An afternoon of talks exploring the links between classical information theory, probability, statistics and their quantum counterparts.

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• Reinhard Werner (Hannover), Fernando Brandao (Microsoft Research), Robert Koenig (TU Munich), Renato Renner (ETH Zurich)
• Wednesday 28 January 2015, 14:00-18:10
• MR15 Centre for Mathematical Sciences.

If you have a question about this talk, please contact HoD Secretary, DPMMS.

An afternoon of talks exploring the links between classical information theory, probability, statistics and their quantum counterparts.

2.00pm Reinhard Werner (Hannover): Quantum Walks

Like random walks, quantum walks are dynamical systems on a lattice with a discrete time step. In contrast to their classical counterparts, however, they are reversible, unitary processes. They move faster, i.e., with a limiting speed, rather than proportional to the square root of the number of steps. I will sketch a proof of the basic limit formula, and give a large deviation estimate for speeds outside the propagation region. Under time-dependent but translation invariant noise the walk typically slows down to the classical, diffusive scaling, whereas with space dependent but stationary disorder (in one dimension) one gets Anderson localization, i.e., no propagation at all. This phenomenon is also typical for quasi-periodic walks, like walks in an external electric field. Finally, I will discuss the recurrence of walks in a scenario, where the return to the initial state is monitored by repeated measurements. It turns out that recurrence has a straightforward characterization in terms of the spectrum of the unitary walk operator.

2.55pm Fernando Brandao (Microsoft Research, Seattle): Hypothesis Testing and Stein’s Lemma for Quantum Systems

I will discuss quantum generalisations of hypothesis testing, in particular of the well-known Stein’s Lemma; the latter shows that the relative entropy is the optimal rate in asymmetric hypothesis testing between two probability measures. I will discuss extensions of the quantum version of Stein’s lemma originally proven by Hiai and Petz in 1991 and show their relevance to the theory of quantum entanglement.

3.50 Coffee Break

4.20pm Robert Koenig (TU Munich) : Entropy Power Inequalities

The classical entropy power inequality, originally proposed by Shannon, is a powerful tool in multi-user information theory. In this talk, I review some of the history of this inequality, as well as Shannon’s original application: such inequalities provide bounds on the capacities of additive noise channels. I then introduce a quantum entropy power inequality which lower bounds the output entropy as two independent signals combine at a beamsplitter. In turn, such inequalities provide upper bounds on the classical capacity of additive bosonic noise channels. This is based on joint work with Graeme Smith.

5.15 pm Renato Renner (ETH Zurich) : Approximate Markov Chains

Three random variables, A, B, and C, are said to satisfy the Markov chain property if A and C are independent of each other conditioned on B. The degree to which this property holds is related to an information-theoretic measure, known as the “conditional mutual information”. More precisely, it can be shown that the Markov chain property holds approximately if and only if the mutual information between A and C conditioned on B is small. In my talk, I will explain how this statement can be extended to the more general setting where A, B, and C are arbitrary quantum systems.

This talk is part of the Peter Whittle Lecture series.

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