D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.blochA (✓ std3).

Citation. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

The real (d² − 1) × (d² − 1) matrix A with A_{ij} = ½ Re tr(λᵢ φ(λⱼ)). For a generalised Bloch basis λ and a trace-preserving, Hermiticity-preserving φ, the Bloch vector r of a state, rᵢ = tr(λᵢρ), is sent to Ar + c, with c the translation part defined next.

Definition 1.4 (The translation part of the Bloch representation).

Formalization. D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.blochC (✓ std3).

Citation. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

The real vector c with cᵢ = Re tr(λᵢ φ(I/d)), the Bloch vector of the image of the maximally mixed state; cast denotes the inclusion of the natural numbers in ℂ.

Definition 1.5 (The coefficient cannot be increased).

Formalization. D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.claim (✓ std3).

Citation. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

Theorem 1 of arXiv:2609.27906v1 states that the Bloch data (A, c) of an entanglement-breaking channel on d × d matrices satisfy ‖A‖*² + (d(d − 1)/2)|c|² ≤ (d − 1)², with ‖A‖* the trace norm and |c| the Euclidean norm, and item 3 of its open questions asks whether the coefficient d(d − 1)/2 is optimal. The inequality with a coefficient κ in place of d(d − 1)/2 becomes stronger as κ grows, so the coefficient is optimal exactly when no larger κ is admissible. The claim states this: for every d ≥ 2, every generalised Bloch basis and every real B > d(d − 1)/2 there is an entanglement-breaking φ with (d − 1)² < ‖A‖_*² + B|c|². Theorem 1 itself is not part of the statement. traceNorm is the frozen trace norm tr √(AᵀA) of D5/S3/Quantum/Foundation/FiniteTraceDistance; here cast denotes the inclusion of the natural numbers in ℝ.

Theorem 1.6 (The coefficient d(d − 1)/2 is optimal).

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

Resolves. Problems/kopel-2026-eb-norm-coefficient (proved) by D5/S3/Quantum/QuantumChannels/EntanglementBreakingNormCoefficient.result.

Source. Repository-derived.

Acknowledgement. E. Kopel (2026). A sharp norm inequality for entanglement-breaking channels. DOI: 10.48550/arXiv.2609.27906. URL: https://arxiv.org/abs/2609.27906v1.

Commentary.

Let P be the matrix unit E₀₀ and φ(X) = tr(X) P. φ has the Holevo form with the one-element measurement {I} and the state P. Every λⱼ is traceless, so φ(λⱼ) = 0 and A = 0, whose trace norm is 0. Since φ(I/d) = P, cᵢ = Re tr(λᵢP) = Re (λᵢ)₀₀, and (λᵢ)₀₀ is real because λᵢ is Hermitian. The d² matrices I/√d and λᵢ/√2 are orthonormal for the Hilbert–Schmidt inner product ⟨M, N⟩ = tr(M†N), hence an orthonormal basis of the d²-dimensional space of d × d matrices, and Parseval’s identity for P gives 1 = tr(P†P) = 1/d + ½ Σᵢ (λᵢ)₀₀². Therefore |c|² = 2(d − 1)/d and ‖A‖_*² + B|c|² = 2(d − 1)B/d, which exceeds (d − 1)² exactly when B > d(d − 1)/2.

References