An entanglement measure for n-qubits

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.Comment: 16 pages, no figure

Pure Even Harmonic Generation from Oriented CO in Linearly Polarized Laser Fields

The first high harmonic spectrum, containing only the odd orders, was observed in experiments 30 years ago (1987). However, a spectrum containing pure even harmonics has never been observed. We investigate the generation of pure even harmonics from oriented CO molecules in linearly polarized laser fields employing the time-dependent density-functional theory. We find that the even harmonics, with no odd orders, are generated with the polarization perpendicular to the laser polarization, when the molecular axis of CO is perpendicular to the laser polarization. Generation of pure even harmonics reveals a type of dipole acceleration originating from the permanent dipole moment. This phenomenon exists in all system with permanent dipole moments, including bulk crystal and polyatomic molecules

SLOCC invariant and semi-invariants for SLOCC classification of four-qubits

We show there are at least 28 distinct true SLOCC entanglement classes for four-qubits by means of SLOCC invariant and semi-invariants and derive the number of the degenerated SLOCC classes for n-qubits.Comment: 22 pages, no figures, 9 tables, submit the paper to a journa