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arXiv:2512.18900

Sufficient conditions for the Kadison--Schwarz property of unital positive maps on $M_3$

Adam Rutkowski

quant-phmath.OASubmitted Dec 21, 2025• Announced Dec 23, 2025• 0 citations• Revised Mar 3, 2026 (v3)
v1 · Dec 21, 2025v2 · Dec 23, 2025v3 · Mar 3, 2026
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Abstract

Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying strictly between positivity and complete positivity. Despite their relevance in operator algebras and quantum dynamics, explicit analytic sufficient conditions for the KS property remain scarce beyond low-dimensional or highly symmetric settings. In this work we analyze unital positive linear maps on $M_3$ within the Bloch--Gell--Mann representation and derive explicit analytic sufficient conditions ensuring the Kadison--Schwarz property. The approach exploits unitary equivalence together with structural properties of the Lie algebra $\mathfrak{su}(3)$ and does not rely on numerical optimization or semidefinite-programming methods. A key mechanism is the cancellation of contributions associated with antisymmetric structure constants, which reduces the problem to estimates governed solely by the symmetric tensor $d_{ijk}$. The results clarify how the Kadison--Schwarz property can hold under assumptions substantially weaker than complete positivity and yield a structural criterion for the KS property on $M_3$ in terms of Bloch parameters.

Comments
1. The title has been reformulated. 3. The proof of the main theorem has been expanded and clarified. 4. Addition of an illustrative two-parameter diagonal family $T(t,s)$, highlighting the relation between positivity, the Kadison--Schwarz property, and complete positivity. 5. Reorganization and structural clarification of the Appendix

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Sufficient conditions for the Kadison--Schwarz property of unital positive maps on M3M_3M3​
Kadison--Schwarz (KS) maps form a natural class of positive linear maps lying strictly between positivity and complete positivity. Despite their relevance in operator algebras and quantum dynamics, explicit analytic sufficient conditions for the KS property remain scarce beyond low-dimensional or highly symmetric settings. In this work we analyze unital positive linear maps on M3M_3M3​ within the Bloch--Gell--Mann representation and derive explicit analytic sufficient conditions ensuring the Kadison--Schwarz property. The approach exploits unitary equivalence together with structural properties of the Lie algebra su(3)\mathfrak{su}(3)su(3) and does not rely on numerical optimization or semidefinite-programming methods. A key mechanism is the cancellation of contributions associated with antisymmetric structure constants, which reduces the problem to estimates governed solely by the symmetric tensor dijkd_{ijk}dijk​. The results clarify how the Kadison--Schwarz property can hold under assumptions substantially weaker than complete positivity and yield a structural criterion for the KS property on M3M_3M3​ in terms of Bloch parameters.
The theory of symmetry of quantum mechanical systems is applied to study the structure and properties of several classes of relevant maps in quantum information theory: CPTP, PPT and Schwarz maps. First, we develop the general structure that equivariant maps Φ:A→BΦ:\mathcal A \to \mathcal BΦ:A→B between C∗C^\astC∗-algebras satisfy. Then, we undertake a systematic study of unital, Hermiticity-preserving maps that are equivariant under natural unitary group actions. Schwarz maps satisfy Kadison's inequality Φ(X∗X)≥Φ(X)∗Φ(X)Φ(X^\ast X) \geq Φ(X)^\ast Φ(X)Φ(X∗X)≥Φ(X)∗Φ(X) and form an intermediate class between positive and completely positive maps. We completely classify U(n)U(n)U(n)-equivariant on Mn(C)M_n(\mathbb C)Mn​(C) and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker DU(n)DU(n)DU(n)-equivariance (diagonal unitary symmetry) and for tensor-product symmetries U(n1)⊗U(n2)U(n_1) \otimes U(n_2)U(n1​)⊗U(n2​). In each case, the parameter regions where ΦΦΦ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the U(n)U(n)U(n)-equivariant family satisfies PPT  ⟺  EB\mathrm{PPT} \iff \mathrm{EB}PPT⟺EB, while the DU(2)DU(2)DU(2), symmetric DU(3)DU(3)DU(3), U(2)⊗U(2)U(2) \otimes U(2)U(2)⊗U(2) and U(2)⊗U(3)U(2) \otimes U(3)U(2)⊗U(3), families obey the PPT2\mathrm{PPT}^2PPT2 conjecture through a direct symmetry argument. These results reveal how group symmetry controls the structure of non-completely positive maps and provide new concrete examples where the PPT2\mathrm{PPT}^2PPT2 property holds.
Quantum entanglement is an important phenomenon in quantum information theory. To detect entanglement theoretically, positive but not completely positive maps are used. The Kadison-Schwarz (KS) inequality interpolates between positivity and complete positivity. KS maps may be key to understanding and detecting entanglement. We provide a description of a subset of KS maps on [Formula: see text] that are unital. This allows for the classification of a wider class of positive maps than the well known bistochastic maps. We derive the conditions for a unital map to be a KS map, and provide nontrivial examples of such a map.
Abstract A class of unital qubit maps displaying diagonal unitary and orthogonal symmetries is analyzed. Such maps have already found a lot applications in quantum information theory. We provide a complete characterization of this class of maps showing intricate relation between positivity, operator Schwarz inequality, and complete positivity. Finally, it is shown how to generalize the entire picture beyond unital case (so called generalized Schwarz maps). Interestingly, the first example of Schwarz but not completely positive map found by Choi belongs to our class. As a case study we provide a full characterization of Pauli maps. Our analysis leads to generalization of seminal Fujiwara–Algoet conditions for Pauli quantum channels.
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.
We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.