#### Ryszard Horodecki, Pawel Horodecki, Michal Horodecki, Karol Horodecki

This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying. In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations. They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon. A basic role of entanglement witnesses in detection of entanglement is emphasized.

For a shorter, less technical article on quantum entanglement see here.

### An introduction to geometric group theory

This PDF may be helpful to some who want to delve more deeply into the subject of geometric group theory. I caution you though, the material presented here is not for the faint of heart, among whom I number myself without hesitation.

From the author’s Preface:

These notes are based on a series of lectures I gave at the Tokyo Institute of Technology from April to July 2005. They constituted a course entitled “An introduction to geometric group theory” totalling about 20 hours. The audience consisted of fourth year students, graduate students as well as several staﬀ members. I therefore tried to present a logically coherent introduction to the subject, tailored to the background of the students, as well as including a number of diversions into more sophisticated applications of these ideas. There are many statements left as exercises. I believe that those essential to the logical developments will be fairly routine. Those related to examples or diversions may be more challenging.

The notes assume a basic knowledge of group theory, and metric and topological spaces. We describe some of the fundamental notions of geometric group theory, such as quasi-isometries, and aim for a basic overview of hyperbolic groups. We describe group presentations from ﬁrst principles. We give an outline description of fundamental groups and covering spaces, suﬃcient to allow us to illustrate various results with more explicit examples. We also give a crash course on hyperbolic geometry. Again the presentation is rather informal, and aimed at providing a source of examples of hyperbolic groups. This is not logically essential to most of what follows. In principle, the basic theory of hyperbolic groups can be developed with no reference to hyperbolic geometry, but interesting examples would be rather sparse.

### The Elusive Neutrino

Not only are neutrinos hard to catch, but they also change form as they travel through space. New experiments hope to understand their chameleonic nature.

The cavernous Super-Kamiokande detector in Japan is lined with 13,000 sensors to pinpoint signs of neutrinos.

Kamioka Observatory, ICRR (Institute for Cosmic Ray Research), The University of Tokyo

### Looking for Neutrinos, Nature’s Ghost Particles

#### To study some of the most elusive particles, physicists have built detectors in abandoned mines, tunnels and Antarctic ice.

• By Ann Finkbeiner
• Smithsonian magazine, November 2010