bibkey: vomende2025twopositivebound authors: F. vom Ende, D. Chruściński, G. Kimura, P. Muratore-Ginanneschi year: 2025 title: “Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps” doi: 10.1016/j.laa.2025.10.022 url: https://arxiv.org/abs/2506.02145v1 claim: “Theorem 1 proves the spectral trace bound for 2-positive trace-preserving maps; question (10) asks for the bound for arbitrary conditionally 2-positive maps.” strata_touched:

• D5/S3/Quantum/QuantumChannels/TwoPositiveTransitionTrace
• D5/S3/Quantum/QuantumChannels/TracePreservingEigenvalueBound
• D5/S3/Quantum/QuantumChannels/FaithfulPerronRegularization
• D5/S3/Quantum/QuantumChannels/ConditionalTwoPositiveSpectralBound license: citation-only triage: anchor

Universal bound for 2-positive maps

Verified locator

DOI: 10.1016/j.laa.2025.10.022

Source: https://arxiv.org/abs/2506.02145v1

Theorem 1 and equation (3) appear on page 4 of arXiv v1. Question (10) and conditional 2-positivity appear in §3, “Outlook”, on page 11. The journal reference is Linear Algebra and its Applications 730 (2026), 262–275. The journal text has not been verified; the quotations below are from arXiv v1.

Source statements

Transition matrix, page 6:

given Φ ∈ L(Cd×d) and any orthonormal basis G := {gj}dj=1 of Cd define TG(Φ) ∈ Cd×d (or TG, for short) via (TG)jk := ⟨gj|Φ(|gk⟩⟨gk|)|gj⟩.

Theorem 1, page 4:

Let $\Phi \in \mathcal{L}({\mathbb{C}}^{d\times d})$ be a 2-positive and trace-preserving linear map. Then

$$\mathrm{tr}(\Phi )\le d\min \Re (\sigma (\Phi ))+(d^2-d).$$

Question (10), page 11:

Hence one may wonder whether all (generators of) completely positive maps, resp.~2-positive maps satisfy

$$\mathrm{tr}(\Phi )\le d\min \Re (\sigma (\Phi ))+(d^2-d)\max \Re (\sigma (\Phi ))?$$

A map $L$ is called conditionally 2-positive if $e^{tL}$ is 2-positive for all $t\ge 0$.

In particular, every completely positive, every conditionally completely positive, and every 2-positive map is conditionally 2-positive.

With this, (10) becomes really a conjecture about the spectrum of arbitrary conditionally 2-positive maps.

The trace is the trace of the linear endomorphism of the matrix algebra. The spectrum is its characteristic-polynomial root multiset. Positivity of $i{d}_2\otimes \Phi$ is 2-positivity. No trace-preservation hypothesis occurs in question (10).

Sharpness and neighbouring questions

Remark 1(iii), equation (5), and Example 2 state that the coefficient d in equation (3) cannot be increased, even for completely positive trace-preserving maps. Those maps have maximal real eigenvalue 1, so the same coefficient optimality holds in the larger class of equation (10). This is a literature-based sharpness consequence; the Example 2 family is not formalized in these modules. Attainment for conditionally 2-positive maps beyond the source’s classes is not addressed. The question of a positive trace-preserving map violating equation (3) for $d\ge 3$ remains open here.