Quantum Channels and Representation Theory

In the study of d-dimensional quantum channels $(d \geq 2)$, an assumption which is not very restrictive, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of SU(n) is a depolarizing channel for all $n$, but for most other representations this is not the case. Since the Bloch sphere is not appropriate here, we develop technology which is a generalization of Bloch's technique. Our method works by representing the density matrix as a polynomial in symmetrized products of Lie algebra generators, with coefficients that are symmetric tensors. Using these tensor methods we prove eleven theorems, derive many explicit formulas and show other interesting properties of quantum channels in various dimensions, with various Lie symmetry algebras. We also derive numerical estimates on the size of a generalized ``Bloch sphere'' for certain channels. There remain many open questions which are indicated at various points through the paper.Comment: 28 pages, 1 figur

Using Functional Near Infrared Spectroscopy (fNIRS) to study dynamic stereoscopic depth perception

The parietal cortex has been widely implicated in the processing of depth perception by many neuroimaging studies, yet functional near infrared spectroscopy (fNIRS) has been an under-utilised tool to examine the relationship of oxy- ([HbO]) and de-oxyhaemoglobin ([HbR]) in perception. Here we examine the haemodynamic response (HDR) to the processing of induced depth stimulation using dynamic random-dot-stereograms (RDS). We used fNIRS to measure the HDR associated with depth perception in healthy young adults (n = 13, mean age 24). Using a blocked design, absolute values of [HbO] and [HbR] were recorded across parieto-occipital and occipital cortices, in response to dynamic RDS. Control and test images were identical except for the horizontal shift in pixels in the RDS that resulted in binocular disparity and induced the percept of a 3D sine wave that 'popped out' of the test stimulus. The control stimulus had zero disparity and induced a 'flat' percept. All participants had stereoacuity within normal clinical limits and successfully perceived the depth in the dynamic RDS. Results showed a significant effect of this complex visual stimulation in the right parieto-occipital cortex (p < 0.01, η(2) = 0.54). The test stimulus elicited a significant increase in [HbO] during depth perception compared to the control image (p < 0.001, 99.99 % CI [0.008-0.294]). The similarity between the two stimuli may have resulted in the HDR of the occipital cortex showing no significant increase or decrease of cerebral oxygenation levels during depth stimulation. Cerebral oxygenation measures of [HbO] confirmed the strong association of the right parieto-occipital cortex with processing depth perception. Our study demonstrates the validity of fNIRS to investigate [HbO] and [HbR] during high-level visual processing of complex stimuli

Rival bishops, rival cathedrals : the election of Cormac, archdeacon of Sodor, as bishop in 1331

Peer reviewedPublisher PD

Lie Algebras and Suppression of Decoherence in Open Quantum Systems

Since there are many examples in which no decoherence-free subsystems exist (among them all cases where the error generators act irreducibly on the system Hilbert space), it is of interest to search for novel mechanisms which suppress decoherence in these more general cases. Drawing on recent work (quant-ph/0502153) we present three results which indicate decoherence suppression without the need for noiseless subsystems. There is a certain trade-off; our results do not necessarily apply to an arbitrary initial density matrix, or for completely generic noise parameters. On the other hand, our computational methods are novel and the result--suppression of decoherence in the error-algebra approach without noiseless subsystems--is an interesting new direction.Comment: 7 page

On the number of representations providing noiseless subsystems

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review

An electron microscopy study of b-FeOOH (akaganeite) nanorods and nanotubes

High-resolution TEM images reveal that samples of b-FeOOH (akaganeite) prepared by quenching of a condensed ferrihydrite gel contain a mixture of rod-like particles and tubes. The tubular particles are usually about 10-15 nm in diameter with a central void that is typically 1/3 of the particle diameter. Dark-field STEM images show that the tubular particles are made up of single crystals that extend across the whole tube diameter, but only rarely extend along the whole length of the tube. Both the solid rods and the tubes appear to be based on subunits of approximately 3-4 nm in diameter, and it is proposed that formation of the akaganeite particles, both tubes and rods, results from secondary nucleation of these subunits at sites on particle edges, followed by rapid linear growth along the c-direction of the akaganeite structure

Zinc sulfide as a solid phase additive for improving the processing characteristics of ferrihydrite residues

The effect of ZnS powders as additives for improving the physical and chemical properties of ferrihydrite residues has been studied in both kinetic and batch experiments. Four ZnS samples were compared: two industrial ZnS concentrates, high purity ZnS pieces, and a sample of synthetic ZnS nanoparticles. The industrial ZnS concentrates were found to be the most effective for promoting the transformation of ferrihydrite to hematite, and this effect was found to be due to their lower surface charge at pH 3, which promotes formation of mixed ZnS/ferrihydrite aggregates. For the high purity ZnS samples, aggregation was not favoured, and only goethite formation was promoted. The effect of particle size of the ZnS additive was also studied, and it was found that samples of smaller average particle size were more effective in promoting phase transformation. Measurement of filtration times and moisture content of ferrihydrite residues precipitated in the presence of ZnS showed only minor improvement in physical handling and indicated that better control of surface properties of the additive would be needed to optimise its effect on phase transformation

Optomechanical self-structuring in cold atomic gases

The rapidly developing field of optomechanics aims at the combined control of optical and mechanical (solid-state or atomic) modes. In particular, laser cooled atoms have been used to exploit optomechanical coupling for self-organization in a variety of schemes where the accessible length scales are constrained by a combination of pump modes and those associated to a second imposed axis, typically a cavity axis. Here, we consider a system with many spatial degrees of freedom around a single distinguished axis, in which two symmetries - rotations and translations in the plane orthogonal to the pump axis - are spontaneously broken. We observe the simultaneous spatial structuring of the density of a cold atomic cloud and an optical pump beam. The resulting patterns have hexagonal symmetry. The experiment demonstrates the manipulation of matter by opto-mechanical self-assembly with adjustable length scales and can be potentially extended to quantum degenerate gases.Comment: 20 pages, 6 figure

The Use of the Decomposition Principle in Making Judgments

One hundred and fifty-one subjects were randomly divided into two groups of roughly equal size. One group was asked to respond to a decomposed version of a problem and the other group was presented with the direct form of the problem. The results provided support for the hypotheses that people can make better judgments when they use the principle of decomposition; and that decomposition is especially valuable for those problems where the subject knows little. The results suggest that accuracy may be improved if the subject provides the data and the computer analyzes it, than if both steps were done implicitly by the subjects

Components of performance in selecting for heterosis in swine

This bulletin is a report on Animal Husbandry Department project number 39 entitled 'Swine Improvement'--P. 6."This report includes much of the material presented by the senior author as a dissertation for the Doctor of Philosophy degree in July, 1952"--P. [7].Digitized 2007 AES.Includes bibliographical references (pages 67-68)