Entanglement Detection Based on the Generalized Landau-Pollak Uncertainty Relation for $(N, M)$-POVMs

In this paper, we introduce a criterion for detecting entanglement based on the uncertainty relation. Our focus is on the generalized Landau-Pollak uncertainty relation and the generalized quantum measurements <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N, M)$</tex>-POVMs for quantum state tomography. We introduce a generalized Landau-Pollak uncertainty relation for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(N, M)$</tex>-POVMs, which is tighter than other inequalities of its kind. Using this relation, we introduce the criterion for detecting entanglement. In this criterion, a relation between the certainty of a bipartite system and the certainty of the subsystem plays the important role. We also demonstrate the ability to detect both the maximally entangled state and the isortopic state under certain parameters.