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Teodor Banica

Publications and source records attributed to Teodor Banica.

At least 55 records · Page 3Linked to original sources
arXiv2014

Truncation and duality results for Hopf image algebras

Associated to an Hadamard matrix $H\in M_N(\mathbb C)$ is the spectral measure $μ\in\mathcal P[0,N]$ of the corresponding Hopf image algebra, $A=C(G)$ with $G\subset S_N^+$. We study here a certain family of discrete measures $μ^r\in\mathcal P[0,N]$, coming from the idempotent state theory of $G$, which converge in Cesàro limit to $μ$. Our main result is a duality formula of type $\int_0^N(x/N)^pdμ^r(x)=\int_0^N(x/N)^rdν^p(x)$, where $μ^r,ν^r$ are the truncations of the spectral measures $μ,ν$ associated to $H,H^t$. We prove as well, using these truncations $μ^r,ν^r$, that for any deformed Fourier matrix $H=F_M\otimes_QF_N$ we have $μ=ν$.

math.OA↗
arXiv2014

The glow of Fourier matrices: universality and fluctuations

The glow of an Hadamard matrix $H\in M_N(\mathbb C)$ is the probability measure $μ\in\mathcal P(\mathbb C)$ describing the distribution of $φ(a,b)= $, where $a,b\in\mathbb T^N$ are random. We prove that $φ/N$ becomes complex Gaussian with $N\to\infty$, and that the universality holds as well at order 2. In the case of a Fourier matrix, $F_G\in M_N(\mathbb C)$ with $|G|=N$, the universality holds up to order 4, and the fluctuations are encoded by certain subtle integrals, which appear in connection with several Hadamard-related questions. In the Walsh matrix case, $G=\mathbb Z_2^n$, we conjecture that the glow is polynomial in $N=2^n$.

math.CO↗
arXiv2013

Analytic aspects of the circulant Hadamard conjecture

We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $Φ=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $Φ\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $Φ$, (2) the study of the critical points of $Φ$, and (3) the computation of the moments of $Φ$. We explore here these questions, with some results and conjectures.

math.CO↗
arXiv2013

First order deformations of the Fourier matrix

The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. The singularity at a point $H\in C_N$ is described by a filtration of cones $T^\times_HC_N\subset T^\circ_HC_N\subset T_HC_N\subset\widetilde{T}_HC_N$, coming from the trivial, affine, smooth and first order deformations. We study here these cones in the case where $H=F_N$ is the Fourier matrix, $(w^{ij})$ with $w=e^{2πi/N}$, our main result being a simple description of $\widetilde{T}_HC_N$. As a consequence, the rationality conjecture $dim_\mathbb R(\widetilde{T}_HC_N)=dim_\mathbb Q(\widetilde{T}_HC_N\cap M_N(\mathbb Q))$ holds at $H=F_N$.

math.CO↗
arXiv2013

The defect of generalized Fourier matrices

The $N\times N$ complex Hadamard matrices form a real algebraic manifold $C_N$. We have $C_N=M_N(\mathbb T)\cap\sqrt{N}U_N$, and following Tadej and Życzkowski we investigate here the computation of the enveloping tangent space $\widetilde{T}_HC_N=T_HM_N(\mathbb T)\cap T_H\sqrt{N}U_N$, and notably of its dimension $d(H)=\dim(\widetilde{T}_HC_N)$, called undephased defect of $H$. Our main result is an explicit formula for the defect of the Fourier matrix $F_G$ associated to an arbitrary finite abelian group $G=\mathbb Z_{N_1}\times...\times\mathbb Z_{N_r}$. We also comment on the general question "does the associated quantum permutation group see the defect", with a probabilistic speculation involving Diaconis-Shahshahani type variables.

math.CO↗
arXiv2012

Almost Hadamard matrices: the case of arbitrary exponents

In our previous work, we introduced the following relaxation of the Hadamard property: a square matrix $H\in M_N(\mathbb R)$ is called "almost Hadamard" if $U=H/\sqrt{N}$ is orthogonal, and locally maximizes the 1-norm on O(N). We review our previous results, notably with the formulation of a new question, regarding the circulant and symmetric case. We discuss then an extension of the almost Hadamard matrix formalism, by making use of the p-norm on O(N), with $p\in[1,\infty]-{2}$, with a number of theoretical results on the subject, and the formulation of some open problems.

math.CO↗
arXiv2012

Block-modified Wishart matrices and free Poisson laws

We study the random matrices of type $\tilde{W}=(id\otimesφ)W$, where $W$ is a complex Wishart matrix of parameters $(dn,dm)$, and $φ:M_n(\mathbb C)\to M_n(\mathbb C)$ is a self-adjoint linear map. We prove that, under suitable assumptions, we have the $d\to\infty$ eigenvalue distribution formula $δm\tilde{W}\simπ_{mnρ}\boxtimesν$, where $ρ$ is the law of $φ$, viewed as a square matrix, $π$ is the free Poisson law, $ν$ is the law of $D=φ(1)$, and $δ=tr(D)$.

math.PR↗
arXiv2012

De Finetti theorems for easy quantum groups

We study sequences of noncommutative random variables which are invariant under "quantum transformations" coming from an orthogonal quantum group satisfying the "easiness" condition axiomatized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups S_n, O_n, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of Köstler and Speicher, and the characterization of operator-valued free semicircular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman.

math.OA↗
arXiv2012

Quantum symmetry groups of C*-algebras equipped with orthogonal filtrations

Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.

math.OA↗
arXiv2012

Almost Hadamard matrices: general theory and examples

We develop a general theory of "almost Hadamard matrices". These are by definition the matrices $H\in M_N(\mathbb R)$ having the property that $U=H/\sqrt{N}$ is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a detailed discussion of the circulant case ($H_{ij}=γ_{j-i}$) and of the two-entry case ($H_{ij}\in{x,y}$), with the construction of several families of examples, and some 1-norm computations.

math.CO↗
arXiv2012

Quantum isometries and group dual subgroups

We study the discrete groups $Λ$ whose duals embed into a given compact quantum group, $\hatΛ\subset G$. In the matrix case $G\subset U_n^+$ the embedding condition is equivalent to having a quotient map $Γ_U\toΛ$, where $F=\{Γ_U|U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon's classification of group dual subgroups $\hatΛ\subset S_n^+$. These results are motivated by Goswami's notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.

math.QA↗
arXiv2012

Asymptotic eigenvalue distributions of block-transposed Wishart matrices

We study the partial transposition ${W}^Γ=(\mathrm{id}\otimes \mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$ of parameters $(dn,dm)$. Our main result is that, with $d\to\infty$, the law of $m{W}^Γ$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half line.

math.PR↗
arXiv2012

Idempotent states and the inner linearity property

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $π:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$-algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Cesàro limit of the linear functional $ϕ=tr\circπ$. We discuss then some consequences of this result, notably to inner linearity questions.

math.QA↗
arXiv2012

Noncommutative homogeneous spaces: the matrix case

Given a quantum subgroup $G\subset U_n$ and a number $k\leq n$ we can form the homogeneous space $X=G/(G\cap U_k)$, and it follows from the Stone-Weierstrass theorem that $C(X)$ is the algebra generated by the last $n-k$ rows of coordinates on $G$. In the quantum group case the analogue of this basic result doesn't necessarily hold, and we discuss here its validity, notably with a complete answer in the group dual case. We focus then on the "easy quantum group" case, with the construction and study of several algebras associated to the noncommutative spaces of type $X=G/(G\cap U_k^+)$.

math.QA↗
arXiv2011

A maximality result for orthogonal quantum groups

We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.

math.QA↗