Hierarchy of entanglement detection criteria for random high-dimensional states

TL;DR

This paper compares various entanglement detection criteria for high-dimensional quantum states, establishing a hierarchy based on effectiveness, and analyzing how state rank and subsystem dimensions influence detection performance.

Contribution

It introduces a hierarchy among entanglement detection criteria for random high-dimensional states and analyzes their effectiveness based on rank and subsystem size.

Findings

Realignment and entropic criteria fail beyond certain state ranks.

Hierarchy among criteria varies with state rank and dimension.

Detection efficiency depends on subsystem dimension asymmetry.

Abstract

Entanglement is a cornerstone in quantum information science, yet detecting it efficiently remains a challenging task. Focusing on non-positive partially transposed (NPT) states, we establish a hierarchy among entropy-based, majorization, realignment, and reduction criteria for Haar uniformly generated random states in finite dimensions, analyzing their performance based on rank and subsystem dimension. We prove lower bounds on the rank of mixed quantum states beyond which the realignment and entropic criteria fail to detect entanglement. We evaluate the relative effectiveness of the considered detection methods using three key indicators -- fraction of detected states, mean detectable entanglement, and minimum required entanglement. Our results provide insights into the entanglement thresholds needed for reliable detection, showing that, beyond a certain level of entanglement, all criteria become equally powerful for low-rank states, while hierarchy among various criteria emerges with moderate to high ranks. Intriguingly, the proposed ordering among the considered criteria in qubit-qudit systems is different from that in higher dimensions. Additionally, we establish that the detection efficiency is influenced by the asymmetry in the subsystem dimensions, by illustrating how the realignment criterion behaves more efficiently than other detection methods when the difference between the subsystem dimensions is small.