A Necessary and Sufficient Entanglement Criterion of N-qubit System Based on Correlation Tensor

TL;DR

This paper introduces a new necessary and sufficient entanglement criterion for N-qubit systems using state decomposition and HOSVD, improving accuracy over existing methods and applicable to complex quantum states.

Contribution

It proposes a novel entanglement criterion based on correlation tensors that is both necessary and sufficient for general N-qubit systems, with demonstrated advantages over prior criteria.

Findings

Accurately determines entanglement thresholds for multi-qubit states.

Tighter than positive partial transpose criterion in examples.

Provides an entanglement measure comparable to concurrence for two-qubit states.

Abstract

Great advances have been achieved in studying characteristics of entanglement for fundamentals of quantum mechanics and quantum information processing. However, even for N-qubit systems, the problem of entanglement criterion has not been well solved. In this Letter, using the method of state decomposition and high order singular value decomposition (HOSVD), we propose a necessary and sufficient entanglement criterion for general N-qubit systems. As an example, we apply our method to study the multi-qubit W state with white noise. We not only obtain the separability critical point, which is tight and thus better than known results, but also the separate states ensemble for decomposition. More examples are presented to show our criterion is accurate, which is tighter than the well-known positive partial transpose criterion. For two-qubit case, we can provide an entanglement measurer which gives similar results with concurrence up to a factor. Our results pave the way to solve the entanglement-separability criterion for more general cases.