Practical and reliable error bars for quantum process tomography

Current techniques in quantum process tomography typically return a single point estimate of an unknown process based on a finite albeit large amount of measurement data. Due to statistical fluctuations, however, other processes close to the point estimate can also produce the observed data with near certainty. Unless appropriate error bars can be constructed, the point estimate does not carry any sound operational interpretation. Here, we provide a solution to this problem by constructing a confidence region estimator for quantum processes. Our method enables reliable estimation of essentially any figure of merit for quantum processes on few qubits, including the diamond distance to a specific noise model, the entanglement fidelity, and the worst-case entanglement fidelity, by identifying error regions which contain the true state with high probability. We also provide a software package, QPtomographer, implementing our estimator for the diamond norm and the worst-case entanglement fidelity. We illustrate its usage and performance with several simulated examples. Our tools can be used to reliably certify the performance of, e.g., error correction codes, implementations of unitary gates, or more generally any noise process affecting a quantum system

Dealing with Food and Eggs in Mouthbrooding Cichlids: Structural and Functional Trade-Offs in Fitness Related Traits

As in any vertebrate, heads of fishes are densely packed with functions. These functions often impose conflicting mechanical demands resulting in trade-offs in the species-specific phenotype. When phenotypical traits are linked to gender-specific parental behavior, we expect sexual differences in these trade-offs. This study aims to use mouthbrooding cichlids as an example to test hypotheses on evolutionary trade-offs between intricately linked traits that affect different aspects of fitness. We focused on the oral apparatus, which is not only equipped with features used to feed and breathe, but is also used for the incubation of eggs. We used this approach to study mouthbrooding as part of an integrated functional system with diverging performance requirements and to explore gender-specific selective environments within a species.Because cichlids are morphologically very diverse, we hypothesize that the implications of the added constraint of mouthbrooding will primarily depend on the dominant mode of feeding of the studied species. To test this, we compared the trade-off for two maternal mouthbrooding cichlid species: a "suction feeder" (Haplochromis piceatus) and a "biter" (H. fischeri). The comparison of morphology and performance of both species revealed clear interspecific and intersex differences. Our observation that females have larger heads was interpreted as a possible consequence of the fact that in both the studied species mouthbrooding is done by females only. As hypothesized, the observed sexual dimorphism in head shape is inferred as being suboptimal for some aspects of the feeding performance in each of the studied species. Our comparison also demonstrated that the suction feeding species had smaller egg clutches and more elongated eggs.Our findings support the hypothesis that there is a trade-off between mouthbrooding and feeding performance in the two studied haplochromine cichlids, stressing the importance of including species-specific information at the gender level when addressing interspecific functional/morphological differences

Associations between Floral Asymmetry and Individual Genetic Variability Differ among Three Prickly Pear (Opuntia echios) Populations

While stress is expected to increase developmental instability (DI), not all studies confirm this. This heterogeneity could in part be due to the use of subtle differences between the left and right side of bilateral symmetrical organisms to quantify DI, leading to large sampling error obscuring associations with DI. Traits that develop simultaneously more than twice (such as flower petals or bird feathers) reflect individual DI more reliably, such that stronger associations are expected to emerge. Furthermore, some studies have shown differences in strengths of associations among populations. We studied the association between individual genetic diversity and DI in flower petals within three Opuntia echios populations inhabiting Galápagos. Quantifying individual DI through variation in length and width of a high number of petals within individual cacti, lead to a strong association between DI and genetic diversity in one population. We conclude that associations between individual DI and genetic diversity can be more easily revealed by measuring traits that develop repeatedly

Associations between floral asymmetry and individual genetic variability differ among three prickly pear (**Opuntia echios**) populations

While stress is expected to increase developmental instability (DI), not all studies confirm this. This heterogeneity could in part be due to the use of subtle differences between the left and right side of bilateral symmetrical organisms to quantify DI, leading to large sampling error obscuring associations with DI. Traits that develop simultaneously more than twice (such as flower petals or bird feathers) reflect individual DI more reliably, such that stronger associations are expected to emerge. Furthermore, some studies have shown differences in strengths of associations among populations. We studied the association between individual genetic diversity and DI in flower petals within three Opuntia echios populations inhabiting Galápagos. Quantifying individual DI through variation in length and width of a high number of petals within individual cacti, lead to a strong association between DI and genetic diversity in one population. We conclude that associations between individual DI and genetic diversity can be more easily revealed by measuring traits that develop repeatedly