Generic behaviour in quantum information theory: applications of the concentration of measure phenomenon
Seminar Room 1, Newton Institute
In that branch of quantum information theory which generalises Shannon's communication theory, it is the rule (just as with Shannon) that information theoretically optimal constructions are not actually "constructive". Instead, the probabilistic method is used to show that some random way to build a code or protocol will succeed with positive probability (so that there actually exists a good code) - in fact, this probability will usually be close to 1.
Thus, it is not surprising that quantitative laws of large numbers play a significant role in the theory, and in fact the most important ones are those which give exponential probability bounds on "large deviations": Cramer's and Azuma's inequality for the convergence of empirical means and martingales, as well as relatives of Talagrand's inequality have been used in classical information theory.
This talk will describe the status of these tools and their applications in quantum information theory: I will show how the concentration of measure phenomenon on Euclidean spheres leads to strong statements for the unitary group, and to strong results regarding state randomisation, remote state preparation, data hiding, and information on the typical entanglement properties of random states. [Several collaborative papers with Abeyesinghe, Bennett, Hayden, Leung, Shor and Smith on quant-ph.]
Furthermore, I will outline operator versions of Cramer's theorem [joint work with Ahlswede, IEEE IT 2002] and of (a simple variant of) Talagrand's isoperimetric inequality [work in progress], and illustrate those with some applications such as the quantum reverse Shannon theorem and a generalisation of a "local converse to the channel coding theorem" due in the classical case to Ahlswede and Dueck.