On Eigenvalues of the sum of two random projections

We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.Comment: 14 page

Exact Minimum Eigenvalue Distribution of an Entangled Random Pure State

A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a random real and a random complex state. Our results are relevant to the entanglement properties of eigenvectors of the orthogonal and unitary ensembles of random matrix theory and quantum chaotic systems. They also provide a rare exactly solvable case for the distribution of the minimum of a set of N {\em strongly correlated} random variables for all values of N (and not just for large N).Comment: 13 pages, 2 figures included; typos corrected; to appear in J. Stat. Phy

Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensemble

The paper studies the spectral properties of large Wigner, band and sample covariance random matrices with heavy tails of the marginal distributions of matrix entries.Comment: This is an extended version of my talk at the QMath 9 conference at Giens, France on September 13-17, 200

Statistical distribution of quantum entanglement for a random bipartite state

We compute analytically the statistics of the Renyi and von Neumann entropies (standard measures of entanglement), for a random pure state in a large bipartite quantum system. The full probability distribution is computed by first mapping the problem to a random matrix model and then using a Coulomb gas method. We identify three different regimes in the entropy distribution, which correspond to two phase transitions in the associated Coulomb gas. The two critical points correspond to sudden changes in the shape of the Coulomb charge density: the appearance of an integrable singularity at the origin for the first critical point, and the detachement of the rightmost charge (largest eigenvalue) from the sea of the other charges at the second critical point. Analytical results are verified by Monte Carlo numerical simulations. A short account of some of these results appeared recently in Phys. Rev. Lett. {\bf 104}, 110501 (2010).Comment: 7 figure

Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble

In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble $GCy$, whose eigenvalues PDF is given by $$\textrm{const}\cdot\prod_{1\leq j<k\leq N}(x_j-x_k)^2\prod_{j=1}^N (1+ix_j)^{-s-N}(1-ix_j)^{-\bar{s}-N}dx_j,$$where $s$ is a complex number such that $\Re(s)>-1/2$ and where $N$ is the size of the matrix ensemble. Using results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that for this ensemble, the largest eigenvalue divided by $N$ converges in law to some probability distribution for all $s$ such that $\Re(s)>-1/2$. Using results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of the largest eigenvalue for fixed $N$, we also express the limiting probability distribution in terms of some non-linear second order differential equation. Eventually, we show that the convergence of the probability distribution function of the re-scaled largest eigenvalue to the limiting one is at least of order $(1/N)$.Comment: Minor changes in this version. Added references. To appear in Journal of Statistical Physic

Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1

For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p >1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.Comment: Merger of arXiv:0707.0402 and arXiv:0707.3291 containing new and improved analysis of counterexamples. 17 page

The evolution of language: a comparative review

For many years the evolution of language has been seen as a disreputable topic, mired in fanciful &quot;just so stories&quot; about language origins. However, in the last decade a new synthesis of modern linguistics, cognitive neuroscience and neo-Darwinian evolutionary theory has begun to make important contributions to our understanding of the biology and evolution of language. I review some of this recent progress, focusing on the value of the comparative method, which uses data from animal species to draw inferences about language evolution. Discussing speech first, I show how data concerning a wide variety of species, from monkeys to birds, can increase our understanding of the anatomical and neural mechanisms underlying human spoken language, and how bird and whale song provide insights into the ultimate evolutionary function of language. I discuss the ‘‘descended larynx’ ’ of humans, a peculiar adaptation for speech that has received much attention in the past, which despite earlier claims is not uniquely human. Then I will turn to the neural mechanisms underlying spoken language, pointing out the difficulties animals apparently experience in perceiving hierarchical structure in sounds, and stressing the importance of vocal imitation in the evolution of a spoken language. Turning to ultimate function, I suggest that communication among kin (especially between parents and offspring) played a crucial but neglected role in driving language evolution. Finally, I briefly discuss phylogeny, discussing hypotheses that offer plausible routes to human language from a non-linguistic chimp-like ancestor. I conclude that comparative data from living animals will be key to developing a richer, more interdisciplinary understanding of our most distinctively human trait: language

Chemical concepts inventory of Grade 12 learners and UP foundation year students

Incoming chemistry students at tertiary institutions have a variety of academic backgrounds that influence their prospects of success at first-year level. The proficiencies of incoming students are currently changing due to the introduction of outcomes-based education and new syllabi for physical science in secondary schools. In order to ensure a smooth transition from secondary to tertiary education, university lecturers should be well informed about the content knowledge, conceptual understanding and skills development of prospective first-year students. This study evaluates the proficiencies of Grade 12 learners in physical science in terms of a number of clearly identified problem areas: conceptual understanding, logical scientific reasoning, basic mathematical ability, knowledge of subject content and scientific process skills. A test instrument was developed that consisted mainly of conceptual questions rather than recall or algorithmic items. Paired questions (two-tier methodology) and pictorial representations were used extensively. A follow-up question about certainty of response was included for all fixed-response items in order to evaluate the influence of guessing on response distributions. The test instrument was administered at the end of the third term to Grade 12 learners taking physical science at three schools in privileged environments (1 English and 2 Afrikaans medium) and four township schools, and to all University of Pretoria Foundation Year (UPFY) students. Analysis of results highlighted the generally poor performance of students from township schools and the significant improvement in performance after one year of intensive instruction of UPFY students, who generally came from similar or more impoverished backgrounds. The poor performance for all cohorts on basic concepts, such as the mole concept, stoichiometry and the limiting reagent, as well as on several special topics, indicates that students lack a sound basis for tertiary chemistry. Of real concern is the evidence of over-confidence obtained from the certainty of response analyses. This result indicates that respondents failed to judge the complexity and level of difficulty of questions accurately. Learners/students from all groups displayed weak understanding of events at molecular level. In order to address this situation, lecturers at tertiary level will have to actively promote conceptual understanding of all basic concepts in chemistry and resist the temptation to teach and assess mainly procedural fluency. Analysis of certainty of response data showed that the guess factor was less serious a complication than anticipated