Taxonomic scale dependency of Bergmann’s patterns: a cross-scale comparison of hawkmoths and birds along a tropical elevational gradient

Bergmann’s rule predicts a larger body size for endothermic organisms in colder environments. The contrasting results from previous studies may be due to the differences in taxonomic (intraspecific, interspecific and community) and spatial (latitudinal vs elevational) scales. We compared Bergmann’s patterns for endotherms (Aves) and ectotherms (Lepidoptera: Sphingidae) along the same 2.6 km elevational transect in the eastern Himalayas. Using a large data spanning 3,302 hawkmoths (76 morpho-species) and 15,746 birds (245 species), we compared the patterns at the intraspecific (hawkmoths only), interspecific and community scales. Hawkmoths exhibited a positive Bergmann’s pattern at the intraspecific and abundance-weighted community scale. Contrary to this, birds exhibited a strong converse Bergmann’s pattern at interspecific and community scales, both with and without abundance. Overall, our results indicate that incorporation of information on intraspecific variation and/or species relative abundances influences the results to a large extent. The multiplicity of patterns at a single location provides the opportunity to disentangle the relative contribution of individual- and species-level processes by integrating data across multiple nested taxonomic scales for the same taxa. We suggest that future studies of Bergmann’s patterns should explicitly address taxonomic and spatial scale dependency, with species relative abundance and intraspecific trait variation as essential ingredients especially at short elevational scales.

Fundamental limitations on distillation of quantum channel resources

Quantum channels underlie the dynamics of quantum systems, but in many practical settings it is the channels themselves that require processing. We establish universal limitations on the processing of both quantum states and channels, expressed in the form of no-go theorems and quantitative bounds for the manipulation of general quantum channel resources under the most general transformation protocols. Focusing on the class of distillation tasks — which can be understood either as the purification of noisy channels into unitary ones, or the extraction of state-based resources from channels — we develop fundamental restrictions on the error incurred in such transformations, and comprehensive lower bounds for the overhead of any distillation protocol. In the asymptotic setting, our results yield broadly applicable bounds for rates of distillation. We demonstrate our results through applications to fault-tolerant quantum computation, where we obtain state-of-the-art lower bounds for the overhead cost of magic state distillation, as well as to quantum communication, where we recover a number of strong converse bounds for quantum channel capacity.

Defining quantum divergences via convex optimization

We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α-Rényi divergence between quantum channels for α>1. Second it allows us to prove a chain rule property for the sandwiched α-Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

Strong Converse Results for Linking Operators and Convex Functions

We consider a family B n , ρ c of operators which is a link between classical Baskakov operators (for ρ = ∞ ) and their genuine Durrmeyer type modification (for ρ = 1 ). First, we prove that for fixed n , c and a fixed convex function f , B n , ρ c f is decreasing with respect to ρ . We give two proofs, using various probabilistic considerations. Then, we combine this property with some existing direct and strong converse results for classical operators, in order to get such results for the operators B n , ρ c applied to convex functions.

Direct and strong converse inequalities for approximation with Fejér means

We present upper and lower estimates of the error of approximation of periodic functions by Fejér means in the Lebesgue spaces {L}_{2{\pi }}^{p}. The estimates are given in terms of a K-functional for 1\le p\le \infty and in terms of the first modulus of continuity in the case 1\lt p\lt \infty . We pay attention to the involved constants.