Schur-Weyl Duality for the Clifford Group with Applications: Property Testing, a Robust Hudson Theorem, and de Finetti Representations

Schur-Weyl duality is a ubiquitous tool in quantum information. At its heart is the statement that the space of operators that commute with the tensor powers of all unitaries is spanned by the permutations of the tensor factors. In this work, we describe a similar duality theory for tensor powers of Clifford unitaries. The Clifford group is a central object in many subfields of quantum information, most prominently in the theory of fault-tolerance. The duality theory has a simple and clean description in terms of finite geometries. We demonstrate its effectiveness in several applications: (1) We resolve an open problem in quantum property testing by showing that "stabilizerness" is efficiently testable: There is a protocol that, given access to six copies of an unknown state, can determine whether it is a stabilizer state, or whether it is far away from the set of stabilizer states. We give a related membership test for the Clifford group. (2) We find that tensor powers of stabilizer states have an increased symmetry group. We provide corresponding de Finetti theorems, showing that the reductions of arbitrary states with this symmetry are well-approximated by mixtures of stabilizer tensor powers (in some cases, exponentially well). (3) We show that the distance of a pure state to the set of stabilizers can be lower-bounded in terms of the sum-negativity of its Wigner function. This gives a new quantitative meaning to the sum-negativity (and the related mana) -- a measure relevant to fault-tolerant quantum computation. The result constitutes a robust generalization of the discrete Hudson theorem. (4) We show that complex projective designs of arbitrary order can be obtained from a finite number (independent of the number of qudits) of Clifford orbits. To prove this result, we give explicit formulas for arbitrary moments of random stabilizer states.Comment: 60 pages, 2 figure

A BVRcIc Survey of W Ursae Majoris Binaries

We report on a BVRcIc survey of field W Ursae Majoris binary stars and present accurate colors for 606 systems that have been observed on at least three photometric nights from a robotic observatory in southern Arizona. Comparison with earlier photometry for a subset of the systems shows good agreement. We investigate two independent methods of determining the interstellar reddening, although both have limitations that can render them less effective than desired. A subset of 101 systems shows good agreement between the two reddening methods.Comment: 10 pages, 6 figures, accepted for publication in A

The non-abelian Born-Infeld action at order F^6

To gain insight into the non-abelian Born-Infeld (NBI) action, we study coinciding D-branes wrapped on tori, and turn on magnetic fields on their worldvolume. We then compare predictions for the spectrum of open strings stretching between these D-branes, from perturbative string theory and from the effective NBI action. Under some plausible assumptions, we find corrections to the Str-prescription for the NBI action at order F^6. In the process we give a way to classify terms in the NBI action that can be written in terms of field strengths only, in terms of permutation group theory.Comment: LaTeX, 31 pages, 30 figure

The Cyborg Astrobiologist: Testing a Novelty-Detection Algorithm on Two Mobile Exploration Systems at Rivas Vaciamadrid in Spain and at the Mars Desert Research Station in Utah

(ABRIDGED) In previous work, two platforms have been developed for testing computer-vision algorithms for robotic planetary exploration (McGuire et al. 2004b,2005; Bartolo et al. 2007). The wearable-computer platform has been tested at geological and astrobiological field sites in Spain (Rivas Vaciamadrid and Riba de Santiuste), and the phone-camera has been tested at a geological field site in Malta. In this work, we (i) apply a Hopfield neural-network algorithm for novelty detection based upon color, (ii) integrate a field-capable digital microscope on the wearable computer platform, (iii) test this novelty detection with the digital microscope at Rivas Vaciamadrid, (iv) develop a Bluetooth communication mode for the phone-camera platform, in order to allow access to a mobile processing computer at the field sites, and (v) test the novelty detection on the Bluetooth-enabled phone-camera connected to a netbook computer at the Mars Desert Research Station in Utah. This systems engineering and field testing have together allowed us to develop a real-time computer-vision system that is capable, for example, of identifying lichens as novel within a series of images acquired in semi-arid desert environments. We acquired sequences of images of geologic outcrops in Utah and Spain consisting of various rock types and colors to test this algorithm. The algorithm robustly recognized previously-observed units by their color, while requiring only a single image or a few images to learn colors as familiar, demonstrating its fast learning capability.Comment: 28 pages, 12 figures, accepted for publication in the International Journal of Astrobiolog

Image Compression for Geological Mapping and Novelty Detection

We describe an image-comparison technique of Heidemann and Ritter [4,5] that uses image compression, and is capable of: (i) detecting novel textures in a series of images, as well as of: (ii) alerting the user to the similarity of a new image to a previously-observed texture. This image-comparison technique has been implemented and tested using our Astrobiology Phone-cam system, which employs Bluetooth communication to send images to a local laptop server in the field for the image-compression analysis. We tested the system in a field site displaying a heterogeneous suite of sandstones, limestones, mudstones and coalbeds. Some of the rocks are partly covered with lichen. The imagematching procedure of this system performed very well with data obtained through our field test, grouping all images of yellow lichens together and grouping all images of a coal bed together, and giving a 91% accuracy for similarity detection. Such similarity detection could be employed to make maps of different geological units. The novelty-detection performance of our system was also rather good (a 64% accuracy). Such novelty detection may become valuable in searching for new geological units, which could be of astrobiological interest. By providing more advanced capabilities for similarity detection and novelty detection, this image-compression technique could be useful in giving more scientific autonomy to robotic planetary rovers, and in assisting human astronauts in their geological exploration