Induced measures in the space of mixed quantum states

We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N x K composite system, induces a unique measure in the space of N x N mixed states (or in the space of K x K mixed states, if the reduction takes place with respect to the first subsystem). For K=N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of $N \times N$ pure states behaves as lnN-1/2.Comment: 12 latex pages, 2 figures in epsf, submited to J. Phys. A. ver.3, some improvements and a few references adde

On the volume of the set of mixed entangled states II

The problem of of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of density matrices describing N--dimensional quantum systems affects the results obtained. We demonstrate that the link between the purity of the mixed states and the probability of entanglement is not sensitive to the measure chosen. Since the criterion of partial transposition is not sufficient to distinguish all separable states for N > 6, we develop an efficient algorithm to calculate numerically the entanglement of formation of a given mixed quantum state, which allows us to compute the volume of separable states for N=8 and to estimate the volume of the bound entangled states in this case.Comment: 14 pages in Latex, Revtex + epsf; 7 figures in .ps included (one new figure in the revised version, several minor changes

Study on Cosmic Ray Background Rejection with a 30 m Stand-Alone IACT using Non-parametric Multivariate Methods in a sub-100 GeV Energy Range

During the last decade ground-based very high-energy gamma-ray astronomy achieved a remarkable advancement in the development of the observational technique for the registration and study of gamma-ray emission above 100 GeV. It is widely believed that the next step in its future development will be the construction of telescopes of substantially larger size than the currently used 10 m class telescopes. This can drastically improve the sensitivity of the ground-based detectors for gamma rays of energy from 10 to 100 GeV. Based on Monte Carlo simulations of the response of a single stand-alone 30 m imaging atmospheric Cherenkov telescope (IACT) the maximal rejection power against background cosmic ray showers for low energy gamma-rays was investigated in great detail. An advanced Bayesian multivariate analysis has been applied to the simulated Cherenkov light images of the gamma-ray- and proton-induced air showers. The results obtained here quantitatively testify that the separation between the signal and background images degrades substantially at low energies, and consequently the maximum overall quality factor can only be about 3.1 for gamma rays in the 10-30 GeV energy range. Various selection criteria as well as optimal combinations of the standard image parameters utilized for effective image separation have been also evaluated.Comment: Accepted for publication in the Journal of Physics