# Probabilistic Methods in Quantum Mechanics

Program

## Titles and Abstracts

Stéphane Attal : "Repeated Quantum Interactions: Old and New"
We propose a review of the so-called model of "repeated quantum interactions". We shall review the physical motivations, the mathematical setup and main properties. We shall also recall some old and new results: continuous-time limit, model for a quantum heat bath, thermalization properties.

Guillaume Aubrun: "Thresholds for Random States"
We consider a model of random mixed states on $C^d \otimes C^d$ obtained by the partial trace of a random pure state on $C^d \otimes C^d \otimes C^s$ (where s corresponds to the dimension of the environment). Intuition tells us that when s is small, the resulting random mixed state is likely to be entangled. On the other hand, when s is large, the resulting state is likely to be almost maximally mixed (hence separable). We obtain quantitative results in this direction by showing that the threshold between generic entanglement and generic separability occurs when s is between d^3 and d^3 log^2 d. We obtain also more precise thresholds for some separability criteria, such as the Positive Partial Transpose criterion and the Realignment criterion. (based on joint works with S. Szarek and D. Ye, and with I. Nechita).

Teodor Banica: "Complex Hadamard Matrices - Probabilisitic Aspects"
One of the main problems in quantum groups and subfactors, raised some time ago by Jones, is the computation of the quantum invariants of the complex Hadamard matrices H\in M_n(C). These invariants are a sequence of positive integers c_1,c_2,c_3,.., which are best understood as being the moments of a certain real probability measure \mu_H associated (in a quite complicated way) to H. I will discuss the construction H->\mu_H, and the few things that are known about it, from a random matrix/probabilistic viewpoint.

Alexander Belton: "Quantum Random Walks and Thermalisation"
The Attal-Pautrat model for repeated interactions is a simple quantum-mechanical model of system-particle interaction. This talk will begin by using the notions of operator spaces and completely bounded maps to obtain a generalisation of it, the quantum random walk. It will then be explained how such walks, subject to suitable scaling, converge to certain quantum stochastic processes, in a manner which is analogous to the convergence of a classical random walk to Brownian motion.Inspired by an example of Attal and Joye, this framework will be extended again to deal with particles in a faithful, normal state. A new scaling, appropriate for this setting, will be introduced and a thermalisation phenomenon is demonstrated: for such walks, fewer quantum noises than might be expected are required to describe the limit process. If time permits, the extension to non-faithful states will be discussed.

Tristan Benoist: "Repeated quantum non-demolition measurements convergence"

François Chapon: "Quantum Random Walks and Minors of Hermitian Brownian Motion"
A result of Adler, Nordenstam and van Moerbeke states that the process of eigenvalues of two consecutive minors of Hermitian Brownian motion is a Markov process, whereas if one considers more than two consecutive minors the Markov property fails. We will see how to understand these results in a noncommutative context by considering quantum random walks which are discrete-time approximations of the eigenvalue processes of minors of Hermitian Brownian motion (joint work with Manon Defosseux).

Benoit Collins: "Non-Commutative Brownian Motions"
Nous étudions la classe des processus non-commutatifs X_t dont les trajectoires sont presque surement continues, tels que X_t et X_t^2-t sont des martingales. Dans le cas commutatif, cette classe est réduite au mouvement Brownien. Nous montrons que dans le cas non-commutatif, cette classe est beaucoup plus grande, qu'elle contient beaucoup de mouvements Browniens non-commutatifs connus (matriciel, q-Brownian Motion, mouvement brownien libre). Par ailleurs nous développons un calcul de Ito non-commutatif simplifie et l'utilisons pour donner une caractérisation algébrique de ces mouvements Browniens, et donner de nouveaux exemples et contre-exemples. Cet expose se base sur un travail en préparation en collaboration avec Marius Junge.

Jason Crann: "Quantum Group Channels and Non-Commutative Convolution"
The recent representation theory of locally compact quantum groups has initiated a new connection between harmonic analysis and quantum information. In this talk, we will explore this connection by generating an intriguing class of quantum channels for every locally compact quantum group. Turning to explicit examples, we will discuss several properties of the resulting channels including fixed points and entanglement preservation as well as provide new counter-examples to two recently solved conjectures. As these quantum group channels are implemented by a convolution type action of trace class operators, we will finish with a discussion of the associated module action along with its applications to quantum information and amenability of quantum groups. This is joint work with Matthias Neufang.

