Benoît Collins

We extend [Q. Chen et al., arXiv:2505.22100 (2025)] by analyzing the complexity of the k-block-positivity testing algorithm that stems from the optimization problem in Definition 5. In this paper, we investigate a symmetry reduction scheme based on rectangular shaped Young diagrams. Connecting the complexity to the dimensions of irreducible representations of U(d), we derive an explicit formula for the complexity, which also clarifies why the semidefinite program hierarchy collapses in the k = d case.

We revisit the quantum de Finetti theorem. We state and prove a couple of variants thereof. In parallel, we introduce an operator version of the Martin boundary on quantum groups and prove generalizations of Biane’s theorem. Our proof of the de Finetti theorem is new in the sense that it is based on an analogy with the theory of operator-valued Martin boundary that we introduce.

We study the problem of testing k -block-positivity via symmetric N -extendibility by taking the tensor product with a k -dimensional maximally entangled state. We exploit the unitary symmetry of the maximally entangled state to reduce the size of the corresponding semidefinite program (SDP). For example, for k = 2, the SDP is reduced from one block of size 2 N + 1 d N + 1 to ⌊ N + 1 2 ⌋ blocks of size ≈ O ( ( N − 1 ) − 1 2 N + 1 d N + 1 ) .

This paper calculates the fluctuations of eigenvalues of polynomials on large Haar unitaries cut by finite rank deterministic matrices. When the eigenvalues are all simple, we can give a complete algorithm for computing the fluctuations. When multiple eigenvalues are involved, we present several examples suggesting that a general algorithm would be much more complex.

We introduce and study the Weingarten calculus for centered random permutation matrices in the symmetric group S_N. After presenting a formulation of the Weingarten calculus on the symmetric group, we derive a formula in the centered case, as well as a sign-respecting formula. Our investigations uncover the fact that a building block of this Weingarten calculus is Kummer's confluent hypergeometric function. It allows us to derive multiple algebraic properties of the Weingarten function and uniform estimate. These results shed a conceptual light on phenomena that take place regarding the algebraic and asymptotic behavior of moments of random permutations in the resolution of Bordenave and Bordenave-Collins of strong convergence. We obtain multiple new non-trivial estimates for moments of coefficients in centered moments.

Consider a finite sequence of independent random permutations, chosen uniformly either among all permutations or among all matchings on n points. We show that, in probability, as n goes to infinity, these permutations viewed as operators on the (n-1) dimensional vector space orthogonal to the vector with all coordinates equal to 1, are asymptotically strongly free. Our proof relies on the development of a matrix version of the non-backtracking operator theory and a refined trace method. As a byproduct, we show that the non-trivial eigenvalues of random n-lifts of a fixed based graphs approximately achieve the Alon-Boppana bound with high probability in the large n limit. This result generalizes Friedman's Theorem stating that with high probability, the Schreier graph generated by a finite number of independent random permutations is close to Ramanujan. Finally, we extend our results to tensor products of random permutation matrices. This extension is especially relevant in the context of quantum expanders.

This paper studies projections of uniform random elements of (co)adjoint orbits of compact Lie groups. Such projections generalize several widely studied ensembles in random matrix theory, including the randomized Horn’s problem, the randomized Schur’s problem, and the orbital corners process. In this general setting, we prove integral formulae for the probability densities, establish some properties of the densities, and discuss connections to multiplicity problems in representation theory as well as to known results in the symplectic geometry literature. As applications, we show a number of results on marginal problems in quantum information theory and also prove an integral formula for restriction multiplicities.