CNRS-PAN Mathematics Summer Institute, Cracow 2015.

Michal Banacki

Title: Quantum symmetries of noncommutative tori

Abstract: tba

Marek Bożejko

Title: Generalized Fock spaces and second quantization as examples of hypercontractive semigroups

Abstract: In my talk we will consider the following topics:

1. Generalized Fock spaces F(T,H) for real contraction T on a real Hilbert space H.

2.Anyonic Fock spaces and type B-Fock spaces.

3.Second quantization Γ(T,S) from the von Neumann algebra G(T,H) generalized by T-Gaussian operators G(f) = a(f) + a+(f) ,where f is in a real Hilbert spcace H.

4.If S = exp(-t) Id , we get so called generalized Ornstein-Uhlenbeck semigroup

U_{t} = Γ(T,S) and in many cases of T we have hypercontactivity ,i.e.

U_{t} maps L^{2} into L^{\infty} , where here L^{2} = the Fock space F(T,H) and L^{\infty} = G(T,H).

The subject of the talk is taken from the papers together with W.Ejsmont, T.Hasebe, E.Lytvynov, I. Rodionova, Q.Xu and J.Wysoczanski.

Raffaella Carbone

Title: Irreducible decompositions and stationary states of quantum channels

Abstract: For a completely positive trace-preserving map (a quantum channel), we prove a generalization to the infinite dimensional case of a result by Baumgartner and Narnhofer ( Rev. Math. Phys. 2012). This result is, in a probabilistic language, a decomposition of a quantum channel into its irreducible recurrent components, with particular attention for the positive recurrent components. This decomposition is related with a communication relation on the reference Hilbert space. This allows us to describe the extremal invariant states, the full structure of general invariant states and of their supports.

The work is in collaboration with Yan Pautrat.

Fabio Cipriani

Abstract: "Amenability of von Neumann algebras and spectrum of Dirichlet forms"

Abstract. The discussion will concern an applications of Noncommutative Potential Theory

to approximation properties of von Neumann algebras: criteria for amenability of von Neumann

algebras and relative amenability of finite inclusions of von Neumann algebras in terms of

spectral properties of Dirichlet forms will be provided.

David Pérez García

Title: Extreme Macroscopic Finite Size Effects

Abstract: Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analysing a sequence of finite size cases ? We present a model for which such an approach gives completely misleading results: a translational invariant local Hamiltonian on a square lattice with constant spectral gap and classical product ground state for system sizes smaller than a threshold, and a ground state with topological degeneracy for larger system sizes. Starting from a minimal case with spins of dimension 5 and threshold size 15, we show that the latter grows extremely quickly as we consider larger dimensions. This shows that taking ever bigger finite system instances is not a viable solution to the general problem of determining thermodynamic properties of a physical system.
(joint work with J. Bausch, T.S. Cubitt, A. Lucia and M.M. Wolf)

Michał Horodecki

Title: Thermodynamics in micro-regime and resource theories

Abstract: Thermodynamics of a microscopic system in presence of macroscopic heat bath  is obtained within a a framework of
so called resource theories.  The constraints for thermodynmaical processing in such reime are derived, which turn out to be more tringent than
the standard second law. In particular, the standard free energy is replaced with two distinct  free energies, one describing the work of formation of a state, and the other
one - extractable work.  One of basic questions of the thermodynamics of micro-systems, is whether a given state can be transformed in to another given state.
While in the case of states diagonal in energy basis, the answer is known, the case of general state is answered only for two-level systems.

Talk will be based mostly on :

Horodecki and Oppenheim http://arxiv.org/abs/1111.3834

Brandao et al http://arxiv.org/abs/1305.5278

Cwiklinski et al, http://arxiv.org/abs/1405.5029

Anna Jencova

Title: $A$-extreme points of generalized state spaces

Abstract:
A generalized state space of a C*-algebra is the set of completely positive maps into the algebra of bounded operators on a Hilbert space $H$. We consider $A$-convex combination of such maps, where instead of elements of the $(0,1)$-interval, the coefficients of the convex combitation are operators in a unital $C*$-subalgebra $A\subseteq B(H)$, forming a resolution of the identity. In this sense, the generalized state spaces are naturally $A$-convex. $A$-convexity is a common generalization of both the usual convexity and C*-convexity.  We characterize the $A$-extreme points of the generalized state space for a finite dimensional Hilbert space $H$. This work is motivated by our investigation of extremal quantum testers, which serve as a mathematical description of measurements on quantum channels. As it turns out, the convex structure of the set of testers is closely related to the $A$-convex structure of POVMs.

Michael J. Kastoryano

Title: Recent advances in ergodic semigroups for preparing Gibbs and ground states of many-body systems. 

