# Oberseminar: Analysis und Zufall

P. Müller (LMU), S. Warzel (TUM)

## Time and location:

Tuesdays, 16:30 - 17:30 Uhr

TUM: Seminarraum 03.10.011

LMU:

## 2.5. at TUM: Felix Krahmer (TUM), Matrix factorization with binary components - uniqueness in a randomized model

Motivated by an application in computational biology, we consider low-rank matrix factorization with {0,1}-constraints on the first of the factors and optionally convex constraints on the second one. For not too large ranks, it has been shown by Hein et al. (2013) that one can provably recover the underlying factorization, provided there exists a unique solution. We show that when choosing a sparse (i.e., biased) Bernoulli random model for the binary factor, there is a unique solution with high probability. For an unbiased Bernoulli model, this corresponds to a result by Odlyzko (1988), but as it turns out, the asymmetry makes the biased case considerably more involved. At the core of our proof is a new asymmetric Littlewood-Offord inequality.

This is joint work with Matthias Hein and David James.

## 9.5. at 17:00 at TUM: Mostafa Sabri (Strasbourg and Cairo), Quantum ergodicity for the Anderson model on regular graphs

In this talk I will discuss a result of delocalization for the Anderson model on the regular tree (Bethe lattice). The Anderson model is a random Schrodinger operator, where we add a random i.i.d. perturbation to the adjacency matrix. Localization at high disorder is well understood today for a wide variety of models, both in the sense of a.s. pure point spectrum with exponentially decaying eigenfunctions, and in a dynamical sense. Delocalization remains a great challenge. For tree models, it is known that for weak disorder, large parts of the spectrum are a.s. purely absolutely continuous, and the dynamical transport is ballistic. In this work, we try to complete the picture by proving that in such AC regime, the eigenfunctions are also delocalized in space, in the sense that if we consider a sequence of regular graphs converging to the regular tree, then the eigenfunctions become asymptotically uniformly distributed. The precise result is a quantum ergodicity theorem, which holds in a much more general framework.

This is a joint work with Nalini Anantharaman.

## 23.5. at TUM: Markus Heyl (Dresden), Dynamical potentials for nonequilibrium quantum phases

Eigenstate order extends the concept of phases of quantum matter beyond the
conventional equilibrium paradigm which is central for inherently dynamical
phenomena such as many-body localization or quantum time crystals. While
eigenstate order is not visible in thermodynamic ensembles, it is rather
imprinted in the properties of single eigenstates. In this talk I discuss how
it is nevertheless possible to construct dynamical potentials capturing the
macroscopic properties of eigenstate phases and which share many formal
analogies to conventional thermodynamic potentials such as Gibbs-Duhem and
Maxwell relations. The presented formalism opens up a route towards a
macroscopic and phenomenological description of eigenstate phases and
potentially also the respective transitions.

## 13.6. at TUM: Ion Nechita (Toulouse and TUM), On some uses of random matrices in quantum information theory

I will review the theory of random quantum states and random quantum channels. I will then discuss recent progress to the question of additivity of the minimum output entropy of quantum channels coming from new random examples.

## 13.7. 15:30-16:30: Sondertermin im TUM Analysis Seminar: Alain Joye (Grenoble): Landauer’s Principle in Repeated Interaction Systems

We study Landauer’s principle for repeated interaction systems consisting of a reference quantum system S in contact with a environment E consisting of a chain of independent quantum probes. The system S interacts with each probe sequentially and the Landauer principle relates the energy variation of E and the decrease of entropy of S by the entropy production of the dynamical process. We address the adiabatic regime where the environment, consisting of T ≫ 1 probes, displays variations of order 1/T between the successive probes. We analyze Landauer’s bound and its refinements at the level of the full statistics associated to a two-time measurement protocol of, essentially, the energy of E.
Joint work with E. Hanson, Y. Pautrat, R. Raquépas

--

SimoneWarzel - 20 Apr 2017