Commutators as Powers in Free Products of Groups

The ways in which a nontrivial commutator can be a proper power in a free product of groups are identified.Comment: AMS-LaTex, 6 pages, no figure

The Chemical Evolution of the Universe I: High Column Density Absorbers

We construct a simple, robust model of the chemical evolution of galaxies from high to low redshift, and apply it to published observations of damped Lyman-alpha quasar absorption line systems (DLAs). The elementary model assumes quiescent star formation and isolated galaxies (no interactions, mergers or gas flows). We consider the influence of dust and chemical gradients in the galaxies, and hence explore the selection effects in quasar surveys. We fit individual DLA systems to predict some observable properties of the absorbing galaxies, and also indicate the expected redshift behaviour of chemical element ratios involving nucleosynthetic time delays. Despite its simplicity, our `monolithic collapse' model gives a good account of the distribution and evolution of the metallicity and column density of DLAs, and of the evolution of the global star formation rate and gas density below redshifts z 3. However, from the comparison of DLA observations with our model, it is clear that star formation rates at higher redshifts (z>3) are enhanced. Galaxy interactions and mergers, and gas flows very probably play a major role.Comment: 36 pages, 11 figures; accepted by MNRA

Experimental quantum verification in the presence of temporally correlated noise

Growth in the complexity and capabilities of quantum information hardware mandates access to practical techniques for performance verification that function under realistic laboratory conditions. Here we experimentally characterise the impact of common temporally correlated noise processes on both randomised benchmarking (RB) and gate-set tomography (GST). We study these using an analytic toolkit based on a formalism mapping noise to errors for arbitrary sequences of unitary operations. This analysis highlights the role of sequence structure in enhancing or suppressing the sensitivity of quantum verification protocols to either slowly or rapidly varying noise, which we treat in the limiting cases of quasi-DC miscalibration and white noise power spectra. We perform experiments with a single trapped $^{171}$Yb$^{+}$ ion as a qubit and inject engineered noise ($\propto \sigma^z$) to probe protocol performance. Experiments on RB validate predictions that the distribution of measured fidelities over sequences is described by a gamma distribution varying between approximately Gaussian for rapidly varying noise, and a broad, highly skewed distribution for the slowly varying case. Similarly we find a strong gate set dependence of GST in the presence of correlated errors, leading to significant deviations between estimated and calculated diamond distances in the presence of correlated $\sigma^z$ errors. Numerical simulations demonstrate that expansion of the gate set to include negative rotations can suppress these discrepancies and increase reported diamond distances by orders of magnitude for the same error processes. Similar effects do not occur for correlated $\sigma^x$ or $\sigma^y$ errors or rapidly varying noise processes, highlighting the critical interplay of selected gate set and the gauge optimisation process on the meaning of the reported diamond norm in correlated noise environments.Comment: Expanded and updated analysis of GST, including detailed examination of the role of gauge optimization in GST. Full GST data sets and supplementary information available on request from the authors. Related results available from http://www.physics.usyd.edu.au/~mbiercuk/Publications.htm

Accumulation is late and brief in preferential choice

Preferential choices are often explained using models within the evidence accumulation framework: value drives the drift rate at which evidence is accumulated until a threshold is reached and an option is chosen. Although rarely stated explicitly, almost all such models assume that decision makers have knowledge at the onset of the choice of all available attributes and options. In reality however, choice information is viewed piece-by-piece, and is often not completely acquired until late in the choice, if at all. Across four eye-tracking experiments, we show that whether the information was acquired early or late is irrelevant in predicting choice: all that matters is whether or not it was acquired at all. Models with potential alternative assumptions were posited and tested, such as 1) accumulation of instantaneously available information or 2) running estimates as information is acquired. These provided poor fits to the data. We are forced to conclude that participants either are clairvoyant, accumulating using information before they have looked at it, or delay accumulating evidence until very late in the choice, so late that the majority of choice time is not time in which evidence is accumulated. Thus, although the evidence accumulation framework may still be useful in measurement models, it cannot account for the details of the processes involved in decision making