One-Shot Rates for Entanglement Manipulation Under Non-entangling Maps

All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory}. But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy. This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding. However, it appeared that this new resource is very complex and difficult to detect. Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure. 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.

We generalize previously proposed conditions each measure of entanglement has to satisfy. We present a class of entanglement measures that satisfy these conditions and show that the Quantum Relative Entropy and Bures Metric generate two measures of this class. We calculate the measures of entanglement for a number of mixed two spin 1/2 systems using the Quantum Relative Entropy, and provide an efficient numerical method to obtain the measures of entanglement in this case. In addition, we prove a number of properties of our entanglement measure which have important physical implications. We briefly explain the statistical basis of our measure of entanglement in the case of the Quantum Relative Entropy. We then argue that our entanglement measure determines an upper bound to the number of singlets that can be obtained by any purification procedure and that distillable entanglement is in general smaller than the entanglement of creation.