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Benoit Collins

Publications and source records attributed to Benoit Collins.

At least 19 recordsLinked to original sources
arXiv2026

A Sudakov--Fernique proof of Lehner-type edge bounds for matrix-valued GUE sums

Let $A_0,A_1,\ldots,A_n\in M_N(\mathbb{C})$ be Hermitian matrices and let $G_1,\ldots,G_n$ be independent $M\times M$ GUE matrices normalized so that $\|M^{-1/2}G_i\|\to 2$ almost surely as $M\to\infty$. We study the spectral edges and operator norm of $H_M = A_0\otimes I_M + \frac{1}{\sqrt{M}}\sum_{i=1}^n A_i\otimes G_i$. Lehner's formula identifies the right and left edges of the corresponding free semicircular operator as $\rho_+ = \inf_{Z\succ 0}\lambda_{\max}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$ and $\rho_- = \sup_{Z\prec 0}\lambda_{\min}(A_0+Z+\sum_{i=1}^n A_iZ^{-1}A_i)$. Assuming $A_i\succeq 0$ for $i\ge 1$ and $M\ge N$, we prove via concentration and minimax duality the finite-dimensional bounds $\mathbb{E}\lambda_{\max}(H_M)\le \rho_+ + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$ and $\mathbb{E}\lambda_{\min}(H_M)\ge \rho_- - 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. With $\rho_* = \max\{\rho_+,-\rho_-\}$, this yields $\mathbb{E}\|H_M\|_{\mathrm{op}}\le \rho_* + 9\sqrt{nN/M}\,\|\sum_{i=1}^n A_i^2\|_{\mathrm{op}}^{1/2}$. For uniformly bounded positive coefficients, bounded $n$, and $N=o(M)$, one obtains $\limsup_{M\to\infty}\mathbb{E}\|H_M\|_{\mathrm{op}}\le\rho$ whenever $\rho_{*,M}\to\rho$. The proof is a matrix-coefficient extension of classical Sudakov--Fernique comparison, combined with a Davidson--Szarek-type singular-value estimate and dual variational formulas for Lehner's edge quantities over density matrices. We also explain why this approach does not extend sharply to signed Hermitian coefficients.

math.PR↗
arXiv2024

Free cumulants and freeness for unitarily invariant random tensors

We address the question of the asymptotic description of random tensors that are local-unitary invariant, that is, invariant by conjugation by tensor products of independent unitary matrices. We consider both the mixed case of a tensor with $D$ inputs and $D$ outputs, and the case where there is a factorization between the inputs and outputs, called pure, which includes the random tensor models extensively studied in the physics literature. The finite size and asymptotic moments are defined using correlations of certain invariant polynomials encoded by $D$-tuples of permutations, up to relabeling equivalence. Finite size free cumulants associated to the expectations of these invariants are defined through invertible finite size moment-cumulants formulas. Two important cases are considered asymptotically: pure random tensors that scale like a complex Gaussian, and mixed random tensors that scale like a Wishart tensor. In both cases, we derive a notion of tensorial free cumulants associated to first order invariants, through moment-cumulant formulas involving summations over non-crossing permutations. The pure and mixed cases involve the same combinatorics, but differ by the invariants that define the distribution at first order. In both cases, the tensorial free-cumulants of a sum of two independent tensors are shown to be additive. A preliminary discussion of higher orders is provided. Tensor freeness is then defined as the vanishing of mixed first order tensorial free cumulants. The equivalent formulation at the level of asymptotic moments is derived in the pure and mixed cases, and we provide an algebraic construction of tensorial probability spaces, which generalize non-commutative probability spaces: random tensors converge in distribution to elements of these spaces, and tensor freeness of random variables corresponds to tensor freeness of the subspaces they generate.

math-ph↗
arXiv2023

Norm of matrix-valued polynomials in random unitaries and permutations

We consider a non-commutative polynomial in several independent $N$-dimensional random unitary matrices, uniformly distributed over the unitary, orthogonal or symmetric groups, and assume that the coefficients are $n$-dimensional matrices. The main purpose of this paper is to study the operator norm of this random non-commutative polynomial. We compare it with its counterpart where the the random unitary matrices are replaced by the unitary generators of the free group von Neumann algebra. Our first result is that these two norms are overwhelmingly close to each other in the large $N$ limit, and this estimate is uniform over all matrix coefficients as long as $n \le\exp (N^\alpha)$ for some explicit $\alpha >0$. Such results had been obtained by very different techniques for various regimes, all falling in the category $n\ll N$. Our result provides a new proof of the Peterson-Thom conjecture. Our second result is a universal quantitative lower bound for the operator norm of polynomials in independent $N$-dimensional random unitary and permutation matrices with coefficients in an arbitrary $C^*$-algebra. A variant of this result for permutation matrices generalizes the Alon-Boppana lower bound in two directions. Firstly, it applies for arbitrary polynomials and not only linear polynomials, and secondly, it applies for coefficients of an arbitrary $C^*$-algebra with non-negative joint moments and not only for non-negative real numbers.

