# Lyon Quantum Information Meeting

Date: 3/12/2019 - 4/12/2019

Place: ENS de Lyon (Site Monod)

How to get around: http://www.ens-lyon.fr/LIP/index.php/useful-informations

List of speakers:

- Mario Berta (London)
- Cécilia Lancien (Toulouse)
- Ion Nechita (Toulouse)
- Andreas Winter (Barcelona)
- Christoph Hirche (Copenhagen)

**Preliminary schedule:**

__Monday__ (**Amphi B****)**

15:00 - 16:00: *Contributions to Quantum Information Theory, *Omar Fawzi

**Tuesday**

**Morning session (****Salle du Conseil du LIP 394 (Nord)****)**

10:00 - 11:00: Andreas Winter

11:00 - 11:30: *Coffee break*

11:30 - 12:30: Christoph Hirche

**Lunch Break (S****alle passerelle (4th floor)****)**

12:30 - 14:30

**Afternoon session (M7.315)**

14:30 - 15:00: Coffee break

15:00 - 16:00: Cecilia Lancien

19:00 - 22:00: Conference Dinner, Tandoor & Wok

**Wednesday**

**Morning session (M7.315)**

10:00 - 11:00: Mario Berta

11:00 - 11:30: *Coffee break*

11:30 - 12:30: Ion Nechita

**Lunch Break (Salle passerelle (4th floor))**

12:30 - 14:30

**Afternoon session (M7.315)**

14:30 - 15:00: *Coffee break*

15:00 - 16:00: *Open problem session*

**Thursday**

Discussion time

**Friday**

Discussion time

**Abstracts:**

*Mario Berta, **Non-Commutative Blahut-Arimoto Algorithms*:

We generalise alternating optimisation algorithms of Blahut-Arimoto type to the quantum setting. In particular, we give iterative algorithms to compute the mutual information of quantum channels, the Holevo quantity of classical-quantum channels, the coherent information of less noisy quantum channels, and the thermodynamic capacity of quantum channels. Our convergence analysis is based on quantum entropy inequalities and leads to a priori additive ε-approximations after O(log(N)/eps) iterations, where N denotes the input dimension of the channel. We complement our analysis with an a posteriori stopping criterion which allows us to terminate the algorithm after significantly fewer iterations compared to the a priori criterion in numerical examples. Finally, we discuss heuristics to accelerate the convergence. arXiv:1905.01286 - joint work with Navneeth Ramakrishnan, Raban Iten, and Volkher B. Scholz

*Christoph Hirche, **Dimension size bounds in quantum information theory from recoverability*

Bounding the dimension of auxiliary quantum random variables is one of the most fundamental problems in quantum information theory. The general lack of tools to determine such bounds often prevents us from actually evaluating important quantities such as squashed entanglement, the quantum information bottleneck, certain amortized measures, different capacities and many more. We present a new approach towards developing such a tool, by relating the problem to the monotonicity of the involved quantities and show that the question is often equivalent to the existence of a recovery map that can approximately reverse the loss of information incurred. We present and investigate several questions, which, if answered in the affirmative, would allow us to approximate a host of different information theoretic quantities using bounded dimension.

__Cecilia Lancien, T____ypical correlations in many-body quantum states__

Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of the work that I will present in this talk is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we will focus on is that of having either rapidly decaying or long-range correlations. In a nutshell, the main result I will state is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. I will show two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap.

All the results that I will present are based on a joint work with David Perez-Garcia, available here: https://arxiv.org/abs/1906.11682

*Ion Nechita, Entanglement criteria from tensor norms*

We discuss several generalizations of the realignment criterion, both in the bipartite and multi-partite entanglement detection problem, and connect them to inequalities between (projective) tensor norms. These are preliminary results obtained with Maria Jivulescu and Cécilia Lancien.

*Andreas Winter, **Towards a fully quantum de Finetti theorem*

Finite de Finetti theorems are powerful statements that leverage permutation symmetry of the distribution of n random variables X_1,...,X_n to obtain an approximation by random variables conditionally i.i.d. on a suitable variable Z. We review a simple proof of such a statement based on mutual information, specifically its chain rule and the existence of recovery maps, to obtain a fully quantum version of a notion of (approximate) de Finetti states, based on the approximate recovery of multiple quantum systems X_i from a single system Z by repeated application of the same channel. As the classical version, we only use the chain rule of quantum mutual information and the Fawzi-Renner recovery map.