Hypercontractivity and Log Sobolev Inequalities in Quantum Information Theory (15w5098)

S (in alphabetic order by speaker surname) Speaker: William Beckner (University of Texas at Austin) Title: Hypercontractivity and the Logarithmic Sobolev Inequality – from Quantum Field Theory to Geometric Inequalities Abstract: This talk will feature a quick historical survey of connections between the log Sobolev inequality and now classical problems in analysis, physics and probability. Speaker: Salman Beigi (Institute for Research in Fundamental Sciences (IPM) ) Title: Measuring quantum correlation via completely bounded norms Abstract: A hypercontractivity ribbon is a measure of bipartite correlation that is dened based on the operator norm of some linear map between Schatten classes that is associated to a given bipartite distribution. This measure of correlation has the intriguing property that the hypercontractivity ribbon of a bipartite distribution is equal to that of its i.i.d. repetitions. This talk is about the generalization of hypercontractivity ribbon for bipartite quantum states. We will discuss that to generalize the above property of the hypercontractivity ribbon to the quantum case we should use completely bounded norms. Then non-commutative vector valued Schatten spaces naturally appear in the study of quantum correlations. Finally we comment that these spaces also appear in the definition of a new generalization of Renyi (conditional) entropy to the quantum case. Speaker: Fernando G.S.L. Brandao (Microsoft Research and UCL) Title: Rapid Mixing versus Clustering for Quantum Systems Abstract: In this talk Ill present recent results relating the time of preparation of thermal states by local quantum dissipative processes (aka Liouvillians) with static properties of the state, namely whether the state is clustering (i.e. has a finite correlation length). In particular I’ll show that for commuting Hamiltonians the Davies master equation (a quantum analogue of Glauber dynamics) has a constant spectral gap if, and only if, a certain strong type of clustering holds. I’ll also discuss the challenges of proving a similar statement with the log-Sovolev constant of the Liouvillian in place of the spectral gap Our goal is to generalize to the quantum case a sequence of beautiful works – by Stroock, Zergalinski, Martinelli and others – in mathematical physics and statistical mechanics showing the equivalence of mixing in time (fast convergence of the Glauber dynamics – meaning a constant spectral gap or even a size-independent log-Sobolev constant) to mixing in space (finite correlation length in the Gibbs state) for classical models. The talk will be based on joint work with Michael Kastoryano (http://arxiv.org/abs/1409.3435). Speaker: Harry Buhrman ( ) Title: Multipartite entanglement in XOR games Speaker: Eric Carlen (Rutgrs University and I.M.A.) Title: Quantitative uniform convexity, fermion hypercontractivity, and related topics Abstract: There is a natural non-commutative analog of the Mehler semigroup, namely the fermionic Mehler semigroup, or fermionic oscillator semigroup. This arises naturally in quantum field theory, and plays a role there for fermionic systems that is entirely analogous to the role played by the usual Mehler semigroup in Edward Nelson’s approach to boson quantum fields.