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The k-positive region of the diagonal-perturbed Tomiyama maps

Abstract

For every 2 ≤ k ≤ d, the k-positive maps of the diagonal-perturbed Tomiyama family on d × d matrices are exactly those whose parameters lie in the quadrilateral spanned by the identity, the two extremal completely positive maps and the k-positive Tomiyama map.

Definition 1.1 (The diagonal-perturbed Tomiyama family).

Formalization. D5/S3/Quantum/QuantumChannels/TomiyamaDiagonalKPositivity.phi (✓ std3).

Citation. A. Bera, B. Bhattacharya, D. Chruściński (2026). Tomiyama-type maps with a diagonal perturbation. DOI: 10.48550/arXiv.2604.18600. URL: https://arxiv.org/abs/2604.18600v1.

Commentary.

Φ_{α,β} = (1 − α − β) id + α τ₀ + β Δ on d × d complex matrices, where τ₀(X) = tr(X) I/d and Δ keeps the diagonal of X, as in Eq. (6) of arXiv:2604.18600v1.

Definition 1.2 (k-positivity).

Formalization. D5/S3/Quantum/QuantumChannels/TomiyamaDiagonalKPositivity.KPositive (✓ std3).

Citation. A. Bera, B. Bhattacharya, D. Chruściński (2026). Tomiyama-type maps with a diagonal perturbation. DOI: 10.48550/arXiv.2604.18600. URL: https://arxiv.org/abs/2604.18600v1.

Commentary.

A map Φ on d × d matrices is k-positive when id_k ⊗ Φ sends every positive semidefinite kd × kd matrix to a positive semidefinite matrix. The tensor product of maps is the frozen kron of D5/S3/Quantum/Foundation/FiniteKrausChannel, with the identity on k × k matrices as the first factor.

Definition 1.3 (The conjectured quadrilateral).

Formalization. D5/S3/Quantum/QuantumChannels/TomiyamaDiagonalKPositivity.quadrilateral (✓ std3).

Citation. A. Bera, B. Bhattacharya, D. Chruściński (2026). Tomiyama-type maps with a diagonal perturbation. DOI: 10.48550/arXiv.2604.18600. URL: https://arxiv.org/abs/2604.18600v1.

Commentary.

The convex hull in the (α, β) plane of the parameter points of Ψ₀ (the identity), Ψ₁ and Ψ₂ (the two completely positive vertices) and the Tomiyama map 𝒯_k, whose α is kd/(kd − 1).

Definition 1.4 (Conjecture 2.4 for 2 ≤ k ≤ d).

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

Citation. A. Bera, B. Bhattacharya, D. Chruściński (2026). Tomiyama-type maps with a diagonal perturbation. DOI: 10.48550/arXiv.2604.18600. URL: https://arxiv.org/abs/2604.18600v1.

Commentary.

Conjecture 2.4 of arXiv:2604.18600v1, “A set of k-positive maps Φ_{α,β} forms a quadrilateral 𝒫_k = conv{Ψ₀, Ψ₁, Ψ₂, 𝒯_k}”, in the range 2 ≤ k ≤ d.

Definition 1.5 (Fourier rows).

Formalization. D5/S3/Quantum/QuantumChannels/TomiyamaDiagonalKPositivity.fourierRows (✓ std3).

Source. Repository-derived.

Commentary.

For positive d and k ≤ d, the k Fourier rows have entries χ_d(ja), where χ_d is the standard additive character on the residues modulo d and a is embedded from Fin k into Fin d.

Theorem 1.6 (Orthogonality of Fourier rows).

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

Source. Repository-derived.

Commentary.

Distinct Fourier rows are orthogonal, and each has squared norm d. At k = d, division by the square root of d gives a unitary Fourier matrix; conjugation gives the negative-character convention.

Theorem 1.7 (The k-positive region is the quadrilateral).

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

Resolves. Problems/bera-2026-tomiyama-diagonal-k-positivity (proved) by D5/S3/Quantum/QuantumChannels/TomiyamaDiagonalKPositivity.result.

Source. Repository-derived.

Commentary.

Write ‖X‖² = Σ_{i,j} |X_ij|² for the squared Frobenius norm and Q(X) = (α/d)‖X‖² + β Σ_i |X_ii|² + (1 − α − β)|tr X|². For vectors u, w in ℂ^k ⊗ ℂ^d with coefficient matrices U, W, the value of (id_k ⊗ Φ_{α,β})(ww*) on u is Q(UW), so k-positivity means Q ≥ 0 on every UW. Four tests give the four edges: the matrix unit E₁₂ gives α ≥ 0; E₁₁ − E₂₂, which needs k ≥ 2, gives α/d + β ≥ 0; the coordinate projection onto k basis vectors gives (kd − 1)α + d(k − 1)β ≤ kd; and the Fourier projection AA*, with A the first k columns of the normalized d × d discrete Fourier matrix, has rank k and every diagonal entry k/d, so it gives (kd − 1)α + k(d − 1)β ≤ kd for every k ≤ d. Conversely the k-positive parameters form a convex set containing the four vertices, where Q ≥ 0 follows from Cauchy–Schwarz on the diagonal and from |tr(UW)|² ≤ k‖UW‖².

References