A FEW ENTANGLEMENT CRITERION FOR TWO-QUBIT AND TWO-QUDIT SYSTEMS BASED ON REALIGNMENT OPERATION

Abstract

It is known that realignment crierion is necessary but not a sufficient criterion even for a two-qubit system. We have derived necessary and sufficient condition based on realignment operation for a particular class of two-qubit system and thus solved this problem partially for two-qubit system. We have shown that the lower bound of the trace norm of realigned form of the particular form of the density matrix exists if and only if the two-qubit state is entangled. The derived necessary and sufficient condition detects two-qubit entangled states, which are not detected by the realignment criterion. Further, we have obtained the upper bound of the minimum singular value of the realigned form of the density matrix for the d ⊗ d dimensional separable states. Moreover, we provide the geometrical interpretation of the derived separability criterion for d ⊗ d dimensional system. Moreover, we show that our criterion may also detect bound entangled state. Our criterion 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.