On the reduction criterion for random quantum states

In this paper we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds

Patterns of Conflict in the Great Lakes Region

The African Great Lakes Region (GLR) has witnessed some of the most intense violence and protracted conflict of the last half-century. There has been spiralling and sometimes over-lapping conflict in Burundi, Rwanda, Uganda and the Democratic Republic of Congo (DRC) (hereinafter Zone 1 conflict states). Yet their neighbours—Kenya, Malawi, Tanzania and Zambia (hereinafter Zone 2 peaceful states)—have remained generally peaceful. This article asks what makes the difference in conflict outcomes between these neighbouring states? It has one goal: to identify a set of structural and historical factors (if any), that differentiate the zone 1 from the zone 2 states and which can explain the incidence of conflicts across time and countries. We set out to document and estimate the impact of a common set of structural factors that underpin the outbreak of wars in this region over the past fifty years, while controlling for time and country specific effects.

A complete characterization of exponential stability for discrete dynamics

For a discrete dynamics defined by a sequence of bounded and not necessarily invertible linear operators, we give a complete characterization of exponential stability in terms of invertibility of a certain operator acting on suitable Banach sequence spaces. We connect the invertibility of this operator to the existence of a particular type of admissible exponents. For the bounded orbits, exponential stability results from a spectral property. Some adequate examples are presented to emphasize some significant qualitative differences between uniform and nonuniform behavior.Comment: The final version will be published in Journal of Difference Equations and Application

Positive reduction from spectra

We study the problem of whether all bipartite quantum states having a prescribed spectrum remain positive under the reduction map applied to one subsystem. We provide necessary and sufficient conditions, in the form of a family of linear inequalities, which the spectrum has to verify. Our conditions become explicit when one of the two subsystems is a qubit, as well as for further sets of states. Finally, we introduce a family of simple entanglement criteria for spectra, closely related to the reduction and positive partial transpose criteria, which also provide new insight into the set of spectra that guarantee separability or positivity of the partial transpose.Comment: Linear Algebra and its Applications (2015

Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid

Correlation with basic differentiation processes of neurons

The development of the spinal cord involves the proliferation of neurons, their migration to well-defined areas, fiber outgrowth and synapse formation. The present study was designed to correlate the spatiotemporal pattern of expression of synaptophysin, an integral membrane protein of small synaptic vesicles, with these basic processes occurring during the embryonic development of the rat spinal cord. Thoracic segments of spinal cords from embryonic days 12, 14, 16, 18, 20 and of adult spinal cords were studied. S1 nuclease protection assays and immunoblots revealed minute amounts of specific mRNA and synaptophysin at embryonic day 12. There was a steep increase of mRNA between embryonic days 14 and 16, after which levels reached a plateau. A rise in the amount of synaptophysin in the spinal cord occurred between embryonic days 12 and 14, and the levels changed only slightly until the end of embryonic development. Even higher levels of synaptophysin, found in the adult spinal cord, may indicate that its biosynthesis continued after birth. In situ hybridization histochemistry revealed the localization of specific synaptophysin mRNA in the neuroepithelium. However, immunocytochemistry failed to detect synaptophysin in the neuroepithelial cells. Following migration of the neuroblasts, synaptophysin was found in neurons concomitantly with the onset of fiber outgrowth. Thus, already at embryonic day 12, outgrowing fibers of the dorsal root sensory neurons and of motoneurons were synaptophysin positive. From embryonic day 14 throughout the prenatal period, strong synaptophysin immunoreactivity was seen in the ventrolateral and dorsal parts of the marginal layer. Most likely this staining pattern indicates transient functional synaptic contacts because, in the adult spinal cord, the corresponding region, the white matter, exhibited only faint synaptophysin immunoreactivity. In the intermediate layer of the embryonic spinal cord, which corresponds to the gray matter of the adult spinal cord, synaptophysin-positive fibers were observed prior to the formation of functional synapses. The latter are most likely permanent, since synaptophysin in the adult spinal cord is mainly confined to the gray matter. Our data (i) show transcription and translation of synaptophysin within the neurons of the spinal cord and correlate these processes with proliferation, migration, fiber outgrowth and the formation of transient or permanent synapses, and (ii) prove that synaptophysin is a marker for fiber outgrowth in addition to synapse formation