A remark on the paper ``Randomizing quantum states: Constructions and applications''

The concept of \e-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly $d \log d$ random unitary operators provide examples of such channels on \C^d. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly $d$

Partial transposition of random states and non-centered semicircular distributions

Let W be a Wishart random matrix of size d^2 times d^2, considered as a block matrix with d times d blocks. Let Y be the matrix obtained by transposing each block of W. We prove that the empirical eigenvalue distribution of Y approaches a non-centered semicircular distribution when d tends to infinity. We also show the convergence of extreme eigenvalues towards the edge of the expected spectrum. The proofs are based on the moments method. This matrix model is relevant to Quantum Information Theory and corresponds to the partial transposition of a random induced state. A natural question is: "When does a random state have a positive partial transpose (PPT)?". We answer this question and exhibit a strong threshold when the parameter from the Wishart distribution equals 4. When d gets large, a random state on C^d tensor C^d obtained after partial tracing a random pure state over some ancilla of dimension alpha.d^2 is typically PPT when alpha>4 and typically non-PPT when alpha<4.Comment: 24 pages. V2 : includes the convergence of extreme eigenvalues. V3 : includes probability estimates showing that p=4d^2 is a sharp threshol

Stochastic domination for iterated convolutions and catalytic majorization

We study how iterated convolutions of probability measures compare under stochastic domination. We give necessary and sufficient conditions for the existence of an integer $n$ such that $\mu^{*n}$ is stochastically dominated by $\nu^{*n}$ for two given probability measures $\mu$ and $\nu$. As a consequence we obtain a similar theorem on the majorization order for vectors in $\R^d$. In particular we prove results about catalysis in quantum information theory

How the News Frames Child Maltreatment: Unintended Consequences

While advocates are usually gratified to see attention paid to their issue in the news, the coverage can often be a mixed blessing, as research by the FrameWorks Institute and others has shown. It is the way that stories are told in the news that affects public thinking, and many of these stories do not guide thinking in constructive directions. A story that seems to convey important information may also have unintended, damaging consequences for public understanding and engagement. This document summarizes some of the major patterns in news coverage of child maltreatment -- the key narratives, frames and causal stories that are conveyed to the public on the issue. The material for the analysis includes a collection of roughly 120 news articles collected by Prevent Child Abuse America and Cultural Logic. Additionally, the review included a collection of several dozen TV news stories assembled by the Center for Communications and Community at UCLA. The premise behind this study is that once advocates have a better idea about the way their issue is portrayed in the media, they can be strategic about choosing which narratives to reinforce, which to challenge, and which to downplay. A close examination of news coverage also gives advocates a window into what they are up against as they try to increase public engagement. This research analysis is part of New FrameWorks Research on Child Abuse and Neglect Prevention, and was conducted in collaboration with the FrameWorks Institute, and commissioned by Prevent Child Abuse America, with funding from the Doris Duke Charitable Foundation. Please visit our website for more information

Tiling Problems on Baumslag-Solitar groups

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.Comment: In Proceedings MCU 2013, arXiv:1309.104

Two Cognitive Obstacles to Preventing Child Abuse: The 'Other Mind' Mistake and the 'Family Bubble'

Following decades of effective publicity about the issue, Americans are now aware of the horrors of child abuse and have an idea (even an exaggerated idea) of the pervasiveness of all types of maltreatment. Making further headway in engaging the public on the issue will have to involve more than raising the volume on awareness campaigns. Such campaigns can even backfire by intensifying the public's media-fed association between abuse and sensational crimes -- which only "sick monsters" could commit and no programs can ever totally eliminate.To take the public to the next step in engagement, communications will need to address counterproductive patterns of reasoning that hinder better understanding of the issue. One of the most pervasive of these is the "Other-Minds" mistake: Lay people misperceive a child as a little mind which develops through abstract processes like learning, memory and choice; or which does not "develop" at all, and exists from the beginning as something like an adult mind which just needs to be "filled" or "guided." This fallacy effectively obscures any scientific understanding of development of biological systems which guide these and all other aspects of behavior. This fallacy is natural, we suggest, because of a highly evolved (and very useful) human mechanism for interpreting the content of Other-Minds (known to psychologists as the "Other-Minds module"). While the "Other-Minds" module is extremely useful for trying to read the minds of other adults, it also leads to a number of distortions that make child maltreatment more likely to happen, and less likely to be prevented. These distortions include a tendency to believe that an infant has an "agenda" that conflicts with ours; an exaggerated sense of children's ability to "get past" abuse through force of will; a sense that even one year-old children can benefit from punishment for breaking moral rules; and a difficulty understanding the concept of "neglect" except as something like "underinvolvement;" among others. An additional cognitive obstacle which communications need to address is the "Family Bubble" -- the default mode of thinking in which events within the family (including child rearing and child maltreatment) take place in a sphere that is separate and different from the public sphere. This default understanding is stronger than a mere belief that families should be autonomous. It means that even thinking about the interaction between child rearing and public policy is difficult for people, and that communications based on reinforcing the "Village," while appealing, can lead to conflictedness rather than change. This research analysis is part of New FrameWorks Research on Child Abuse and Neglect Prevention, and was conducted in collaboration with the FrameWorks Institute, and commissioned by Prevent Child Abuse America, with funding from the Doris Duke Charitable Foundation

Catalytic majorization and ℓp\ell_p norms

An important problem in quantum information theory is the mathematical characterization of the phenomenon of quantum catalysis: when can the surrounding entanglement be used to perform transformations of a jointly held quantum state under LOCC (local operations and classical communication) ? Mathematically, the question amounts to describe, for a fixed vector $y$, the set $T(y)$ of vectors $x$ such that we have $x \otimes z \prec y \otimes z$ for some $z$, where $\prec$ denotes the standard majorization relation. Our main result is that the closure of $T(y)$ in the $\ell_1$ norm can be fully described by inequalities on the $\ell_p$ norms: $\|x\|_p \leq \|y\|_p$ for all $p \geq 1$. This is a first step towards a complete description of $T(y)$ itself. It can also be seen as a $\ell_p$-norm analogue of Ky Fan dominance theorem about unitarily invariant norms. The proofs exploits links with another quantum phenomenon: the possibiliy of multiple-copy transformations ($x^{\otimes n} \prec y^{\otimes n}$ for given $n$). The main new tool is a variant of Cram\'er$ theorem on large deviations for sums of i.i.d. random variables