Let (σ1,...,σ
d
) be 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 → ∞, ...these permutations viewed as operators on the n–1 dimensional vector space {(x
1,..., xn
) ∈ ℂ
n
, ∑xi = 0}, 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.
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.
Free Probability Theory (FPT) provides rich knowledge for handling mathematical difficulties caused by random matrices appearing 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 freeness of the layerwise Jacobian has not been proven mathematically 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 to 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.
For a compact quantum group \mathbb{G} of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra L^\infty (\mathbb{G}) into an ultrapower of the hyperfinite ...II _1-factor (the Connes embedding property for L^\infty (\mathbb{G})). We establish a connection between the Connes embedding property for L^\infty (\mathbb{G}) and the structure of certain quantum subgroups of \mathbb{G} and use this to prove that the II _1-factors L^\infty (O_N^+) and L^\infty (U_N^+) associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all N \ge 4. As an application, we deduce that the free entropy dimension of the standard generators of L^\infty (O_N^+) equals 1 for all N \ge 4. We also mention an application of our work to the problem of classifying the quantum subgroups of O_N^+.
We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in Collins et al. (Linear Algebra Appl 555:398–411, 2018). These maps satisfy an algebraic property which ...makes it easy to study their positivity or
k
-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices. In particular, we focus on a subfamily that we call (
a
,
b
)-unitarily equivariant. They can be seen as both a generalization of maps invariant under unitary conjugation as studied by Bhat (Banach J Math Anal 5(2):1–5, 2011) and as a generalization of the equivariant maps studied in Collins et al. (2018). Using representation theory, we fully compute them and study their graphical representation and show that they are basically enough to study all equivariant maps. We finally apply them to the problem of entanglement detection and prove that they form a sufficient (infinite) family of positive maps to detect all
k
-entangled density matrices.
The spectral edge of unitary Brownian motion Collins, Benoît; Dahlqvist, Antoine; Kemp, Todd
Probability theory and related fields,
02/2018, Volume:
170, Issue:
1-2
Journal Article
Peer reviewed
The Brownian motion
(
U
t
N
)
t
≥
0
on the unitary group converges, as a process, to the free unitary Brownian motion
(
u
t
)
t
≥
0
as
N
→
∞
. In this paper, we prove that it converges
strongly
as a ...process: not only in distribution but also in operator norm. In particular, for a fixed time
t
>
0
, we prove that the unitary Brownian motion has a
spectral edge
: there are no outlier eigenvalues in the limit. We also prove an extension theorem: any strongly convergent collection of random matrix ensembles independent from a unitary Brownian motion also converge strongly
jointly
with the Brownian motion. We give an application of this strong convergence to the Jacobi process.
In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact ...set
K
k
,
t
. We also showed that the set
K
k
,
t
is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of
ℓ
p
norms on the set
K
k
,
t
and prove that the maximum is attained on a vector of shape (
a
,
b
, . . . ,
b
) where
a
>
b
. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any
ε
>
0
, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as
log
2
-
ε
. Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.
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