Completely entangled subspaces from Moore-like matrices

Abstract

Abstract Construction of completely entangled subspaces (CES) has gained a considerable attention, recently. These subspaces which contain only entangled states are of great importance for entanglement theory and also provide a valuable resource for quantum information processing tasks. The results of [Proc. Math. Sci., 114, 365 (2004)] and in particular using the properties of certain matrix, namely Vandermonde matrix, to build CES motivated us to search for new distinct CES's. Mainly, the stimulating question of whether there are other matrices that can lead to building CESs emerged. In the current paper we give an affirmative answer to this question by providing a method for constructing CESs using the properties of Moore-like matrices. In addition, we give few examples for the proposed subspaces in case of 3-qubit and 2-qutrit systems. Then a comparison between the resulted subspaces and those constructed from Vandermonde matrix has been given for the systems understudy. The results shows that the two methods give identically the same subspaces in case of multiqubit systems. However, for multipartite systems with local dimensions d ≥ 3 the two methods gave unequivalent CES subspaces. Interestingly, the properties of the proposed Moore-like matrices provided a far rich way for constructing CES subspaces. It leads to generating as many distinct CES's as we want for each multipartite quantum system. This is in contrary to Vandermonde-based method which can give only one CES per system. In addition, the basis for each of the given examples has been obtained in a simple form. Moreover, we evaluated the entanglement of uniformly mixed states over the obtained subspaces in terms of concurrence and geometric measure of entanglement. Since different parameters of a Moore-like matrix lead to distinct CESs for the same system, the realized results can open the door for more investigations and/or applications.