Families of $k$-positive maps and Schmidt number witnesses from generalized equiangular measurements

TL;DR

This paper introduces a new family of $k$-positive maps and Schmidt number witnesses derived from generalized equiangular measurements, enabling more efficient entanglement detection and quantification in quantum states.

Contribution

It presents a novel method to construct $k$-positive maps and Schmidt number witnesses using generalized equiangular measurements, improving entanglement quantification.

Findings

Constructed witnesses detect two-parameter entangled states with Schmidt number $k+1$ in any dimension.

The approach offers more efficient entanglement quantification than existing methods.

Provides examples of $k$-positive maps linked to generalized equiangular measurements.

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 measured using the Schmidt number. Witnesses of the Schmidt number 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 linear maps and the corresponding Schmidt number witnesses. We present examples of witnesses that detect two-parameter entangled states with Schmidt number $k+1$ in any dimension. Our approach allows for a more efficient entanglement quantification than other known Schmidt number witnesses constructed from symmetric measurement operators.