Mickael de la Salle: "Non-commutative Lp spaces without the completely bounded approximation property"

Julien Deschamps: "Continuous-Time Limit of Classical Repeated Interactions"

Yoann Dabrowski: "Random microstates approach to free entropy"

Uwe Franz: "Quantum Probabilistic Construction of Spectral triples on Compact Quantum Groups"

Motohisa Fukuda: "Additivity questions in quantum communication"
In this talk, first we overview some additivity questions which arose from the study on quantum channels and see how random matrix theory came into this research. Secondly, we understand how those additivity questions are violated. Some of the consequences, the main idea of Hasitngs' proof and related works by us are presented. Final part of this talk is about open problems."

Dragi Karevski: "Quantum Non-Equilibrium Steady States Induced by Repeated Interactions"

Claus Koestler: "Distributional symmetries for Large Quantum Systems"
In classical probability, de Finetti type results infer conditional independence from exchangeability or spreadability of an infinite sequence of random variables. Recently we have transferred such results to an operator algebraic framework and shown that a certain kind of noncommutative independence emerges from the exchangeability or spreadability of an infinite sequence of quantum subsystems. In particular we have found new distributional symmetries, called 'braidability' and 'quantum exchangeability'. My talk will introduce into some of these new developments and is based in parts on joint work with Rolf Gohm and Roland Speicher.

Camille Male: "Distribution of traffics and their free product"

Ion Nechita: "Random Subspaces of a Tensor Product and the Additivity Problem"
We study singular values (or Schmidt coefficients) of vectors in a random subspace of a tensor product. The set of singular values of unit norm vectors in a random subspace is shown to converge to a deterministic limit that we characterize with the help of a norm arising in free probability. We show how these results are related to the additivity question for the minimum output entropy of random quantum channels.

Clément Pellegrini: "From discrete to continous quantum trajectories and estimation"

Gilles Pisier: "Strong Continuity of the Reduced Free Product of States"
We give a different proof of Paul Skoufranis's very recent resultshowing that the strong convergence (in Camille Male's sense) of possibly non-commutative random variables $X^{(k)}\to X$  is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\{X_i^{(k)}\}$ and $Y^{(k)}=\{Y_j^{(k)}\}$ are $*$-free as well as their   limits (in moments)  $X =\{X_i \}$ and $Y =\{Y_j\}$, the strong convergences $X^{(k)}\to X$ and $Y^{(k)}\to Y$ imply that of  $\{X^{(k)},Y^{(k)} \}$   to $\{X ,Y\}$. Phrased in more striking language: the reduced free product is `continuous" with respect to strong convergence. The analogue for weak convergence (i.e. convergence of all moments) is obvious.

René Schott: "Operator calculus and invertible Clifford Appell systems: theory and application to the n-particle fermion algebra"
Appell systems can be interpreted as polynomial solutions of generalized heat equations. In probability theory, they are also used to obtain non-central limit theorems.Their analogues have been defined on Lie groups by P. Feinsilver et al., and on quantum groups by U. Franz et al.Motivated by evolution equations on Clifford algebras and illustrated with the n-particle fermion algebra, we have developed a theory of invertible left and right Appell systems for Clifford algebras of an arbitrary quadratic form. A direct connection is also shown between blade factorization algorithms and the construction of Appell systems in these algebras.(joint work with G. Stacey STAPLES)

Yanqi Qiu: "The UMD constants for a class of itereted $L_p(L_q)$ spaces"
Let $1 < p \neq q < \infty$ and $(D, \mu) = (\{\pm 1\}, \frac{1}{2} \delta_{-1} + \frac{1}{2}\delta_1)$. Define by recursion: $X_0 = \C$ and $X_{n+1} = L_p(\mu; L_q(\mu; X_n))$. We show that there exist $c_1=c_1(p, q)>1$ and $c_2 = c_2(p, q, s) > 1$, such that the $\text{UMD}_s$ constants of $X_n$'s satisfy $c_1^n \leq C_s(X_n) \leq c_2^n$ for all $1 < s < \infty$. Our results yield an elementary construction of super-reflexive non-$\text{UMD}$ Banach lattices.

Karol Życzkowski: "Probabilistic approach to quantum states: Numerical Shadow of an operator"
For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A),  induced by the unitarily invariant Fubini--Study measure on the complex projective manifold of pure quantum states of size N. The notions of numerical range and numerical shadow can be extended for operators acting on a Hilbert space with a tensor product structure. Restricting the set of pure states to the set of product states or maximally entangled states we introduce restricted numerical range and restricted numerical shadow of an operator. Analyzing restricted shadows of operators of a fixed size N=KM we analyze the geometry of sets of separable and maximally entangled states of  the K times M composite quantum system.