Fractional $k$-positivity: a continuous refinement of the $k$-positive scale

TL;DR

This paper introduces a continuous refinement of the $k$-positivity hierarchy for maps between matrix algebras, revealing new structural insights and thresholds that interpolate between classical integer levels.

Contribution

It defines fractional $k$-positivity cones, extends the Kraus theorem to fractional levels, and uncovers structural transitions at non-integer parameters.

Findings

Fractional $k$-positivity cones interpolate between classical levels.

Extended Kraus theorem characterizes fractional superpositive maps.

Identified structural transition and thresholds at non-integer parameters.

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 $\alpha$-admissible unit vectors ($\alpha\in[1,d]$), we define closed cones $\mathsf K_\alpha$ of bipartite positive operators that interpolate strictly between successive Schmidt-number cones, together with their dual witness cones. Via the Choi--Jamio\l{}kowski correspondence this yields a matching filtration of map cones $\mathsf P_\alpha$, 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}: $\alpha$-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 $\alpha$ the cones $\mathsf P_\alpha$ 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.