Benoit Collins

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type.

Product of random projections, Jacobi ensembles and universality problems arising from free probability

Abstract.We consider the product of two independent randomly rotated projectors. The square of its radial part turns out to be distributed as a Jacobi ensemble. We study its global and local properties in the large dimension scaling relevant to free probability theory. We establish asymptotics for one point and two point correlation functions, as well as properties of largest and smallest eigenvalues.

Free Bessel Laws

We introduce and study a remarkable family of real probability measures that we call free Bessel laws. These are related to the free Poisson law. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

Integration over Compact Quantum Groups

We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.

Integration over quantum permutation groups

Abstract We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters.

Random Quantum Channels I: Graphical Calculus and the Bell State Phenomenon

This paper is the first of a series where we study quantum channels from the random matrix point of view. We develop a graphical tool that allows us to compute the expected moments of the output of a random quantum channel.As an application, we study variations of random matrix models introduced by Hayden [7], and show that their eigenvalues converge almost surely.In particular we obtain, for some models, sharp improvements on the value of the largest eigenvalue, and this is shown in further work to have new applications to minimal output entropy inequalities.

Generating random density matrices

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.

Random quantum channels II: Entanglement of random subspaces, Renyi entropy estimates and additivity problems

In this paper we obtain new bounds for the minimum output entropies of random quantum channels. These bounds rely on random matrix techniques arising from free probability theory. We then revisit the counterexamples developed by Hayden and Winter to get violations of the additivity equalities for minimum output Renyi entropies. We show that random channels obtained by randomly coupling the input to a qubit violate the additivity of the

Gaussianization and eigenvalue statistics for Random quantum channels (III)

p

Integration over the Pauli quantum group

-Renyi entropy. For some sequences of random quantum channels, we compute almost surely the limit of their Schatten