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

Decision by Sampling and Memory Distinctiveness: Range Effects from Rank-Based Models of Judgment and Choice

To what extent are preferences for risk – and for other economic quantities – stable, and to what extent are they malleable and context-dependent? Judgments and choices are strongly influenced by the context of available options in both the laboratory and the real world (e.g., Parducci, 1995; Sharpe et al., 2008), and this applies both to choices between risky options and more generally (Stewart et al., 2003). What cognitive processes underpin these contextual influences? According to the decision by sampling model (DbS: Stewart et al., 2006), judgments of a stimulus in a context depend solely on the relative ranked position of the stimulus within the remembered or experienced context of judgment. The claim that only relative ranked position matters appears, however, to contradict both empirical data and an earlier model of judgment, range frequency theory (RFT: Parducci, 1965, 1995), according to which the position of a stimulus with respect to the highest and lowest stimuli in the context (its range position) also matters. Here we show that a purely rank-based approach can account for apparent range effects when the relative memorability of contextual items, as independently determined by a memory model (Brown et al., 2007), is taken into account. Such a demonstration is important for several reasons. In particular, it is important to understand whether the skew of a distribution (e.g., the degree of inequality of an income distribution) influences the judgments of items (e.g., individuals ’ own incomes) within that distribution. If judgments are based solely on relative rank (as DbS claims), there should be no effects of distribution skewness – yet such effects are frequently observed. For example, there is a tension between the claim that income inequality within a society influences various indices of societal well-being (e.g., Wilkinson and Pickett, 2009) and the claim that individuals are primarily or solely concerned with the ranked position of their income (Boyce et al., 2010). Here we address this tension directly

A low cost scheme for high precision dual-wavelength laser metrology

A novel method capable of delivering relative optical path length metrology with nanometer precision is demonstrated. Unlike conventional dual-wavelength metrology which employs heterodyne detection, the method developed in this work utilizes direct detection of interference fringes of two He-Ne lasers as well as a less precise stepper motor open-loop position control system to perform its measurement. Although the method may be applicable to a variety of circumstances, the specific application where this metrology is essential is in an astrometric optical long baseline stellar interferometer dedicated to precise measurement of stellar positions. In our example application of this metrology to a narrow-angle astrometric interferometer, measurement of nanometer precision could be achieved without frequency-stabilized lasers although the use of such lasers would extend the range of optical path length the metrology can accurately measure. Implementation of the method requires very little additional optics or electronics, thus minimizing cost and effort of implementation. Furthermore, the optical path traversed by the metrology lasers is identical with that of the starlight or science beams, even down to using the same photodetectors, thereby minimizing the non-common-path between metrology and science channels.Comment: 17 pages, 4 figures, accepted for publication in Applied Optic

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

Bostonia. Volume 16

Founded in 1900, Bostonia magazine is Boston University's main alumni publication, which covers alumni and student life, as well as university activities, events, and programs

Measurements of complex permittivity of microwave substrates in the 20 to 300 K temperature range from 26.5 to 40.0 GHz

A knowledge of the dielectric properties of microwave substrates at low temperatures is useful in the design of superconducting microwave circuits. Results are reported for a study of the complex permittivity of sapphire (Al2O3), magnesium oxide (MgO), silicon oxide (SiO2), lanthanum aluminate (LaAlO3), and zirconium oxide (ZrO2), in the 20 to 300 Kelvin temperature range, at frequencies from 26.5 to 40.0 GHz. The values of the real and imaginary parts of the complex permittivity were obtained from the scattering parameters, which were measured using a HP-8510 automatic network analyzer. For these measurements, the samples were mounted on the cold head of a helium gas closed cycle refrigerator, in a specially designed vacuum chamber. An arrangement of wave guides, with mica windows, was used to connect the cooling system to the network analyzer. A decrease in the value of the real part of the complex permittivity of these substrates, with decreasing temperature, was observed. For MgO and Al2O3, the decrease from room temperature to 20 K was of 7 and 15 percent, respectively. For LaAlO3, it decreased by 14 percent, for ZrO2 by 15 percent, and for SiO2 by 2 percent, in the above mentioned temperature range

The relationship between extracurricular lessons and general achievement, self-perception of ability, and persistence of elementary and middle school students with learning disabilities across SES groups

Working with the Special Education Elementary Longitudinal Study (SEELS) database, this study gathered evidence regarding the relationship between extracurricular lessons and general achievement, self-perception of ability, and persistence of elementary and middle school students with learning disabilities across low and high socioeconomic groups. There was not enough viable data to test for statistical significance on the relationship between extracurricular lessons and self-perceived ability. Findings showed that participation in extracurricular lessons did not have a statistically significant effect on general achievement and persistence in both SES groups. The low amount of viable data supports the need for further examination using a primary data collection method. With the growing number of children with learning disabilities, it is critical that we examine the relationship between learning disabilities and self-efficacy, persistence, and general achievement