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.
Let $\Phi:M_d(\C)\to M_d(\C)$ be a completely positive map with a transpose-symmetric Kraus representation. Its Choi operator determines the rank-two constant \[ \beta_2(\Phi) =\frac12\sup_{\substack{\norm z=1\\ \SR(z)\le2}} \ip z{J(\Phi)z}. \] We prove that, for every
r≥1, \[ \Phi\circ\tau+(r-1)\beta_2(\Phi) \left(\Delta-\frac1r\id\right) \] is
r-positive, where
τ is transposition and $\Delta(X)=\Tr(X)I$. The proof turns the negative spectrum of a partially transposed rank-one operator into the edges of the complete graph
Kr. Regrouping by vertices costs its degree
r−1, rather than its
(2r) edges. Transpose symmetry protects one direction and is exactly what produces the term $-\id/r$. For
Φ=ΛU⊗ΛV, with $\Lambda_E(X)=\Tr(X)I-X^T$, a sharp double-skew singular-value estimate gives
β2≤1, with equality when both local dimensions are at least two. A map identity then yields, for every operator
C of rank at most
r on
U⊗V, \[ \norm{\Tr_U C}_2^2+\norm{\Tr_V C}_2^2 \le r\norm C_2^2+\frac1r\abs{\Tr C}^2, \] with dimension-uniform coefficients that are sharp whenever one local dimension is at least
r and the other is at least two. A second family, built from the edges of an arbitrary graph, has the same local constant and attains the universal coefficient
r−1 whenever the graph contains an
r-clique. The Choi lifting chain through the partial-trace and graph results, together with its sharpness witnesses, is formalized in Lean~4.
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.
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.
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.