math.PR↗
arXiv2022

The spectrum of local random Hamiltonians

The spectrum of a local random Hamiltonian can be represented generically by the so-called $\epsilon$-free convolution of its local terms' probability distributions. We establish an isomorphism between the set of $\epsilon$-noncrossing partitions and permutations to study its spectrum. Moreover, we derive some lower and upper bounds for the largest eigenvalue of the Hamiltonian.

math-ph↗
arXiv2022

Moment Methods on compact groups: Weingarten calculus and its applications

A fundamental property of compact groups and compact quantum groups is the existence and uniqueness of a left and right invariant probability -- the Haar measure. This is a natural playground for classical and quantum probability, provided it is possible to compute its moments. Weingarten calculus addresses this question in a systematic way. The purpose of this manuscript is to survey recent developments, describe some salient theoretical properties of Weingarten functions, as well as applications of this calculus to random matrix theory, quantum probability, and algebra, mathematical physics and operator algebras.

math.OA↗
arXiv2021

The Weingarten Calculus

This is a short introduction to Weingarten Calculus. Weingarten Calculus is a method to compute the joint moments of matrix variables distributed according to the Haar measure of compact groups.

math-ph↗
arXiv2021

Asymptotic Freeness of Layerwise Jacobians Caused by Invariance of Multilayer Perceptron: The Haar Orthogonal Case

Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices that appear in research related to deep neural networks (DNNs), such as the dynamical isometry, Fisher information matrix, and training dynamics. FPT suits these researches because the DNN's parameter-Jacobian and input-Jacobian are polynomials of layerwise Jacobians. However, the critical assumption of asymptotic freenss of the layerwise Jacobian has not been proven completely so far. The asymptotic freeness assumption plays a fundamental role when propagating spectral distributions through the layers. Haar distributed orthogonal matrices are essential for achieving dynamical isometry. In this work, we prove asymptotic freeness of layerwise Jacobians of multilayer perceptron (MLP) in this case. A key of the proof is an invariance of the MLP. Considering the orthogonal matrices that fix the hidden units in each layer, we replace each layer's parameter matrix with itself multiplied by the orthogonal matrix, and then the MLP does not change. Furthermore, if the original weights are Haar orthogonal, the Jacobian is also unchanged by this replacement. Lastly, we can replace each weight with a Haar orthogonal random matrix independent of the Jacobian of the activation function using this key fact.

stat.ML↗
arXiv2021

A metric characterization of freeness

Let $\mathcal{M}$ be a finite von Neumann algebra and $u_1,\dots,u_N$ be unitaries in $\mathcal{M}$. We show that $u_1,\dots,u_N$ freely generate $L(\mathbb{F}_N)$ if and only if $$\left\|\sum_{i=1}^N u_i \otimes (u_i^{\mathrm{op}})^* + u_i^*\otimes u_i^{\mathrm{op}}\right\|_{\mathcal{M}\overline{\otimes}\mathcal{M}^{\mathrm{op}}} = 2\sqrt{2N - 1}.$$

math.OA↗
arXiv2020

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature $\rho$, i.e. two finite sequences of non-increasing natural numbers, and for $n$ large enough, consider the irreducible representation $V_{n,\rho}$ of $\mathbb{U}_n$ associated to the signature $\rho$. We consider the quotient $\mathbb{U}_{n,\rho}$ of $\mathbb{U}_n$ viewed as a matrix subgroup of $\mathbb{U}(V_{n,\rho})$, and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. In order to achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator valued non-backtracking theory in a general $C^*$-algebra, and finally, (iv) combinatorics of tensor moment matrices.

math.PR↗
arXiv2019

Gelfand-Tsetlin polytopes and random contractions away from the limiting shapes

In this paper, we consider a sequence of selfadjoint matrices $A_n$ having a limiting spectral distribution as $n\to \infty$, and we consider a sequence of full flags $\{0\le p_1^n\le\ldots\le p_i^n\le\ldots\le 1_n\}$ chosen at random according to the uniform measure on full flag manifolds. We are interested in the behaviour of the extremal eigenvalues of $p_i^nA_np_i^n$. This problem is known to be equivalent to the study of uniform probability measures on Gelfand-Tsetlin polytopes. Our main results consist in explicit uniform estimates for extremal eigenvalues, and the fact that an outlier behavior has an exponentially small probability. This problem is of intrinsic interest in random matrix theory, but it has also a strong motivation and some applications in quantum information, which we discuss. The proofs rely on a reinterpretation of the problem with the help of determinantal point processes and the techniques are based on steepest descent analysis.

