Equivariant quantization of spin systems

We investigate the geometric and conformally equivariant quantizations of the supercotangent bundle of a pseudo-Riemannian manifold $(M,g)$, which is a model for the phase space of a classical spin particle. This is a short review of our previous works.Comment: 7 pages. From a talk given at the Workshop on Geometric Methods in Physics XXI

Conformally Equivariant Quantization - a Complete Classification

Conformally equivariant quantization is a peculiar map between symbols of real weight $\delta$ and differential operators acting on tensor densities, whose real weights are designed by $\lambda$ and $\lambda+\delta$. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight $\delta$. Later, Silhan has determined the critical values of $\delta$ for which unique existence is lost, and conjectured that for those values of $\delta$ existence is lost for a generic weight $\lambda$. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of $\delta$ and $\lambda$. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight $\delta$, and (ii) in that case the conformally equivariant quantization exists only for a finite number of $\lambda$, corresponding to nontrivial conformally invariant differential operators on $\lambda$-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization

On group theory for quantum gates and quantum coherence

Finite group extensions offer a natural language to quantum computing. In a nutshell, one roughly describes the action of a quantum computer as consisting of two finite groups of gates: error gates from the general Pauli group P and stabilizing gates within an extension group C. In this paper one explores the nice adequacy between group theoretical concepts such as commutators, normal subgroups, group of automorphisms, short exact sequences, wreath products... and the coherent quantum computational primitives. The structure of the single qubit and two-qubit Clifford groups is analyzed in detail. As a byproduct, one discovers that M20, the smallest perfect group for which the commutator subgroup departs from the set of commutators, underlies quantum coherence of the two-qubit system. One recovers similar results by looking at the automorphisms of a complete set of mutually unbiased bases.Comment: 10 pages, to appear in J Phys A: Math and Theo (Fast Track Communication

Coseismic surface deformation from air photos: The Kickapoo step over in the 1992 Landers rupture

Coseismic deformation of the ground can be measured from aerial views taken before and after an earthquake. We chose the area of the Kickapoo-Landers step over along the 1992 Landers earthquake zone, using air photos (scale 1:40,000) scanned at 0.4 m resolution. Two photos acquired after the earthquake are used to assess the accuracy and to evaluate various sources of noise. Optical distortions, film deformation, scanning errors, or errors in viewing parameters can yield metric bias at wavelength larger than 1 km. Offset field at shorter wavelength is more reliable and mainly affected by temporal decorrelation of the images induced by changes in radiometry with time. Temporal decorrelation and resulting uncertainty on offsets are estimated locally from the correlation degree between the images. Relative surface displacements are measured independently every about 15 m and with uncertainty typically below 10 cm (RMS). The offset field reveals most of the surface ruptures mapped in the field. The fault slip is accurate to about 7 cm (RMS) and measured independently every 200 m from stacked profiles. Slip distribution compares well with field measurements at the kilometric scale but reveals local discrepancies suggesting that deformation is generally, although not systematically, localized on the major fault zone located in the field. This type of data can provide useful insight into the fault zone's mechanical properties. Our measurements indicate that elastic coseismic strain near the fault zone can be as large as 0.5 × 10^(−3), while anelastic yielding was attained for strain in excess of about 1–2 × 10^(−3)