SIC-POVMs and Compatibility among Quantum States

An unexpected connection exists between compatibility criteria for quantum states and symmetric informationally complete POVMs. Beginning with Caves, Fuchs and Schack's "Conditions for compatibility of quantum state assignments" [Phys. Rev. A 66 (2002), 062111], I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.Comment: 15 pages, 4 MUBs, 2 errata for CFS (2002), 1 graph with chromatic number 4. v4: journal versio

Cosmology with photometric redshift surveys

We explore the utility of future photometric redshift imaging surveys for delineating the large-scale structure of the Universe, and assess the resulting constraints on the cosmological model. We perform the following two complementary types of analysis. (i) We quantify the statistical confidence and the accuracy with which such surveys will be able to detect and measure characteristic features in the clustering power spectrum such as the acoustic oscillations and the turnover, in a 'model-independent' fashion. We show for example that a 10 000-deg2 imaging survey with depth r= 22.5 and photometric redshift accuracy δz/(1 +z) = 0.03 will detect the acoustic oscillations with 99.9 per cent confidence, measuring the associated preferred cosmological scale with 2 per cent precision. Such a survey will also detect the turnover with 95 per cent confidence, determining the corresponding scale with 20 per cent accuracy. (ii) By assuming a Λ cold dark matter (ΛCDM) model power spectrum we calculate the confidence with which a non-zero baryon fraction can be deduced from such future galaxy surveys. We quantify 'wiggle detection' by calculating the number of standard deviations by which the baryon fraction is measured, after marginalizing over the shape parameter. This is typically a factor of 4 more significant (in terms of number of standard deviations) than the above 'model-independent' result. For both analyses, we quantify the variation of the results with magnitude depth and photometric redshift precision, and discuss the prospects for obtaining the required performance with realistic future surveys. We conclude that the precision with which the clustering pattern may be inferred from future photometric redshift surveys will be competitive with contemporaneous spectroscopic redshift surveys, assuming that systematic effects can be controlled. We find that for equivalent wiggle detection power, a photometric redshift survey requires an area approximately 12[δz/(1 +z)]/0.03 times larger than a spectroscopic survey, for a given magnitude limit. We also note that an analysis of luminous red galaxies in the Sloan Digital Sky Survey may yield a marginal detection of acoustic oscillations in the imaging survey, in addition to that recently reported for the spectroscopic component