Institut Camille Jordan

Résumé : 

In his solution of Hilbert’s 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin’s result can always be chosen as an N-th power of a specific quadratic form and gave explicit bounds on N. By using concepts from quantum information theory (such as partial traces and an identity due to Chiribella) we present a simpler proof of a complex version of this result and we will mention applications to quantum information theory.

 

Résumé : Deux observateurs effectuant des mesures binaires sur des sous-parties d'un système global peuvent obtenir des résultats plus fortement corrélés lorsqu'ils partagent un état quantique intriqué que lorsqu'ils ne partagent que de l'aléa commun. Ce phénomène bien connu, dit de violation d'inégalités de Bell, peut précisément se caractériser mathématiquement. En effet, être une matrice de corrélations classique ou quantique correspond exactement à être dans la boule unité de certaines normes tensorielles. Je commencerai par expliquer tout cela en détail. Ensuite, je m'intéresserai au problème suivant: étant donnée une matrice aléatoire de taille n, peut-on estimer la valeur typique de ses normes "classique" et "quantique", lorsque n devient grand? Pour une large classe de matrices aléatoires, la réponse est oui, et montre une séparation entre les deux valeurs. Ce résultat peut s'interpréter de la façon suivante: dans une direction typique, les frontières des ensembles de corrélations classique et quantique ne se touchent pas, ou encore: dans une direction typique, les ensembles duaux des ensembles de corrélations classique et quantique ont des largeurs différentes.

Résumé : On expliquera aujourd'hui (1) le lemme de Yannakakis reliant rang positif et taille des formulations étendues en programmation linéaire/semi-définie (2) la preuve combinatoire simple de l'article http://arxiv.org/abs/1307.3543

Résumé : Preuve du lemme de Yannakakis sur la relation entre complexité linéaire d'un polytope et rang positif de sa matrice des écarts

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Continuation from last time. We explain basics of hypercontractivity on the hypercube and apply it to obtain lower bounds on the extension complexity.

Résumé : 

We illustrate the fact that Semi Definite Programming is more powerful than Linear Programming by proving the following theorem: any polyhedral cone which approximates the cone of n x n positive matrices has extension complexity at least exp(c sqrt(n)). The proof uses notably the hypercontractivity inequality on the discrete cube. The talk will be self-contained. Reference: arXiv:1811.09649

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Résumé : 

Given two convex finite-dimensional cones, one can naturally define their minimal and their maximal tensor product. We show that both coincide exactly when one of the cones is isomorphic to the positive orthant, as was conjectured by Barker (1976). An interpretation is as follows: the phenomenon of entanglement is universal among probabilistic theories which are not classical (quantum mechanics being a particular case).

Résumé : 

The Kitaev chain model exhibits a phase where two  Majorana fermions at the edge of the chain pair up non-locally. From the pair emerges a zero-energy excitation with topological order justifying the application of anyon statistics to the Majorana fermions. An arbitrary large yet finite number of such Majorana fermion pairs can  constitute a lattice of anyons, then referred to as Ising anyons, which have been extensively studied with respect to quantum transport and quantum computation. We study quantum teleportation in such fermionic systems of Majorana fermions without considering anyon statistics. Generally, quantum teleportation suggests the transfer of a quantum state by means of a bipartite resource state,
local operations and classical channels. While it is well-understood in qubit systems, many essential concepts, like quantum entanglement of the bipartite resource state, cannot be directly translated to fermionic systems. We present our own quantum teleportation protocol on the basis of fermionic linear optics and fermionic Gaussian resource states, which allow for an unambiguous  definition of entanglement. This approach is unique to the best of our knowledge, and we argue that it supersedes other approaches as it includes the teleportation via a single non-local Majorana pair transferring only ’half an electron’. With the proposed protocol, we explore the capacity of quantum teleportation depending on a given fermionic Gaussian resource state and reproduce results found for qubit systems by Horodecki. In particular, we show that the maximal fidelity of a quantum teleportation channel only depends on the best attainable bipartite resource state by means of local operations and classical communication.