Finding decompositions of a class of separable states

By definition a separable state has the form \sum A_i \otimes B_i, where 0 \leq A_i, B_i for each i. In this paper we consider the class of states which admit such a decomposition with B_1, ..., B_p having independent images. We give a simple intrinsic characterization of this class of states, and starting with a density matrix in this class, describe a procedure to find such a decomposition with B_1, ..., B_p having independent images, and A_1, ..., A_p being distinct with unit trace. Such a decomposition is unique, and we relate this to the facial structure of the set of separable states. A special subclass of such separable states are those for which the rank of the matrix matches one marginal rank. Such states have arisen in previous studies of separability (e.g., they are known to be a class for which the PPT condition is equivalent to separability). The states investigated also include a class that corresponds (under the Choi-Jamio{\l}kowski isomorphism) to the quantum channels called quantum-classical and classical-quantum by Holevo

Complete positivity of the map from a basis to its dual basis

The dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of the n by n matrices to its dual basis is a complete order isomorphism and complete co-order isomorphism. In the case of the standard matrix units this map is a complete order isomorphism and this is a restatement of the correspondence between completely positive maps and the Choi matrix. However, we exhibit natural orthonormal bases for the matrices such that this map is an order isomorphism, but not a complete order isomorphism. Some bases yield complete co-order isomorphisms. Included among such bases is the Pauli basis and tensor products of the Pauli basis. Consequently, when the Pauli basis is used in place of the the matrix unit basis, the analogue of Choi's theorem is a characterization of completely co-positive maps

A Qualitative Study of Student-Centered Learning Practices in New England High Schools

In early 2015, the Nellie Mae Education Foundation (NMEF) contracted with the UMass Donahue Institute (UMDI) to conduct a qualitative study examining the implementation of student-centered learning (SCL) practices in select public high schools in New England. This study extends lines of inquiry explored through a prior (2014) project that UMDI conducted for NMEF. The 2014 study employed survey methodology to examine the prevalence of student-centered practices in public high schools across New England. The present study builds upon the investigation, using a variety of qualitative methods to further probe the richness and complexity of SCL approaches in use across the region. Specifically, this study was designed to address what student-centered practices "look like" in an array of contexts. The study also addresses the perceived impacts that SCL approaches have on students, staff, and schools. Additionally, it highlights the broad array of factors within and beyond school walls that reportedly foster and challenge the implementation of SCL practices. This study seeks to help NMEF understand the intricacies of SCL and provides strategic considerations for how Nellie Mae can promote the adoption and development of student-centered practices in the region.Nellie Mae organizes student-centered learning by four tenets: (1) learning is personalized; (2) learning is competency-based; (3) learning takes place anytime, anywhere; and (4) students take ownership.Specifically, the study addresses five research questions:What are the characteristics of student-centered practices in relation to the four SCL tenets? How are SCL approaches implemented?What are the salient contextual factors (e.g., systems, structures, policies, procedures) associated with the implementation of SCL practices? How do they support, impede, and otherwise shape the adoption, development, and implementation of SCL approaches?How are schools with moderate and high levels of SCL implementation organized to foster SCL practices? What mechanisms are in place to promote student-centered learning?What is the role of SCL approaches in schools and classrooms? In what ways, if at all, are they embedded in the goals and practices of schools and classrooms?What is the quality of SCL instructional practices in study schools? What relationships, if any, do administrators and educators perceive between these approaches and student learning

Unique decompositions, faces, and automorphisms of separable states

Let S_k be the set of separable states on B(C^m \otimes C^n) admitting a representation as a convex combination of k pure product states, or fewer. If m>1, n> 1, and k \le max(m,n), we show that S_k admits a subset V_k such that V_k is dense and open in S_k, and such that each state in V_k has a unique decomposition as a convex combination of pure product states, and we describe all possible convex decompositions for a set of separable states that properly contains V_k. In both cases we describe the associated faces of the space of separable states, which in the first case are simplexes, and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras. As an application of these results, we characterize all affine automorphisms of the convex set of separable states, and all automorphisms of the state space of B(C^m otimes C^n). that preserve entanglement and separability.Comment: Since original version:Cor. 6 revised and renamed Thm 6, some definitions added before Cor. 11, introduction revised and references added, typos correcte