Homological Error Correction: Classical and Quantum Codes

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.Comment: Revtex4 fil

How Sample Completeness Affects Gamma-Ray Burst Classification

Unsupervised pattern recognition algorithms support the existence of three gamma-ray burst classes; Class I (long, large fluence bursts of intermediate spectral hardness), Class II (short, small fluence, hard bursts), and Class III (soft bursts of intermediate durations and fluences). The algorithms surprisingly assign larger membership to Class III than to either of the other two classes. A known systematic bias has been previously used to explain the existence of Class III in terms of Class I; this bias allows the fluences and durations of some bursts to be underestimated (Hakkila et al., ApJ 538, 165, 2000). We show that this bias primarily affects only the longest bursts and cannot explain the bulk of the Class III properties. We resolve the question of Class III existence by demonstrating how samples obtained using standard trigger mechanisms fail to preserve the duration characteristics of small peak flux bursts. Sample incompleteness is thus primarily responsible for the existence of Class III. In order to avoid this incompleteness, we show how a new dual timescale peak flux can be defined in terms of peak flux and fluence. The dual timescale peak flux preserves the duration distribution of faint bursts and correlates better with spectral hardness (and presumably redshift) than either peak flux or fluence. The techniques presented here are generic and have applicability to the studies of other transient events. The results also indicate that pattern recognition algorithms are sensitive to sample completeness; this can influence the study of large astronomical databases such as those found in a Virtual Observatory.Comment: 29 pages, 6 figures, 3 tables, Accepted for publication in The Astrophysical Journa

GUT-Scale Primordial Black Holes: Consequences and Constraints

A population of very light primordial black holes which evaporate before nucleosynthesis begins is unconstrained unless the decaying black holes leave stable relics. We show that gravitons Hawking radiated from these black holes would source a substantial stochastic background of high frequency gravititational waves ($10^{12}$ Hz or more) in the present universe. These black holes may lead to a transient period of matter dominated expansion. In this case the primordial universe could be temporarily dominated by large clusters of "Hawking stars" and the resulting gravitational wave spectrum is independent of the initial number density of primordial black holes.Comment: 4 pages; grey body factors included in graviton emission calculations, and a couple of references added, but the conclusions are unchanged. v3 Minor changes to references and wording; final versio

Dilution effects in Ho2−x_{2-x}Yx_xSn2_2O7_7: from the Spin Ice to the single-ion magnet

A study of the modifications of the magnetic properties of Ho$_{2-x}$Y$_x$Sn$_2$O$_7$ upon varying the concentration of diamagnetic Y$^{3+}$ ions is presented. Magnetization and specific heat measurements show that the Spin Ice ground-state is only weakly affected by doping for $x\leq 0.3$, even if non-negligible changes in the crystal field at Ho$^{3+}$ occur. In this low doping range $\mu$SR relaxation measurements evidence a modification in the low-temperature dynamics with respect to the one observed in the pure Spin Ice. For $x\to 2$, or at high temperature, the dynamics involve fluctuations among Ho$^{3+}$ crystal field levels which give rise to a characteristic peak in $^{119}$Sn nuclear spin-lattice relaxation rate. In this doping limit also the changes in Ho$^{3+}$ magnetic moment suggest a variation of the crystal field parameters.Comment: 4 pages, 5 figures, proceedings of HFM2008 Conferenc

Stabilization of single-electron pumps by high magnetic fields

We study the effect of perpendicular magnetic fields on a single-electron system with a strongly time-dependent electrostatic potential. Continuous improvements to the current quantization in these electron pumps are revealed by high-resolution measurements. Simulations show that the sensitivity of tunnel rates to the barrier potential is enhanced, stabilizing particular charge states. Nonadiabatic excitations are also suppressed due to a reduced sensitivity of the Fock-Darwin states to electrostatic potential. The combination of these effects leads to significantly more accurate current quantization

The discovery of polarization in the afterglow of GRB 990510 with the ESO Very Large Telescope

Following a BeppoSAX alert (Piro 1999a) and the discovery of the OT at SAAO (Vreeswijk et al. 1999a), we observed GRB 990510 with the FORS instrument on ESO's VLT Unit 1 (`Antu'). The burst is unremarkable in gamma rays, but in optical is the first one to show good evidence for jet-like outflow (Stanek et al. 1999, Harrison et al. 1999). We report the detection of significant linear polarization in the afterglow: it is (1.6 +/- 0.2)% 0.86 days after trigger, and after 1.81 days is consistent with that same value, but much more uncertain. The polarization angle is constant on a time scale of hours, and may be constant over one day. We conclude that the polarization is intrinsic to the source and due to the synchrotron nature of the emission, and discuss the random and ordered field geometries that may be responsible for it.Comment: submitted to ApJ Lett., 5 pages including 2 figures, uses emulateapj.st