Realigning random states

TL;DR

This paper analyzes the effectiveness of the realignment criterion in detecting entanglement in large-dimensional random quantum states, revealing its limitations compared to the partial transposition criterion.

Contribution

It provides an asymptotic threshold for the realignment criterion's success in detecting entanglement in high-dimensional random states.

Findings

Realignment detects entanglement if and only if ancilla dimension s is below a specific threshold.

The criterion is asymptotically weaker than the partial transposition criterion.

Threshold for detection scales as approximately 0.85 times the square of the system dimension.

Abstract

We study how the realignment criterion (also called computable cross-norm criterion) succeeds asymptotically in detecting whether random states are separable or entangled. We consider random states on $\C^d \otimes \C^d$ obtained by partial tracing a Haar-distributed random pure state on $\C^d \otimes \C^d \otimes \C^s$ over an ancilla space $\C^s$. We show that, for large $d$, the realignment criterion typically detects entanglement if and only if $s \leq (8/3\pi)^2 d^2$. In this sense, the realignment criterion is asymptotically weaker than the partial transposition criterion.