Average fidelity between random quantum states

We analyze mean fidelity between random density matrices of size N, generated with respect to various probability measures in the space of mixed quantum states: Hilbert-Schmidt measure, Bures (statistical) measure, the measures induced by partial trace and the natural measure on the space of pure states. In certain cases explicit probability distributions for fidelity are derived. Results obtained may be used to gauge the quality of quantum information processing schemes.Comment: 15 revtex pages with 4 figures; Ver. 2: another distribution derived, an extra figure included, Ver. 3: comments in introduction and conclusion added ver 4. minor improvment

Truncations of Random Orthogonal Matrices

Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two components: finite fraction of eigenvalues are real, while the remaining part of the spectrum is located inside the unit disk symmetrically with respect to the real axis. In the case of strong non--orthogonality, $M/N=$const, the behavior typical to real Ginibre ensemble is found. In the case $M=N-L$ with fixed $L$, a universal distribution of resonance widths is recovered.Comment: 4 pages, final revised version (one reference added, minor changes in Introduction

Subnormalized states and trace-nonincreasing maps

We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its volume with respect to the Hilbert-Schmidt (Euclidean) measure is computed explicitly for an arbitrary N. The spectra of partially reduced rescaled dynamical matrices associated with trace-nonincreasing completely positive maps belong to the N-cube inscribed in the set of subnormalized states of size N. As a by-product we derive the measure in M_N induced by partial trace of mixed quantum states distributed uniformly with respect to HS-measure in $M_{N^2}$.Comment: LaTeX, 21 pages, 4 Encapsuled PostScript figures, 1 tabl

Distribution of Scattering Matrix Elements in Quantum Chaotic Scattering

Scattering is an important phenomenon which is observed in systems ranging from the micro- to macroscale. In the context of nuclear reaction theory the Heidelberg approach was proposed and later demonstrated to be applicable to many chaotic scattering systems. To model the universal properties, stochasticity is introduced to the scattering matrix on the level of the Hamiltonian by using random matrices. A long-standing problem was the computation of the distribution of the off-diagonal scattering-matrix elements. We report here an exact solution to this problem and present analytical results for systems with preserved and with violated time-reversal invariance. Our derivation is based on a new variant of the supersymmetry method. We also validate our results with scattering data obtained from experiments with microwave billiards.Comment: Published versio