bibkey: bera2026tomiyama authors: A. Bera, B. Bhattacharya, D. Chruściński year: 2026 title: “Tomiyama-type maps with a diagonal perturbation” doi: 10.48550/arXiv.2604.18600 url: https://arxiv.org/abs/2604.18600v1 claim: “Conjecture 2.4: the k-positive maps of the family Φ_{α,β} = (1−α−β) id + α τ₀ + β Δ on M_d form the quadrilateral conv{Ψ₀, Ψ₁, Ψ₂, 𝒯_k}; proved by the authors for β ≤ 0, for k dividing d and for k = d − 1, and stated as open in general.” strata_touched:

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Bera, Bhattacharya and Chruściński, Tomiyama-type maps with a diagonal perturbation

A. Bera, B. Bhattacharya and D. Chruściński, Tomiyama-type maps with a diagonal perturbation, arXiv:2604.18600v1 (9 April 2026; math.RA, cross-listed math.OA and quant-ph).

Verified locator

DOI: 10.48550/arXiv.2604.18600. Primary version: https://arxiv.org/abs/2604.18600v1 (the only arXiv version). The TeX source Tomiyama_plus.tex of v1 supplies Eq. (6) (the family), §2.1 (positivity, complete positivity and the parameter points), §2.2 (Conjecture 2.4 and Propositions 2.5, 2.6 and 2.8) and §4. Numbering is the printed one; the caption of Figure 2 calls the conjecture “Conjecture 3.1”.

Source statements

Eq. (6): “${\Phi }_{\alpha ,\beta }=(1-\alpha -\beta )\mathcal{I}+\alpha {\tau }_0+\beta \Delta ,\alpha ,\beta \in \mathbb{R}$”, with “\tau_0(X) = \frac{\I}{d} \Tr\, X” and “$\Delta (X)$ denotes a projection to the diagonal of X \in \Md”.

Parameter points (§2.1): “${\Psi }_0\leftrightarrow (0,0),{\Psi }_1\leftrightarrow \left(0,\frac{d}{d-1}\right),{\Psi }_2\leftrightarrow \left(\frac{d}{d-1},-\frac{1}{d-1}\right)$”; “${\mathcal{T}}_1\leftrightarrow \left(\frac{d}{d-1},0\right),\mathcal{P}\leftrightarrow \left(\frac{d}{d-1},-\frac{2}{d-1}\right)$”; and from Tomiyama’s theorem “${\alpha }_k:=\frac{kd}{kd-1}$”, “${\mathcal{T}}_k:={\Phi }_{{\alpha }_k}$”.

Corollary 2.3: “a set of positive maps forms a quadrilateral containing CP triangle ${\mathcal{P}}_1=\mathrm{conv}\{{\Psi }_0,{\Psi }_1,{\mathcal{T}}_1,\mathcal{P}\}$”.

Conjecture 2.4: “A set of $k$-positive maps ${\Phi }_{\alpha ,\beta }$ forms a quadrilateral ${\mathcal{P}}_k=\mathrm{conv}\{{\Psi }_0,{\Psi }_1,{\Psi }_2,{\mathcal{T}}_k\}$.” Followed by: “Since CP maps $\{{\Psi }_0,{\Psi }_1,{\Psi }_2\}$ are $k$-positive the above set contains $k$-positive maps only. Our goal is to show that if ${\Phi }_{\alpha ,\beta }$ is $k$-positive it must belong to ${\mathcal{P}}_k$. For $k=1$ it reduces to Corollary \ref{cor1}.”

§4: “The full proof of the conjectured characterization of $k$–positive maps for arbitrary $k$ and $d$ is still missing.”

Scope

The authors prove the conjecture for $\beta \le 0$ (Proposition 2.5, using the vector ${\sum }_{i\le k}{e}_i\otimes e_i$), for $k\mid d$ (Proposition 2.6, using block vectors of $\ell =d/k$ coordinates) and for $k=d-1$ (Proposition 2.8). The open part is $\beta \ge 0$ for the other $k$. The sentence about $k=1$ does not hold as printed: $\mathcal{P}\in {\mathcal{P}}_1$ violates $\alpha /d+\beta \ge 0$, so ${\mathcal{P}}_1\ne \mathrm{conv}\{{\Psi }_0,{\Psi }_1,{\Psi }_2,{\mathcal{T}}_1\}$.