Abstract: As the advent of quantum computers and quantum simulators is approaching,

it has become increasingly pressing to have efficient algorithms for preparing relevant states

of matter on quantum computer. I will review some recent advances on the problem of preparing

Gibbs states and ground states of interacting many-body systems as the stationary states of quantum

dynamical semigroups. The relation between exponential decay of correlations and the mixing time

of the semigroup will be characterized. In particular, I will show that following classes of states can be

prepared efficiently:

(i) Gibbs states of commuting Hamiltonians in 1D,

(ii) general Gibbs states at high enough temperature,

(iii) ground states of gapped frustration-free Hamiltonians, (iv) matrix-product states.

Marcin Marciniak

Title: On new classes of extremal positive maps

Abstract: In the first part of my talk I will formulate some general properties of extremal and exposed positive maps. Then, two new examples of extremal positive maps will be presented. The first one is motivated by Miler-Olkiewicz work, the second is a based on a modified transposition.

Anilesh Mohari

Title: Isomorphism theorem for Kolmogorov states in quantum spin chain


Abstract: We prove translation dynamics on a two-sided quantum spin chain
with Kolmogorov states are weakly isomorphic. For a class of Kolmogorov
states of two sided quantum spin chain, we prove further that they give
isomorphic translation dynamics once their restrictions to the two sided
infinite tensor product of diagonal matrices with respect to an
orthonormal basis are isomorphic. We also give an overview sketching how
these problems are related to various properties of translation invariant
ground states of Heisenberg type of Hamiltonian in one dimensional quantum
spin chain.

Cambyse Rouzé

Second order asymptotics for quantum hypothesis testing in settings beyond i.i.d.

Abstract: Quantum Stein’s Lemma is a cornerstone of quantum statistics and concerns the problem of correctly identifying a quantum state, given the promise that it is one of two specific states (ρ or σ). It was originally derived in the asymptotic i.i.d. setting, in which arbitrarily many (say, n) identical copies of the state (ρ⊗n or σ⊗n) are considered to be available. In this setting, the lemma states that, for any given upper bound on the probability αn of erroneously inferring the state to be ρ, the probability βn of erroneously inferring the state to be σ decays exponentially in n, with the rate of decay converging to the relative entropy of the two states. The second order asymptotics for quantum hypothesis testing, which establishes the speed of convergence of this rate of decay to its limiting value, was derived in the i.i.d. case independently in [TH13] and [L14]. Using methods developed by Jaksic et al. [JOPS12], we extend this to settings beyond i.i.d..

References

[JOPS12] V. Jaksic and Y. Ogata and C.-A. Pillet and R. Seiringer, Quantum Hypothesis Testing and Non-Equilibrium Statistical Mechanics, Rev. Math. Phys. 24, 1230002 (2012)

[TH13] Marco Tomamichel and Masahito Hayashi, A Hierarchy of Information Quanti- ties for Finite Block Length Analysis of Quantum Tasks, IEEE Transactions on Information Theory 59 (11), p. 7693-7710 (2013)

[L14] Ke Li, Second-order asymptotics for quantum hypothesis testing, Ann. Statist. Volume 42, Number 1 (2014)

Andrzej Sitarz

Title: The Moyal Sphere

Abstract: The Moyal plane is usually considered as a deformation of the flat plane, however,

the algebra could equally well describe a compact noncommutative manifold.

I will review Moyal deformation and demonstrate that there exists a family of metrics

on the Moyal "plane" which have a constant scalar curvature thus yielding yet another

quantum sphere.

Piotr Sołtan

Title: Subgroups of quantum groups, the center and inner automorphisms

Abstract: I will introduce the notion of a closed subgroup of a locally compact quantum

group and show how to define the center of such a locally compact quantum group. Then I will

define the quantum group of inner automorphisms of a locally compact quantum group and describe

its relationship with the center.

Tomasz Tylec

Title: Non-signalling theories and tensor products of quantum logics

Abstract: Recently the so-called "non-signalling theoriesŤ become a popular model in the context of quantum information theory. They are supposed to describe "the most general physical systemsŤ still compatible with the principle of relativistic causality. The widely used definition of non-signalling models is somewhat naive and its interpretation, in our opinion, might be questionable. We provide rigorous discussion of such models in the framework of quantum logics (understood as partially ordered structures) and discuss relation with the various notions of tensor products of quantum logics. Among the structural results about non-signalling models obtained in this way, we are able to conclude, that contrary to common belief, discussed non-signalling models cannot be considered as a ŞgeneralisationsŤ of quantum mechanics (including models requiring only finite dimensional Hilbert space).

Yin Zhi

Title: Function spaces on quantum tori

Abstract: We study Sobolev, Besov and Triebel–Lizorkin spaces on quantum tori. These spaces share many properties with their classical counterparts. The results announced include: Besov and Sobolev embedding theorems; Littlewood–Paley-type characterizations of Besov and Triebel–Lizorkin spaces; an explicit description of the K-functional of (L_p(T^d_(θ )),W_p^k(T^d_(θ ) ));descriptions of completely bounded Fourier multipliers on these spaces.