Measuring the degree of unitarity for any quantum process

Quantum processes can be divided into two categories: unitary and non-unitary ones. For a given quantum process, we can define a \textit{degree of the unitarity (DU)} of this process to be the fidelity between it and its closest unitary one. The DU, as an intrinsic property of a given quantum process, is able to quantify the distance between the process and the group of unitary ones, and is closely related to the noise of this quantum process. We derive analytical results of DU for qubit unital channels, and obtain the lower and upper bounds in general. The lower bound is tight for most of quantum processes, and is particularly tight when the corresponding DU is sufficiently large. The upper bound is found to be an indicator for the tightness of the lower bound. Moreover, we study the distribution of DU in random quantum processes with different environments. In particular, The relationship between the DU of any quantum process and the non-markovian behavior of it is also addressed.Comment: 7 pages, 2 figure

Abundance of moderate-redshift clusters in the Cold + Hot dark matter model

Using a set of \pppm simulation which accurately treats the density evolution of two components of dark matter, we study the evolution of clusters in the Cold + Hot dark matter (CHDM) model. The mass function, the velocity dispersion function and the temperature function of clusters are calculated for four different epochs of $z\le 0.5$. We also use the simulation data to test the Press-Schechter expression of the halo abundance as a function of the velocity dispersion $\sigma_v$. The model predictions are in good agreement with the observational data of local cluster abundances ($z=0$). We also tentatively compare the model with the Gunn and his collaborators' observation of rich clusters at $z\approx 0.8$ and with the x-ray luminous clusters at $z\approx 0.5$ of the {\it Einstein} Extended Medium Sensitivity Survey. The important feature of the model is the rapid formation of clusters in the near past: the abundances of clusters of \sigma_v\ge 700\kms and of \sigma_v\ge 1200 \kms at $z=0.5$ are only 1/4 and 1/10 respectively of the present values ($z=0$). Ongoing ROSAT and AXAF surveys of distant clusters will provide sensitive tests to the model. The abundance of clusters at $z\approx 0.5$ would also be a good discriminator between the CHDM model and a low-density flat CDM model both of which show very similar clustering properties at $z=0$.Comment: 21 pages + 6 figures (uuencoded version of the PS files), Steward Preprints No. 118

Transmission statistics and focusing in single disordered samples

We show in microwave experiments and random matrix calculations that in samples with a large number of channels the statistics of transmission for different incident channels relative to the average transmission is determined by a single parameter, the participation number of the eigenvalues of the transmission matrix, M. Its inverse, M-1, is equal to the variance of relative total transmission of the sample, while the contrast in maximal focusing is equal to M. The distribution of relative total transmission changes from Gaussian to negative exponential over the range in which M-1 changes from 0 to 1. This provides a framework for transmission and imaging in single samples.Comment: 9 pages, 4 figure

Tunnelling Effect and Hawking Radiation from a Vaidya Black Hole

In this paper, we extend Parikh' work to the non-stationary black hole. As an example of the non-stationary black hole, we study the tunnelling effect and Hawking radiation from a Vaidya black hole whose Bondi mass is identical to its mass parameter. We view Hawking radiation as a tunnelling process across the event horizon and calculate the tunnelling probability. We find that the result is different from Parikh's work because $\frac{dr_{H}}{dv}$ is the function of Bondi mass m(v)