Families of $k$-positive maps and Schmidt number witnesses from generalized equiangular measurements
Katarzyna Siudzińska
TL;DR
The paper addresses entanglement detection via Schmidt numbers by constructing Schmidt-number witnesses from k-positive maps. It introduces generalized equiangular measurements (GEAM) as a flexible POVM framework built from equiangular tight frames, with explicit trace relations and an equidistant, conical 2-design structure. From GEAM, a family of linear maps $\,\Phi^{(k)}$ is built by mixing depolarizing and signed GEAM components; a closed-form $A_k$ is derived to guarantee $k$-positivity under Mehta’s criterion, yielding corresponding Schmidt-number witnesses via the Choi–Jamiołkowski isomorphism. The work unifies several known POVM classes under GEAM and provides a constructive route to entanglement detection through explicit, computable positivity conditions.
Abstract
Quantum entanglement is an important resource in many modern technologies, like quantum computation or quantum communication and information processing. Therefore, most interest is given to detect and quantify entangled states. Entanglement degree of bipartite mixed quantum states can be quantified using the Schmidt number. Witnesses of the Schmidt numbers are closely related to $k$-positive linear maps, for which there is no general construction. Here, we use the generalized equiangular measurements to define a family of $k$-positive maps and the corresponding Schmidt number witnesses.