We study and extend the semidefinite programming (SDP) hierarchies introduced in [Phys. Rev. Lett. 115, 020501] for the characterization of the statistical correlations arising from finite dimensional quantum systems. First, we introduce the dimensio
n-constrained noncommutative polynomial optimization (NPO) paradigm, where a number of polynomial inequalities are defined and optimization is conducted over all feasible operator representations of bounded dimensionality. Important problems in device independent and semi-device independent quantum information science can be formulated (or almost formulated) in this framework. We present effective SDP hierarchies to attack the general dimension-constrained NPO problem (and related ones) and prove their asymptotic convergence. To illustrate the power of these relaxations, we use them to derive new dimension witnesses for temporal and Bell-type correlation scenarios, and also to bound the probability of success of quantum random access codes.
We describe a simple method to derive high performance semidefinite programming relaxations for optimizations over complex and real operator algebras in finite dimensional Hilbert spaces. The method is very flexible, easy to program and allows the us
er to assess the behavior of finite dimensional quantum systems in a number of interesting setups. We use this method to bound the strength of quantum nonlocality in bipartite and tripartite Bell scenarios where the dimension of a subset of the parties is bounded from above. We derive new results in quantum communication complexity and prove the soundness of the prepare-and-measure dimension witnesses introduced in [Phys. Rev. Lett. 105, 230501 (2010)]. Finally, we propose a new dimension witness that can distinguish between classical, real and complex two-level systems.
Chloroplast microsatellites are becoming increasingly popular markers for population genetic studies in plants, but there has been little focus on their potential for demographic inference. In this work the utility of chloroplast microsatellites for
the study of population expansions was explored. First, we investigated the power of mismatch distribution analysis and the F(S) test with coalescent simulations of different demographic scenarios. We then applied these methods to empirical data obtained for the Canary Island pine (Pinus canariensis). The results of the simulations showed that chloroplast microsatellites are sensitive to sudden population growth. The power of the F(S) test and accuracy of demographic parameter estimates, such as the time of expansion, were reduced proportionally to the level of homoplasy within the data. The analysis of Canary Island pine chloroplast microsatellite data indicated population expansions for almost all sample localities. Demographic expansions at the island level can be explained by the colonization of the archipelago by the pine, while population expansions of different ages in different localities within an island could be the result of local extinctions and recolonization dynamics. Comparable mitochondrial DNA sequence data from a parasite of P. canariensis, the weevil Brachyderes rugatus, supports this scenario, suggesting a key role for volcanism in the evolution of pine forest communities in the Canary Islands.
