Mutually Unbiased Bases and The Complementarity Polytope

A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a geometrical object such a polytope exists for all values of N, while it is unknown whether it can be made to lie within the body of density matrices unless N=p^k, where p is prime. We investigate the polytope in order to see if some values of N are geometrically singled out. One such feature is found: It is possible to select N^2 facets in such a way that their centers form a regular simplex if and only if there exists an affine plane of order N. Affine planes of order N are known to exist if N=p^k; perhaps they do not exist otherwise. However, the link to the existence of MUBs--if any--remains to be found.Comment: 18 pages, 3 figure

Mubs and Hadamards of Order Six

We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organise our results. Finally we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.Comment: 33 pages, 3 figures. References added in v

The public economic burden of suboptimal type 2 diabetes control upon taxpayers in Sweden:Looking beyond health costs

Aim To estimate the fiscal burden for taxpayers in Sweden associated with type 2 diabetes (T2D) attributed to diabetes-related complications in patients failing to meet HbA1c targets. Material and Methods We developed a public economic framework to assess how changes in diabetes-related complications influenced projected tax contributions and government disability payments for people with T2D. The analysis applied accepted disease-modelling practices to estimate different rates of diabetes-related complications based on an HbA1c of 6.9% (52 mmol/mol) and of 6.0% (42 mmol/mol). We adjusted the employment activity rates for those experiencing T2D-related events, applying age-specific earnings to estimate lifetime tax losses. Furthermore, the likelihood of receiving payments for health-related employment inactivity was estimated. Direct healthcare costs are excluded from this analysis. Results The estimated per person earnings loss for immediate and delayed HbA1c control was Swedish krona (SEK) 42 299 and SEK 44 157, respectively, over 10 years. The lost employment activity of people with T2D translates to lost tax revenues of SEK 23 265 and SEK 24 287 for immediate and delayed control, respectively. The estimated difference in disability payments was SEK 538. Combining the tax revenue loss and excess disability payments defines the broader fiscal costs, where we observe combined fiscal losses that favour immediate and sustained control by SEK 1560 over 10 years. Conclusions We show that conducting fiscal analysis of diabetes interventions offers an enriched perspective capturing a range of costs that fall on government in relation to lost tax revenue and disability payments. Tax-financed health systems may benefit from broadening the consideration of costs and benefits when evaluating new interventions and treatment practices

Cost Effectiveness of Budesonide/Formoterol in a Single Inhaler for COPD Compared with Each Monocomponent Used Alone

Objective: To compare the healthcare costs and effects of budesonide/formoterol in a single inhaler with those of budesonide and formoterol monotherapies, and placebo, in a multinational study in patients with chronic obstructive pulmonary disease (COPD), National Heart, Lung and Blood Institute (NHLBI)/WHO Global Initiative for Chronic Obstructive Lung Disease (GOLD) stages III or IV. Previous analysis of the clinical data from the study had shown that budesonide/formoterol was associated with better lung function and improved health-related QOL compared with the monocomponents or placebo and lower frequency of exacerbations compared with formoterol and placebo. Method: Patients (n = 1022) were randomised to twice-daily treatment with two inhalations of budesonide/formoterol (160mug/4.5mug) in a single inhaler, budesonide 200mug, formoterol 4.5mug or placebo for 12 months. Data on medication and healthcare use were combined with Swedish unit cost data to estimate the total annual healthcare cost per patient from the Swedish healthcare payer perspective. Costs were valued in Swedish kronor (SEK) [2001 values] and converted to euros (SEK1 = _0.11, 25th April 2003). Results: This evaluation estimated the total annual healthcare costs per patient to be numerically lower for budesonide/formoterol (_2518) than for budesonide (_3194), formoterol (_3653) or placebo (_3213). Cost-effectiveness acceptability curves suggest that budesonide/formoterol may be cost effective compared with formoterol, even if the decision maker is not willing to pay anything for the additional clinical effects, and that budesonide/formoterol is cost effective compared with placebo if a decision maker is willing to pay about _2 per day, per avoided exacerbation. Conclusion: This economic analysis suggests that the clinical benefits of using budesonide/formoterol in a single inhaler are achieved at a numerically lower total healthcare cost than either monocomponent or placebo. Budesonide/formoterol in patients with severe COPD (GOLD stages III or IV) may be cost effective, from the healthcare provider perspective, compared with either monocomponent.Budesonide, Budesonide/formoterol, Chronic-obstructive-pulmonary-disease, Cost-effectiveness, Formoterol, Quality-of-life

Factors affecting chronic obstructive pulmonary disease (COPD)-related costs: a multivariate analysis of a Swedish COPD cohort.

Chronic obstructive pulmonary disease (COPD) is an increasing public health problem, generating considerable costs. The objective of this study was to identify factors affecting COPD-related costs. A cohort of 179 subjects with COPD was interviewed over the telephone on four occasions about their annual use of COPD-related resources. The data set and explanatory variables were analysed by means of multivariate regression techniques for six different types of cost: societal (or total), direct (health care) and indirect (productivity), and three subcomponents of direct costs-hospitalisation, outpatient and medication. Poor lung function, dyspnoea and asthma were independently associated with higher costs. Poor lung function (severity of COPD) significantly increased all six examined cost types. Dyspnoea (breathing problems) also increased costs, though to a varying extent. The presence of reported asthma increased total, direct, outpatient and medication costs. Poor lung function and, to a lesser extent, extent of dyspnoea and concomitant asthma, were all strongly associated with higher COPD-related costs. Strong efforts should be made to prevent the progression of COPD and its symptoms