We present a general characterization of k-positivity for a positive map in terms of the estimation of the Ky Fan norm of the matrix constructed from the Kraus operators of the associated completely positive map. Combining this with the result given by Takasaki and Tomiyama we construct a family of positive maps between matrix algebras of different dimensions depending on a parameter. The estimate bounds on the parameter to obtain the
k-positivity are better than those derived from the spectral conditions considered by Chruściński and Kossakowski. We further look with special attention at the case where we give the precise bound for the regions of decomposability.
A linear map between matrix algebras corresponds to the Choi matrix in the tensor product of two matrix algebras whose definition depends on matrix units. Paulsen and Shultz [J. Math. Phys. 54, 072201 (2013)] considered the question if one can replace matrix units by another basis of matrix algebras in the definition of the Choi matrix to retain the correspondence between the complete positivity of maps and the positivity of Choi matrices and gave a sufficient condition on basis under which this is true. In this note, we provide necessary and sufficient conditions to see that the Paulsen–Shultz condition is also necessary.
Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.
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 introduce a real-parameter refinement of the classical integer hierarchies underlying Schmidt number, block-positivity, and
k-positivity for maps between matrix algebras. Starting from a compact family of
α-admissible unit vectors (
α∈[1,d]), we define closed cones
Kα of bipartite positive operators that interpolate strictly between successive Schmidt-number cones, together with their dual witness cones. Via the Choi--Jamiołkowski correspondence this yields a matching filtration of map cones
Pα, recovering the usual
k-positive/
k-superpositive classes at integer parameters and complete positivity at the top endpoint.
Two results show that the fractional levels capture genuinely new structure. First, we prove a \emph{fractional Kraus theorem}:
α-superpositive maps are precisely the completely positive maps admitting a Kraus decomposition whose Kraus operators satisfy an explicit singular-value (Ky--Fan) constraint, extending the classical rank-
k characterization. Second, for non-integer
α the cones
Pα fail stability under CP post-composition, highlighting a sharp structural transition away from the integer theory. Finally, we derive sharp thresholds on canonical symmetric families (including the depolarizing ray and the isotropic slice), turning familiar stepwise criteria into continuous, computable profiles.
We show that, as the Choi matrix of a completely positive perturbation direction approaches a singular limit, the shift required for
k-positivity can diverge more slowly than that required for complete positivity. For every
k≥2, we construct a
(k−1)-positive map
Φ and a completely positive map
Γ on
Mk+1 with singular Choi matrix
CΓ. Regularizing
Γ by
Γε=Γ+εΔ, where
Δ(X)=Tr(X)I, we determine the least
t≥0 for which
Φ+tΓε is
j-positive. More precisely,
sj,Γε(Φ)=0 for
j0 is determined explicitly. The fractional exponent is obtained through singular-value estimates and an explicit construction of rank-
k perturbations of a nilpotent Jordan block.