On the comparison of volumes of quantum states

This paper aims to study the \a-volume of \cK, an arbitrary subset of the set of $N\times N$ density matrices. The \a-volume is a generalization of the Hilbert-Schmidt volume and the volume induced by partial trace. We obtain two-side estimates for the \a-volume of \cK in terms of its Hilbert-Schmidt volume. The analogous estimates between the Bures volume and the \a-volume are also established. We employ our results to obtain bounds for the \a-volume of the sets of separable quantum states and of states with positive partial transpose (PPT). Hence, our asymptotic results provide answers for questions listed on page 9 in \cite{K. Zyczkowski1998} for large $N$ in the sense of \a-volume. \vskip 3mm PACS numbers: 02.40.Ft, 03.65.Db, 03.65.Ud, 03.67.M

Statistics of conductance and shot-noise power for chaotic cavities

We report on an analytical study of the statistics of conductance, $g$, and shot-noise power, $p$, for a chaotic cavity with arbitrary numbers $N_{1,2}$ of channels in two leads and symmetry parameter $\beta = 1,2,4$. With the theory of Selberg's integral the first four cumulants of $g$ and first two cumulants of $p$ are calculated explicitly. We give analytical expressions for the conductance and shot-noise distributions and determine their exact asymptotics near the edges up to linear order in distances from the edges. For $0<g<1$ a power law for the conductance distribution is exact. All results are also consistent with numerical simulations.Comment: 7 pages, 3 figures. Proc. of the 3rd Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, Poland, May 25-27, 200

Characteristic polynomials in real Ginibre ensembles

We calculate the average of two characteristic polynomials for the real Ginibre ensemble of asymmetric random matrices, and its chiral counterpart. Considered as quadratic forms they determine a skew-symmetric kernel from which all complex eigenvalue correlations can be derived. Our results are obtained in a very simple fashion without going to an eigenvalue representation, and are completely new in the chiral case. They hold for Gaussian ensembles which are partly symmetric, with kernels given in terms of Hermite and Laguerre polynomials respectively, depending on an asymmetry parameter. This allows us to interpolate between the maximally asymmetric real Ginibre and the Gaussian Orthogonal Ensemble, as well as their chiral counterparts

Correlation functions of impedance and scattering matrix elements in chaotic absorbing cavities

Wave scattering in chaotic systems with a uniform energy loss (absorption) is considered. Within the random matrix approach we calculate exactly the energy correlation functions of different matrix elements of impedance or scattering matrices for systems with preserved or broken time-reversal symmetry. The obtained results are valid at any number of arbitrary open scattering channels and arbitrary absorption. Elastic enhancement factors (defined through the ratio of the corresponding variance in reflection to that in transmission) are also discussed.Comment: 10 pages, 2 figures (misprints corrected and references updated in ver.2); to appear in Acta Phys. Pol. A (Proceedings of the 2nd Workshop on Quantum Chaos and Localization Phenomena, May 19-22, 2005, Warsaw

Extremely High Energy Neutrinos and their Detection

We discuss in some detail the production of extremely high energy (EHE) neutrinos with energies above 10^18 eV. The most certain process for producing such neutrinos results from photopion production by EHE cosmic rays in the cosmic background photon field. However, using assumptions for the EHE cosmic ray source evolution which are consistent with results from the deep QSO survey in the radio and X-ray range, the resultant flux of neutrinos from this process is not strong enough for plausible detection. A measurable flux of EHE neutrinos may be present, however, if the highest energy cosmic rays which have recently been detected well beyond 10^20 eV are the result of the annihilation of topological defects which formed in the early universe. Neutrinos resulting from such decays reach energies of the grand unification (GUT) scale, and collisions of superhigh energy neutrinos with the cosmic background neutrinos initiate neutrino cascading which enhances the EHE neutrino flux at Earth. We have calculated the neutrino flux including this cascading effect for either massless or massive neutrinos and we find that these fluxes are conceivably detectable by air fluorescence detectors now in development. The neutrino-induced showers would be recognized by their starting deep in the atmosphere. We evaluate the feasibility of detecting EHE neutrinos this way using air fluorescence air shower detectors and derive the expected event rate. Other processes for producing deeply penetrating air showers constitute a negligible background.Comment: 33 pages, including 12 eps figures, LaTe