A few entanglement criterion for two-qubit and two-qudit system based on realignment operation

TL;DR

This paper develops new entanglement criteria for two-qubit and higher-dimensional systems based on realignment operations, providing necessary and sufficient conditions for certain classes and a computationally efficient detection method.

Contribution

It introduces a necessary and sufficient condition for a class of two-qubit states and a necessary condition for higher-dimensional states, improving entanglement detection efficiency.

Findings

Detects two-qubit entangled states not identified by realignment criterion

Provides a geometric interpretation of the separability criterion for $d\otimes d$ systems

Offers a computationally simpler entanglement detection method

Abstract

It is known that realignment crierion is necessary but not a sufficient criterion for lower as well as higher dimensional system. In this work, we first consider a two-qubit system and derived the necessary and sufficient condition based on realignment operation for a particular class of two-qubit system. Thus we solved the problem of if and only if condition partially for a particular class of two-qubit state. We have shown that the derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Next, we discuss the higher dimensional system and obtained the necessary condition on the minimum singular value of the realigned matrix of $d\otimes d$ dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for $d\otimes d$ dimensional system. Furthermore, we show that our criterion may also detect bound entangled state. The entanglement detection criterion studied here is beneficial in the sense that it requires to calculate only minimum singular value of the realigned matrix while on the other hand realignment criterion requires all singular values of the realigned matrix. Thus, our criterion has computational advantage over the realignment criterion.