Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable, and present a complete description of anti-degradable unital qubit channels with a new proof. For higher output dimensions we explore the relationship between the output and environment dimensions (dB and dE, respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with an environment that is “small” in the sense of ΦC. Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with “large” environments, in the sense that the minimal dimension dE>dB. Indeed, one can have dE>14d2B. These examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels. In the case of channels with diagonal Kraus operators, we describe the subclasses that are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.
Alternate version on arXiv: https://arxiv.org/abs/0802.1360
In a talk aimed at the general public, Professor Hawking discusses theories on the origin of the universe. He explains how time can have a beginning, and addresses the progress made by cosmologists in an area which has traditionally belonged to theologists and philosophers.
Stephen Hawking holds the prestigious Lucasian chair at Cambridge University, once held by Sir Isaac Newton. He is one of the early developers of the theory of black holes and author of the international best-selling book A Brief History of Time.
PLEASE NOTE: This event is free, but tickets will be required. General admission tickets will be distributed on the morning of the lecture only. Please carefully review the complete ticketing procedures, available in a PDF file here.
Tuesday, April 4, 2006
8:00pm to 10:00pm
Arrived around 7:15 to get in line and ended up with a nice seat about 10 rows back from the stage. He was entertaining and even a tad inspirational, but it was definitely a “public” lecture and disappointingly had absolutely no technical content in the least.