Reflection positivity originates from one of the Osterwalder–Schrader axioms for constructive quantum field theory. It serves as a bridge between euclidean and relativistic quantum field theory. In mathematics, more specifically, in representation theory, it is related to the Cartan duality of symmetric Lie groups (Lie groups with an involution) and results in a transformation of a unitary repr...

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for ...

The eigenstate thermalization hypothesis provides to date the most successful description of in isolated quantum systems by conjecturing statistical properties matrix elements typical operators (quasi)energy eigenbasis. Here we study distribution for a class dual-unitary circuits dependence frequency associated with corresponding eigenstates. We provide an exact asymptotic expression spectral f...

Quantum circuits that are classically simulatable tell us when quantum computation becomes less powerful than or equivalent to classical computation. Such of importance because they illustrate what makes universal different from computers. In this work, we propose a novel family by making use dual-unitary (DUQCs), which have been recently investigated as exactly solvable models non-equilibrium ...

Recent works have investigated the emergence of a new kind random matrix behaviour in unitary dynamics following quantum quench. Starting from time-evolved state, an ensemble pure states supported on small subsystem can be generated by performing projective measurements remainder system, leading to <mml:m...

We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple generating vectors. The dual frame v...

In this paper, we discuss the positivity of the Hermitian form ð; Þp introduced by Li in Invent. Math. 27 (1989) 237–255. Let ðG1;G2Þ be a type I dual pair with G1 the smaller group. Let p be an irreducible unitary representation in the semistable range of yðMG1;MG2Þ (see Communications in Contemporary Mathematics, Vol. 2, 2000, pp. 255–283). We prove that the invariant Hermitian form ð; Þp is ...