09:00 to 09:50 Registration 09:50 to 10:00 Welcome by John Toland, Director of the Institute 10:00 to 11:00 M Christandl & M Walter (ETH Zürich)Welcome and Overview Talk by the Organisers INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 The classical entropy of quantum states Co-author: Elliott Lieb (Princeton University) To quantify the inherent uncertainty of quantum states Wehrl ('79) suggested a definition of their classical entropy based on the coherent state transform. He conjectured that this classical entropy is minimized by states that also minimize the Heisenberg uncertainty inequality, i.e., Gaussian coherent states. Lieb ('78) proved this conjecture and conjectured that the same holds when Euclidean Glauber coherent states are replaced by SU(2) Bloch coherent states. This generalized Wehrl conjecture has been open for almost 35 years until it was settled last year in joint work with Elliott Lieb. Recently we simplified the proof and generalized it to SU(N) for general N. I will present this here. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 K Mulmuley (University of Chicago)The complexity of Kronecker Coefficients The Kronecker coefficients are of fundamental importance in representation theory and quantum physics. Their complexity turns out to be of fundamental importance in the Geometric Complexity Theory (GCT) program towards that P vs. NP and related problems. In this talk we will describe some results related to the complexity of Kronecker coefficients. INI 1 15:00 to 15:30 Afternoon Tea 15:30 to 16:30 Quantum marginals: the generic entanglement regime Co-authors: Stanislaw SZAREK (Universite Paris 6 & Case Western Reserve University), Deping YE (Memorial University of Newfoundland) Consider N qubits in a generic pure state on $(C^2)^{\otimes N}$. Give a fraction $p \in (0,1/2)$ of them to Alice, another fraction $p$ of them to Bob, and assume the remaining qubits disappear in the environment. Do Alice and Bob share some entanglement ? When N is large, this problem features a threshold phenomenon around the critical value $p=1/5$. In this talk I give an elementary proof of the "easy" half of this result: for $p>1/5$, entanglement is generic. The general question, and a discussion of the threshold phenomenon, will be addressed in the talk by S. Szarek. I will introduce background from high-dimensional convex geometry and prove some key estimates on the size (specifically the mean width) of the set of separable states. The talk is a variation on arxiv:1106.2264 (joint with S. Szarek and D. Ye). INI 1 17:00 to 18:00 Quantum Shannon Theory: Rothschild Distinguished Visiting Fellow Lecture The notions of channel and capacity are central to the classical Shannon theory. "Quantum Shannon theory" denotes a subfield of quantum information science which uses operator analysis, convexity and matrix inequalities, asymptotic techniques such as large deviations and measure concentration to study mathematical models of communication channels and their information-processing performance. From the mathematical point of view quantum channels are normalized completely positive maps of operator algebras, the analog of Markov maps in the noncommutative probability theory, while the capacities are related to certain norm-like quantities. In applications noisy quantum channels arise from irreversible evolutions of open quantum systems interacting with environment-a physical counterpart of a mathematical dilation theorem. It turns out that in the quantum case the notion of channel capacity splits into the whole spectrum of numerical information-processing characteristics depending on the kind of data transmitted (classical or quantum) as well as on the additional communication resources. An outstanding role here is played by quantum correlations - entanglement - inherent in tensor-product structure of composite quantum systems. This talk presents a survey of basic coding theorems providing analytical expressions for the capacities of quantum channels in terms of entropic quantities. We also touch upon some open mathematical problems, such as additivity and Gaussian optimizers, concerning the entropic characteristics of both theoretically and practically important Bosonic Gaussian channels. INI 1 18:00 to 19:00 Welcome Wine Reception
 09:00 to 10:00 Marginal Entropies for Causal Inference and Quantum Non-Locality Co-authors: Rafael Chaves (University of Freiburg), Lukas Luft (University of Freiburg) The fields of quantum non-locality in physics, and causal discovery in machine learning, both face the problem of deciding whether observed data is compatible with a presumed causal relationship between the variables (for example a local hidden variable model). Traditionally, Bell inequalities have been used to describe the restrictions imposed by causal structures on marginal distributions. However, some structures give rise to non-convex constraints on the accessible data, and it has recently been noted that linear inequalities on the observable entropies capture these situations more naturally. In this talk, I will introduce the machine learning background, advertise the program of investigating entropic marginals, and present some recent results. INI 1 10:00 to 11:00 A Harrow (Massachusetts Institute of Technology)Group representations and quantum information theory The method of types---classifying strings according to letter frequencies---is a fundamental tool of classical information theory. I will discuss a natural quantum generalisation, known as the Schur basis, in which quantum states are decomposed into irreps of the unitary and symmetric groups. First, I'll explain the analogy to the classical method of types and will review past work that applies the Schur basis to quantum information theory. Then I'll discuss applications to spectrum estimation, channel coding problems and mention some open problems. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Eigencones and Levi movability Let $g_{\lambda\mu\nu}$ denote the Kronecker coefficient. It is a multiplicity in the decomposition of the tensor product of two irreducible representations of the symmeric group. The set of triples $(\lambda,\mu,\nu)$ of bounded length such that $g_{\lambda\mu\nu}$ is nonzero generate a closed convex polyhedral cone. In this talk, we will give a description of these cones and more generaly of branching cones. Some branching cones have an interpretation in terms of eigenvalues of Hermitian matrices known as the addive Horn problem. We will also give an answer of the so called multiplicative Horn problem. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Operator norm convergence for sequence of matrices and application to QIT Random matrix theory is the study of probabilistic quantities associated to random matrix models, in the large dimension limit. The eigenvalues counting measure and the eigenvalues spacing are amongst the most studied and best understood quantities. The purpose of this talk is to focus on a quantity that was less understood until recently, namely the operator norm of random matrices. I will state recent results in this direction, and mention three applications to quantum information theory: -a- convergence of the collection of images of pure states under typical quantum channels (joint with Fukuda and Nechita) -b- thresholds for random states to have the absolute PPT property (joint with Nechita and Ye), -c- new examples of k-positive maps (ongoing, joint with Hayden and Nechita). INI 1 15:00 to 15:30 Afternoon Tea 15:30 to 16:30 Threshold phenomena for quantum marginals Co-authors: Guillaume Aubrun (U. Lyon 1), Deping Ye (Memorial U. of Newfoundland) Consider a quantum system consisting of N identical particles and assume that it is in a random pure state (i.e., uniformly distributed over the sphere of the corresponding Hilbert space). Next, let A and B be two subsystems consisting of k particles each. Are A and B likely to share entanglement? Is the AB-marginal typically PPT? As it turns out, for many natural properties there is a sharp "phase transition" at some threshold depending on the property in question. For example, there is a threshold K asymptotically equivalent to N/5 such that - if k > K then A and B typically share entanglement - if k The first statement was (essentially) shown in the talk by G. Aubrun. Here we present a general scheme for handling such questions and sketch the analysis specific to entanglement. The talk is based on arxiv:1106.2264v3; a less-technical overview is in arxiv:1112.4582v2. INI 1 16:30 to 17:30 Poster Session