Fractional $k$-positivity: a continuous refinement of the $k$-positive scale
Mohsen Kian
TL;DR
This work introduces a real-parameter refinement of the classical Schmidt-number/k-positivity hierarchies by restricting positivity tests to an α-admissible family of vectors, producing nested cones $\mathsf{K}_\alpha$ and $\mathsf{BP}_\alpha$ and their map counterparts $\mathsf{P}_\alpha$ and $\mathsf{SP}_\alpha$ that interpolate the integer levels. A central contribution is a fractional Kraus theorem, which characterizes $\alpha$-superpositive maps via Kraus operators constrained by a Ky--Fan type singular-value bound, generalizing the rank-$k$ story. The paper also shows that non-integer $\alpha$ breaks CP-post-composition stability, highlighting genuine structural differences from the integer theory, and provides sharp, computable thresholds for canonical symmetric families such as the depolarizing channel and the isotropic slice. These results offer a continuous, computable profile of positivity-cone membership, enriching the operational understanding of quantum entanglement witnesses and completely positive maps. Overall, the fractional framework seamlessly recovers the standard $k$-positive/$k$-superpositive hierarchies at integers while revealing new, analyzable structure in the fractional regime.
Abstract
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 ($α\in[1,d]$), we define closed cones $\mathsf K_α$ 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 $\mathsf P_α$, 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 $\mathsf P_α$ 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.