Related papers: Are almost-symmetries almost linear?
Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into…
Wigner's celebrated theorem, which is particularly important in the mathematical foundations of quantum mechanics, states that every bijective transformation on the set of all rank-one projections of a complex Hilbert space which preserves…
Representations of the Poincar\'{e} symmetry are studied by using a Hilbert space with a phase space content. The states are described by wave functions ( quasi amplitudes of probability) associated with Wigner functions (quasi probability…
We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if…
This paper presents a study of the properties of a matrix model that was introduced to describe transitions between all Wigner surmises of Random Matrix theory. New results include closed-form exact analytical expressions for the…
Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then…
Whether an almost-commuting pair of operators must be close to a commuting pair is a central question in operator and matrix theory. We investigate this problem for pairs of $C^*$-subalgebras $\mathcal{A}$ and $\mathcal{B}$ of…
We show that the transformations of Grassmannians (of complex Hilbert spaces) induced by linear or conjugate-linear isometries can be characterized as transformations preserving some of principal angles (corresponding to the orthogonality,…
In this article we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability condition. We provide an explicit…
In the work by M. C. Lee, A. Naber, and R. Neumayer a beautiful $\varepsilon$-regularity theorem is proved under small negative scalar curvature and entropy bounds. In that paper, the $d_p$ distance for Riemannian manifolds is introduced…
Consider the $n!$ different unitary matrices that permute $n$ $d$-dimensional quantum systems. If $d\geq n$ then they are linearly independent. This paper discusses a sense in which they are approximately orthogonal (with respect to the…
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result…
Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are…
A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with…
We study a generalization of the Wigner function to arbitrary tuples of hermitian operators, which is a distribution uniquely characterized by the property that the marginals for all linear combinations of the given operators agree with the…
Wigner symmetry in Nuclear Physics provides a unique example of a non-perturbative long distance symmetry, a symmetry strongly broken at short distances. We analyse the consequences of such a concept within the framework of One Boson…
Given a real-valued phase-space function, it is a nontrivial task to determine whether it corresponds to a Wigner distribution for a physically acceptable quantum state. This topic has been of fundamental interest for long, and in a modern…
The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its…
We continue the analysis of random series associated to the multidimensional harmonic oscillator $-\Delta + |x|^2$ on $\mathbb{R}^d$ with d \geq 2$$. More precisely we obtain a necessary and sufficient condition to get the almost sure…
We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces H . More precisely, we determine whether a given frame…