math.PR↗
arXiv2018

Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement

We consider a bipartite transformation that we call self-embezzlement and use it to prove a constant gap between the capabilities of two models of quantum information: the conventional model, where bipartite systems are represented by tensor products of Hilbert spaces; and a natural model of quantum information processing for abstract states on C*-algebras, where joint systems are represented by tensor products of C*-algebras. We call this the C*-circuit model and show that it is a special case of the commuting-operator model (in that it can be translated into such a model). For the conventional model, we show that there exists a constant $\epsilon_0 > 0$ such that self-embezzlement cannot be achieved with precision parameter less than $\epsilon_0$ (i.e., the fidelity cannot be greater than $1 - \epsilon_0$); whereas, in the C*-circuit model -- as well as in a commuting-operator model -- the precision can be $0$ (i.e., fidelity~$1$).

quant-ph↗
arXiv2018

On a family of a linear maps from $M_{n}(\mathbb{C})$ to $M_{n^{2}}(\mathbb{C})$

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $\Phi_{\alpha, \beta,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $\Phi_{\alpha, \beta,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,3}$ is $2$-positive and not completely positive.

math-ph↗
arXiv2016

Highly entangled, non-random subspaces of tensor products from quantum groups

In this paper we describe a class of highly entangled subspaces of a tensor product of finite dimensional Hilbert spaces arising from the representation theory of free orthogonal quantum groups. We determine their largest singular values and obtain lower bounds for the minimum output entropy of the corresponding quantum channels. An application to the construction of $d$-positive maps on matrix algebras is also presented.

math-ph↗
arXiv2016

Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\text{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| > 2\cos\big(\frac{\pi}{k+1}\big)$. In particular, this yields a new formula for the structure coefficients of the Jones-Wenzl projection in $\text{TL}_k(d)$. The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation in terms of a certain graph structure we place on non-crossing pairings, and these coefficients turn out to have the remarkable property that they either always positive integers or always negative integers. As an application, we answer affirmatively a question of Vaughan Jones, asking whether every Temperley-Lieb diagram appears with non-zero coefficient in the expansion of each dual basis element in $\text{TL}_k(d)$ (when $d \in \mathbb R \backslash [-2\cos\big(\frac{\pi}{k+1}\big),2\cos\big(\frac{\pi}{k+1}\big)]$). Specializing to Jones-Wenzl projections, this result gives a new proof of a result of Ocneanu, stating that every Temperley-Lieb diagram appears with non-zero coefficient in a Jones-Wenzl projection. Our methods establish a connection with the Weingarten calculus on free quantum groups, and yield as a byproduct improved asymptotics for the free orthogonal Weingarten function.

math.QA↗
arXiv2016

$*$-Freeness in Finite Tensor Products

In this paper, we consider the following question and variants thereof: given $\mathbf D:=\big(a_{1;i}\otimes\cdots\otimes a_{K;i}:i\in I\big)$, a collection of elementary tensor non-commutative random variables in the tensor product of probability spaces $(\mathcal A_1\otimes\cdots\otimes\mathcal A_K,\phi_1\otimes\cdots\otimes\phi_K)$, when is $\mathbf D$ $*$-free? (See Section 1.2 for a precise formulation of this problem.) Settling whether or not freeness occurs in tensor products is a recurring problem in operator algebras, and the following two examples provide a natural motivation for the above question: (A) If $(a_{1;i}:i\in I)$ is a $*$-free family of Haar unitary variables and $a_{k,i}$ are arbitrary unitary variables for $k\geq2$, then the $*$-freeness persists at the level of the tensor product $\mathbf D$. (B) A converse of (A) holds true if all variables $a_{k;i}$ are group-like elements (see Corollary 1.7 of Proposition 1.6). It is therefore natural to seek to understand the extent to which such simple characterizations hold true in more general cases. While our results fall short of a complete characterization, we make notable steps toward identifying necessary and sufficient conditions for the freeness of $\mathbf D$. For example, we show that under evident assumptions, if more than one family $(a_{k,i}:i\in I)$ contains non-unitary variables, then the tensor family fails to be $*$-free (see Theorem 1.8 (1)).

math.OA↗
arXiv2015

Free probability for purely discrete eigenvalues of random matrices

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distribution in Voiculescu's sense and are globally rotationally invariant. We assume that each monomial constituting this polynomial contains at least one variable of type (a), and show that this random matrix model has a set of eigenvalues that almost surely converges to a deterministic set of numbers that is either finite or accumulating to only zero in the large dimension limit. For this purpose we define a framework (cyclic monotone independence) for analyzing discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of discrete eigenvalues of our model.

math.PR↗
arXiv2015

Random matrix techniques in quantum information theory

The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.

quant-ph↗