ar X iv : 0 71 0 . 34 25 v 2 [ qu an t - ph ] 1 4 N ov 2 00 7 Properties of the residual entanglement for n - qubits 1

X iv :0 71 0. 34 25 v2 [ qu an tph ] 1 4 N ov 2 00 7 Properties of the residual entanglement for n-qubits Dafa Li, Xiangrong Li, Hongtao Huang, Xinxin Li a Dept of mathematical sciences, Tsinghua University, Beijing 100084 CHINA b Department of Mathematics, University of California, Irvine, CA 92697-3875, USA c Electrical Engineering and Computer Science Department University of Michigan, Ann Arbor, MI 48109, USA d Dept. of computer science, Wayne State University, Detroit, MI 48202, USA Abstract Recently, Coffman, Kundu, and Wootters introduced the residual entanglement for three qubits to quantify the three-qubit entanglement in Phys. Rev. A 61, 052306 (2000). In Phys. Rev. A 65, 032304 (2007), we defined the residual entanglement for n qubits, whose values are between 0 and 1. In this paper, we want to show that the residual entanglement for n qubits is a natural measure of entanglement by demonstrating the following properties. (1). It is SL-invariant, especially LU-invariant. (2). It is an entanglement monotone. (3). It is invariant under permutations of the qubits. (4). It vanishes or is multiplicative for product states. Pacs: 03.67.Mn, 03.65.Ud.