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
M3 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) 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
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
M3 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→B between
C∗-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) and form an intermediate class between positive and completely positive maps. We completely classify
U(n)-equivariant on
Mn(C) and determine those that are completely positive and Schwarz. Partial classifications are then obtained for the weaker
DU(n)-equivariance (diagonal unitary symmetry) and for tensor-product symmetries
U(n1)⊗U(n2). In each case, the parameter regions where
Φ is Schwarz or completely positive are described by explicit algebraic inequalities, and their geometry is illustrated. Finally, we further show that the
U(n)-equivariant family satisfies
PPT⟺EB, while the
DU(2), symmetric
DU(3),
U(2)⊗U(2) and
U(2)⊗U(3), families obey the
PPT2 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 property holds.
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 apply random matrix and free probability techniques to the study of linear maps of interest in quantum information theory. Random quantum channels have already been widely investigated with spectacular success. Here, we are interested in more general maps, asking only for k-positivity instead of the complete positivity required of quantum channels. Unlike the theory of completely positive maps, the theory of k-positive maps is far from being completely understood, and our techniques give many new parametrized families of such maps. We also establish a conceptual link with free probability theory and show that our constructions can be obtained to some extent without random techniques in the setup of free products of von Neumann algebras. Finally, we study the properties of our examples and show that for some parameters, they are indecomposable. In particular, they can be used to detect the presence of entanglement missed by the partial transposition test, that is, positive partial transposition entanglement. As an application, we considerably refine our understanding of positive partial transposition states in the case where one of the spaces is large, whereas the other one remains small.
In this paper, we discuss some general connections between the notions of positive map, weak majorization and entropic inequalities in the context of detection of entanglement among bipartite quantum systems. First, basing on the fact that any positive map can be written as the difference between two completely positive maps Λ=Λ 1 −Λ 2 , we propose a possible way to generalize the Nielsen–Kempe majorization criterion. Then, we present two methods of derivation of some general classes of entropic inequalities useful for the detection of entanglement. While the first one follows from the aforementioned generalized majorization relation and the concept of Schur-concave decreasing functions, the second is based on some functional inequalities. What is important is that, contrary to the Nielsen–Kempe majorization criterion and entropic inequalities, our criteria allow for the detection of entangled states with positive partial transposition when using indecomposable positive maps. We also point out that if a state with at least one maximally mixed subsystem is detected by some necessary criterion based on the positive map Λ, then there exist entropic inequalities derived from Λ (by both procedures) that also detect this state. In this sense, they are equivalent to the necessary criterion [ I ⊗Λ](ϱ AB )⩾0. Moreover, our inequalities provide a way of constructing multi-copy entanglement witnesses and therefore are promising from the experimental point of view. Finally, we discuss some of the derived inequalities in the context of the recently introduced protocol of state merging and the possibility of approximating the mean value of a linear entanglement witness.